Preface

This book discusses several topics in modern physics, including spinor fields, key outcomes of 5D relativity, dissipative quantum system and energy balance, centrifugal acceleration in relativistic astrophysics, and time-dependent photoluminescence and photoluminescence excitation in exciton systems and related phenomena.

Chapter 1, "Spinor Fields", proposes a spinor representation of the generalized energy-momentum density 4-vector. In geometry and physics, spinors are elements of a complex number-based vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to an infinitesimal rotation, but unlike geometric vectors and tensors, a spinor transforms to its negative when the space rotates through 360 degrees. It takes a rotation of 720 degrees for a spinor to go back to its original state. This property characterizes spinors. Spinors can be viewed as the "square roots" of vectors, although this is inaccurate and may be misleading. They are better viewed as "square roots" of sections of vector bundles; in the case of the exterior algebra bundle of the cotangent bundle, they thus become "square roots" of differential forms. It is also possible to associate a substantially similar notion of spinor to Minkowski space, in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s, physicists discovered that spinors are essential to describe the intrinsic angular momentum or "spin" of the electron and other subatomic particles.

The proposed spinor representation corresponds to the classical representation of the particle's own rotation, which is described by the diagonal matrix of the moment of inertia. The concept of self-angular rotation of a particle is defined as a spatial characteristic of the field, at each point of which there is a local vortex rotation with an angular velocity Ω – a spinor field. The matrix representation of the vortex rotation Ω (spinor) and the values of the components of such a representation are derived from the matrix representation of the Lorentz transformation. The traditional concept of spin–orbit interaction, as the interaction of the magnetic moment of a particle with the magnetic field of orbital motion, is presented as the interaction of a charged particle with a spinor field. Solutions to the problems of particle motion in an external spinor field in the case of a hydrogen-like atom and planetary motion, splitting of the electron energy levels of an atom in an external magnetic field, deflection of a photon by the gravitational field, and representations in metric spaces are presented.

Chapter 2, "Key Outcomes of 5D Relativity", discusses Kaluza–Klein theory, which is general relativity extended to 5D. The gravitational constant is substituted by a scalar field, making it variable. This scalar field is predicted to change under strong dynamic electromagnetic fields. Deriving the equation of motion from this 5D metric predicts a fifth force. In physics, Kaluza–Klein theory is a classic unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string

theory. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential and writing the Maxwell equations in five dimensions.

pulsars have rotating magnetospheres; therefore, they potentially can drive charged particles to high and ultra-high energies. It is a proposed explanation for ultra-highenergy cosmic rays (UHECRs) and extreme-energy cosmic rays (EECRs) exceeding the Greisen–Zatsepin–Kuzmin limit. The author gives a brief review of the theoretical ideas and results related to Machabeli and Rogava gedankenexperiment (1994) and their astrophysical implications. In particular, three astrophysical cases, including acceleration of particles by rotating magnetospheres in AGNs, centrifugal acceleration and gamma flares in Crab nebula, and self-trapping as a beaming mechanism for

Although direct centrifugal acceleration has limitations, analysis shows the effects of rotation still might play an important role in the processes of acceleration of charged particles. Generally speaking, it is believed that the centrifugal relativistic effects may induce plasma waves, which under certain conditions might be unstable efficiently pumping energy from the background flow. In the second stage, the energy of wave modes can be transformed into the energy of plasma particles, leading to consequent acceleration. In rotating magnetospheres, the centrifugal force acts differently in different locations, leading to the generation of Langmuir waves or plasma oscillations via the parametric instability. One can show that this mechanism efficiently works in

Chapter 5 is titled "Time-Dependent Photoluminescence and Photoluminescence Excitation in Exciton Systems and Related Phenomena". When an electron is excited into a higher energy state, either through absorption of a photon or another excitation method (such as in electroluminescence), this creates a positively charged space in the lower energy level known as a "hole." This results in the formation of an electron–hole pair. In some cases, these two particles exist in a bound state, forming a single quasi-particle known as an exciton. Within an exciton, the electron and hole are bound together by Coulombic interactions, and the strength of this bond is quantified by its exciton binding energy. Excitons can be grouped into two categories: Frenkel excitons and Wannier–Mott excitons. Frenkel excitons are tightly bound excitons that have a radius similar in magnitude to the crystal unit cell or atomic radii. These are often found in insulators or organic semiconductors, as they are often bound to specific atoms or molecules. Alternatively, Wannier–Mott excitons have a larger excitonic radius. The exciton therefore encompasses many unit cells and thus can move more freely through the crystal structure. These are commonly found in inorganic semiconductors, and their excitonic bonds can be overcome at room temperature. Generally, organic matter such as fullerenes and insulators tend to exhibit Frenkel excitons with a larger binding energy. These are sometimes known as bound electrons. Most inorganic or hybrid semiconductors (such as perovskites) have Wannier–Mott excitons and free excitons that can travel through the crystal structure. Investigations of exciton systems represent one of the fastest-growing, multifaceted, and productive areas of modern solid-state physics and material science. Particularly noteworthy in this regard is the deployment of state-of-the-art spectroscopic techniques (e.g., tr-ARPES and femtosecond optical spectroscopy) to better elucidate the structural, dynamical, and quantum mechanical properties of

Selected topics presented in the chapters come from very different fields, including theoretical physics, astrophysics, and solid-state physics. The analysis of the chapters

Fast Radio Bursts, are discussed.

the magnetospheres of AGN and pulsars.

exciton systems.

The 5D theory was developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are identified with the 4D space-time metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton." Correspondingly, the 5D Einstein equations yield the 4D Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the "cylinder condition" hypothesis that no component of the 5D metric depends on the fifth dimension. Without this restriction, terms are introduced that involve derivatives of the fields with respect to the fifth coordinate, and this extra degree of freedom makes the mathematics of the fully variable 5D relativity enormously complex. Standard 4D physics seems to manifest this "cylinder condition" and, along with it, simpler mathematics.

The Pioneer 10 and 11 spacecraft were not where they were supposed to be. These missions, launched in 1972 and 1973, have covered hundreds of millions of kilometers, heading toward the edge of our solar system. But something was holding them back. Each year, they fell behind in their projected travel by about 5000 kilometers (3000 miles). In the end, after the recovery of the data and years of painstaking work, the researchers concluded that anisotropic thermal radiation (heat radiating from the spacecraft unevenly in many directions) explained the mystery of the Pioneers' deceleration. No revolution in physics was needed.

Chapter 3, "Dissipative Quantum System and Energy Balance", discusses how various parts of a quantum many-body system exchange energies at thermal equilibrium. The author assumes a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. The author uses the Quantum Langevin equation to show such energy exchange at equilibrium and concludes that different parts of a physical system can exchange energies even at absolute zero temperature. The author computed the instantaneous power supplied by the fluctuating (random) force, which provides information about the work done by the random force on the quantum subsystem of interest. The quantum formalism is used here to verify that, at equilibrium, the work done by the fluctuating force balances the energy lost by the quantum subsystem to the heat bath. The quantum subsystem chosen to couple to the heat bath is the charged oscillator in a magnetic field. The calculations were done using the Drude regularized spectral density of bath oscillators instead of using a strict ohmic spectral density that gives memoryless damping.

Chapter 4, "Centrifugal Acceleration in Relativistic Astrophysics", is devoted to the centrifugal acceleration of astro-particles to relativistic energies that might take place in rotating astrophysical objects. It is strongly believed that active galactic nuclei and

pulsars have rotating magnetospheres; therefore, they potentially can drive charged particles to high and ultra-high energies. It is a proposed explanation for ultra-highenergy cosmic rays (UHECRs) and extreme-energy cosmic rays (EECRs) exceeding the Greisen–Zatsepin–Kuzmin limit. The author gives a brief review of the theoretical ideas and results related to Machabeli and Rogava gedankenexperiment (1994) and their astrophysical implications. In particular, three astrophysical cases, including acceleration of particles by rotating magnetospheres in AGNs, centrifugal acceleration and gamma flares in Crab nebula, and self-trapping as a beaming mechanism for Fast Radio Bursts, are discussed.

Although direct centrifugal acceleration has limitations, analysis shows the effects of rotation still might play an important role in the processes of acceleration of charged particles. Generally speaking, it is believed that the centrifugal relativistic effects may induce plasma waves, which under certain conditions might be unstable efficiently pumping energy from the background flow. In the second stage, the energy of wave modes can be transformed into the energy of plasma particles, leading to consequent acceleration. In rotating magnetospheres, the centrifugal force acts differently in different locations, leading to the generation of Langmuir waves or plasma oscillations via the parametric instability. One can show that this mechanism efficiently works in the magnetospheres of AGN and pulsars.

Chapter 5 is titled "Time-Dependent Photoluminescence and Photoluminescence Excitation in Exciton Systems and Related Phenomena". When an electron is excited into a higher energy state, either through absorption of a photon or another excitation method (such as in electroluminescence), this creates a positively charged space in the lower energy level known as a "hole." This results in the formation of an electron–hole pair. In some cases, these two particles exist in a bound state, forming a single quasi-particle known as an exciton. Within an exciton, the electron and hole are bound together by Coulombic interactions, and the strength of this bond is quantified by its exciton binding energy. Excitons can be grouped into two categories: Frenkel excitons and Wannier–Mott excitons. Frenkel excitons are tightly bound excitons that have a radius similar in magnitude to the crystal unit cell or atomic radii. These are often found in insulators or organic semiconductors, as they are often bound to specific atoms or molecules. Alternatively, Wannier–Mott excitons have a larger excitonic radius. The exciton therefore encompasses many unit cells and thus can move more freely through the crystal structure. These are commonly found in inorganic semiconductors, and their excitonic bonds can be overcome at room temperature. Generally, organic matter such as fullerenes and insulators tend to exhibit Frenkel excitons with a larger binding energy. These are sometimes known as bound electrons. Most inorganic or hybrid semiconductors (such as perovskites) have Wannier–Mott excitons and free excitons that can travel through the crystal structure. Investigations of exciton systems represent one of the fastest-growing, multifaceted, and productive areas of modern solid-state physics and material science. Particularly noteworthy in this regard is the deployment of state-of-the-art spectroscopic techniques (e.g., tr-ARPES and femtosecond optical spectroscopy) to better elucidate the structural, dynamical, and quantum mechanical properties of exciton systems.

Selected topics presented in the chapters come from very different fields, including theoretical physics, astrophysics, and solid-state physics. The analysis of the chapters requires the reader to have very advanced knowledge of mathematics, theoretical physics, and so on. However, even a reader unfamiliar with the topic can learn about new directions or tendencies in contemporary physics.

## **Zbigniew Piotr Szadkowski**

**Chapter 1**

**Abstract**

Spinor Fields

*Vahram Mekhitarian*

A spinor representation of the generalized energy-momentum density 4-vector is proposed, and examples of such representations for various particles and fields are given. This representation corresponds to the classical representation of the particle's own rotation, which is described by the diagonal matrix of the moment of inertia. The concept of self-angular rotation of a particle is defined as a spatial characteristic of the field, at each point of which there is a local vortex rotation with an angular velocity **Ω** – a spinor field. The matrix representation of the vortex rotation **Ω** (spinor) and the values of the components of such a representation are derived from the matrix representation of the Lorentz transformation. The traditional concept of spin-orbit interaction, as the interaction of the magnetic moment of a particle with the magnetic field of orbital motion, is presented as the interaction of a charged particle with a spinor field. Solutions to the problems of particle motion in an external spinor field in the case of a hydrogen-like atom and planetary motion, splitting of the electron energy levels of an atom in an external magnetic field, deflection of a photon by the gravita-

**Keywords:** spinor fields, relativistic and quantum mechanics, spin concept, Lorentz

The material presented in this chapter is based on the new approaches of relativis-

tic and quantum mechanics developed in the works [1–4]. Equations, which are obtained by applying the invariance principle for the total four-dimensional momentum of the system "field + particle," have some significant advantages as compared with its analog equations such as the Klein-Fock-Gordon and Dirac equations. For instance, the problem of a hydrogen-like atom has solutions for an arbitrary value of the interaction constant not restricted to whatever the atomic number of the nucleus (we recall that for the Dirac equation the atomic number is restricted to Z < 137). In contrast to the well-known equations of relativistic and quantum mechanics, the energy levels of the ground state of the particle for the considered equation prove to be limited by the size of the spatial characteristic. This property directly reflects the

Attributed to Werner Heisenberg (1901–1976).

tional field, and representations in metric spaces are presented.

transformation, spinor fields representation

*When I meet God, I am going to ask Him two questions: Why relativity? And why turbulence? I really believe He will have an answer for the first.*

**1. Introduction**

Faculty of Physics and Applied Informatics, Department of Intelligent Systems, University of Lodz, Lodz, Poland

## **Chapter 1** Spinor Fields

*Vahram Mekhitarian*

## **Abstract**

A spinor representation of the generalized energy-momentum density 4-vector is proposed, and examples of such representations for various particles and fields are given. This representation corresponds to the classical representation of the particle's own rotation, which is described by the diagonal matrix of the moment of inertia. The concept of self-angular rotation of a particle is defined as a spatial characteristic of the field, at each point of which there is a local vortex rotation with an angular velocity **Ω** – a spinor field. The matrix representation of the vortex rotation **Ω** (spinor) and the values of the components of such a representation are derived from the matrix representation of the Lorentz transformation. The traditional concept of spin-orbit interaction, as the interaction of the magnetic moment of a particle with the magnetic field of orbital motion, is presented as the interaction of a charged particle with a spinor field. Solutions to the problems of particle motion in an external spinor field in the case of a hydrogen-like atom and planetary motion, splitting of the electron energy levels of an atom in an external magnetic field, deflection of a photon by the gravitational field, and representations in metric spaces are presented.

**Keywords:** spinor fields, relativistic and quantum mechanics, spin concept, Lorentz transformation, spinor fields representation

## **1. Introduction**

*When I meet God, I am going to ask Him two questions: Why relativity? And why turbulence? I really believe He will have an answer for the first.*

Attributed to Werner Heisenberg (1901–1976).

The material presented in this chapter is based on the new approaches of relativistic and quantum mechanics developed in the works [1–4]. Equations, which are obtained by applying the invariance principle for the total four-dimensional momentum of the system "field + particle," have some significant advantages as compared with its analog equations such as the Klein-Fock-Gordon and Dirac equations. For instance, the problem of a hydrogen-like atom has solutions for an arbitrary value of the interaction constant not restricted to whatever the atomic number of the nucleus (we recall that for the Dirac equation the atomic number is restricted to Z < 137).

In contrast to the well-known equations of relativistic and quantum mechanics, the energy levels of the ground state of the particle for the considered equation prove to be limited by the size of the spatial characteristic. This property directly reflects the

uncertainty principle in that, irrespective of the well depth value, the particle can be localized in a bound state only if the well width is larger than the half-wavelength of the particle.

For the problem of the passage of a particle through a potential barrier, if the energy of the particle does not exceed the height of the potential barrier, then the transmission coefficient is equal to zero regardless of the height of the barrier. In this case, there is no contradiction like Klein's paradox.

The equations are applicable for different types of particles and interactions. The analysis of the solutions shows full compliance with the principles of relativistic and quantum mechanics, and the solutions are devoid of any restrictions on the nature and magnitude of the interactions.


**Table 1.**

*Old and new [1–4] concepts and equations of relativistic and quantum mechanics.*

#### *Spinor Fields DOI: http://dx.doi.org/10.5772/intechopen.105569*

However, this new theory was presented without an explicit representation of the spinor properties of fields and systems (**Table 1**). In this chapter, based on the representation of local relativistic rotation by the Lorentz transformation matrix, the representations of the four-dimensional energy-momentum vector for various spinor fields and systems, the results of corresponding solutions of the new equations of relativistic and quantum mechanics are given.

Since within the framework of the new, generalized relativistic theory, there is an exact correspondence between the representations of relativistic, quantum mechanics, and general relativity, in the chapter spinor properties and equations are also presented in metric spaces.

## **2. The spin concept**

Although in physics the concept of spin arose as a property of the proper rotation of a particle (electron, G. E. Uhlenbeck, S. Goudsmit), as a result of quantization of the self-angular momentum, further development by W. Pauli and P. Dirac led to the description of spin as a property of space itself, in which we describe particles. Their interactions corresponded to the data of the physical experiment, when they were presented in spinor spaces.

One of the first, the Dirac equation, which describes systems in spaces with spin 1/ 2, the solution in the case of the hydrogen atom gives a very good match with the real spectrum of the hydrogen atom. But the hydrogen atom problem (**Figure 1**),

which consists of a proton and an electron, is solved for a generalized particle with the reduced mass *m* of an electron *m*<sup>1</sup> and a proton *m*<sup>2</sup> (the two-body problem), whose coordinates **r** do not coincide with either the coordinates of the electron **r**<sup>1</sup> or the coordinates of the proton **r**2, and the spin of this generalized particle (the sum of the spins of the proton and electron) can only have values of 0 or 1.

Obviously, Dirac's spin 1/2 refers to the properties of the space in which the hydrogen atom is described, not to an electron, a proton, or a hydrogen atom. Similarly, regardless of the spin of the nucleus and the spin of the electron shell of hydrogen-like atoms, the spectrum, and fine splitting is described by solving the Dirac equation with spin 1/2. The same is true for other problems, regardless of the properties of the components of the physical system themselves—solutions to the Dirac equations describe systems with only a spin of 1/2.

**Figure 2.**

*The concept of spin in quantum mechanics does not describe the physical properties of the vortex motion or proper rotation of bodies in the classical sense.*

**Figure 3.**

*The projections of the self-angular momentum (spin) of nonspherical nuclei have different values in the directions of the main axes of inertia.*

Naturally, the spinor properties of space do not in any way describe the physical properties of the self-angular momentum of a particle or a vortex (turbulent) field (the flow of a liquid or gas) in the classical sense, and therefore, the statement arose that spin has no classical analog (**Figure 2**). And the problem of describing the self-angular momentum of the particles and the vortex field remained unresolved.

Note that the description of the motion of an asymmetric spinning top, represented by the tensor of the moment of inertia, has no analog in quantum mechanics since the modern concept of spin does not imply any representations of the concept of the moment of inertia (**Figure 2**). But in the case of nonspherical nuclei (**Figure 3**), the projection of the self-angular momentum must have different spin values relative to the main axes of inertia (rotation) of the nucleus.

The main, fundamental physical variable, for which the variational principles and equations of relativistic and quantum mechanics are formulated, is the energymomentum density 4-vector, so all the properties of the system, also spinor, must be initially reflected in the representation of the energy-momentum 4-vector **P** ¼ ð Þ *ε*, **p** .

## **3. Orbital and vortex motions of the continuous media**

Imagine holding a bicycle wheel by the axle at a distance *l* and rotating around its axis with an angular velocity of **Ω** (**Figure 4**).

Since the wheel rotates freely around its axis, there is no moment of own rotation of the wheel itself, and it makes an only translational motion (movement without its own rotation). Energy *W* is expressed by the usual formula of kinetic energy through the mass *<sup>m</sup>* and velocity **<sup>v</sup>** of the wheel movement *<sup>W</sup>* <sup>¼</sup> *<sup>m</sup>***v**<sup>2</sup>*=*<sup>2</sup> <sup>¼</sup> *ml*<sup>2</sup> **Ω**2 *=*2.

If there is any friction of the axle, then the rotational motion of the axle will gradually be transmitted to the wheel and eventually, the wheel will rotate with the same angular speed. Note that in this case, we considered the option of a rigidly fixed

**Figure 4.** *Bicycle wheel at a distance l rotating around axis with an angular velocity of* **Ω***.*

axis (rigidly fixed center of mass) wheel. This corresponds to a rigid spin-orbit interaction when the angular velocities of rotation are the same.

In this case, the energy is equal to the sum of the kinetic energy of the wheel and the energy of the wheel's own rotation with the moment of inertia *I* (a sample conclusion of the Huygens-Stern theorem)

$$\mathbf{W} = m\mathbf{v}^2/2 + I\mathbf{\Omega}^2/2 = (ml^2 + I)\mathbf{\Omega}^2/2. \tag{1}$$

In other cases, in terms of energy, there will be other ratios of translational energy and energy of its own rotation (spin). In particular, the motion of the Moon corresponds to the case of a rigidly fixed center of mass (the Moon is constantly facing the Earth on the same side), and the movement of the Earth around the Sun corresponds to the case of a freely fixed center of mass (the tilt of the axis, the period of its own rotation is in no way related to the movement around the Sun).

Note that regardless of the distance from the center of rotation, the angular velocity of its own rotation and energy is constant. If the system is represented as a medium with distributed local rotation, then it can be described by the energy density of vortex rotation with an angular velocity **Ω**. A good example of such a system is a permanent magnet, where at each point of the medium there are eddy currents (rotating electric field) and, accordingly, the magnetic field of these currents (rotating electric field).

To illustrate the vortex motion of the distributed systems, let us consider an example of a large, thin hard disk on which much identical small metal (heavy) spinners of mass *m* and moment of inertia *I* are attached perpendicularly and densely (**Figure 5**). The mass of the disk relative to the total mass of spinners Σ*m* can be neglected. If the spinners do not have axial friction, then when the disk rotates, the spinners will rotate around the axis of the disk, but will not rotate around their own axis—the arrows will always be turned toward the original direction (**Figure 5**). In this case, the rotation energy of the system *W* is determined only by the sum of the kinetic energies of the orbital rotation of the spinners *W* ¼ Σ*m***v***<sup>i</sup>* <sup>2</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>Σ</sup>*mri* <sup>2</sup>**Ω**<sup>2</sup> *=*2 relative to the axis of the disk, and when the disk stops, the energy of the system becomes zero.

In the case when the spinners have some small axial friction (like the spin-orbit interaction of an electron in an atom or the connection of the moon's rotation with the Earth's rotation due to tides), then when the disk rotates, due to friction, after some

**Figure 5.** *Spinners of mass* m *and moment of inertia* I *are perpendicularly and densely fixed on the thin hard disk.*

**Figure 6.** *Spinners rotate around their axis with the angular velocity of the disk* **Ω***.*

time the spinners will begin to rotate around its own axis and with the angular speed of rotation of the disk **Ω** (**Figure 6**). In this case, during rotation, the spinners are at any moment oriented toward the center of the disk axis (like a Moon), which corresponds to the rotation of the disk with the axes of the spinners fixed rigidly—at each point, the spinners rotate around their own axis with the angular speed of rotation of the disk. The energy of the system will be represented as the sum of the kinetic energy of the orbital rotation and the energy of the own rotation of the spinners in the form of *W* ¼ Σ*mri* <sup>2</sup>**Ω**<sup>2</sup> *<sup>=</sup>*<sup>2</sup> <sup>þ</sup> <sup>Σ</sup>*I***Ω**<sup>2</sup> *=*2.

If we stop the rotation of the disk in such an established stationary state, then the energy of the orbital motion of the spinners will be reset, but the energy of its own rotation Σ*I***Ω**<sup>2</sup> *=*2 will be preserved (**Figure 7**).

The picture, obtained after stopping the disk, represents a distributed system with local vortex motion. Regardless of the distance from the center of the disk, the angular velocity of rotation and energy of the spinners are constant. If the system is represented as a medium with distributed local rotation, then it can be described by the energy density of vortex rotation with an angular velocity **Ω**.

**Figure 7.** *The spinners keep spinning after the disk has stopped.*

Let us emphasize that although the described system does not have orbital momentum, the generalized angular momentum and the total energy of the system are not zero and have minimal internal angular momentum and energy. It should be noted that in the ground state, both the angular momentum and the energy of the system have corresponding minimum values due to the internal rotation—spin.

A good example of such a system is a permanent magnet, where eddy currents (rotating electric field) and, accordingly, the magnetic field of these currents (rotating electric field) exist at each point of the medium (**Figure 8**).

Vortex and circular fields should be distinguished: in vortex fields, the rotor is nonzero at any point in the field, and in circular fields, it is zero (**Figure 9**). Such is the electric field outside the alternating current solenoid, where the magnetic field is zero.

A time-varying magnetic field in the solenoid generates, induces an electric field **E**, which is described by Maxwell's equations. Such a field is described in cylindrical coordinates and is represented by the vector potential **A** ¼ **A r**ð Þ , *t* in the form [5].

$$\mathbf{A}(\mathbf{r},t) = \begin{cases} \frac{1}{2} [\mathbf{B}(t) \times \mathbf{r}], & r \le R\_{\mathcal{S}}; \\\\ \frac{R\_{\mathcal{S}}}{2(\mathbf{x}^2 + \mathbf{y}^2)} [\mathbf{B}(t) \times \mathbf{r}], & r > R\_{\mathcal{S}}; \end{cases} \tag{2}$$

where *RS* is a radius of the solenoid.

**Figure 9.** *Vortex and circular fields of the solenoid.*

For the fields of the solenoid, we have

$$\mathbf{B} = \begin{array}{c} \mathbf{rotA} = \begin{cases} \mathbf{B}, & r \le R\_{\mathcal{S}}, \\ 0, & r > R\_{\mathcal{S}}. \end{cases} \qquad \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} = \begin{cases} \frac{1}{2c} \left[ \mathbf{r} \times \frac{\partial \mathbf{B}}{\partial t} \right], & r \le R\_{\mathcal{S}}, \\\\ \frac{R\_{\mathcal{S}}}{2c(\mathbf{x}^2 + \mathbf{y}^2)} \left[ \mathbf{r} \times \frac{\partial \mathbf{B}}{\partial t} \right], & r > R\_{\mathcal{S}}. \end{cases} \end{array} \tag{3}$$

Inside the solenoid, the electric field is vortex – rot**E** 6¼ 0, and outside the solenoid it is circular and the magnetic field is zero – rot**E** ¼ 0.

## **4. Representation of the spinor fields in Minkowski spaces**

When representing continuously distributed systems, one should average the energy of the spinners over the occupied volume and describe the continuous medium with the energy-momentum distribution density.

If we want to describe the spatial properties of the vortex fields, then rotation angular velocity **Ω** at a given point in space can serve as such a kinematic variable. The occurrence of vortex rotation distributed in space in the framework of the relativistic theory is described by the Lorentz transformation, calculating the energy-momentum distribution density of local rotations.

For fields, we have [6]

$$\begin{aligned} \mathbf{A}(\boldsymbol{\varrho}',\mathbf{A}') &= \begin{pmatrix} \boldsymbol{\varrho} + \boldsymbol{\mathfrak{f}} \cdot \mathbf{A} & \boldsymbol{\varrho}\boldsymbol{\mathfrak{f}} + \boldsymbol{q}\mathbf{A}\_{\parallel} \\ \sqrt{1 - \boldsymbol{\beta}^2} & \sqrt{1 - \boldsymbol{\beta}^2} \end{pmatrix} + \boldsymbol{q}\mathbf{A}\_{\perp} \\ \boldsymbol{\varrho} &= \begin{pmatrix} \boldsymbol{\varrho}, & \mathbf{A} \end{pmatrix} + \begin{pmatrix} (\boldsymbol{\varrho} - \mathbf{1})\boldsymbol{\varrho} + \boldsymbol{\chi}\boldsymbol{\mathfrak{f}} & \boldsymbol{\mathfrak{f}}\boldsymbol{\mathfrak{g}}\boldsymbol{\mathfrak{f}} + (\boldsymbol{\chi} - \mathbf{1})(\mathbf{A} \cdot \boldsymbol{\mathfrak{f}})\boldsymbol{\mathfrak{f}}/\boldsymbol{\beta}^2 \end{pmatrix}. \end{aligned} \tag{4}$$

where *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>β</sup>*<sup>2</sup> <sup>p</sup> is a Lorentz factor. This transformation can be presented in matrices form

$$(\phi', \quad \mathbf{A}') \quad = (\boldsymbol{\wp}, \quad \mathbf{A}) \quad + \hat{\mathbf{T}} \ (\boldsymbol{\wp}, \quad \mathbf{A}).\tag{5}$$

where a Lorentz transformation has a form [1].

$$
\hat{\mathbf{T}} = \mathbf{y} \left\| \begin{array}{c|ccccc} \mathbf{0} & \boldsymbol{\beta}\_{1} & \boldsymbol{\beta}\_{2} & \boldsymbol{\beta}\_{3} \\ \boldsymbol{\beta}\_{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \boldsymbol{\beta}\_{2} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \boldsymbol{\beta}\_{3} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{array} \right\|\_{2} + (\boldsymbol{\gamma} - \mathbf{1}) \left\| \begin{array}{c|ccccc} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \boldsymbol{\beta} & \frac{\boldsymbol{\beta}\_{1}\boldsymbol{\beta}\_{1}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{1}\boldsymbol{\beta}\_{2}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{1}\boldsymbol{\beta}\_{3}}{\boldsymbol{\beta}^{2}} \\ \mathbf{0} & \frac{\boldsymbol{\beta}\_{2}\boldsymbol{\beta}\_{1}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{2}\boldsymbol{\beta}\_{2}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{2}\boldsymbol{\beta}\_{3}}{\boldsymbol{\beta}^{2}} \\ \mathbf{0} & \frac{\boldsymbol{\beta}\_{3}\boldsymbol{\beta}\_{1}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{3}\boldsymbol{\beta}\_{2}}{\boldsymbol{\beta}^{2}} & \frac{\boldsymbol{\beta}\_{3}\boldsymbol{\beta}\_{3}}{\boldsymbol{\beta}^{2}} \end{array} \right\|\_{2} \tag{6}
$$

$$\mathbf{y} = \mathbf{1}/\sqrt{\mathbf{1} - \left(\mathbf{v}/c\right)^2} = \mathbf{1}/\sqrt{\mathbf{1} - \left[\mathbf{A} \times \mathbf{r}\right]^2/c^2} = \mathbf{1}/\sqrt{\mathbf{1} - \left(r/r\_h\right)^2},\tag{7}$$

$$\dot{\mathbf{T}} = \gamma \dot{\boldsymbol{\theta}} \begin{vmatrix} \mathbf{0} & \cos\left(\boldsymbol{\phi}\right) & \sin\left(\boldsymbol{\phi}\right) & \mathbf{0} \\ \cos\left(\boldsymbol{\phi}\right) & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \sin\left(\boldsymbol{\phi}\right) & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{vmatrix} + \begin{vmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \cos^2\left(\boldsymbol{\phi}\right) & \sin\left(\boldsymbol{\phi}\right)\cos\left(\boldsymbol{\phi}\right) & \mathbf{0} \\ \mathbf{0} & \sin\left(\boldsymbol{\phi}\right)\cos\left(\boldsymbol{\phi}\right) & \sin^2\left(\boldsymbol{\phi}\right) & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{vmatrix},\tag{8}$$

$$\begin{aligned} \hat{\mathbf{T}} &= \frac{1}{V} \int\_{V} \hat{\mathbf{T}} dv = \\ & \frac{1}{\pi r\_h^2 d} \left| \int\_0 dx \cdot \left| r \left( \frac{1}{\sqrt{1 - r^2/r\_h^2}} - 1 \right) dr \cdot \int\_0^{2\pi} \begin{vmatrix} 1 & 0 & 0 \\ 0 & \cos^2(\phi) & \sin(\phi)\cos(\phi) \\ \sin(\phi)\cos(\phi) & 0 & \sin^2(\phi) \end{vmatrix} d\phi = 0 \right| \\ & \left| \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/2 \end{vmatrix} \right| , \end{aligned}$$

where

$$\frac{2\pi}{2\pi r\_h^2} \int\_0^{r\_h} r \left(\frac{1}{\sqrt{1 - r^2/r\_h^2}} - 1\right) dr = \int\_0^1 \left(\frac{1}{\sqrt{1 - x^2}} - 1\right) 2x dx = -2\sqrt{1 - x^2} - x^2 \Big|\_0^1 = 1. \tag{10}$$

Averaging over the entire volume to the event horizon *rh*, we obtain a result independent of the speed of angular rotation **Ω** This seemingly unexpected result is due to the inverse dependence of the event horizon *rh* ¼ *c=*Ω on the angular velocity of rotation. The greater the speed of angular rotation, the greater the relativistic compression and, accordingly, the increase in energy density due to a decrease in the volume of integration, so the integral does not depend on the angular velocity **Ω** This means that own rotation, spin is an invariant for any reference frames, in the approaches of general relativity.

Based on the principle of superposition and additivity, such elementary excitations of the generating field in a unit of volume can be any integer, so that in the general case *n*

$$
\hat{\mathbf{Q}} = \left\| \begin{array}{ccccc} & \mathbf{1}/2 & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & \mathbf{1}/2 & \mathbf{0} \\ & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{array} \right\|\_{\sharp} \quad \rightarrow \quad \pm \frac{\mathbf{1}}{2} \times \mathfrak{n} = \mathfrak{s}\_{\mathfrak{x}} \quad \rightarrow \quad \left\| \begin{array}{ccccc} & \mathfrak{s}\_{\mathfrak{x}} & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & \mathfrak{s}\_{\mathfrak{x}} & \mathbf{0} \\ & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{array} \right\|\_{\sharp}, \tag{11}
$$

where is a **<sup>Ω</sup>**^ representation matrix of the rotation (spinor), and <sup>Ω</sup>*<sup>x</sup>* <sup>¼</sup> <sup>Ω</sup>*<sup>y</sup>* <sup>¼</sup> *sz* – the value of components for the **x**^, **y**^ axis's when rotating around an axis **z**^. As we see, the rotation along the selected axis generates two equal perpendicular spatial components of the generalized momentum with half-integer coefficients (**Figure 10**).

Accordingly, taking into account the independent rotations in all axes, for the matrix of the spinor representation we have

#### **Figure 10.**

*Rotation along the selected axis generates two equal perpendicular spatial components with half-integer coefficients.*

*Spinor Fields DOI: http://dx.doi.org/10.5772/intechopen.105569*

$$
\hat{\mathbf{O}} = \left\| \begin{array}{ccccc} \mathbf{0} & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & s\_{\mathbf{x}} & \mathbf{0} \\ & & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & s\_{\mathbf{x}} \end{array} \right\|\_{\hat{\mathbf{x}}} + \left\| \begin{array}{ccccc} s\_{\mathbf{y}} & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & \mathbf{0} & s\_{\mathbf{y}} \end{array} \right\|\_{\hat{\mathbf{y}}} + \left\| \begin{array}{ccccc} s\_{\mathbf{z}} & \mathbf{0} & \mathbf{0} \\ & \mathbf{0} & s\_{\mathbf{z}} & \mathbf{0} \\ & & \mathbf{0} & \mathbf{0} \end{array} \right\|\_{\hat{\mathbf{z}}} = \\ \left\| \begin{array}{ccccc} s\_{\mathbf{y}} + s\_{\mathbf{z}} & \mathbf{0} \\ & \mathbf{0} \\ & s\_{\mathbf{x}} + s\_{\mathbf{z}} & \mathbf{0} \\ & \mathbf{0} & \mathbf{0} \\ & & \mathbf{0} \end{array} \right\|\_{\hat{\mathbf{x}}}. \end{split} \tag{12}
$$

Thus, the spinor, the averaged Lorentz transformation for local rotations, is represented by a diagonal matrix or column as

$$\begin{aligned} \hat{\mathbf{Q}} &= \{\mathbf{s}\_{\mathcal{Y}} + \mathbf{s}\_{\mathfrak{x}}, \ \mathbf{s}\_{\mathfrak{x}} + \mathbf{s}\_{\mathfrak{x}}, \ \mathbf{s}\_{\mathfrak{x}} + \mathbf{s}\_{\mathfrak{y}}\}, \quad \mathbf{s}\_{\mathfrak{x}}, \ \mathbf{s}\_{\mathfrak{y}}, \ \mathbf{s}\_{\mathfrak{x}} = \mathbf{0}, \ \pm\frac{1}{2}, \ \pm\mathbf{1}, \ \pm\frac{3}{2}, \ \dots, \ \pm\frac{n}{2}, \ \dots \end{aligned} \tag{13}$$
 
$$\hat{\mathbf{Q}}^2 = 2\{\mathbf{s}\_{\mathfrak{x}}^2 + \mathbf{s}\_{\mathfrak{y}}^2 + \mathbf{s}\_{\mathfrak{x}}^2 + \mathbf{s}\_{\mathfrak{x}}\mathbf{s}\_{\mathfrak{y}} + \mathbf{s}\_{\mathfrak{x}}\mathbf{s}\_{\mathfrak{x}} + \mathbf{s}\_{\mathfrak{y}}\mathbf{s}\_{\mathfrak{x}}\}. \tag{14}$$

The index symbols of the components of the matrix are specifically selected to indicate from rotation around which axes these components originated:

$$
\boldsymbol{\Omega}\_{\mathbf{x}} = \mathfrak{s}\_{\mathfrak{y}} + \mathfrak{s}\_{\mathbf{z}}, \quad \boldsymbol{\Omega}\_{\mathfrak{y}} = \mathfrak{s}\_{\mathbf{x}} + \mathfrak{s}\_{\mathbf{z}}, \quad \boldsymbol{\Omega}\_{\mathbf{z}} = \mathfrak{s}\_{\mathbf{x}} + \mathfrak{s}\_{\mathfrak{y}}.\tag{14}
$$

Negative signs of the projections of the spinor matrix **Ω**^ appeared during the transformation of reflections and inversions (antiparticles).

$$\hat{\mathbf{T}} = \begin{vmatrix} \pm m & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & s\_{\mathcal{V}} + s\_{\mathcal{x}} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & s\_{\mathcal{x}} + s\_{\mathcal{x}} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & s\_{\mathcal{x}} + s\_{\mathcal{y}} \\\\ \mathbf{T} = \left(\pm m, \quad s\_{\mathcal{V}} + s\_{\mathcal{x}}, \quad s\_{\mathcal{X}} + s\_{\mathcal{x}}, \quad s\_{\mathcal{X}} + s\_{\mathcal{y}}\right) . \end{vmatrix},\tag{15}$$

where *m* is integer.

If a scalar field *φ* is given in space, at each point of which the vortices are excited, then the vector potential of such a field can be represented as **A** ¼ *φ***Ω**, and the generalized energy-momentum **P** as (**Table 2**)

$$\begin{aligned} \mathbf{P} &= (\boldsymbol{\varrho}, \ \mathbf{A}) = (\boldsymbol{\varrho}, \ \boldsymbol{\varrho}\boldsymbol{\Omega}), \ \mathbf{P}^2 = \boldsymbol{\varrho}^2 (\mathbf{1} - \boldsymbol{\Omega}^2), \\\mathbf{1} - \boldsymbol{\Omega}^2 &= \mathbf{1} - 2(\boldsymbol{\varrho}\_x{}^2 + \boldsymbol{\varrho}\_y{}^2 + \boldsymbol{\varrho}\_x{}^2 + \boldsymbol{\varsigma}\_x \boldsymbol{\varsigma}\_y + \boldsymbol{\varsigma}\_x \boldsymbol{\varsigma}\_x + \boldsymbol{\varsigma}\_y \boldsymbol{\varsigma}\_x). \end{aligned} \tag{16}$$

If for qualitative evaluation, we assume that the energy of such elementary excitation corresponds to the quantum of the rotator energy *<sup>E</sup>* <sup>¼</sup> *mc*<sup>2</sup> <sup>¼</sup> <sup>ℏ</sup>Ω, then we get

$$
\hbar = \hbar/m\\\mathfrak{c} = \mathfrak{c}/\mathfrak{\Omega} = r\_{\hbar},\tag{17}
$$

that is, the event horizon radius ("particle size") is the wavelength of the particle *rh* ¼ ƛ.

Since the spin of the field has been determined locally, its direction from point to point can change, provided that the internal structure is preserved. The spinor field, as in the general case of any vector potential **A** ¼ *φ***Ω**, must satisfy the basic equations for the fields [2].

*Recent Topics and Innovations in Quantum Field Theory*


#### **Table 2.**

*Possible structures and invariants for spinor fields (component signs are arbitrary).*

Spin is invariant from the point of view of general relativity since it does not depend on the state of motion—the speed and rotation of the reference frame, in which we describe the spinor field.

The spinor properties of fields are unchanged and do not depend on the method and sources of their creation. For each case of motion of a particle in a given field, the spinor properties of the field are determined by the physical nature of the field itself. Spin is a fundamental, unchanging characteristic of the field. Whichever way it is created, it will be with the same back.

Spin in the expression of potential energy describes the system of particle + field, the spinor properties of which are described by the above expressions, regardless of which components are included in the system. The new field of interaction is also represented by one of the above configurations. Whether the spinor field of interaction is a simple sum of spinor fields or another configuration can be found by comparing with experimental data. For each system, a spin configuration should be selected so that the results of the calculation coincide with the experimental data or with the already known characteristics of the system.

The spin of the nucleus and the presence of other electrons with their spins do not determine the spin of the electromagnetic field created by it in atoms. Fine splitting, which directly depends on the spin of the interaction field, does not depend on the spin of the nuclei and electron shell of the atoms. For example, the value of fine splitting of isotope atoms Potassium 39K, 41K, and 40K with spines 3/2, 3/2, and 4; Rubidium 87Rb and 85Rb with spines 3/2 и 5/2; Hydrogen <sup>1</sup> H, <sup>2</sup> H, <sup>3</sup> H with spines 1/2, 1, 1/2 [7].

Also in the solar system, the spin of the gravitational field does not depend on the positions of celestial bodies and their rotation.

Saying that a particle has spin, within the framework of the foregoing, means that the particle has a spinor field, and this property manifests itself at any point in space during any interactions. Spin is a spatial characteristic and is not attributed to any point particle. Therefore, the spin of elementary particles is not determined by their internal structure, but is determined by the spin of the interaction fields created by these particles. The phrase "spin-orbit interaction" in this case means the interaction of the orbital moment with the spinor field.

## **5. Examples of spinor fields representation**

#### **5.1 Electromagnetic wave: photon**

The spin of an electromagnetic wave with potential *φ* is represented as a rotating electric field with a vector potential **A** ¼ **Ω***φ*. Given that the magnetic field **B** of the wave is defined as the rotor of the vector potential

$$\mathbf{B} = \text{rot}(\rho \mathbf{\hat{Q}}) = -\left[\mathbf{\hat{Q}} \times \frac{\partial \rho}{\partial \mathbf{r}}\right] = [\mathbf{\hat{Q}} \times \mathbf{E}],\tag{18}$$

and must satisfy the condition divð Þ¼ *<sup>φ</sup>***<sup>Ω</sup> <sup>Ω</sup>** � **<sup>E</sup>** <sup>¼</sup> 0, get **<sup>B</sup>**<sup>2</sup> <sup>¼</sup> **<sup>Ω</sup>**<sup>2</sup> **E**2 . So how for an electromagnetic wave **<sup>B</sup>**<sup>2</sup> <sup>¼</sup> **<sup>E</sup>**<sup>2</sup> , **<sup>Ω</sup>**<sup>2</sup> <sup>¼</sup> 1 and the invariant is zero, then the only structure corresponding to these conditions is **Ω** ¼ ð Þ 1, 0, 0 (**Table 2**) with a single projection on the axis of propagation (spin) 1. Thus, the four-dimensional vector of the electromagnetic wave is represented as

$$\mathbf{P} = (\boldsymbol{\varrho}, \quad \boldsymbol{\varrho}\mathbf{Q}) = (\boldsymbol{\varrho}, \ \boldsymbol{\varrho}, \ \mathbf{0}, \ \mathbf{0}) = (\boldsymbol{\varrho}, \ \boldsymbol{\varrho}\mathbf{n}), \ \mathbf{P}^2 = \mathbf{0},\tag{19}$$

where n is the unit vector in the direction of wave propagation.

For a monochromatic wave propagating in the direction of the axis **z**^ with a potential *φ*ð Þ *x*, *y* and wavelength *λ*, we have

$$\mathbf{P} = (\varepsilon, \ \mathbf{p}) = \rho(\mathbf{x}, y) \exp\left(\frac{\pi - z}{\lambda} i\right) (\mathbf{1}, \quad \hat{\mathbf{z}}), \ \mathbf{P}^2 = \mathbf{0}, \ \frac{\partial \varepsilon}{\partial \tau} + \text{div} \mathbf{p} = \mathbf{0}. \tag{20}$$

For transverse waves of other fields, you can select one-component structures with other integers in the form **Ω** ¼ ð Þ *m*, 0, 0 , *m* ¼ 1, 2, 3, … (**Table 2**).

#### **5.2 Stationary magnetic field**

$$\begin{split} \mathbf{P} &= (\mathbf{0}, \ \rho \boldsymbol{\Omega}); \quad \mathbf{B} = \operatorname{rot}(\rho \boldsymbol{\Omega}) = \left[ \mathbf{\Omega} \times \frac{\partial \rho}{\partial \mathbf{r}} \right] = \\ & \left( \boldsymbol{\Omega}\_{\mathcal{V}} \frac{\partial \rho}{\partial \mathbf{z}} - \boldsymbol{\Omega}\_{\mathbf{z}} \frac{\partial \rho}{\partial \mathbf{y}}, \ - \, \boldsymbol{\Omega}\_{\mathbf{x}} \frac{\partial \rho}{\partial \mathbf{z}} + \boldsymbol{\Omega}\_{\mathbf{z}} \frac{\partial \rho}{\partial \mathbf{x}}, \ \boldsymbol{\Omega}\_{\mathbf{x}} \frac{\partial \rho}{\partial \mathbf{y}} - \boldsymbol{\Omega}\_{\mathbf{y}} \frac{\partial \rho}{\partial \mathbf{x}} \right) = (\mathbf{0}, \ \mathbf{0}, \ \mathbf{B}). \end{split} \tag{21}$$

For solenoidal magnetic fields having cylindrical symmetry, we have Ω*<sup>z</sup>* ¼ 0 and Ω*<sup>x</sup>* ¼ Ω*<sup>y</sup>* ¼ 1*=*2. The solution for such a system is the vector potential in the form of a structure. **<sup>Ω</sup>** <sup>¼</sup> ð Þ <sup>1</sup>*=*2, 1*=*2, 0 , **<sup>Ω</sup>**<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2 (**Table 2**) with one of the linear functions as an electric field potential

$$
\rho \clubsuit \mathbf{0} = \rho(\mathbf{1}/2, \ \mathbf{1}/2, \ \mathbf{0}); \ \rho = \rho\_0(\mathbf{x} - \mathbf{y}); \ \mathbf{B} = [\mathbf{Q} \times \mathbf{E}] = (\mathbf{0}, \ \mathbf{0}, \ E). \tag{22}
$$

## **5.3 Spinor field with potential** *φ* ¼ *α=r*

For the potential in the form of *φ* ¼ *α=r* we have

$$\mathbf{P} = \left(\frac{a}{r}, \quad \frac{a}{r}\mathbf{\Omega}\right) = \frac{a}{r}(\mathbf{1}, \quad \mathbf{\Omega}), \qquad \mathbf{P}^2 = \frac{a^2}{r^2}\left(\mathbf{1} - \mathbf{\Omega}^2\right) \tag{23}$$

The spinor field of this potential can have different structures and invariant values 1 � **<sup>Ω</sup>**<sup>2</sup> . For Coulomb interaction **<sup>Ω</sup>** <sup>¼</sup> ð Þ 1, 1, 0 , 1 � **<sup>Ω</sup>**<sup>2</sup> ¼ �1, and gravity – **<sup>Ω</sup>** <sup>¼</sup> ð Þ 2, 2, 0 , 1 � **<sup>Ω</sup>**<sup>2</sup> ¼ �7 (**Table 2**). Such structures and values of the spinor representation provide an accurate description of the spectrum of hydrogenlike atoms, the value of fine splitting, the secular shift of Mercury's perihelion, and the angle of deflection of the light beam by the gravitational field of the Sun.

## **6. Equations of relativistic and quantum mechanics with spinor fields**

Adding spinor fields in the equations of relativistic and quantum mechanics for a particle in an external field (**Table 1**) is not associated with any difficulties, since the spin properties of the field are clearly represented as the vector potential of interaction **A** ¼ *φ***Ω**. Equations can be represented as

$$
\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 = \frac{\left(mc^2 + q\rho\right)^2 - \left(q\rho\right)^2 \mathbf{\Omega}^2}{c^2}.\tag{24}
$$

$$\frac{\partial^2 \Psi}{\partial \mathbf{r}^2} - \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} = -\frac{\left(mc^2 + q\rho\right)^2 - \left(q\rho\right)^2 \Omega^2}{\hbar^2 c^2} \Psi. \tag{25}$$

Accordingly, the Hamiltonian of the system can be represented as an expression (4)

$$\mathbf{H} = \frac{E^2 - m^2 c^4}{2mc^2} = \frac{\mathbf{p}^2}{2m} + q\rho + \frac{q^2 \rho^2}{2mc^2} (\mathbf{1} - \mathbf{Q}^2). \tag{26}$$

By presenting momentum as **p** ¼ **p**<sup>0</sup> þ *qφ***Ω***=c*, the Hamiltonian can be represented is as

$$\mathbf{H} = \frac{1}{2m}\mathbf{p}'^2 + \frac{q\rho}{mc}\mathbf{p}'\cdot\mathbf{\Omega} + q\rho + \frac{q^2\rho^2}{2mc^2},\tag{27}$$

and we see that the angular momentum and energy of the particle in the ground state have additional angular momentum and spin state energy. And the expression *qφ***p**<sup>0</sup> � **Ω***=mc*is "spin-orbit interaction"—the interaction of a particle with a spinor field.

If other external fields are present, such as a field with a vector potential **A**, then the equations are represented as

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 = \frac{\left(mc^2 + q\rho\right)^2 - q^2(\mathbf{A} + q\mathbf{B})^2}{c^2} \tag{28}$$

$$\frac{\partial^2 \Psi}{\partial \mathbf{r}^2} - \frac{\partial^2 \Psi}{\partial \mathbf{r}^2} = -\frac{(mc^2 + q\rho)^2 - q^2(\mathbf{A} + \mathbf{\Omega}\rho)^2}{\hbar^2 c^2} \Psi. \tag{29}$$

The Hamiltonian of the system can be represented as

$$\mathbf{H} = \frac{E^2 - m^2 c^4}{2mc^2} = \frac{\mathbf{p}^2}{2m} - q\rho \rho - \frac{q^2 \rho^2}{2mc^2} \left(\mathbf{1} - \mathbf{\Omega}^2\right) - \frac{q\rho}{mc^2} \mathbf{\Omega} \cdot \mathbf{A} - \frac{q^2}{2mc^2} \mathbf{A}^2 \tag{30}$$

*Spinor Fields DOI: http://dx.doi.org/10.5772/intechopen.105569*

By presenting momentum as **p** ¼ **p**<sup>0</sup> þ *q*ð Þ *φ***Ω** þ *q***A** *=c*, the Hamiltonian can be represented is as

$$\mathbf{H} = \frac{1}{2m}\mathbf{p}'^2 + \frac{q}{mc}\mathbf{p}' \cdot (q\mathbf{\Omega} + \mathbf{A}) + q\rho + \frac{q^2\rho^2}{2mc^2}.\tag{31}$$

## **7. Some results of solutions of equations of relativistic and quantum mechanics with spinor fields**

### **7.1 Secular shift of Mercury's perihelion**

Let us consider the motion of a particle with the mass *m* in the field of a point body of mass *m*<sup>0</sup> . Then the problem reduces to an investigation of the motion of the particle in the centrally symmetric gravitational field with the potential �*Gm*<sup>0</sup> *=r*, where *G* is the gravitational constant, *rg* ¼ *Gm*<sup>0</sup> *=c*2.

Choosing the polar coordinates ð Þ *r*, *φ* in the plane of motion, we obtain the Hamilton-Jacobi (24) equation in the form

$$
\left(\frac{\partial \mathbf{S}}{\partial \tau}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 = (mc)^2 \left(\mathbf{1} - \frac{2r\_g}{r} - \left(\mathbf{1}\mathbf{2}^2 - \mathbf{1}\right) \frac{r\_g^2}{r^2}\right). \tag{32}
$$

Let us represent the action *S* in the form

$$S = -\epsilon \tau + M\rho + mcf(r),\tag{33}$$

where *ε* and *M* are the constant energy and angular momentum, respectively. As a result, we obtain

$$\left(\frac{\partial f(r)}{\partial r}\right)^2 = \left(\frac{\varepsilon}{mc}\right)^2 - \mathbf{1} - \frac{2r\_{\rm g}}{r} - \frac{(M/mc)^2 - (\mathbf{1}^2 - \mathbf{1})r\_{\rm g}^{-2}}{r^2} \tag{34}$$

and

$$f(r) = \int \sqrt{\left(\frac{\varepsilon}{mc}\right)^2 - 1 + \frac{2r\_\text{g}}{r} - \frac{\left(M/mc\right)^2 - \left(\Omega^2 - 1\right)r\_\text{g}r\_\text{-}^2}{r^2}} dr.\tag{35}$$

We find trajectories from the condition *<sup>∂</sup>S=∂<sup>M</sup>* <sup>¼</sup> 0, with the use of which we obtain,

$$\rho = -\int \frac{M / m c r\_{\rm g}}{\sqrt{\left(\frac{c}{mc}\right)^2 - \mathbf{1} + 2\frac{r\_{\rm g}}{r} - \left(\left(M / m c r\_{\rm g}\right)^2 - \left(\mathbf{1}^2 - \mathbf{1}\right)^2\right) \frac{r\_{\rm g}^2}{r^2}}} d \frac{r\_{\rm g}}{r},\tag{36}$$

which results in the solution

$$r = r\_{\mathfrak{g}} \left( \left( \frac{M}{mcr\_{\mathfrak{g}}} \right)^2 + 1 \right) \times \underbrace{1}\_{\left( \begin{array}{c} 1 \\ mcr\_{\mathfrak{g}} \end{array} \right)}\_{\left( \begin{array}{c} 1 \\ mcr\_{\mathfrak{g}} \end{array} \right)} \frac{1}{\left( \begin{array}{c} 1 \\ mcr\_{\mathfrak{g}} \end{array} \right)^2 - \left( \begin{array}{c} M \\ mcr\_{\mathfrak{g}} \end{array} \right)^2} \times \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \begin{array}{c} \left( \end{array} \right) \end{array} \right) \end{array} \right) \end{array} \right) \right)} \right) \right) \right) \right) \right)} \right) \right)} \right) \right) \right)} \right)} \right) \right)}\tag{37}$$

The secular shift for the gravitational field (23) with **<sup>Ω</sup>** <sup>¼</sup> ð Þ 2, 2, 0 , 1 � **<sup>Ω</sup>**<sup>2</sup> ¼ �<sup>7</sup> is calculated by the formula

$$\Delta\rho = 2\pi - \frac{2\pi}{\sqrt{1 - \left(\Omega^2 - 1\right) \left(G \frac{mm'}{c\mathcal{M}}\right)^2}} \approx -7\pi \left(G \frac{mm'}{c\mathcal{M}}\right)^2. \tag{38}$$

For the Schwarzschild metric, the formula is: �6*<sup>π</sup> Gmm*<sup>0</sup> ð Þ *<sup>=</sup>cJ* <sup>2</sup> [6].

### **7.2 Photon deflection by the gravitational field of the sun**

For a photon whit energy *<sup>E</sup>* and equivalent mass*<sup>m</sup>* <sup>¼</sup> *<sup>E</sup>=c*2, from (20) the Eq. (24) represented in the form

$$\begin{split} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 &= (mc)^2 \left( \left(\mathbf{1} - \frac{r\_\mathbf{g}}{r}\right)^2 - \left(\boldsymbol{\Omega}\_p + \frac{r\_\mathbf{g}}{r}\boldsymbol{\Omega}\_\mathbf{g}\right)^2 \right) = \\ \left((mc)^2 \left(-\frac{2r\_\mathbf{g}}{r} + \boldsymbol{\Omega}\_p \cdot \boldsymbol{\Omega}\_\mathbf{g} \frac{2r\_\mathbf{g}}{r} + \left(\mathbf{1} - \boldsymbol{\Omega}\_\mathbf{g}\right)^2 \frac{r\_\mathbf{g}}{r^2}\right) .\end{split} \tag{39}$$

Choosing the polar coordinates ð Þ *r*, *φ* in the plane of motion, we obtain the Hamilton-Jacobi (24) equation in the form

$$
\left(\frac{\partial \mathbf{S}}{\partial \tau}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 = (mc)^2 \left(-\frac{2r\_\mathbf{g}}{r} - \left(\mathbf{\Omega}\_\mathbf{g}^2 - \mathbf{1}\right) \frac{r\_\mathbf{g}}{r^2}\right) \tag{40}
$$

In this case, we used the choice of special directions of the spins of the particle and the field when **Ω***<sup>p</sup>* � **Ω***<sup>g</sup>* ¼ ð Þ� 0, 0, 1 ð Þ¼ 2, 2, 0 0(spin-spin interactions are absent).

Let us represent the action *S* in the form

$$\mathcal{S} = -m c \tau + M \rho + f(r) = -m c \tau + m c R \rho + f(r), \qquad M = m c R,\tag{41}$$

where *M* and *R* are the constant angular momentum and impact parameter, respectively. As a result, we obtain

$$f(r) = mc \int \sqrt{1 + \frac{2r\_{\rm g}}{r} - \frac{\mathbf{R}^2 - \left(\mathbf{\Omega\_{\rm g}}^2 - \mathbf{1}\right)r\_{\rm g}^2}{r^2}} dr. \tag{42}$$

We find trajectories from the condition *<sup>∂</sup>S=∂<sup>M</sup>* <sup>¼</sup> 0, with the use of which we obtain,

$$\rho = -\int \frac{R}{\sqrt{\mathbf{1} + \frac{2r\_\varepsilon}{r} - \frac{R^2 - \left(\mathbf{\Omega}\_\pi^2 - 1\right)r\_\varepsilon^2}{r^2}}} d\frac{\mathbf{1}}{r}.\tag{43}$$

The photon deflection is calculated by the formula

$$\Delta\rho = 2\arcsin\frac{1}{\sqrt{1-\left(\mathbf{Q}\_{\mathbf{g}}^{2}-\mathbf{1}\right)\left(\frac{r}{R}\right)^{2}}}\approx -2\sqrt{7}\frac{r\_{\mathbf{g}}}{R} = -2\sqrt{7}Gm'/c^{2}R = -5.3Gm'/c^{2}R.\tag{44}$$

For the Schwarzschild metric, the formula is: Δ*φ* ¼ �4*Gm*<sup>0</sup> *=c*2*R* [6].

#### **7.3 The hydrogen atom problem**

For the problem of a hydrogen-like atom without external fields, we have [4].

$$\begin{split} \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \boldsymbol{\mu}}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial \boldsymbol{\mu}}{\partial \theta} \right) + \\\frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 \boldsymbol{\mu}}{\partial \phi^2} + \frac{1}{\hbar^2 c^2} \left( E^2 - \left( mc^2 - \frac{Ze^2}{r} \right)^2 + \frac{Z^2 e^4}{r^2} \boldsymbol{\Omega}^2 \right) \boldsymbol{\mu} = \mathbf{0} \end{split} \tag{45}$$

For the radial part of the solution get

$$\frac{d^2R}{d\rho^2} + \frac{2}{\rho} \frac{dR}{d\rho} - \frac{l(l+1) + (1-\Omega^2)Z^2\alpha^2}{\rho^2}R = -\left(\frac{N}{\rho} - \frac{1}{4}\right)R,\tag{46}$$

Energy levels of the hydrogen atom are (**Figure 11**a)

$$E\_{n,l} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n\_\mathrm{r} + 1/2 + \sqrt{\left(l + 1/2\right)^2 - Z^2 a^2}\right)^2}}.\tag{47}$$

The fine splitting value

$$E\_{2,1} - E\_{1,0} \simeq -\frac{Z^2 a^2}{12}.\tag{48}$$

The ground states are

$$E\_{n,j} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(1/2 + \sqrt{(l+1/2)^2 - Z^2 a^2}\right)^2}}, \qquad l+1/2 > \sqrt{Z^2 a^2 + \sqrt{Za - 1/2}}.\tag{49}$$

Ground state with *Z* > 68 has a *l* more than 0, and the greater the atomic number, the discretely it increases in increments of 1.

**Figure 11.** *Fine splitting structures.*

From the Dirac equation with **<sup>Ω</sup>** <sup>¼</sup> ð Þ 1, 1, 0 , **<sup>Ω</sup>**<sup>2</sup> <sup>¼</sup> 2 we get [4].

$$E\_{nj} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(n\_r + \sqrt{\left(j + 1/2\right)^2 - Z^2 a^2}\right)^2}}.\tag{50}$$

The fine splitting value

$$E\_{2,3/2} - E\_{1,1/2} \simeq -\frac{Z^2 a^2}{32}.\tag{51}$$

The ground states are (**Figure 11a**)

$$E\_{n,j} = mc^2 \sqrt{1 - \frac{Z^2 a^2}{\left(j + 1/2\right)^2 - Z^2 a^2}}, \qquad j + 1/2 > \sqrt{2}Za. \tag{52}$$

Ground state with *Z* >1*=* ffiffi 2 <sup>p</sup> *<sup>α</sup>* <sup>≃</sup>97 has a *<sup>j</sup>* more than 1*=*2, and the greater the atomic number, the discretely it increases in increments of 1.

As we can see, the magnitude of the spin-orbit interaction is obtained with a reverse sign and is twice as large as the relativistic splitting of the electron energy levels, which leads to a shift and a change in the order of the splitting levels (**Figure 11a**).

Dirac's solution prompt that if 1/2 is added to the orbital momentum *l* ) *l* þ 1*=*2 ¼ *j* and 1 to *nr* ) *nr* � 1*=*2 in solutions without spatial spin (47), then the results of the solutions will be the same. That is, it should be taken into account that in the expression of the orbital moment and energy there are initially present (26–27), independent of the state and fields of the system, a spatial spin with the corresponding value *j* ¼ *l* þ 1*=*2and energy *n* ¼ *nr* þ *l* þ 1 ) *n* þ *j* þ 1*=*2.

#### **7.4 Hydrogen atom in a magnetic field**

For the hydrogen atom problem in a constant homogeneous magnetic field **B** we have (30)

$$\frac{q^2}{2m^2c^4} \left( \frac{Q}{r} \mathbf{\hat{\Omega}} - \frac{1}{2} [\mathbf{B} \times \mathbf{r}] \right)^2 = a^2 \left( 2Z^2 \frac{\mathbb{X}^2}{r^2} + \frac{1}{2} \left( \frac{r}{r\_h} \right)^2 \sin^2 \theta + Z \frac{\mathbb{X}}{r\_h} \sin \theta \right), \tag{53}$$

where

$$\mathbb{X} = \frac{\hbar}{mc}, \qquad a = \frac{e^2}{\hbar c}, \qquad r\_h = \frac{c}{o\nu}; \qquad o = \frac{eB}{2mc}. \tag{54}$$

As we can see, the splitting of levels in a constant magnetic field due to the linear dependence on the field *<sup>Z</sup>α*<sup>2</sup>ð Þ <sup>ƛ</sup>*=rh* sin *<sup>θ</sup>*does not depend on the state of the atom.

#### **7.5 About hyperfine splitting**

On hyperfine splitting: why when an electron interacts with nuclear spin, there are only two levels, and a complex structure manifests itself only in the external magnetic field (**Figure 12**).

#### **Figure 12.**

*Theoretical magnetic field dependence of* Fg *= 2, 3 ground hyperfine levels of 85Rb. Red lines: Calculations by the coupled basis theory; black lines: Calculations as given by Eq. (12) (HPB regime). Ground levels for the transitions 4–9 are indicated as (4)g* � *(9)g [8].*

As has been shown in solutions to the hydrogen-like atom problem for spinor fields, the magnitude of the spin-orbit interaction **L** � **S**is obtained with a reverse sign and is twice as large as the relativistic splitting of the electron energy levels, which leads to a shift and a change in the order of the splitting levels (**Figure 11**a). In the ground state, when the nuclear spin **I** interacts with the spinor field (**Figure 11**b), the spin-spin interaction of the nucleus **I** � **S**, there will be two times less orbital interaction and relativistic splitting (convergence points of black lines in **Figure 12**) will be accurately compensated by the magnitude of the spin-spin interaction of the nucleus (red lines).

## **8. Representation of the relativistic and quantum mechanics equations in metric spaces**

The representation of the equations of relativistic and quantum mechanics in metric spaces means that coordinate transformations have been proposed that bring the equations from **Table 1** to equations with a constant, unit invariant on the right part

$$\begin{aligned} \left(\frac{\partial \mathbb{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbb{S}}{\partial \mathbf{r}}\right)^2 &= I^2 \Rightarrow \quad \mathbf{g}\_{ik}\sqrt{-\mathbf{g}}\frac{\partial \mathbb{S}}{\partial \mathbf{x}^i}\frac{\partial \mathbb{S}}{\partial \mathbf{x}^k} = \mathbf{1},\\ \frac{\partial^2 \mathbb{W}}{\partial \mathbf{r}^2} - \frac{\partial^2 \mathbb{W}}{\partial \mathbf{r}^2} &= -\frac{I^2}{\hbar^2}\mathbb{W} \Rightarrow \quad \frac{1}{\sqrt{-\mathbf{g}}}\frac{\partial}{\partial \mathbf{x}^i}\left(\mathbf{g}\_{ik}\sqrt{-\mathbf{g}}\frac{\partial \mathbb{W}}{\partial \mathbf{x}^k}\right) &= -\mathbb{W}.\end{aligned} \tag{55}$$

For clarity, consider the problems with spherical symmetry. For the Hamilton-Jacobi equation in spherical coordinates we have

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \frac{1}{r^2 \sin^2 \theta} \left(\frac{\partial \mathbf{S}}{\partial \theta}\right)^2 = I(r)^2. \tag{56}$$

and after the coordinates are transformed *τ*, **r** ! *τ*<sup>0</sup> , **r**<sup>0</sup> , they must be represented in metric spaces as

$$\left(\frac{\partial \mathbf{S}}{\partial r'}\right)^2 - g^2(r')\left(\frac{\partial \mathbf{S}}{\partial r'}\right)^2 - \frac{1}{r'^2}\left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \frac{1}{r'^2 \sin^2 \theta} \left(\frac{\partial \mathbf{S}}{\partial \theta}\right)^2 = 1. \tag{57}$$

To do this, divide the Eq. (56) by *I r*ð Þ<sup>2</sup>

$$\frac{1}{I^2} \left(\frac{\partial \mathbf{S}}{\partial \tau}\right)^2 - \frac{1}{I^2} \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{I^2 r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \frac{1}{I^2 r^2 \sin^2 \theta} \left(\frac{\partial \mathbf{S}}{\partial \theta}\right)^2 = 1. \tag{58}$$

and we immediately find an implicit transformation for*r*

$$I(r)r = r'.\tag{59}$$

and

$$\frac{r^2}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial \tau}\right)^2 - \frac{r^2}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{\mathbf{1}}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \frac{\mathbf{1}}{r'^2 \sin^2 \theta} \left(\frac{\partial \mathbf{S}}{\partial \theta}\right)^2 = \mathbf{1}.\tag{60}$$

It is convenient to choose a new coordinate system associated with a particle, where the velocity is zero *dr*<sup>0</sup> *=dτ*<sup>0</sup> � 0. Then,

$$d\tau = \frac{\partial \tau}{\partial \tau'} d\tau' + \frac{\partial \tau}{\partial r'} dr' = \left(\frac{\partial \tau}{\partial \tau'} + \frac{\partial \tau}{\partial r'} \frac{dr'}{d\tau'}\right) d\tau' = \frac{\partial \tau}{\partial \tau'} d\tau'.$$

$$\frac{\partial \mathcal{S}}{\partial \tau} = \frac{\partial \mathcal{S}}{\partial \tau'} \frac{d\tau'}{d\tau} + \frac{\partial \mathcal{S}}{\partial r'} \frac{dr'}{d\tau'} = \frac{\partial \mathcal{S}}{\partial r'} \frac{\partial \tau'}{\partial \tau}.\tag{61}$$

and we get

$$\frac{r^2}{r r^2 \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 - \frac{r^2}{r'^2 \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 - \frac{1}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\rho}}\right)^2 - \frac{1}{r'^2 \sin^2(\boldsymbol{\theta})} \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\theta}}\right)^2 = \mathbf{1}.\tag{62}$$
 
$$\mathbf{v} \quad \text{d}\mathbf{r}$$

$$\log(r') = \frac{r}{r'} / \frac{\partial r}{\partial r'} \tag{63}$$

Note that if the *I r*ð Þ¼ *const*, then *g r*<sup>0</sup> ð Þ¼ 1 – only the linear scale changes.

## **8.1 Field with scalar potential** *φ*ð Þ¼� *r rg=r*

For an invariant in the form *I r*ð Þ¼ 1 � *rg=r* (Coulomb, gravitational field) we find

$$r' = rI(r) = r - r\_{\mathfrak{g}}, \qquad r = r' + r\_{\mathfrak{g}}, \quad \frac{\partial r}{\partial r'} = \mathbf{1}, \tag{64}$$

$$
\tau = \frac{r}{r'}\tau' = \left(\mathbf{1} + \frac{r\_\mathbf{g}}{r'}\right)\tau', \quad \frac{\partial \tau}{\partial \tau'} = \frac{r}{r'} = \mathbf{1} + \frac{r\_\mathbf{g}}{r'}, \tag{65}
$$

$$\mathbf{g}(r') = \frac{r}{r'} / \frac{\partial r}{\partial r'} = \mathbf{1} + \frac{r\_{\mathbf{g}}}{r'}. \tag{66}$$

*Spinor Fields DOI: http://dx.doi.org/10.5772/intechopen.105569*

From (43), (67), we get

$$\left(\frac{\partial \mathbf{S}}{\partial r'}\right)^2 - \left(\mathbf{1} + \frac{r\_{\mathbf{g}}}{r'}\right)^2 \left(\frac{\partial \mathbf{S}}{\partial r'}\right)^2 - \frac{\mathbf{1}}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\rho}}\right)^2 - \frac{\mathbf{1}}{r'^2 \sin^2(\boldsymbol{\theta})} \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\theta}}\right)^2 = \mathbf{1}.\tag{67}$$

If the field is spinor with invariant *I* <sup>2</sup> <sup>¼</sup> <sup>1</sup> � *rg=<sup>r</sup>* � �<sup>2</sup> � **<sup>Ω</sup>***rg=<sup>r</sup>* � �<sup>2</sup> , then from getting

$$
\tau = r\_{\mathfrak{g}} + \sqrt{r'^2 + \Omega^2 r\_{\mathfrak{g}}^2}, \quad \tau = \frac{r}{r'} \tau' = \left(\frac{r\_{\mathfrak{g}}}{r'} + \sqrt{\mathbf{1} + \Omega^2 r\_{\mathfrak{g}}^2 / r'^2}\right) \tau'. \tag{68}
$$

and

$$\log^2(r') = \left(\mathbf{1} + \mathfrak{Q}^2 \frac{r\_\mathcal{g}^2}{r'^2}\right) \left(\mathbf{1} + \frac{r\_\epsilon}{r'} + \frac{\mathfrak{Q}^2 r\_\mathcal{g}^2 / r'^2}{\mathbf{1} + \sqrt{\mathbf{1} + \mathfrak{Q}^2 r\_\mathcal{g}^2 / r'^2}}\right)^2. \tag{69}$$

#### **8.2 Stationary magnetic field**

For a magnetic field with invariant *I* <sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>r</sup>*<sup>2</sup>*=rB* 2, where *rB* the magnetic event horizon (54), is in cylindrical coordinates we have

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial r}\right)^2 - \frac{1}{r^2} \left(\frac{\partial \mathbf{S}}{\partial \rho}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{z}}\right)^2 = I^2 = \mathbf{1} - \left(\frac{r}{r\_{\mathcal{B}}}\right)^2,\tag{70}$$

and we get (*τ* and *z* are transforming similarly)

$$\left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 - 2\frac{\mathbf{1} - \left(2\mathbf{r}'/r\_B\right)^2}{\mathbf{1} + \sqrt{\mathbf{1} - \left(2\mathbf{r}'/r\_B\right)^2}} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 - \frac{\mathbf{1}}{r'^2} \left(\frac{\partial \mathbf{S}}{\partial \boldsymbol{\rho}}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{z}'}\right)^2 = \mathbf{1}.\tag{71}$$

$$\log^2(r') = 2\frac{\mathbf{1} - \left(2r'/r\_B\right)^2}{\mathbf{1} + \sqrt{\mathbf{1} - \left(2r'/r\_B\right)^2}}\tag{72}$$

#### **8.3 Representations of the space-time interval and the wave equation**

The results of the metric representations are easily portable to represent the spacetime interval (it should be noted that for the space-time interval, the metric tensor is inverse). For example, for a gravitational field with a potential *rg=r*<sup>0</sup> , we have (68)

$$\begin{split} ds^2 &= I^2(d\tau^2 - dr^2 - r^2 d\rho^2 - r^2 \sin^2(\theta) d\theta^2) = \\ \sigma &= d\tau'^2 - \frac{dr'^2}{g^2(r')} - r'^2 d\rho^2 - r'^2 \sin^2(\theta) d\theta^2, \\ ds^2 &= ds'^2 = d\tau'^2 - \frac{1}{\left(1 + r\_\xi/r'\right)^2} dr'^2 - r'^2 d\rho^2 - r'^2 \sin^2(\theta) d\theta^2. \end{split} \tag{73}$$

and the wave equation in cylindrical and spherical coordinates (*τ* and *z* are transforming similarly)

$$\frac{\partial^2 \boldsymbol{\Psi}}{\partial \boldsymbol{r}^2} - \left(\mathbf{1} + \frac{r\_\xi}{\rho'}\right)^2 \frac{\partial^2 \boldsymbol{\Psi}}{\partial \rho'^2} - \frac{1}{\rho'} \left(\mathbf{1} + \frac{r\_\xi}{\rho'}\right) \frac{\partial \boldsymbol{\Psi}}{\partial \rho'} - \frac{\mathbf{1}}{\rho'^2} \frac{\partial^2 \boldsymbol{\Psi}}{\partial \rho^2} - \frac{\partial^2 \boldsymbol{\Psi}}{\partial \mathbf{z}^2} = \boldsymbol{\Psi},\tag{74}$$
 
$$\frac{\partial^2 \boldsymbol{\Psi}}{\partial \boldsymbol{r}'^2} - \left(\mathbf{1} + \frac{r\_\xi}{\rho'}\right)^2 \frac{\partial^2 \boldsymbol{\Psi}}{\partial \boldsymbol{r}'^2} - \frac{2}{r'} \left(\mathbf{1} + \frac{r\_\xi}{r'}\right) \frac{\partial \boldsymbol{\Psi}}{\partial \boldsymbol{r}'} - \frac{\mathbf{1}}{r'^2} \frac{\partial^2 \boldsymbol{\Psi}}{\partial \rho^2} - \frac{1}{r'^2 \sin\left(\theta\right)} \frac{\partial}{\partial \theta} \left(\sin\left(\theta\right) \frac{\partial \boldsymbol{\Psi}}{\partial \theta}\right) = \boldsymbol{\Psi} \tag{75}$$

For more complex cases, we will only point out that first, you need to convert the coordinates to bring the expression of the invariant to a diagonal (spherically or cylindrically symmetric) form, and then transform the resulting equation and invariant in the above way.

#### **8.4 Inverse problem**

If an equation is given in metric space that has spherical symmetry, it is easy to perform inverse transformations and find the corresponding invariant (field) from (59), (64)

$$g(r') = \frac{r}{r'} \frac{dr'}{dr}; \qquad \frac{dr}{r} = \frac{dr'}{r'g(r')}; \qquad r = e^{\int \frac{dr'}{r'g(r')}}, \qquad I(r) = \frac{r'}{r}. \tag{76}$$

Within the framework of developed approaches, there is an exact correspondence of representations of equations in metric spaces. This is important because it is possible to unambiguously find out which fields correspond to given metric spaces and vice versa.

Naturally, all solutions of equations in metric spaces correspond to solutions of the original equations of relativistic and quantum mechanics having corresponding trajectories of motion and describing the quantum properties of systems.

#### **8.5 Equations in the gravitation field**

From the point of view of general relativity, the gravitational field is universal, and all physical processes are described already in metric space. In general, the equations can be represented as follows:

$$\mathcal{g}\_{ik}\sqrt{-\mathcal{g}}\,\frac{\partial \mathcal{S}}{\partial \mathbf{x}^i}\,\frac{\partial \mathcal{S}}{\partial \mathbf{x}^k} = I^2,\tag{77}$$

$$\frac{1}{\sqrt{-\overline{\mathbf{g}}}} \frac{\partial}{\partial \mathbf{x}^i} \left( \mathbf{g}\_{ik} \sqrt{-\overline{\mathbf{g}}} \frac{\partial \boldsymbol{\mu}}{\partial \mathbf{x}^k} \right) \\ \quad = -\frac{I^2}{\hbar^2} \boldsymbol{\mu} . \tag{78}$$

where *gik* is the metric tensor determined by a given gravitational field. After the appropriate conversion *x* ! *x*<sup>0</sup>

$$\begin{aligned} \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 - \left(\frac{\partial \mathbf{S}}{\partial \mathbf{r}'}\right)^2 &= I\_{\mathbf{g}}^2(\mathbf{x}')I^2(\mathbf{x}(\mathbf{x}')) \\ \frac{\partial^2 \mathbf{y}}{\partial \mathbf{r}'^2} - \frac{\partial^2 \mathbf{y}}{\partial \mathbf{r}'^2} &= -\frac{I\_{\mathbf{g}}^2(\mathbf{x}')I^2(\mathbf{x}(\mathbf{x}'))}{\hbar^2} \mathbf{y}. \end{aligned} \tag{79}$$

For example, if a hydrogen-like atom is in a constant homogeneous gravitational spinor field *φ*<sup>0</sup> ð Þ , *φ*0**Ω** , in the field in which the particle has an invariant *Ig* ¼ 1 � 2*φ*<sup>0</sup> � *φ*<sup>0</sup> <sup>2</sup> **<sup>Ω</sup>**<sup>2</sup> � <sup>1</sup> is considered, we get

$$\begin{split} & \left(\frac{\partial \mathbb{S}}{\partial \boldsymbol{r}'}\right)^2 - \left(\frac{\partial \mathbb{S}}{\partial \boldsymbol{r}'}\right)^2 - \frac{1}{r'^2} \left(\frac{\partial \mathbb{S}}{\partial \boldsymbol{q}}\right)^2 - \frac{1}{r'^2 \sin^2(\boldsymbol{\theta})} \left(\frac{\partial \mathbb{S}}{\partial \boldsymbol{\theta}}\right)^2 = I\_{\mathbb{S}}^2 I\_{\boldsymbol{\epsilon}}^2 = \\ & \left(1 - 2\rho\_0 - \rho\_0^2 \left(\boldsymbol{\Omega}\_{\mathbb{S}}^2 - 1\right)\right) \left(1 - \frac{2r\_{\boldsymbol{\epsilon}}}{r(r')} - \left(\boldsymbol{\Omega}\_{\mathbb{S}}^2 - 1\right) \frac{r\_{\boldsymbol{\epsilon}}^2}{r^2 (r')}\right). \end{split} \tag{80}$$

A constant homogeneous gravitational field changes the scale of the coordinates by a constant coefficient.

## **9. Conclusion**

Spinor representation of the generalized energy-momentum density 4-vector is proposed, based on the representation of local relativistic rotation by the Lorentz transformation matrix. This representation corresponds to the classical representation of the particle's own rotation, which is described by the diagonal matrix of the moment of inertia.

The spin of the field is invariant from the point of view of general relativity, it does not depend on the state of the system and the sources of its creation.

Spin is a fundamental, unchanging characteristic of the field. Whichever way it is created, it will be with the same back. The spin of particles is not determined by their internal structure, but is determined by the spin of the interaction fields created by these particles.

Spin is a spatial characteristic and is not attributed to any point particle. The phrase "spin-orbit interaction" in this case means the interaction of the orbital moment with the spinor field.

Solutions to the problems of the motion of particles in various external spinor fields are presented. The results of solving these problems show that the developed approach correctly describes the physical properties of the interaction of particles and fields.

The proposed representation of spinor fields applies to the equations of relativistic and quantum mechanics and their representation in metric spaces.

On the other hand, these proposed approaches are still new and need more detailed theoretical and experimental studies and a more developed and rigorous formulation of the mathematical foundations of the new theory.

All aspects and a more complete presentation of the new theory will be presented shortly in the author's book "Relativistic and Quantum Mechanics – with new formulations of principles and theory".

## **Conflict of interest**

The author declares no conflict of interest.

## **Author details**

Vahram Mekhitarian1,2

1 Institute for Physical Research of Armenian National Academy of Sciences, Ashtarak, Armenia

2 Russian–Armenian University, Yerevan, Armenia

\*Address all correspondence to: vm@ipr.sci.am; vahram.mekhitarian@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Spinor Fields DOI: http://dx.doi.org/10.5772/intechopen.105569*

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[2] Mekhitarian V. Canonical solutions of variational problems and canonical equations of mechanics. Journal of Contemporary Physics. 2013;**48**:1-11. DOI: 10.3103/S1068337213010015

[3] Mekhitarian V. Equations of relativistic and quantum mechanics and exact solutions of some problems. Journal of Contemporary Physics. 2018;**53**:1-21. DOI: 10.3103/S1068337218020123

[4] Mekhitarian V. Equations of relativistic and quantum mechanics (without spin). In: Bracken P, editor. Quantum Mechanics. London: IntechOpen; 2020. pp. 107-137. DOI: 10.5772/intechopen.93336

[5] Mekhitarian V. The faraday law of induction for an arbitrarily moving charge. Journal of Contemporary Physics. 2016;**51**:108-126. DOI: 10.3103/ S1068337216020031

[6] Landau LD, Lifshitz EM. The Classical Theory of Fields. 4th ed. Vol. 2. Oxford: Butterworth-Heinemann; 1980. p. 444

[7] Alkali D. Line Data [Online]. 2022. Available from: https://steck.us/alkalidata/

[8] Sargsyan A, Hakhumyan G, Leroy C, Pashayan-Leroy Y, Papoyan A, Sarkisyan D, et al. Hyperfine Paschen-Back regime in alkali metal atoms: Consistency of two theoretical considerations and experiment. Optical Society of America B. 2014;**31**(5):1046-1053. DOI: 10.1364/ JOSAB.31.001046

## **Chapter 2** Key Outcomes of 5D Relativity *Detlef Hoyer*

## **Abstract**

Interstellar travel needs enormous amounts of energy to accelerate payload, structure, and propellant to high speeds. For a travel to a distant star all three have to be huge. Chemical propellant needs more than 90% of the whole start weight, even from an orbit around the Moon. Most of the energy is necessary to increase kinetic energy. Because kinetic energy depends on inertial mass, a reduction of effective energy could reduce the amount of energy needed. As inertial mass and gravitational mass are the same following the equivalence principle of General Relativity and gravity depends on the gravitational constant, a local variation of the gravitational constant might result in variation of the effective mass. A varying scaling between mass/energy and spacetime could also cause a new force as known form the Pioneer anomaly. Kaluza-Klein Theory is General Relativity extended to 5 dimensions (5D). The gravitational constant is substituted by a scalar field making it variable. This scalar field is predicted to change under strong dynamic electromagnetic fields. Deriving the equation of motion from this 5D-metric predicts a fifth force.

**Keywords:** 5D relativity, 5D electromagnetism, scalar field, fifth force, interstellar travel

## **1. Introduction**

The basic idea of classical Kaluza-Theory [1] is briefly presented. Many following introductory and basic passages and also figures are taken from a conference paper published with the ASCEND 2021 conference [2]. The focus here is on the didactic presentation of the results in the form of 10 key statements, which are listed and explained in different sections. This 5D Relativity theory is named "induced matter theory," because no extra 5D stress-energy-momentum tensor is necessary, but the known 4D stress-energy-momentum tensor is generated or "induced" by curvature of an empty 5D space. The key results are as follows:

	- ix. there is an inhomogeneous wave equation for the scalar field with excitation by variation of the ten 4D gravitiy-potentials with the 5th dimension
	- x. there is a new rectilinear component of the Lorentz force that causes acceleration that could look like a reduction in inertia.

## **2. Classical electrodynamics retrospected**

Maxwell's eqs. (3D plus time) specify sources and curls for electric field **E** and magnetic field **B** by which the whole fields are determined according to Helmholtz's theorem [3]. As basic mathematical laws they do not reveal which quantity is a cause and which is a consequence. This way the 3rd Maxwell equation

$$
\nabla \times \mathbf{E} = -\mu\_0 \frac{\partial \mathbf{B}}{\partial t} \tag{1}
$$

only declares two simultaneously occurring effects always as equal (non-causal). The cause for all electric and magnetic fields are always charges and currents (charges at rest and charges in motion) [4]. The coupling of the fields **E** and **B** by Maxwell's equations means that both appear as a dual entity. So both are parts of one electromagnetic field with six components (3 electric and 3 magnetic) and instead of electricity and magnetism we speak about electromagnetism.

#### **2.1 Electric and magnetic field derived from four scalar potentials**

From Maxwell's equations follows that both fields can be derived from potentials. Maxwell's 2nd equation states that there are no sources and only curls for magnetic fields; thus, the magnetic field **B** can be written as rotation of another vector field **A**(**r**, *t*) which depends on position and time:

$$
\nabla \cdot \mathbf{B} = \mathbf{0} \Leftrightarrow \mathbf{B} = \nabla \times \mathbf{A} \quad \text{with} \quad \mathbf{A} = (A\_1, A\_2, A\_3) \tag{2}
$$

The vector potential **A** can be used in the 3rd Maxwell equation (Eq. (1)) and we get:

$$\mathbf{O} = \nabla \times \mathbf{E} + \mu\_0 \frac{\partial}{\partial t} \nabla \times \mathbf{A} = \nabla \times \left(\mathbf{E} + \mu\_0 \frac{\partial \mathbf{A}}{\partial t}\right) \tag{3}$$

Because the term in the brackets has a curl of zero, it can be written as a gradient of a scalar potential *φ*:

$$\mathbf{E} = -\mu\_0 \frac{\partial \mathbf{A}}{\partial t} + \nabla \varrho(\mathbf{r}, t) \tag{4}$$

Thus, it is possible to calculate **B** and **E** also from **A** and *φ*. The electromagnetic potentials introduced in classical theory—formally as helping quantities—appear in other theories as real necessary fields with physical meaning (e.g., Schrödingerequation, QED, Proca-equation [3]).

## **2.2 Electromagnetic field tensor: a dual entity derived from 4-dimensional rotation of four vector potential**

In special Relativity (4D) the electromagnetic potentials **A** and *φ* are combined to a 4-vector, setting *A*<sup>0</sup> or *A*<sup>4</sup> to *φ*. The Faraday tensor or field tensor is defined by the 16 components [5]:

$$F\_{ik} = \frac{\partial A\_k}{\partial \mathbf{x}\_i} - \frac{\partial A\_i}{\partial \mathbf{x}\_k} \quad \text{with} \quad i, k = \mathbf{1}, 2, 3, 4 \tag{5}$$

These are the partial derivatives of 4-*dimensional* (4D) *rotation* of *Ai*. They also occur in 3D rotation and in the gradient, that way the Faraday-tensor contains directly electric and magnetic field components. In Cartesian coorinates they represent Eqs. (1) and (2):

$$\begin{aligned} B\_{\mathbf{x}} &= \frac{\partial A\_3}{\partial \mathbf{x}\_2} - \frac{\partial A\_2}{\partial \mathbf{x}\_3} \\ B\_{\mathbf{y}} &= \frac{\partial A\_1}{\partial \mathbf{x}\_3} - \frac{\partial A\_3}{\partial \mathbf{x}\_1} \\ B\_{\mathbf{z}} &= \frac{\partial A\_2}{\partial \mathbf{x}\_1} - \frac{\partial A\_1}{\partial \mathbf{x}\_2} \end{aligned} \qquad \begin{aligned} E\_{\mathbf{x}} &= \dot{\mathbf{c}} \left( \frac{\partial A\_4}{\partial \mathbf{x}\_1} - \frac{\partial A\_1}{\partial \mathbf{x}\_4} \right) \\ E\_{\mathbf{y}} &= \dot{\mathbf{c}} \left( \frac{\partial A\_4}{\partial \mathbf{x}\_2} - \frac{\partial A\_2}{\partial \mathbf{x}\_4} \right) \\ E\_{\mathbf{z}} &= \dot{\mathbf{c}} \left( \frac{\partial A\_4}{\partial \mathbf{x}\_3} - \frac{\partial A\_3}{\partial \mathbf{x}\_4} \right) \end{aligned} \tag{6}$$

which were derived from 2nd and 3rd Maxwell's equation. The tensor *Fik* is the 4D rotation of four vectors *Ai*. It can be written this way with use of the electromagnetic field quantities:

$$\mathbf{F} = (F\_{ik}) = \begin{bmatrix} \mathbf{0} & B\_x & -B\_y & -\frac{i}{c}E\_x \\\\ -B\_x & \mathbf{0} & -B\_x & -\frac{i}{c}E\_y \\\\ B\_y & B\_x & \mathbf{0} & -\frac{i}{c}E\_x \\\\ -\frac{i}{c}E\_x & -\frac{i}{c}E\_y & -\frac{i}{c}E\_x & \mathbf{0} \end{bmatrix} \tag{7}$$

This tensor is a physical quantity, which unifies electric and magnetic field components into one dual entity: the electromagnetic field tensor F.

#### **2.3 Stress-energy-momentum tensor**

The *Maxwell stress tensor σ* (3D) is used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum ([6], p. 72, 931). Starting with the Lorentz force law **F** ¼ *q*ð Þ **E** þ **v** � **B** the force per unit volume is **f** ¼ *ρ* **E** þ **j** � **B**. Replacing sources *ρ* and curls **j** with Maxwell's equations by functions of the fields **E** and **B**, eliminating the curls by vector calculus identities [7] and writing the result in a more compact way one gets:

$$
\sigma\_{\vec{\eta}} = \varepsilon\_0 E\_i E\_j + \frac{1}{\mu\_0} B\_i B\_j - \frac{1}{2} \delta\_{\vec{\eta}} \left( \varepsilon\_0 E^2 + \frac{1}{\mu\_0} B^2 \right) \quad \text{in} \qquad \text{SI units} \tag{8}
$$

Which is explicitly in cartesian coordinates

$$
\sigma = \begin{pmatrix}
\varepsilon\_{0}E\_{x}E\_{x} + \frac{1}{\mu\_{0}}B\_{x}B\_{x} - \frac{1}{2}(E^{2} + H^{2}) & \varepsilon\_{0}E\_{x}E\_{y} + \frac{1}{\mu\_{0}}B\_{x}B\_{y} & \varepsilon\_{0}E\_{x}E\_{x} + \frac{1}{\mu\_{0}}B\_{x}B\_{x} \\
\varepsilon\_{0}E\_{j}E\_{x} + \frac{1}{\mu\_{0}}B\_{j}B\_{x} & \varepsilon\_{0}E\_{j}E\_{j} + \frac{1}{\mu\_{0}}B\_{j}B\_{j} - \frac{1}{2}(E^{2} + B^{2}) & \varepsilon\_{0}E\_{j}E\_{x} + \frac{1}{\mu\_{0}}B\_{j}B\_{x} \\
\varepsilon\_{0}E\_{z}E\_{x} + \frac{1}{\mu\_{0}}B\_{z}B\_{x} & \varepsilon\_{0}E\_{z}E\_{z} + \frac{1}{\mu\_{0}}B\_{z}B\_{y} & \varepsilon\_{0}E\_{z}E\_{z} + \frac{1}{\mu\_{0}}B\_{z}B\_{z} - \frac{1}{2}(E^{2} + B^{2}) \\
\end{pmatrix} \tag{9}
$$

This stress tensor can be used to derive the *Lorentz force* per unit volume by the *divergence* of the stress tensor as

$$\mathbf{f} = \text{div } \boldsymbol{\sigma} + \varepsilon\_0 \mu\_0 \frac{\partial \mathbf{S}}{\partial t} \quad \text{with Poynting vector} \quad \mathbf{S} = \frac{1}{\mu\_0} \mathbf{E} \times \mathbf{B} \tag{10}$$

With potentials instead of fields (Eq. (6)) and using the definition of the field tensor *Fμλ* (Eq. (5)) one gets the relativistic electromagnetic stress–energy-momentum tensor *Tμν* (4D):

$$T\_{\mu\nu} = \frac{1}{\mu\_0} \left[ F\_{\mu\lambda} F\_{\lambda\nu} + \frac{1}{4} \delta\_{\mu\nu} (F\_{\kappa\lambda} F\_{\kappa\lambda}) \right] \tag{11}$$

which is explicitly in matrix form (in SI units):

$$\mathbf{T} = (T\_{ik}) = \begin{bmatrix} \frac{1}{2} \left( \varepsilon\_0 E^2 + \frac{1}{\mu\_0} B^2 \right) & \mathbf{S}\_x / c & \mathbf{S}\_y / c & \mathbf{S}\_z / c \\\\ \mathbf{S}\_x / c & -\sigma\_{11} & -\sigma\_{12} & -\sigma\_{13} \\\\ \mathbf{S}\_y / c & -\sigma\_{21} & -\sigma\_{22} & -\sigma\_{23} \\\\ \mathbf{S}\_z / c & -\sigma\_{31} & -\sigma\_{32} & -\sigma\_{33} \end{bmatrix} \tag{12}$$

where **<sup>S</sup>** <sup>¼</sup> <sup>1</sup> *μ*0 **<sup>E</sup>** � **<sup>B</sup>** is the Poynting vector, *<sup>σ</sup>ij* <sup>¼</sup> *<sup>ε</sup>*0*EiEj* <sup>þ</sup> <sup>1</sup> *μ*0 *BiBj* � 1 <sup>2</sup> *<sup>ε</sup>*0*E*<sup>2</sup> <sup>þ</sup> <sup>1</sup> *μ*0 *<sup>B</sup>*<sup>2</sup> � �*δij* is the Maxwell stress tensor, and the speed of light is *<sup>c</sup>* <sup>¼</sup> ffiffiffiffiffiffiffi 1 *μ*0*ε*<sup>0</sup> q . Eq. (10)—the Lorentz force per unit volume - can be rewritten as:

*Key Outcomes of 5D Relativity DOI: http://dx.doi.org/10.5772/intechopen.110241*

$$\mathbf{f} = \text{div } \mathbf{T} \tag{13}$$

## **3. Electromagnetism from 5D**

Kaluza found that the relativistic electromagnetic stress–energy-momentum tensor **T** from Eqs. (11) and (12) is mathematically structural equal to a part of a 5D curvature tensor. To see this, the electromagnetic vector potential **A**, which had initially 3 spatial components *Ax*,*Ay*,*Az* and was extended by a fourth (time) component *At*, now is extended again by a fifth component *Al* to a 5-vector

$$\mathbf{A} = \left(\mathbf{A}\_l, \mathbf{A}\_x, \mathbf{A}\_y, \mathbf{A}\_z, \mathbf{A}\_l\right). \tag{14}$$

The component along the ℓ-axis shall have constant length and may be slanted giving a picture (**Figure 1**) like the hairs of a horses coat or fur, in a pelt or like the tufts of a carpet:

**Figure 1.**

*4-brane (x,y,z,t) with upright 5th component* ℓ *(axes of a pyramid).*

Φ is the component orthogonal to all spacetime components, which are also perpendicular to each other, so the sum

$$A = \sqrt{A\_t^2 + A\_x^2 + A\_y^2 + A\_z^2 + \Phi^2} \tag{15}$$

is the length of **A**. To show the capabilities of the Ricci curvature tensor in 5D we take the Minkowski metric and add the 5-vector **A** as fifth row and column.

$$\mathbf{g}^{AB} = \begin{pmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & A\_t \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & A\_x \\ \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} & A\_y \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{1} & A\_x \\ A\_t & A\_x & A\_y & A\_x & -\mathbf{0}^2 \end{pmatrix} \tag{16}$$

With *Al* <sup>¼</sup> 1 and *At* <sup>¼</sup> *<sup>φ</sup>* <sup>¼</sup> *<sup>q</sup> <sup>r</sup>* the fifth diagonal element becomes *Recent Topics and Innovations in Quantum Field Theory*

$$\Phi^2 = A\_l^2 - A\_t^2 = 1 - \rho^2 = 1 - \frac{q^2}{r^2} \tag{17}$$

and setting *Ax* <sup>¼</sup> *Ay* <sup>¼</sup> *Az* <sup>¼</sup> 0 (no magnetic field) we get with *<sup>r</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>p</sup> :

$$\mathbf{g}^{AB} = \begin{pmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \frac{\mathbf{q}}{\mathbf{r}}\\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} \\ \frac{\mathbf{q}}{\mathbf{r}} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\left(\mathbf{1} - \frac{\mathbf{q}^2}{\mathbf{r}^2}\right) \end{pmatrix} \tag{18}$$

Applying the 5D Einstein tensor *<sup>G</sup>AB* <sup>¼</sup> *<sup>R</sup>AB* � <sup>1</sup> <sup>2</sup> *gAB R* multiplied by two we get (Maple-instruction: 2\*Einstein[�]):

$$\begin{aligned} 2 \cdot \mathbf{G}^{AB} &= \begin{bmatrix} -\frac{q^2}{r^4} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\frac{\sqrt{2}q^3}{r^5} \\ \mathbf{0} & \left(\frac{qx}{r^5}\right)^2 - \frac{q^2}{2r^4} & \frac{qx}{r^3}\frac{qy}{r^3} & \frac{qx}{r^3}\frac{qx}{r^3} & \mathbf{0} \\\\ \mathbf{0} & \frac{qy}{r^3}\frac{qx}{r^3} & \left(\frac{qy}{r^5}\right)^2 - \frac{q^2}{2r^4} & \frac{qy}{r^3}\frac{qx}{r^3} & \mathbf{0} \\\\ \mathbf{0} & \frac{qx}{r^3}\frac{qx}{r^3} & \frac{qx}{r^3}\frac{qy}{r^3} & \left(\frac{qx}{r^5}\right)^2 - \frac{q^2}{2r^4} & \mathbf{0} \\\\ \frac{\sqrt{2}q^3}{r^5} & \mathbf{0} & \mathbf{0} & \mathbf{0} & -\frac{2q^2}{r^3} + \mathbf{1} \end{bmatrix} \tag{19}$$

With: *e* ! *<sup>r</sup>* <sup>¼</sup> *<sup>x</sup>*^*i*þ*y*^*j*þ*z*^ *k* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>x</sup>*2þ*y*2þ*z*<sup>2</sup> <sup>p</sup> , *<sup>E</sup>* ! <sup>¼</sup> *<sup>q</sup> r e* ! *<sup>r</sup>* <sup>¼</sup> *q x*^*i*þ*y*^*j*þ*z*^ ð Þ*<sup>k</sup> <sup>x</sup>*2þ*y*2þ*z*<sup>2</sup> ð Þ3*=*<sup>2</sup> . an identification of the electric field components in (19) is possible:

$$E^2 = \frac{q^2(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z})}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^3} = \frac{q^2}{r^4}$$

$$E\_\mathbf{x} = \hat{i} \cdot \vec{E} = \frac{q\mathbf{x}}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^{3/2}} = \frac{q\mathbf{x}}{r^3}$$

$$E\_\mathbf{y} = \hat{j} \cdot \vec{E} = \frac{q\mathbf{y}}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^{3/2}} = \frac{q\mathbf{y}}{r^3} \tag{20}$$

$$E\_\mathbf{z} = \hat{k} \cdot \vec{E} = \frac{q\mathbf{z}}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^{3/2}} = \frac{q\mathbf{z}}{r^3}$$

$$\mathbf{z}^2 - \frac{1}{2}E^2 = \frac{q^2\mathbf{x}^2}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^3} - 1/2 \frac{q^2}{\left(\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{z}^2\right)^2}$$

and it becomes obvious, that the first 4 rows and columns represent the electromagnetic energy-stress-momentum tensor *Tik* of (11) and (12).

*E*2

**Key result (i)**: This was only a demonstration which shows that the 5th row and 5th column are related to the electromagnetic 4-vector. Next has to be taken into account, that energy causes a gravitational field.

## **4. Field of a charged mass at rest in 5D**

In the previous section was shown how the 5D Einstein tensor produces the electromagnetic energy-stress-momentum tensor *Tμν*. Because the 5D Einstein tensor already includes the electromagnetic energy-stress-momentum tensor, there is no need for anything besides the Ricci curvature tensor and solutions shall fulfill (symbols with a tilde mark 5D quantities) the 5D equation *<sup>R</sup>*~*AB* <sup>¼</sup> 0 which infers *<sup>R</sup>*<sup>~</sup> <sup>¼</sup> 0 and *<sup>G</sup>*<sup>~</sup> *AB* <sup>¼</sup> 0.

We start with a 5D line element (with capital S as mark for a 5D quantity) in a form which can be found in ([8], p. 128):

$$d\mathbf{S}^2 = \left(\frac{B}{E} - EA^2\right)d(ct)^2 - \frac{1}{B}\left.dr^2 - r^2d\theta^2 - r^2\sin\theta^2 - 2\text{ EA }dt\ \,dl - E\ \,dl^2\right.\tag{21}$$

To be a solution of *RAB* ¼ 0 the functions *A*, *B*, and *E* have to be:

$$E(r) = 1 + \frac{q^2}{2mr}, \quad B(r) = \mathcal{E} - \frac{2m}{r}, \quad A(r) = -\frac{q}{\mathcal{E}}\tag{22}$$

Inserting the solution into the the line element gives:

$$dS^2 = \left(\frac{1 + \frac{q^2}{2Mr} - \frac{2M}{r}}{1 + \frac{q^2}{2Mr}} - \frac{q^2}{1 + \frac{q^2}{2Mr}}\right) d(ct)^2 - \frac{1}{1 + \frac{q^2}{2Mr} - \frac{2M}{r}} dr^2 - r^2 d\theta^2 - r^2 \sin\theta^2 \tag{23}$$

$$-2 \begin{array}{c} \frac{q}{r} \quad dt \quad dl - \left(1 + \frac{q^2}{2Mr}\right) dl^2 \end{array} \tag{23}$$

In matrix form this is:

$$
\bar{g}\_{AB} = \begin{pmatrix}
1 - \frac{2M}{r} & 0 & 0 & 0 & \frac{q}{r} \\
0 & \frac{1}{1 - \frac{2M}{r} + \frac{q^2}{2Mr}} & 0 & 0 & 0 \\
& \mathbf{1} - \frac{2M}{r} + \frac{q^2}{2Mr} & & & \\
0 & 0 & -r^2 & 0 & 0 \\
0 & 0 & 0 & -r^2(\sin(\theta))^2 & 0 \\
\frac{q}{r} & 0 & 0 & 0 & -\mathbf{1} - \frac{q^2}{2Mr}
\end{pmatrix} \tag{24}
$$

Calculating the Ricci curvature with computer algebra software one gets:

$$
\tilde{R}\_{AB} = \begin{pmatrix}
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0}
\end{pmatrix} \tag{25}
$$

The 5D-Monopole metric <sup>~</sup>*gAB* of Eq. (24) thus is a solution of equation *<sup>R</sup>*~*AB* <sup>¼</sup> 0.

#### **4.1 Projection of 5D solutions onto 4D space**

We start with a common fully covariant 5D line element and use a latin letters to mark 5D components ([8], p. 118)

$$d\mathbf{S}^2 = \chi\_{AB} \, d\mathbf{x}^A d\mathbf{x}^B \tag{26}$$

To get the 4D line element and 4D metric tensor from 5D, the part along the fifth coordinate is split into an orthogonal part and a tangent part in relation to the 4D hyper-surface. A unit vector along the fifth coordinate is (because *<sup>γ</sup>*<sup>55</sup> <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup> <sup>51</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> <sup>52</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> <sup>53</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> <sup>54</sup> <sup>þ</sup> <sup>Φ</sup>2):

$$\Psi^A \equiv \frac{\delta^A\_{\mathfrak{S}}}{\sqrt{\varepsilon \gamma\_{\mathfrak{S}}}} \gamma\_{AB} = \frac{1}{\sqrt{\varepsilon \mathfrak{S} \mathfrak{S}}} (\gamma\_{\mathfrak{S}1}, \gamma\_{\mathfrak{S}2}, \gamma\_{\mathfrak{S}3}, \gamma\_{\mathfrak{S}4}, \Phi) \tag{27}$$

This will enable us to split the 5D metric into a part parallel to Ψ*<sup>A</sup>* and a 4D part orthogonal to it. We define a projector:

$$\mathcal{g}\_{AB} \equiv \chi\_{AB} - \varepsilon \Psi^A \Psi^B \tag{28}$$

where by definition of Ψ follows

$$\mathbf{g}\_{AB} = \boldsymbol{\gamma}\_{AB} - \frac{\boldsymbol{\gamma}\_{5A}\boldsymbol{\gamma}\_{5B}}{\boldsymbol{\gamma}\_{55}}\tag{29}$$

This has only 4 rows and 4 columns:

$$\mathbf{g\_{55}} = \mathbf{g\_{54}} = \mathbf{0} \tag{30}$$

so the 5D line element becomes the sum of a 4D line element and an extra part:

$$d\mathbf{S}^2 = \mathbf{g}\_{\mu\nu} d\mathbf{x}^\mu d\mathbf{x}^\nu + \chi\_{55} \left( d\mathbf{x}^5 + \frac{\chi\_{5a}}{\chi\_{55}} d\mathbf{x}^a \right)^2 \tag{31}$$

$$ds^2 \equiv \mathbf{g}\_{\mu\nu} dx^{\mu} dx^{\nu} \tag{32}$$

Let us introduce a 4-vector and a scalar field

$$A\_{\mu} \equiv \frac{\chi\_{5a}}{\chi\_{55}}\tag{33}$$

$$
\Phi^2 = \varepsilon \gamma\_{\mathfrak{S}} \tag{34}
$$

With these we get

$$d\mathbf{S}^2 = ds^2 + \varepsilon \Phi^2 \left(d\mathbf{x}^4 + A\_\mu d\mathbf{x}^\mu\right)^2 \tag{35}$$

This line element rewritten in matrix form is ([8], p. 135):

$$\tilde{\mathbf{g}}^{AB} = \begin{bmatrix} \mathbf{g}^{\mu\nu} & -A^{\mu} \\ -A^{\nu} & -\frac{1}{\mathbf{g}^2} + A^{\mu}A\_{\alpha} \end{bmatrix}, \quad \tilde{\mathbf{g}}\_{AB} = \begin{bmatrix} \mathbf{g}\_{\mu\nu} - \Phi^2 A\_{\mu} A\_{\nu} & -\Phi^2 A\_{\mu} \\ -\Phi^2 A\_{\nu} & -\Phi^2 \end{bmatrix} \tag{36}$$

With the assumption that the ~*gAB* are functions only of *x<sup>μ</sup>* and do not depend on *x*5, meaning

$$\frac{\partial}{\partial \mathbf{x}^5} \; \tilde{\mathbf{g}}\_{AB}(\mathbf{x}^\mu) = \mathbf{0} \tag{37}$$

we get the following expression

$$\tilde{\mathbf{G}}^{a\beta} \equiv \left[ \tilde{\mathbf{R}}^{a\beta} - \frac{1}{2} \tilde{\mathbf{g}}^{a\beta} \tilde{\mathbf{R}} \right] \tag{38}$$

$$=\left[R^{a\theta}-\frac{1}{2}g^{a\theta}R\right]+\frac{1}{2}\Phi^2\left(F^{a}{}\_{\lambda}F^{\theta\lambda}-\frac{1}{4}g^{a\theta}F^{\mu\nu}F\_{\mu\nu}\right)-\frac{1}{\Phi}\left(\Phi^{a;\beta}-g^{a\theta}\Phi^{\mu}{}\_{;\mu}\right)\tag{39}$$

$$=\mathbf{G}^{a\beta} + \frac{\mathbf{1}}{2}\boldsymbol{\Phi}^2 \ \ \boldsymbol{T}\_{em}^{a\beta} + \boldsymbol{T}\_{sc}^{a\beta} \tag{40}$$

$$\equiv \mathbf{0} \tag{41}$$

**Key result (ii)**: here the scalar field plays the role of the gravity constant *via* <sup>1</sup> <sup>2</sup> Φ2. For the components with *A* ¼ 5 or *B* ¼ 5 two more equations can be derived: a Maxwell-like equation, where the source depends on the scalar field

$$F^{l}\_{\
u;\lambda} = -\frac{3}{\Phi} \Phi^{\dot{\lambda}} F\_{\lambda a} \tag{42}$$

$$\Phi^{a}\_{\ ;a} = -\frac{1}{4} \Phi^{3} F^{\mu \nu} F\_{\mu \nu} \tag{43}$$

and a wave-like equation for the scalar field itself—**Key result (iii)**.

## **5. Projection of a 5D-monopole onto 4D**

To get the 4D-metric of the 5D-Monpole solution one has to perform the projection of Eq. (29). Because only the first 4 rows need a projection and components *γ*52, *γ*<sup>53</sup> and *γ*<sup>54</sup> are zero, the only projection which has to be done is the one with *γ*51:

$$\mathbf{g\_{11}} = \mathbf{y\_{11}} - \frac{\mathbf{y\_{11}^2}}{\mathbf{y\_{55}}} \tag{44}$$

We have *<sup>γ</sup>*<sup>11</sup> <sup>¼</sup> *<sup>B</sup> <sup>E</sup>* � *EA*<sup>2</sup> , *γ*<sup>51</sup> ¼ �*EA* and *γ*<sup>55</sup> ¼ �*E*, so

$$\log\_{11} = \frac{B}{E} - EA^2 - \frac{E^2 A^2}{-E} = \frac{B}{E} = 1 - \frac{2m}{rE} = 1 - \frac{2m}{r\left(1 + \frac{q^2}{2mr}\right)} = 1 - \frac{4m^2}{2mr + q^2} \tag{45}$$

The 4D line element is then

$$ds^2 = \frac{B}{E} \left( d(ct)^2 - \frac{1}{B} \right. \left. dr^2 - r^2 d\theta^2 - r^2 \sin \theta^2 \tag{46}$$

or as matrix

$$\mathbf{g}\_{\mu\nu} = \begin{pmatrix} -\frac{4m^2 + 2mr + q^2}{2mr + q^2} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \frac{2mr}{4m^2 - 2mr - q^2} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & -r^2 & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & -r^2(\sin(\theta))^2 \end{pmatrix} \tag{47}$$

Now that the 4D metric for the 5D-Monopole is known, one can calculate the 4D-Einstein tensor:

$$G\_{q\theta} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & q^2 & 0 & 0 \\ 0 & \frac{q^2(2mr+q^2)}{r^2(2mr+q^2)} & 0 & 0 \\ 0 & 0 & \frac{q^2(20m^3r+4m^2q^2-4r^2m^2-4mq^2r-q^4)}{4rm(2mr+q^2)^2} & 0 \\\\ 0 & 0 & 0 & \frac{q^2(\sin(\theta))^2(20m^3r+4m^2q^2-4r^2m^2-4mq^2r-q^4)}{4rm(2mr+q^2)^2} \\ \end{pmatrix} \tag{48}$$

which after (38) is equal to <sup>1</sup> <sup>2</sup> <sup>Φ</sup><sup>2</sup> *<sup>T</sup>αβ em* <sup>þ</sup> *<sup>T</sup>αβ sc* with <sup>1</sup> <sup>2</sup> <sup>Φ</sup><sup>2</sup> <sup>¼</sup> *<sup>g</sup>*.

As stated, the scalar field plays the role of the gravity constant *<sup>g</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> <sup>2</sup>*mr* � �.

**Key result (iv)**: with smaller radius *r* the gravity constant *g* increases and the electromagnetic energy generates a stronger gravity field in the vicinity of the charge than from pure 4D theory.

**Key result (v)**: Because *G*<sup>00</sup> is the energy component and is zero, this means scalar and electric energy cancel out and there are only pressure terms left to generate force and field lines.

#### **5.1 Comparison with Reissner-Nordström metric**

The 4D line element of the Reissner-Nordström metric is known as

$$ds^2 = \left(1 - \frac{2m}{r} + \frac{q^2}{r^2}\right)d(ct)^2 - \frac{1}{1 - \frac{2m}{r} + \frac{q^2}{r^2}} \left. dr^2 - r^2 d\theta^2 - r^2 \sin\theta^2 \tag{49}$$

which is as matrix

$$\mathcal{g}\_{\mu\nu} = \begin{pmatrix} 1 - \frac{2m}{r} + \frac{q^2}{r^2} & 0 & 0 & 0\\ 0 & \left(1 - \frac{2m}{r} + \frac{q^2}{r^2}\right)^{-1} & 0 & 0\\ 0 & 0 & -r^2 & 0\\ 0 & 0 & 0 & -r^2 \end{pmatrix} \tag{50}$$

Both are solution of

$$\mathbf{G}\_{\mu\nu} = R\_{\mu\nu} - \frac{1}{2}R = \frac{q^2}{r^4} \text{ g}\_{\mu\nu} \left(=T\_{\mu\nu}\right) \tag{51}$$

The electromagnetic stress-energy-momentum tensor as matrix:

$$T\_{\mu\nu} = \begin{pmatrix} \frac{-2mq^2r + q^4 + q^2r^2}{r^6} & 0 & 0 & 0\\ 0 & \frac{q^2}{2mr^3 - q^2r^2 - r^4} & 0 & 0\\ 0 & 0 & -\frac{q^2}{r^2} & 0\\ 0 & 0 & 0 & -\frac{q^2}{r^2}\sin^2(\theta) \end{pmatrix} \tag{52}$$

**Key result (vi)**: The Reissner-Nordström solution has a negative inverse *r* term <sup>2</sup>*<sup>m</sup> r* and a positive inverse *r*<sup>2</sup> term *<sup>q</sup>*<sup>2</sup> *<sup>r</sup>*<sup>2</sup> which can result in 2 event horizons. The 4D projection of the 5D Monopole can have only one event horizon.

Both differ only in the near field. The expansion of *g*<sup>00</sup> of the 4D projection of the 5D Monopole shows a vanishing difference to *g*<sup>00</sup> of the Reissner-Nordström solution with increasing *r*:

$$\mathbf{1} - \frac{2\mathbf{m}}{\mathbf{r}} + \frac{\mathbf{q}^2}{\mathbf{r}^2} - \frac{1}{2}\frac{q^4}{mr^3} + \frac{1}{4}\frac{q^6}{m^2r^4} - \frac{1}{8}\frac{q^8}{m^3r^5} + O\left(r^{-6}\right) \tag{53}$$

## **6. Neutral soliton solution with a static scalar field**

Setting the electromagnetic 4-potential components *g*5*<sup>α</sup>* ¼ 0 results in an electrically neutral solution. Only *g*<sup>55</sup> is nonzero and varies with distance *r*. With a function

$$A(r) = 1 - \frac{2m}{r} \tag{54}$$

the line-element of the solution is

$$ds^2 = A^a dt^2 - A^{-(a+b)} dr^2 - A^{1-a-b} r^2 \left(d\theta^2 + \sin\theta^2\right) - A^b d\ell^2 \tag{55}$$

which is in matrix form

$$
\begin{pmatrix}
\left(1-\frac{2m}{r}\right)^{a} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & -\left(1-\frac{2m}{r}\right)^{-a-b} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & -r^{2}\left(1-\frac{2m}{r}\right)^{1-a-b} & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & -r^{2}\sin\left(\theta\right)^{2}\left(1-\frac{2m}{r}\right)^{1-a-b} & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \left(1-\frac{2m}{r}\right)^{b}
\end{pmatrix} \tag{56}
$$

Calculating the Ricci tensor for this metric results in

$$
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
0 & \frac{2m^2(a^2 + 2ab + b^2 - 1)}{r^2(-r + 2m)^2} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix}
\tag{57}
$$

and requests the condition:

$$a^2 + ab + b^2 - \mathbf{1} = \mathbf{0} \tag{58}$$

Since *a* 6¼ *b* in general, the metric (52) is bivalent in the sense that it has gravitational and scalar contributions to the energy. Calculating the total energy of a soliton gives (a + b/2) M [7] where M is the mass seen from infinity.

**Key result (vii)**: since there are no electromagnetic sources in the metric (56) and all of its associated matter outside of the central source comes from the last term, the logical consequence is that the cloud of radiation around a soliton is not photons but scalerons, or quanta of the scalar field (radions, dilatons, axions).

#### **6.1 Neutral soliton solution with a dynamic scalar field**

There is also a dynamic (isotropic) solution of a soliton, where the scalar field quanta vanish to infinity whith an analog of a Hubble-constant leading to a Black Hole without a scalar field:

$$\begin{split} ds^2 &= \left(\frac{ar-1}{ar+1}\right)^{\frac{4}{\sqrt{3}}} dt^2 - \left(\frac{a^2r^2-1}{a^2r^2}\right) \left(\frac{ar-1}{ar+1}\right)^{\frac{2}{\sqrt{3}}} (1+Ht) d\sigma^2 \\ &- \left(\frac{a^2r^2-1}{a^2r^2}\right) \left(\frac{ar-1}{ar+1}\right)^{\frac{2}{\sqrt{3}}} (1+Ht)^{-1} \, d\ell^2 \end{split} \tag{59}$$

## **7. Induced matter**

In the preceding section, there was no explicit dependence on ℓ. Now, we allow all potentials *gAB* to depend on ℓ and again work with electric neutral matter (see matrix form in Eq. (36) with condition Eq. (37)):

$$\mathbf{g}\_{a\emptyset} = \mathbf{g}\_{a\emptyset}(\mathbf{x}^A) \qquad \mathbf{g}\_{5a} = \mathbf{0} \qquad \mathbf{g}\_{55} = \Phi(\mathbf{x}^A)^2 \tag{60}$$

$$d\mathbb{S}^2 = \mathbf{g}\_{AB} \, d\mathfrak{x}^A d\mathfrak{x}^B \tag{61}$$

Next, we can split the 5D Ricci tensor in its 4D analog and terms belonging to the 5th dimension, which can be taken as terms of a stress-energy-momentum tensor. Derivations of the 5th dimension with respect to ℓ are denoted with a star:

*Key Outcomes of 5D Relativity DOI: http://dx.doi.org/10.5772/intechopen.110241*

$$\begin{split} \tilde{R}\_{a\beta} &= R\_{a\beta} + \left(\Gamma^{4}\_{a\beta}\right)\_{,4} - \left(\Gamma^{4}\_{a4}\right)\_{,\beta} + \Gamma^{i}\_{a\beta}\Gamma^{4}\_{\lambda\lambda} + \Gamma^{4}\_{a\beta}\Gamma^{D}\_{aD} - \Gamma^{4}\_{a\lambda}\Gamma^{\lambda}\_{\beta\mu} - \Gamma^{D}\_{a4}\Gamma^{4}\_{\beta D} \\ &= R\_{a\beta} - \frac{\Phi\_{a;\beta}}{\Phi} + \frac{1}{2\Phi^{2}} \left(\frac{\Phi^{\*}\_{a\beta}}{\Phi} - \stackrel{\*}{\mathbf{g}}^{\*} + \mathcal{g}^{\mu\nu} \stackrel{\*}{\mathbf{g}}\_{a} \stackrel{\*}{\mathbf{g}} - \frac{\mathcal{g}^{\mu\nu}}{2} \frac{\stackrel{\*}{\mu}}{\beta}\right) \\ &= R\_{a\beta} - T\_{a\beta} + \frac{1}{2} \left. \mathcal{g}\_{a\beta} \right| \ T \equiv 0 \end{split} \tag{62}$$

**Key result (viii)**: the dependence of the elements on the coordinate can generate mass, pressure, momentum and energy (tensor **T***αβ*).

The fifth diagonal element of the 5D-Ricci tensor is also required to be zero:

$$\ddot{R}\_{\text{\ $\$ }} = \Phi \square \Phi - \frac{\ast \text{g}^{\ast \cdot \lambda \beta}}{2} - \frac{\text{g}^{\dot{\lambda} \dot{\beta}}}{2} - \frac{\text{g}^{\dot{\lambda} \dot{\beta}}}{2} - \frac{\ast \Phi \text{ g}^{\lambda \theta} \cdot \ast \text{g}\_{\dot{\lambda} \theta}}{2\Phi} - \frac{\text{g}^{\mu \theta} \text{ g}^{\lambda \sigma} \cdot \ast \text{g}\_{\dot{\lambda} \theta} \cdot \ast \text{g}\_{\mu \sigma}}{4} \equiv 0 \tag{63}$$

**Key result (ix)**: and yields a wave equation for the gravitational constant.

## **8. Geodesic motion in 5D and extension of Lorentz force**

In the preceding section, a new energy-momentum-tensor *Tαβ sc* occurred. Its divergence should yield in force density (force per unit inertial mass). For that reason, the equation of motion is interesting. Using the Lagrangian approach minimizing the distance between two points in 5D, the equation of geodesic motion becomes ([7], p. 154):

$$\frac{d\boldsymbol{u}^{\mu}}{d\boldsymbol{s}} + \Gamma^{\mu}\_{\beta\gamma}\boldsymbol{u}^{\beta}\boldsymbol{u}^{\gamma} + \left(\boldsymbol{g}^{\mu\alpha} - \frac{1}{2}\frac{d\boldsymbol{x}^{\mu}}{d\boldsymbol{s}}\frac{d\boldsymbol{x}^{\alpha}}{d\boldsymbol{s}}\right)\frac{d\boldsymbol{\ell}}{d\boldsymbol{s}}\frac{d\boldsymbol{x}^{\beta}}{d\boldsymbol{s}}\frac{\partial\mathbf{g}\_{\alpha\beta}}{\partial\boldsymbol{\ell}} = \mathbf{0} \tag{64}$$

The first two terms describe geodesic motion in 4D, so we split the equation and define a force density:

$$\frac{d\boldsymbol{u}^{\mu}}{d\boldsymbol{s}} + \Gamma^{\mu}\_{\beta\gamma}\boldsymbol{u}^{\beta}\boldsymbol{u}^{\gamma} = \boldsymbol{f}^{\mu} \tag{65}$$

$$f^{\mu} \equiv \left( -\text{g}^{\mu a} + \frac{1}{2} \frac{d\mathbf{x}^{\mu}}{ds} \frac{d\mathbf{x}^{a}}{ds} \right) \frac{d\ell}{ds} \frac{d\mathbf{x}^{\beta}}{ds} \frac{\partial \mathbf{g}\_{a\beta}}{\partial \ell} = \mathbf{0} \tag{66}$$

The new force density is nonzero if the 4D metric depends on the fifth coordinate *<sup>∂</sup>gαβ* ∂ℓ � � and there is motion in the fifth dimension *<sup>d</sup>*<sup>ℓ</sup> *ds* � �. A force *<sup>F</sup><sup>μ</sup>* can be split into a normal component *N<sup>μ</sup>* in relation to the 4D velocity and a tangent or parallel component *<sup>P</sup><sup>μ</sup>* in relation to the 4D velocity, thus *<sup>F</sup><sup>μ</sup>* <sup>¼</sup> *<sup>N</sup><sup>μ</sup>* <sup>þ</sup> *<sup>P</sup><sup>μ</sup>* giving force densities:

$$n^{\mu} = (-\mathbf{g}^{\mu \alpha} + \mathbf{u}^{\mu}\mathbf{u}^{\alpha})\mathbf{u}^{\beta}\frac{\partial \mathbf{g}\_{\alpha\beta}}{\partial \ell} \tag{67}$$

$$p^{\mu} = -\frac{1}{2} u^{\mu} \left( u^{a} u^{\beta} \frac{\partial \mathbf{g}\_{a\beta}}{\partial \ell} \right) \frac{d\ell}{ds} \tag{68}$$

**Key result (x)**: the force **P** *parallel* to 4D velocity **u** is new and occurs only if there is fifth dimension dependency ([7], p. 157). In the preceding section, we had condition (37) wherefore there a fifth force does not occur.

## **9. Conclusions**

From five dimensional metrics with unrestricted dependence from all five coordinates resolving the equation *RAB* ¼ 0 many inductively gained physical laws can be deduced: Newton's gravity law, Coulomb law, Faraday induction law, Ampère's circuital law, Maxwell Equations, Einstein's Gravity law, Einstein-Maxwell equation. Because of further degrees of freedom a scalar wave equation for the gravity constant is gained, yielding in an additional energy and an additional force. For 4D assemblies a reduction of inertia is predicted. The gravity constant corresponds to a scalar field which is part of the 5D metric as the 15th potential.

This 15th component of the metric varies in the vincinity of a charge and seems to behave like the potentials of the electromagnetic 4-vector-potential which is able detach from the sources, giving reason to the assumption that changes of the gravity constant could emerge into the surrounding spacetime.

Aiming at the reduction of inertia and an accelerating fifth force, for interstellar travel or asteroid deflection missions a change from analytical to numerical calculation methods will be necessary.

Recent experiments investigate the possible effects of a fifth force with neutron beam scattering on silicon crystals [9]. Neutrons have internally two negative charges and a double positive charge in the smallest space and in the near field the scalar potential causes the greatest deviations.

## **Acknowledgements**

The author would like to thank the American Institute of Aeronautics and Astronautics (AIAA) for kindly allowing parts of this chapter to be adapted from a conference paper published in the ASCEND 2021 conference. Since most of the content presented is introductory and basic, many passages and figures are taken from that paper. The focus here is on the didactic presentation of the results in the form of 10 key statements.

## **Nomenclature**


*Key Outcomes of 5D Relativity DOI: http://dx.doi.org/10.5772/intechopen.110241*


## **Author details**

Detlef Hoyer† Technical University of Hamburg, Germany

\*Address all correspondence to: d.hoyer@hamburg.de

†Alumnus.

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Kaluza T. Zum Unitätsproblem der Physik, Sitzungsberichte der Preussischen Akademie der Wissenschaft. Berlin: Phys.-Math. Klasse; 1921. p. 966

[2] Hoyer D. Fast interstellar space travel by reduced mass via 5D Electromagnetism. [Internet]. ASCEND 2021. American Institute of Aeronautics and Astronautics; 2021. DOI: 10.2514/ 6.2021-4002

[3] Lehner G. Elektromagnetische Feldtheorie für Ingenieure und Physiker. Berlin, Heidelberg: Springer; 1990

[4] Jefimenko O. Causality, Electromagnetic Induction and Gravitation. Star City: Electric Scientific Company; 1992. p. 16

[5] Jackson JD. Klassische Elektrodynamik. 2. Auflage ed. Berlin New York: de Gruyter; 1983

[6] Simonyi K. Introduction. In: Theoretische Elektrotechnik. Berlin: VEB Deutscher Verlag der Wissenschaften; 1989. p. 929

[7] Spiegel MR. Ch. VII. In: Vectoranalysis. Düsseldorf: McGraw-Hill; 1977. p. 17

[8] Wesson P, Overduin J. Space-Time-Matter. Singapore: World Scientific Publishing Co.; 2019

[9] Heacock B. Pendellösung interferometry probes the neutron charge radius, lattice dynamics, and fifth forces. Science. 2021;**373**:1239

## **Chapter 3**

## Dissipative Quantum System and Energy Balance

*Jishad Kumar*

## **Abstract**

We discuss how various parts of a quantum many-body system exchange energies at thermal equilibrium. To show this, we assume a quantum system is coupled to a many-body environment (at thermal equilibrium with a bigger environment) consisting of a large number of independent and non-interacting quantum harmonic oscillators above a stable ground state. Once the coupling to a large environment is switched on, the system dissipates its energy continuously to the environment until it reaches equilibrium with the latter. We use the Quantum Langevin equation to show such energy exchange at equilibrium. We conclude that different parts of a physical system can exchange energies even at absolute zero temperature.

**Keywords:** open systems, quantum dissipation, fluctuations, instantaneous power, the charged oscillator in a magnetic field

## **1. Introduction**

Isolating a quantum system from its environment is not possible since the coupling energy plays a pivotal role in the low-temperature properties of the system. Moreover, a complete understanding or control of the huge environment is also not feasible. How one will study the properties of the quantum system? A working method is to partition the whole system into two different parts, viz., the *system* and the *environment or heat bath*, and eliminate the bath degrees of freedom after carefully considering the effects of the heat bath on the system parameters. This makes the system essentially an *open* one and the study of open systems has been very crucial in many applications of quantum mechanics [1]. There are consequences due to the establishment of a coupling between the system and the heat bath. Firstly, there may be an irreversible transfer of energy from the system to the environment (*dissipation*), next there is *Brownian motion* - fluctuations in the system's degrees of freedom due to the noisy force exerted on the system by the environment. There is *decoherence*, a purely quantum mechanical phenomenon where the system-bath coupling destroys the coherent superposition of states. The first two processes have classical counterparts which have been extensively studied by many authors in the literature. A more detailed understanding shows that the fluctuating force from the environment induces decoherence and damping in the system properties.

#### **Figure 1.**

*(a) Schematic representation of Brownian motion. The jittering motion of a large mass immersed in the medium containing a large number of particles/molecules is the phenomenon of Brownian motion. The random kicks the mass received from its environmental particles make the motion stochastic. (b) The famous system-plus-bath arrangement. The test system with one or few degrees of freedom is assumed to be in contact with its environment containing a large number of independent and identical harmonic oscillators. The individual masses of the oscillators in the environment m*1, *m*2, *m*3*:* … *mN are assumed to be smaller than the mass M of the test system.*

The well-known model of the system-plus-bath approach to open quantum systems stems from classical physics. The underlying phenomenon is the classical Brownian motion where a test system undergoes random motion due to the "kicks" received by the former from the surrounding particles when immersed in a suitable medium (**Figure 1**). In the classical setting, dissipation is introduced in the system "by hand" by inserting a time-dependent damping term into the equation of motion. However, this naive approach never works in the quantum domain where everything is governed by certain principles like Heisenberg's uncertainty relation. Moreover, the physical quantities are operators in quantum mechanics and these operators must satisfy certain commutation relations. The damping term inserted in the equation of motion violates the uncertainty principle. The role of fluctuating/random forces is crucial in order to preserve the canonical structure. The knowledge of the details of the processes including dissipation in the system may not be explicitly known so sometimes the dissipation mechanism is globally described by friction, resistance, viscosity, etc. These parameters are introduced in order to compensate for the information loss due to dissipation. In this microscopic system-bath model, *friction* comes about by the irreversible transfer of energy from the system to its environment. The environment is modeled such that no energy, which is transferred, may come back to the system within any physically relevant time periods. That means the so-called Poincaré recurrence time is infinity. A really necessary condition for the full Hamiltonian is that, under certain conditions, the known classical results must be recovered. Quantum dynamics at arbitrarily low temperatures and (or) with strong damping can be studied within the system-plus-bath approach, regardless of whether the bath is ohmic, sub-ohmic, or super-ohmic. The key thermodynamical quantity of a quantum dissipative system is the reduced density operator *ρ*ðÞ¼ *t trBρT*ð Þ*t* , i.e., the partial trace of the total system plus bath density operator *ρ<sup>T</sup>* over the bath degrees of freedom. Here *t* denotes time. Quantum dissipation theory describes not only the evolution of *ρ*ð Þ*t* but also the equilibrium behavior of the reduced system as *ρ*ð Þ¼ *t* ! ∞ *<sup>ρ</sup>eq*ð Þ *<sup>T</sup>* <sup>∝</sup>*trBe*�H*T=kBT*, where *<sup>T</sup>* is the temperature, *kB* is the Boltzmann constant and <sup>H</sup>*<sup>T</sup>* is the total Hamiltonian. The latter property is also referred as the detailed balance

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

relation of the quantum dissipation theory [2]. The description of the dissipative system is recovered by the reduced density matrix obtained by eliminating the bath degrees of freedom which imparts damping and fluctuations.

The system plus reservoir (bath) approach to open quantum systems, was originally introduced by many authors [3–9] and popularized later in the literature by many others [10–12]. The idea here is to couple a system with a finite degree of freedom (system under study) with a reservoir consisting of an infinite number of independent and non-interacting harmonic oscillators. This model has been discussed in the literature both for harmonic systems [7, 8, 13, 14] and anharmonic systems [15]. Once the coupling is established the reservoir imparts fluctuations in the system coordinates which thereby causes the system under observation to lose energy rapidly (and is irrevocable) to the bath. Because of this fluctuating or random force, the system undergoes Brownian motion. The reservoir is commonly known as a heat bath because the system dissipates its energy continuously and the former distributes this dissipated energy it its various energy-infusing modes. The relevant variables of the heat bath are averaged out later from the larger Hilbert space of the full system-plusbath setup, to obtain an effective description of the test system alone. The projected dynamics of the test quantum system belonging to the truncated Hilbert space appear dissipative due to the bath-induced decoherence effects. Usually, either a formal path integral approach in the Schrödinger picture [16] or the quantum Langevin equation in the Heisenberg picture [12] is used to eliminate the heat bath degrees of freedom.

## **2. Theoretical framework**

The system-plus-bath model for dissipative quantum systems is described as follows. A quantum system of a finite degree of freedom is coupled to a heat bath consisting of independent and non-interacting harmonic excitations above a stable ground state. The interaction between the quantum system and an individual oscillator of the heat bath is inversely proportional to the total volume V of the bath, thereby ensuring that the individual coupling is a linear function of bath coordinates. This nice property further allows one to eliminate the bath degrees of freedom easily. Because the number of oscillators in the bath is very large, the weak perturbation of any individual bath oscillator on the quantum system does not necessarily mean that the coupling of the system and the bath is weak. This model, even if the individual oscillators of the bath couple weakly to the system, allows the inclusion of strong damping also [11].

We write the total Hamiltonian for the "full" system as

$$H = H\_{\mathcal{S}} + H\_{\mathcal{B}} + H\_{\mathcal{SB}},\tag{1}$$

where the system Hamiltonian is given by

$$H\_S = \frac{p^2}{2M} + V(q),\tag{2}$$

where *M* is the mass of the quantum system which is moving in a potential *V q*ð Þ, with *q* being the coordinate of the system. The heat bath Hamiltonian is written as the sum of *N* non-interacting oscillators

*Recent Topics and Innovations in Quantum Field Theory*

$$H\_B = \sum\_{j=1}^{N} \left( \frac{p\_j^2}{2m\_j} + \frac{m\_j}{2} \alpha\_j^2 \mathbf{x}\_j^2 \right). \tag{3}$$

The possibility of revival of the initial state after a course of time, since one can pass on to the normal coordinates with the heat bath consisting of harmonic oscillators and *V q*ð Þ as a harmonic potential, can be overcome with heat bath having sufficiently many oscillators so that the Poincaré recurrence time becomes infinity [17]. The third contribution, the interaction Hamiltonian can be written as

$$H\_{SB} = -q\sum\_{j=1}^{N} \mathbf{C}\_{j}\mathbf{x}\_{j} + q^{2} \sum\_{j=1}^{N} \frac{\mathbf{C}\_{j}^{2}}{2m\_{j}a\_{j}^{2}},\tag{4}$$

which is bilinear in the system and bath coordinates. The last term (which depends on the coupling constants *Cj* and only contains an operator in the system Hilbert space) in the interaction Hamiltonian is included as a counter term to ensure that the dissipation is homogeneous in all spaces. Since the coupling is via the position variables, if this term is not included, then the coupling becomes different wherever the quantum particle is located. Or in other words, the model is not translationally invariant and that will result in an unphysical renormalization of the potential. Therefore, one must understand that the counter term in the interaction Hamiltonian is included to make sure of the fact that dissipation has been introduced solely by the coupling to the reservoir not by a renormalization of *V q*ð Þ. If it was not included, then the minimum of the potential surface of the global system for a given *q* is at *xj* ¼ *Cjq=mjω*<sup>2</sup> *<sup>j</sup>* for all *j* . This result in an 'effective' potential renormalized by the coupling which is given by

$$V\_{\rm eff}(q) = V(q) - \sum\_{j=1}^{N} \frac{C\_j^2 q^2}{2m\_j \alpha\_j^2}.\tag{5}$$

This becomes clear if we consider the minimum of the Hamiltonian with respect to the system and environment coordinates. From the requirement

$$\frac{\partial H}{\partial \mathbf{x}\_j} = m\_j a\_j^2 \mathbf{x}\_j - \mathbf{C}\_j q = \mathbf{0},\tag{6}$$

we obtain

$$\mathbf{x}\_{j} = \frac{\mathbf{C}\_{j}}{m\_{j}\alpha\_{j}^{2}}\mathbf{q}.\tag{7}$$

Using this result, we determine the minimum of the Hamiltonian with respect to the system coordinate and is given by

$$\frac{\partial H}{\partial q} = \frac{\partial V}{\partial q} - \sum\_{j=1}^{N} C\_j \mathbf{x}\_j + q \sum\_{j=1}^{N} \frac{C\_j^2}{m\_j \alpha\_j^2} = \frac{\partial V}{\partial q}. \tag{8}$$

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

The second term in Eq. (4) thus ensures that this minimum is determined by the potential *V q*ð Þ only. Before we start writing equations of motion, for simplicity, we write the full Hamiltonian as

$$\mathcal{H} = \frac{p^2}{2\mathcal{M}} + V(q) + \sum\_{j=1}^{N} \left\{ \frac{p\_j^2}{2m\_j} + \frac{1}{2} m\_j \alpha\_j^2 \left( \mathbf{x}\_j - \frac{\mathbf{C}\_j q}{m\_j \alpha\_j^2} \right)^2 \right\}. \tag{9}$$

#### **2.1 Quantum mechanical derivation**

We need to look at the reduced equation of motion for the system coordinate. In this section, we derive the quantum Langevin equation for our test system coordinate under the influence of the heat bath induced fluctuation effects. In the quantum domain, all the parameters are quantum variables and are operators. In the literature, Magalinskii*^* [14] showed that the elimination of the environmental degrees of freedom leads to a damped equation of motion for the system coordinate. The time evolution of an operator *A*, in the Heisenberg picture, is given by

$$\frac{dA}{dt} = \frac{i}{\hbar}[H, A]. \tag{10}$$

From Eq. (9) we obtain the equations of motion for the bath degrees of freedom

$$
\dot{p}\_j = -m\_j a\_j^2 \mathbf{x}\_j + \mathbf{C}\_j q \tag{11}
$$

$$
\dot{\mathbf{x}}\_{j} = \frac{p\_{j}}{m\_{j}},
\tag{12}
$$

and, similarly, the equations of motion for the system degree of freedom are given by

$$\dot{p} = -\frac{\partial V}{\partial q} + \sum\_{j=1}^{N} \mathbf{C}\_{j} \mathbf{x}\_{j} - q \sum\_{j=1}^{N} \frac{\mathbf{C}\_{j}^{2}}{m\_{j} a\_{j}^{2}},\tag{13}$$

$$
\dot{q} = \frac{p}{M}.\tag{14}
$$

We treat the system coordinate *q t*ð Þ as if it were a given function of time, we then solve the environmental equations of motion and it turns out to be an ordinary second order linear inhomogeneous differential equation with the solution of the form

$$\mathbf{x}\_{j}(t) = \mathbf{x}\_{j}(\mathbf{0})\cos\left(\alpha\_{\text{j}}t\right) + \frac{p\_{j}(\mathbf{0})}{m\_{\text{j}}\alpha\_{\text{j}}}\sin\left(\alpha\_{\text{j}}t\right) + \frac{C\_{\text{j}}}{m\_{\text{j}}\alpha\_{\text{j}}}\int\_{0}^{t}d\boldsymbol{s}\sin\left(\alpha\_{\text{j}}(t-s)\right)q(s). \tag{15}$$

Inserting Eq. (15) into Eq. (13), we obtain an effective equation of motion for the system coordinate

*Recent Topics and Innovations in Quantum Field Theory*

$$M\ddot{q} - \int\_0^t ds \sum\_{j=1}^N \frac{C\_j^2}{m\_j \alpha\_j} \sin\left(\alpha\_j(t-s)\right) q(s) + \frac{\partial V}{\partial q} + q \sum\_{j=1}^N \frac{C\_j^2}{m\_j \alpha\_j^2} \tag{16}$$

$$= \sum\_{j=1}^N C\_j \left[ \varkappa\_j(0) \cos\left(\alpha\_j t\right) + \frac{p\_j(0)}{m\_j \alpha\_j} \sin\left(\alpha\_j t\right) \right].$$

This is further simplified by partially integrating the second term of the LHS and that yields

$$M\ddot{q} + M\int\_0^t ds \gamma(t-s)\dot{q}(s) + \frac{\partial V}{\partial q} = f(t),\tag{17}$$

where the damping kernel is given by

$$\gamma(t) = \frac{1}{M} \sum\_{j=1}^{N} \frac{C\_j^2}{m\_j \alpha\_j^2} \cos \left(\alpha\_l t\right). \tag{18}$$

It can also be expressed alternatively as

$$\gamma(t - t') = \Theta(t - t') \frac{1}{M} \sum\_{j=1}^{N} \frac{C\_j^2}{m\_j \alpha\_j^2} \cos\left[\alpha\_j(t - t')\right]. \tag{19}$$

The operator-valued random force in Eq. (17) takes the form

$$f(t) = \sum\_{j=1}^{N} \mathbf{C}\_{j} \left[ \left( \mathbf{x}\_{j}(\mathbf{0}) - \frac{\mathbf{C}\_{j}}{m\_{j}\alpha\_{j}^{2}} q(\mathbf{0}) \right) \cos \left( \alpha\_{j} t \right) + \frac{p\_{j}(\mathbf{0})}{m\_{j}\alpha\_{j}} \sin \left( \alpha\_{j} t \right) \right]. \tag{20}$$

The statistical average of this fluctuating force vanishes when the average is taken over the total density matrix of the bath degrees of freedom and the coupling. That is

$$\left< f(t) \right>\_{\rho\_{(\mathcal{B}+\mathcal{SB})}} = \frac{Tr\_B[f(t) \exp\left(-\beta(\mathcal{H}\_{\mathcal{B}} + \mathcal{H}\_{\mathcal{SB}})\right)]}{Tr\_B[\exp\left(-\beta(\mathcal{H}\_{\mathcal{B}} + \mathcal{H}\_{\mathcal{SB}})\right)]} = \mathbf{0},\tag{21}$$

where *ρ*ð Þ *<sup>B</sup>*þ*SB* is the shifted canonical equilibrium distribution of the heat bath which is given by

$$\rho\_B = Z\_B^{-1} \exp\left[ -\beta \sum\_j \left( \frac{p\_j(\mathbf{0})^2}{2m\_j} + \frac{m\_j \alpha\_j^2}{2} \left( \mathbf{x}\_j(\mathbf{0}) - \frac{\mathbf{C}\_j}{m\_j \alpha\_j^2} q(\mathbf{0}) \right)^2 \right) \right],\tag{22}$$

where *ZB* is the partition function of the bath oscillators. Also

$$\left. \langle f(t)f(t') \rangle\_{\rho\_{(\mathcal{B}+\mathcal{SB})}} = \mathcal{M}k\_B T \gamma(t - t'). \tag{23}$$

This is the fluctuation-dissipation relation. For weak coupling, we seperate the transient term (or the initial slip) *Mγ*ð Þ*t q*ð Þ 0 which is of second order in the coupling constant *Cj* and write the random force as [18].

$$f(t) = \zeta(t) - m\gamma(t)q(0). \tag{24}$$

Therefore the generalized Langevin equation takes the form

$$M\ddot{q}(t) + M\int\_0^t dt'\chi(t-t')\dot{q}(t') + V'(q) = \zeta(t) - M\gamma(t)q(0),\tag{25}$$

where

$$\mathcal{L}(t) = \sum\_{j} \mathbf{C}\_{j}(\mathbf{0}) \cos \left( a \boldsymbol{\uprho} t \right) + \frac{p\_{j}(0)}{m\_{j} \boldsymbol{\uprho}\_{j}} \sin \left( a \boldsymbol{\uprho} t \right). \tag{26}$$

The integration in the forward direction of time in Eq. (25) ensures that it breaks the time reversal invariance explicitly thereby introducing irreversibility in the problem. When taken an average of the Eq. (26) with respect to the canonical classical equilibrium density operator of the unperturbed bath

$$\rho\_B(\mathbf{0}) = Z\_B^{-1} \exp\left[ -\beta \sum\_j \left( \frac{p\_j(\mathbf{0})^2}{2m\_j} + \frac{m\_j \alpha\_j^2}{2} \mathbf{x}\_j(\mathbf{0})^2 \right) \right],\tag{27}$$

we obtain

$$\langle \zeta(t) \rangle\_{B} = \frac{Tr\_{B}[\zeta(t) \exp\left(-\beta \mathcal{H}\_{B}\right)]}{Tr\_{B}[\exp\left(-\beta \mathcal{H}\_{B}\right)]} = \mathbf{0},\tag{28}$$

and the unequal time correlation

$$
\langle \zeta(t)\zeta(t')\rangle\_{\rho\_B(0)} = Mk\_BT\gamma(t-t').\tag{29}
$$

From Eq. (20), it is clear that the force operator depends explicitly on the initial conditions of the bath positions and momenta and also on an inhomogeneous slip term *Mγ*ð Þ*t q*ð Þ 0 . Usually, this term is neglected under the Markovian/Ohmic limit, when the friction assumes the ohmic form *γ*ðÞ!*t* 2*γδ*ð Þ*t* [18].

Now, we calculate the correlation function of the random force. We may use either *f t*ð Þ with respect to *ρ*ð Þ *<sup>B</sup>*þ*SB* or equivalently *ζ*ð Þ*t* with respect to *ρB*. Eqs. (20) and (24) gives

$$\langle \zeta(t)\zeta(0)\rangle\_{\rm B} = \sum\_{j,l} \mathbf{C}\_{j} \mathbf{C}\_{l} \left\langle \left(\mathbf{x}\_{j}(0)\cos\left(o\_{l}t\right) + \frac{p\_{j}(0)}{m\_{j}o\_{l}}\sin\left(o\_{l}t\right)\right) \mathbf{x}\_{l}(0) \right\rangle\_{\rm B} \tag{30}$$

At thermal equilibrium, the second moments of the position and momentum of the bath are calculated and yields

$$
\langle \mathfrak{x}\_{\dot{j}}(\mathbf{0}) \mathfrak{x}\_{l}(\mathbf{0}) \rangle\_{B} = \delta\_{\dot{j}l} \frac{\hbar}{2m\_{\dot{j}}a\dot{\imath}\_{\dot{j}}} \coth\left(\frac{\hbar\beta a\dot{\jmath}\_{\dot{j}}}{2}\right), \tag{31}
$$

$$\left< p\_j(\mathbf{0}) \varkappa\_l(\mathbf{0}) \right>\_{\mathcal{B}} = -\frac{i\hbar}{2} \delta\_{jl}.\tag{32}$$

Incorporating Eqs. (31) and (32), the noise correlation in Eq. (30) can be expressed as

$$
\langle \zeta(t)\zeta(0)\rangle\_{B} = \sum\_{j=1}^{N} \frac{\hbar C\_{j}^{2}}{2m\_{j}\alpha\_{j}} \left[ \coth\left(\frac{\hbar \beta a\_{j}}{2}\right) \cos\left(\alpha\_{j}t\right) - i\sin\left(\alpha\_{j}t\right) \right].\tag{33}
$$

It is to be noted that the noise correlation contains an imaginary part. This is due to the fact that *ζ*ð Þ*t* and *ζ*ð Þ 0 are now operators in quantum mechanics and, in general, do not commute with each other. We have obtained all the relations corresponding to the classical counterparts, here. Now, we need the reduced description of the quantum system alone and to get such an expression we have to eliminate the bath degrees of freedom from the total picture. The formal way of doing that is to introduce the "spectral density" of the bath oscillators which contains all the information about the heat bath. The spectral density is given by

$$J(\boldsymbol{\alpha}) = \pi \sum\_{j=1}^{N} \frac{C\_j^2}{2m\_j \alpha\_j} \delta(\boldsymbol{\alpha} - \boldsymbol{\alpha}\_j),\tag{34}$$

so that the damping kernel takes a form

$$\gamma(t) = \frac{1}{M} \sum\_{j=1}^{N} \frac{C\_j^2}{m\_j \alpha\_j^2} \cos\left(\alpha\_j t\right) = \frac{2}{M} \int\_0^\infty \frac{d\alpha}{\pi} \frac{J(\alpha)}{\alpha} \cos\left(\alpha t\right). \tag{35}$$

The most widely used form of the spectral density is of the following form

$$J(\alpha) = M\gamma\alpha,\tag{36}$$

which is called the *ohmic* and it apparently produces a memoryless friction

$$
\gamma(\mathbf{t}) = 2\gamma \delta(\mathbf{t}).\tag{37}
$$

But this is an ideal situation. In real physical situations, the spectral density falls off in the *ω* ! ∞ limit. The above form (cf. Eq. (36)) of the spectral density, in the large *ω* limit, gives divergences in certain physical quantities like the momentum dispersion. It is then customary to introduce a cut-off to the spectrum with which the specturm vanishes above that cutoff. Such a spectrum is known as a Drude regularized spectral density [1] and is given by

$$J(\alpha) = M\gamma\alpha \frac{\alpha\_D^2}{\alpha^2 + \alpha\_D^2},\tag{38}$$

where *ω<sup>D</sup>* is a cutoff to the spectrum of bath oscillators above which the spectral density vanishes. From Eq. (35), for positive arguments *t*>0, the damping or memory kernel takes a form

$$
\gamma(t) = \gamma a\_D \exp\left(-a\_D t\right). \tag{39}
$$

We need ~*γ ω*ð Þ, the Fourier transform of Eq. (39), for our forthcoming calculations and is written as

$$\tilde{\gamma}(o) = \frac{\gamma o\_D}{o o\_D - i o}, \quad \text{and } \Re[\tilde{\gamma}(o)] = \frac{\gamma o\_D^2}{o o\_D^2 + o^2}. \tag{40}$$

The "tilde" sign is used to denote the Fourier transform of a function throughout the chapter. One can still use the terminology "ohmic damping" even if the Eq. (38)

does not hold above a critical frequency, provided all the typical frequencies appearing in the dynamics should be much lower than this critical frequency. In the strict 'Ohmic' limit the generalized Langevin equation becomes memoryless and corresponds to the classical Langevin equation.

## **3. The model and the calculation of instantaneous power**

We discussed the theoretical descriptions to understand how a dissipative system behaves when connected to a heat bath at equilibrium. The quantum system was considered to be not in equilibrium prior to the coupling. Once the coupling is established the quantum system continuously transfers its energy to the equilibrium bath and eventually, the system reaches thermal equilibrium (asymptotically) with the heat bath. Thermal equilibrium is said to have reached when a quantum system explores its phase-space fully. From here onwards we discuss the various energy exchanges that happen in a many-body system at equilibrium. The thermal properties of the quantum system can be calculated by assuming that the entire system-plus-bath arrangement is embedded in an infinitely large environment which provides the working temperature. Therefore, from here onwards we denote the quantum system as a subsystem of the bigger bath. Of course, the heat bath with *N* oscillator modes can also be considered as a subsystem of the bigger bath. Put it differently, the quantum system of our interest and the heat bath with which it is connected became the constituents of a large environment (**Figure 2**). Now we discuss how the energy exchange processes within this 'envelope'. We show here how the random force balances the energy lost by the quantum subsystem to the heat bath. It is enough to calculate the work done by the random force to that it compensates for the energy lost from the subsystem. Moreover, this work done by the random force is necessary to maintain equilibrium.

To proceed further, we need a working model for the subsystem. We choose the charged oscillator in a magnetic field as our quantum subsystem. Hence our system of

#### **Figure 2.**

*Pictorial representation of various constituents of a bigger bath. The quantum system as well as the heat bath are now two different parts of the larger environment. The terminology "subsystem" makes better sense in this context.*

#### **Figure 3.**

*(a) Electron motion under a perpendicular uniform magnetic field. The trajectories are helical in nature, but the projection of an individual electron's trajectory onto a two-dimensional plane shows a circular motion around the magnetic line of force. (b) The mexican-hat potential. We take this potential to confine the electron under perpendicular magnetic field and the whole arrangement can be simply called a "charged oscillator in a magnetic field". This real physical model is very useful and studied extensively in the condensed matter realm in various contexts. Studies on quantum dots and wires rely on this model heavily.*

study is the dissipative charged oscillator in a magnetic field not the uncoupled charged oscillator in a magnetic field. A damped harmonic oscillator was used as a quantum subsystem in the literature [19, 20]. Magnetic field effects in the domain of dissipative quantum physics are of great interest in various phenomena including the quantum Hall effect [21] and superconductivity [22]. Studies on the dissipative charged oscillator in a magnetic field using the system-plus-reservoir approach were originally carried out by Li et al. [23, 24]. This model, later, was used by many others [25–30] in different contexts (**Figure 3**).

H<sup>S</sup> represents the Hamiltonian of the charged magneto-oscillator. It is given by

$$\mathcal{H}\_{\rm S} = \frac{1}{2M} \left(\mathbf{p} - \frac{e\mathbf{A}}{c}\right)^2 + \frac{1}{2} M o\_0^2 \mathbf{r}^2,\tag{41}$$

where *M* is the mass of the quantum subsystem. The two dimensional vectors **p** and **r** represent the momentum and position coordinates of the subsystem respectively. Here **A** is denotes the magnetic vector potential, *e* is the charge of the electron, and *c* is the velocity of light. The Hamiltonian for the bath and the coupling, i.e., H<sup>B</sup> þ HSB, can be expressed as

$$\mathcal{H}\_{\rm B} + \mathcal{H}\_{\rm SB} = \sum\_{j=1}^{N} \left\{ \frac{\mathbf{p}\_j^2}{2m\_j} + \frac{1}{2} m\_j \alpha\_j^2 \left( \mathbf{x}\_j - \frac{\mathbf{C}\_j \mathbf{r}}{m\_j \alpha\_j^2} \right)^2 \right\}. \tag{42}$$

*mj*s and *ωj*s are the masses and frequencies of the individual bath oscillators respectively. *Cj*s are the coupling between the system and the heat bath oscilators. The two dimensional vectors **p***<sup>j</sup>* s and **x***j*s represent the momentum and the position coordinates of the bath oscillators respectively. The position and momentum vectors of the subsystem and the heat bath are operators and they satisfy the following commutation relations

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

$$\left[\mathbf{r}\_{i},\mathbf{p}\_{k}\right] = i\hbar\delta\_{ik}, \quad \left[\mathbf{x}\_{ji},\mathbf{p}\_{lk}\right] = i\hbar\delta\_{jl}\delta\_{ik}.\tag{43}$$

Following the steps given in the previous section, we use the Heisenberg equations of motion from the total Hamiltonian, we obtain the equations of motion for the subsystem and the bath coordinates. Eliminating the bath variables yields the generalized equation for the system coordinate (which is an operator) and that is given by [30]:

$$M\ddot{\mathbf{r}} + \int\_0^t dt' \dot{\mathbf{r}}(t')\chi(t - t') - \frac{e}{c}(\dot{\mathbf{r}} \times \mathbf{B}) + M o\_0^2 \mathbf{r} = -M\chi(t)\mathbf{r}(0) + \mathbf{F}(t). \tag{44}$$

This is an initial value equation and has an exact solution as well [30, 31]. Like we saw in the previous section, the spurious initial slip term is here as well. We may define an auxiliary random force KðÞ¼� *t Mγ*ð Þ*t* **r**ð Þþ 0 **F**ð Þ*t* just for esthetics. The Langevin equation is shown to be independent of any particular choice of gauge since it does not explicitly contain the magnetic vector potential **A**. The appearance of external magnetic field is via the quantum version of the Lorentz force term in the equation. The *memory friction function γ*ð Þ*t* is defined already in the previous section. The stochastic Brownian noise **F**ð Þ*t* , depends explicitly on initial coordinates and the momenta of the bath oscillators, is given by

$$\mathbf{F}(t) = \sum\_{j} \left\{ \mathbf{C}\_{j} \mathbf{x}\_{j}(\mathbf{0}) \cos \left( \alpha\_{j} t \right) + \frac{\mathbf{C}\_{j} \mathbf{p}\_{j}(\mathbf{0})}{m\_{j} \alpha\_{j}} \sin \left( \alpha\_{j} t \right) \right\}. \tag{45}$$

It is necessary to note that the damping *γ*ð Þ*t* and the operator valued Gaussian random force remain unchanged by the external magnetic field. The random force (45) satisfies the following relations [23, 24, 26, 28–30].

$$
\langle \langle F\_{\alpha}(t), F\_{\kappa}(t') \rangle \rangle = \delta\_{\alpha \kappa} \frac{2}{\pi} \int\_{0}^{\infty} d\alpha \Re[\check{\mathbf{y}}(\omega + i\mathbf{0}^{+})] \hbar \alpha$$

$$
\times \coth \left( \frac{\hbar \alpha}{2k\_{B}T} \right) \cos \left[ \alpha(t - t') \right], \tag{46}$$

$$
\langle [F\_{\alpha}(t), F\_{\kappa}(t')] \rangle = \delta\_{\alpha \kappa} \frac{2}{i\pi} \int\_{0}^{\infty} d\alpha \Re[\check{\mathbf{y}}(\omega + i\mathbf{0}^{+})] \hbar \alpha$$

$$
\times \sin \left[ \alpha(t - t') \right], \tag{47}$$

where *<sup>α</sup>*, *<sup>κ</sup>* <sup>¼</sup> *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* and <sup>~</sup>*γ*ð Þ¼ *<sup>z</sup>* <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> *dt* exp ð Þ *izt γ*ð Þ*t* , ð Þ *z*>0 . For the Drude regularized Ohmic spectral density, using Eq. (38), the Eqs. (46) and (47) can be written as

$$
\langle \{F\_a(t), F\_\kappa(t')\} \rangle = \delta\_{a\kappa} \frac{2\gamma a\_D^2}{\pi} \int\_0^\infty d\nu \frac{\hbar \nu}{a\_D^2 + \alpha^2}
$$

$$
\times \coth \left(\frac{\hbar \nu}{2k\_B T}\right) \cos \left[\omega(t - t')\right],
\tag{48}
$$

$$
\langle [F\_a(t), F\_\kappa(t')] \rangle = \delta\_{a\kappa} \frac{2\gamma a\_D^2}{\pi} \int\_0^\infty d\nu \frac{\hbar \nu}{a\_D^2 + \alpha^2}
$$

$$
\langle [F\_a(t), F\_\kappa(t')] \rangle = \delta\_{a\kappa} \frac{\omega\_f \omega\_D}{i\pi} \int\_0 d\phi \frac{\tau \omega}{w\_D^2 + \alpha^2}
$$

$$
\times \sin\left[\omega(t - t')\right]. \tag{49}
$$

The anti-symmetric correlation can also be written as

$$\langle\langle F\_a(t), F\_k(t')\rangle\rangle = 2\delta\_{\rm ox} \int\_0^\infty d\alpha \mathcal{G}(\alpha) \cos\left[\alpha(t - t')\right],\tag{50}$$

where Gð Þ *ω* is called the power spectrum, and is given by

$$\mathcal{G}(\boldsymbol{\alpha}) = \frac{\gamma \boldsymbol{\alpha}\_{\mathrm{D}}^{2}}{\pi} \frac{\hbar \boldsymbol{\alpha}}{\boldsymbol{\alpha}\_{\mathrm{D}}^{2} + \boldsymbol{\alpha}^{2}} \coth\left(\frac{\hbar \boldsymbol{\alpha}}{2k\_{\mathrm{B}}T}\right)$$

$$= \frac{2\gamma \boldsymbol{\alpha}\_{\mathrm{D}}^{2}}{\pi \left(\boldsymbol{\alpha}\_{\mathrm{D}}^{2} + \boldsymbol{\alpha}^{2}\right)} \left(\frac{\hbar \boldsymbol{\alpha}}{2} + \frac{\hbar \boldsymbol{\alpha}}{\boldsymbol{\epsilon}^{\hbar \boldsymbol{\alpha}/k\_{\mathrm{B}}T} - 1}\right). \tag{51}$$

The fluctuation dissipation theorem ensures that the symmetric combination and the commutator structure of the random force are (i) proportional to the friction constant *γ*, and (ii) are independent of the external potential *V*ð Þ**r** . From Eq. (52), we observe that the power spectrum of the random force is the Planck spectrum, with an extra term due to the zero point fluctuations. Needless to say, in the limit of ℏ ! 0 the spectrum becomes the flat spectrum of the white noise.

We now calculate the expectation value of the instantaneous power supplied by the random force. Since we work with operators, we use the symmetric form of the power. The power can be written as

$$\mathcal{P}\_F = \frac{1}{2} \langle \mathbf{v}(t)\mathbf{F}(t) + \mathbf{F}(t)\mathbf{v}(t) \rangle,\tag{52}$$

with **v**ðÞ¼ *t* **r**\_ð Þ*t* . We define the variables *Z* ¼ *x* þ *iy*, *F* ¼ *Fx* þ *iFy*, and *<sup>γ</sup>*ðÞ¼ *<sup>t</sup> <sup>γ</sup>*ð Þ*<sup>t</sup> <sup>m</sup>* þ *iωc*, to re-write the Langevin equation given in Eq. (44) as [30].

$$\ddot{Z} + \int\_0^t dt' \overline{\gamma}(t - t') \dot{Z}(t') + a\_0^2 Z = -\gamma(t) Z(\mathbf{0}) + \frac{F(t)}{M},\tag{53}$$

where *ω<sup>c</sup>* ¼ *eB=mc* is the magnetic cyclotron frequency. The new position coordinate is *Z* and its derivative with respect to time represents the velocity operator. Using Laplace transform and with the aid of fundamental solutions, we find the solution of the Langevin equation given in Eq. (53) as [30, 31].

$$Z(t) = M\dot{\chi}(t)Z(\mathbf{0}) + M\chi(t)\dot{Z}(\mathbf{0}) + \int\_0^t d\tau \chi(t-\tau)F(\tau),\tag{54}$$

where *χ*ð Þ*t* is called the response function of the system and is defined as [30, 31].

$$\chi(t-\tau) = \frac{1}{2\pi i} \int\_{-i\infty}^{+i\infty} ds \hat{\chi}(s) \exp\left(s(t-\tau)\right). \tag{55}$$

The Laplace transform of the response function is written as

$$\hat{\chi}(\mathfrak{s}) = \frac{1}{M} \frac{1}{\mathfrak{s}^2 + a\_0^2 + \mathfrak{s}\overline{\mathfrak{y}}(\mathfrak{s})},\tag{56}$$

with the Fourier transform

$$\chi(t-\tau) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} d\alpha \tilde{\chi}(\alpha) \exp\left(-i\alpha(t-\tau)\right),\tag{57}$$

where the dynamical susceptibility is given by

$$\tilde{\chi}(o) = \frac{1}{M} \frac{1}{-o^2 + o\_0^2 - i o \tilde{\eta}(o)}. \tag{58}$$

We have, here, *γ ω*ð Þ¼ *iω<sup>c</sup>* þ ~*γ ω*ð Þ. The memory kernel *γ*ð Þ*t* typically falls to zero in the bath relaxation time. As a result, the initial slip term in Eq. (44) (and subsequently in Eq. (53)) vanishes for very long times (can be experimental time scales) compared to the system's characteristic decay time. Therefore, for longer times compared to the system's relaxation time, the quantum Langevin equation (cf. Eq. (53)) becomes a stationary one with the lower limit of integration �∞, i.e., [30]

$$
\ddot{Z} + \int\_{-\infty}^{t} dt' \overline{\gamma}(t - t') \dot{Z}(t') + o\_0^2 Z = \frac{F(t)}{M},\tag{59}
$$

with the solution, which is a stationary process, is given by

$$Z(t) = \int\_{-\infty}^{t} d\tau \chi(t-\tau)F(\tau). \tag{60}$$

We note from the expressions (Eqs. (53) and (57)) that, so long as *ω*<sup>0</sup> remains nonzero, the response function *χ*ð Þ *t* � *τ* vanishes exponentially for longer times. This, in fact, is in close proximity with the Tauberian theorem which states the asymptotic behavior of a function depends on the low frequency behavior of its Fourier transform. Hence for very longer times, the dependence of *Z t*ð Þ on the initial values (cf. Eq. (54)) completely disappears and that yield [30, 31].

$$Z(t) = \int\_0^t d\tau \chi(t-\tau)F(\tau). \tag{61}$$

Upon comparing Eq. (61) with Eq. (60), we conclude that *Z t*ð Þ in Eq. (61) is the solution of the stationary Langevin equation in Eq. (59). With these remarks in mind, we write the Langevin equation in Eq. (44) as

$$M\ddot{\mathbf{r}} + M\int\_{-\infty}^{t} dt'\dot{\mathbf{r}}(t')\gamma(t - t') - \frac{e}{c}(\dot{\mathbf{r}} \times \mathbf{B}) + M\alpha\_0^2 \mathbf{r} = \mathbf{F}(t).\tag{62}$$

Using Eqs. (52) and (62), we write the power as

$$\mathcal{P}\_F = \frac{d}{dt} \left\langle \frac{1}{2M} \left(\mathbf{P} - \frac{e\mathbf{A}}{c}\right)^2 + \frac{1}{2} M \omega\_0^2 \mathbf{r}^2 \right\rangle$$

$$+ \int\_{-\infty}^t dt' \chi(t - t') \frac{1}{2} \langle \dot{\mathbf{r}}(t) \dot{\mathbf{r}}(t') + \dot{\mathbf{r}}(t') \dot{\mathbf{r}}(t) \rangle. \tag{63}$$

Since the Langevin equation describes a stationary process the first term is zero because for a stationary process the expectation values of time dependent quantities must be time translational invariant or, in other words, they are constant. This becomes clear when we evaluate the expectation value of an arbitrary time dependent operator *W t*ð Þ . We write [19].

$$\langle W(t)\rangle = \frac{\text{Tr}\left\{e^{-\beta H}e^{i\text{Ht}/\hbar}W(\mathbf{0})e^{-i\text{Ht}/\hbar}\right\}}{\text{Tr}e^{-\beta H}} = \frac{\text{Tr}\{W(\mathbf{0})\}}{\text{Tr}e^{-\beta H}},\tag{64}$$

where *β* ¼ 1*=kBT*. The cyclic property of the Trace operation is used to get the desired result. Thus it becomes clear that h i *W t*ð Þ is time independent for a canonical ensemble and the derivative of it must be zero. Our task is just to find the second term in Eq. (64). We use Eq. (61) to compute the velocity autocorrelation function. It is to be noted that the real part of the quantity <sup>1</sup> <sup>2</sup> *Z t* \_ð Þ, *<sup>Z</sup>*\_ † *<sup>t</sup>* <sup>0</sup> ð Þ D E n o is

equivalent to <sup>1</sup> **r**\_ð Þ*t* **r**\_ *t* ð Þþ **r**\_ *t* h i ð Þ**r**\_ð Þ*t* .

We write [30].

$$\frac{1}{2}\langle\left\{Z(t),Z^{\dagger}(t')\right\}\rangle = \frac{1}{2m^2}\int\_0^t d\tau \int\_0^{t'} d\tau' \chi(t-\tau)\chi^\*\left(t'-\tau'\right)\langle\left\{F(\tau),F^{\dagger}(\tau')\right\}\rangle,\tag{65}$$

where, using (47), we write [30].

$$
\left\langle \left\{ F(\tau), F^\dagger(\tau') \right\} \right\rangle = \left\langle \left\{ F\_x(\tau), F\_x(\tau') \right\} \right\rangle + \left\langle \left\{ F\_\mathcal{V}(\tau), F\_\mathcal{V}(\tau') \right\} \right\rangle,
$$

$$
= \frac{4}{\pi} \int\_0^\infty da \Re[\bar{\jmath}(\varkappa + i\mathbf{0}^+)] \hbar \varkappa \times \coth\left(\frac{\hbar \varkappa}{2k\_B T}\right) \cos\left[ \wp(\tau - \tau') \right].
$$

After some algebra, we obtain [30].

$$\frac{1}{2}\langle\left\{Z(t),Z^{\dagger}(t')\right\}\rangle = \frac{\hbar}{M\pi}\int\_{-\infty}^{+\infty} d\alpha \,\alpha \Re\left[\frac{\check{\gamma}(\alpha)}{m}\right] \check{\chi}(\alpha)\check{\chi}^\*\left(\alpha\right)$$

$$\times \coth\left(\frac{\hbar\alpha}{2k\_BT}\right)e^{-i\alpha(t-t')},\tag{67}$$

where

$$\tilde{\chi}(\boldsymbol{\alpha}) = -\frac{1}{M} \frac{(\boldsymbol{\alpha} + i\boldsymbol{\alpha}\_{\rm D})}{(\boldsymbol{\alpha} + i\boldsymbol{\lambda}\_{1})(\boldsymbol{\alpha} + i\boldsymbol{\lambda}\_{2})(\boldsymbol{\alpha} + i\boldsymbol{\lambda}\_{3})},\tag{68}$$

Here *λjs* are the roots of the cubic equation

$$i a^3 + i a^2 (a o\_\mathcal{D} + i a\_\mathcal{c}) - a \left(a\_\mathcal{O}^2 + \chi a o\_\mathcal{D} + i a\_\mathcal{c} a o\_\mathcal{D}\right) - i a\_\mathcal{O}^2 a o\_\mathcal{D} = \mathbf{0}.\tag{69}$$

Similarly, the complex conjugate <sup>~</sup>*<sup>χ</sup>* <sup>∗</sup> ð Þ *<sup>ω</sup>* of Eq. (68) is given by

$$\tilde{\chi}^\*(\boldsymbol{\alpha}) = -\frac{1}{M} \frac{(\boldsymbol{\alpha} - i\boldsymbol{\alpha}\_{\rm D})}{(\boldsymbol{\alpha} - i\boldsymbol{\lambda}\_{1'})(\boldsymbol{\alpha} - i\boldsymbol{\lambda}\_{2'})(\boldsymbol{\alpha} - i\boldsymbol{\lambda}\_{3'})},\tag{70}$$

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

with the *λ*<sup>0</sup> *j* s are the complex conjugates of *λj*s. Since Eqs. (68) and (70) satisfy the following relation

$$\mathcal{M}o\mathfrak{R}[\check{\chi}(o)]\check{\chi}(o)\check{\chi}^\*\left(o\right) = \frac{1}{2i}[\check{\chi}(o) - \check{\chi}^\*\left(o\right)] = \check{\chi}''(o),\tag{71}$$

we obtain [30].

$$\frac{1}{2}\langle\left\{Z(t),Z^{\dagger}(t')\right\}\rangle = \frac{\hbar}{M\pi}\int\_{-\infty}^{+\infty} d\alpha \,\tilde{\chi}^{\prime\prime}(\alpha)\coth\left(\frac{\hbar\alpha}{2k\_BT}\right)e^{-i\alpha(t-t')},\tag{72}$$

where ~*χ*00ð Þ *ω* is given by

$$\tilde{\chi}''(o) = \frac{1}{M} \frac{o\tilde{\gamma}(o)}{\left(o\_0^2 - o^2 + oo o\_\varepsilon\right)^2 + o^2\tilde{\gamma}^2(o)}.\tag{73}$$

Taking the derivatives of Eq. (72) with respect to *t* and *t* <sup>0</sup> yields the velocity autocorrelation

$$\frac{1}{2}\left\langle \left\{ \dot{Z}(t), \dot{Z}^{\dagger}(t') \right\} \right\rangle = \frac{\hbar}{M\pi} \int\_{-\infty}^{\infty} d\phi \,\alpha^{2} \,\bar{\chi}^{\prime\prime}(\alpha) \coth\left(\frac{\hbar\alpha}{2k\_{B}T}\right) e^{-i\alpha(t-t')}.\tag{74}$$

Substituting the real part of the velocity autocorrelation into the equation for the power yields

$$\mathcal{P}\_F = \int\_{-\infty}^t dt' \chi(t - t') \frac{\hbar}{M\pi} \int\_{-\infty}^\infty d\alpha \,\alpha^2 \,\bar{\chi}''(\alpha)$$

$$\times \coth\left(\frac{\hbar\alpha}{2k\_B T}\right) \cos\left[\alpha(t - t')\right]. \tag{75}$$

Since the damping *<sup>γ</sup>*ð Þ*<sup>t</sup>* is 0 for negative *<sup>t</sup>* and <sup>~</sup>*γ ω*ð Þ¼ <sup>Ð</sup> <sup>∞</sup> <sup>0</sup> *dtγ*ð Þ*<sup>t</sup> <sup>e</sup><sup>i</sup>ω<sup>t</sup>* , *ω*>0, the upper limit of the integral in Eq. (44) can be replaced with þ∞. Therefore

$$\mathcal{P}\_F = \frac{\hbar}{M\pi} \int\_{-\infty}^{+\infty} d\alpha \int\_{-\infty}^{\infty} dt' \,\chi(t - t') \cos\left[\omega(t - t')\right]$$

$$\times \omega^2 \,\tilde{\chi}''(\omega) \coth\left(\frac{\hbar\omega}{2k\_B T}\right). \tag{76}$$

It is possible to write

$$\int\_{-\infty}^{+\infty} dt' \gamma(t - t') \cos \left[ \alpha(t - t') \right] = \Re \left\{ \int\_{-\infty}^{+\infty} dt' \gamma(t - t') e^{i\alpha(t - t')} \right\}.\tag{77}$$

Since the bracketed term is just the Fourier transform ~*γ ω*ð Þ of *γ t* � *t* <sup>0</sup> ð Þ, we write Eq. (77) as

$$\mathcal{P}\_F = \frac{\hbar}{M\pi} \int\_{-\infty}^{\infty} d\alpha \,\alpha^2 \,\tilde{\chi}''(\alpha) \Re[\tilde{\jmath}(\alpha)] \coth\left(\frac{\hbar\alpha}{2k\_B T}\right). \tag{78}$$

It is immediately observed that only the even part of the above integral contributes. We know that <sup>~</sup>*γ*ð Þ �*<sup>ω</sup>* <sup>∗</sup> <sup>¼</sup> <sup>~</sup>*γ ω*ð Þ and <sup>~</sup>*χ*ð Þ¼ �*<sup>ω</sup>* <sup>~</sup>*χ ω*ð Þ, so that <sup>~</sup>*χ*00ð Þ *<sup>ω</sup>* is an odd function of *ω* and ℜ½ � ~*γ ω*ð Þ is an even function of the frequency *ω*. Upon using the real part of ~*γ ω*ð Þ, we write

$$\mathcal{P}\_F = \frac{\hbar}{M\pi} \int\_{-\infty}^{\infty} d\alpha \,\,\alpha^2 \,\,\tilde{\chi}''(\alpha) \,\frac{\gamma \alpha\_D^2}{\alpha\_D^2 + \alpha^2} \coth\left(\frac{\hbar \alpha}{2k\_B T}\right). \tag{79}$$

This is the instantaneous power supplied by the random force for our particular model and the above expression is positive quantity always.

To check the result, we evaluate the power supplied by the random force at high temperatures (or in the classical case). After some tedious mathematical manipulations we obtain [32].

$$\mathcal{P}\_F = \frac{4k\_B T \chi}{M} \frac{\left(2 + \alpha\_0^2 / \alpha\_\mathrm{D}^2 + \chi / \alpha\_\mathrm{D}\right)}{\left[\left(2 + \alpha\_0^2 / \alpha\_\mathrm{D}^2 + \chi / \alpha\_\mathrm{D}\right)^2 + \left(2 \alpha\_\mathrm{c} / \alpha\_\mathrm{D}\right)^2\right]},\tag{80}$$

which in the large cutoff (strict ohmic) limit (*ω<sup>D</sup>* ! ∞) yields the form

$$\mathcal{P}\_F = \frac{4k\_B T \gamma}{M}.\tag{81}$$

This clearly indicates to us that the rate of work done by the random force on the quantum subsystem is indeed proportional to the damping/dissipation strength *γ*. This in turn tells us that, at equilibrium, the energy lost by the quantum subsystem due to dissipation is compensated by an amount of energy received from the random force. As a consequence of the fluctuation-dissipation theorem, the Eq. (81) does not contain any term from the external potential (and magnetic field), in the strict ohmic limit.

## **4. Discussion of the result**

The full quantum many-body system has an infinite number of degrees of freedom each with its corresponding zero-point oscillations. The full system must be in the ground state at the absolute zero of temperature (*T* ¼ 0). This further implies that no work is done on or by the system. However, for any finite coupling (irrespective of the strength of the coupling), H*<sup>S</sup>* does not commute with H. That means the ground state of H is not the ground state of H*S*. Therefore, even at absolute zero temperature, the energy of the charged oscillator in a magnetic field must fluctuate. Mathematically, this statement can be expressed as

$$
\Delta \mathcal{H}\_{\mathbb{S}}^2 = \left< \mathcal{H}\_{\mathbb{S}}^2 \right> - \left< \mathcal{H}\_{\mathbb{S}} \right>^2 \neq \mathbf{0},\tag{82}
$$

the mean square fluctuations of H*<sup>S</sup>* is not equal to zero. No matter how weak the coupling between the subsystem and the bath, the fluctuation of the subsystem Hamiltonian does not vanish at *T* ¼ 0. This fluctuation is obviously driven by the random force. Therefore, the work done per unit time by this random force is, indeed, balanced by the dissipative loss of the subsystem. Hence there is no net work done on the subsystem. To evaluate the RHS of Eq. (82) we resort to Eq. (41). We write

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

$$
\langle \mathcal{H}\_{\rm S} \rangle = \frac{1}{2\mathcal{M}} \left\langle \left( \mathbf{p} - \frac{e\mathbf{A}}{c} \right)^{2} \right\rangle + \frac{1}{2} \mathcal{M} o\_{0}^{2} \langle \mathbf{r}^{2} \rangle,\tag{83}
$$

In the new variable *Z* ¼ *x* þ *iy*, the subsystem Hamiltonian can be written as [30].

$$\mathcal{H}\_{\rm S} = \frac{1}{2} \dot{M} \dot{Z} \dot{Z}^{\dagger} - \frac{1}{2} \hbar \alpha\_{\rm c} + \frac{1}{2} \mathcal{M} \alpha\_{0}^{2} Z \mathcal{Z}^{\dagger}. \tag{84}$$

Therefore

$$
\langle \mathcal{H}\_{\rm S} \rangle = \frac{1}{2} \mathbf{M} \Big\langle \dot{Z} \dot{Z}^{\dagger} \Big\rangle - \frac{1}{2} \hbar o\_{\rm c} + \frac{1}{2} M o\_{\rm 0}^{2} \langle Z Z^{\dagger} \rangle. \tag{85}
$$

We need to compute the unequal time correlation functions *Z t*ð Þ*Z*† *<sup>t</sup>* <sup>0</sup> ð Þ � � and *Z t* \_ð Þ*Z*\_ † *<sup>t</sup>* <sup>0</sup> ð Þ D E. To get this, we only evaluate the symmetric combination of *Z t*ð Þ and *Z*† *t* <sup>0</sup> ð Þ. The anti-symmetric combination vanishes in the analytic continuation *t* <sup>0</sup> ¼ *t* to obtain the equilibrium values [30].

Using Eqs. (72) and (74), at *T* ¼ 0, we compute Eq. (85). We get [30].

$$
\begin{split}
\langle\mathcal{H}\_{\mathcal{S}}\rangle\_{T=0} &= \frac{\hbar\chi}{\pi} \ln\left(\frac{\alpha\nu\_{\rm D}}{\alpha\_{0}}\right) + \frac{\hbar\nu\_{0}^{2}}{\pi} \left\{\frac{1}{\Lambda} \ln\left(\frac{\lambda\_{1}}{\lambda\_{2}}\right) + \frac{1}{\Lambda'} \ln\left(\frac{\lambda\_{1}'}{\lambda\_{2}'}\right)\right\} \\ &- \frac{\hbar}{4\pi} \left\{\frac{\overline{\mathcal{T}}^{2}}{\Lambda} \ln\left(\frac{\lambda\_{1}}{\lambda\_{2}}\right) + \frac{\overline{\mathcal{T}}^{\*2}}{\Lambda'} \ln\left(\frac{\lambda\_{1}'}{\lambda\_{2}'}\right)\right\},
\end{split} \tag{86}
$$

where Λ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>γ</sup>*<sup>2</sup> � <sup>4</sup>*ω*<sup>2</sup> 0 q and Λ<sup>0</sup> ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>γ</sup>* <sup>∗</sup> <sup>2</sup> � <sup>4</sup>*ω*<sup>2</sup> 0 q . Also *λ*1,2 ¼ *γ=*2 � *i* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 <sup>0</sup> � *<sup>γ</sup>*<sup>2</sup>*=*<sup>4</sup> q and *λ*0 1,2 <sup>¼</sup> *<sup>γ</sup>* <sup>∗</sup> *<sup>=</sup>*2∓*<sup>i</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 <sup>0</sup> � *<sup>γ</sup>* <sup>∗</sup> <sup>2</sup>*=*<sup>4</sup> q . For weak dissipation *γ*, *γ* <sup>∗</sup> ≪ 2*ω*0,

$$\begin{split} \langle \mathcal{H}\xi \rangle\_{T=0} &= \frac{\hbar \gamma}{\pi} \ln \left( \frac{\omega \alpha\_{0}}{\omega\_{0}} \right) - \frac{\hbar \gamma \alpha\_{c}}{2 \pi \alpha\_{0}} \ln \left\{ \frac{(\overline{\gamma}/2 + i\alpha o)(\overline{\gamma}^{\*}/2 - i\alpha o)}{(\overline{\gamma}/2 - i\alpha\_{0})(\overline{\gamma}^{\*}/2 + i\alpha\_{0})} \right\} \\ &+ \frac{i\hbar \left( \chi^{2} + \alpha\_{c}^{2} - 4\alpha\_{0}^{2} \right)}{4\pi \alpha\_{0}} \ln \left\{ \frac{(\chi^{2} + \alpha\_{c}^{2} - 4\alpha\_{0}^{2}) + 4i\gamma \alpha\_{0}}{(\chi^{2} + \alpha\_{c}^{2} - 4\alpha\_{0}^{2}) - 4i\gamma \alpha\_{0}} \right\}. \end{split} \tag{87}$$

In the absence of the magnetic field (*ω<sup>c</sup>* ¼ 0) we get

$$
\langle \mathcal{H}\_{\rm S} \rangle\_{T=0} = \hbar a \rho\_0 + \frac{\hbar \gamma}{\pi} \ln \left( \frac{a \rho\_D}{a \rho\_0} \right),
\tag{88}
$$

which is nothing but the result for a two dimensional isotropic oscillator. Since **r** and **r**\_ (or *Z* and *Z*\_ ) are Gaussian operators with zero mean values, H<sup>2</sup> *S* � � (cf, Eq. (41)) can be readily expressed in terms of products of **r**<sup>2</sup> � � and **r**\_ <sup>2</sup> � � (or using *Z*<sup>2</sup> and *Z*\_ <sup>2</sup> with Eq. (84)). We evaluate the H<sup>2</sup> *S* � � in the absence of magnetic field to just show the concerned point. Magnetic field has no significant role in the present evaluation to show that the mean square fluctuation of the subsystem Hamiltonian does not vanish. With the aid of Eq. (88), we write

$$
\Delta \mathcal{H}\_{\rm S}^{2} = \langle \mathcal{H}\_{\rm S} \rangle \frac{\hbar \chi}{\pi} \ln \left( \frac{\alpha\_{\rm D}}{\alpha\_{0}} \right) . \tag{89}
$$

Even for a very weak damping, the mean of the subsystem energy is above its ground state energy and that the fluctuations in this energy do not vanish. This tells us that one part of a physical system in its ground state can and does exchange energy with another part. The formalism we have chosen to show the energy balance is an exact one. We use the Langevin equation to calculate the expectation values after taking the time derivative of the subsystem Hamiltonian.

$$\frac{d}{dt}\langle \mathcal{H}\_{\mathcal{S}} \rangle + \chi \langle \dot{\mathbf{r}}^2(t) \rangle = \frac{1}{2} \langle \dot{\mathbf{r}}(t)\mathbf{F}(t) + \mathbf{F}(t)\dot{\mathbf{r}}(t) \rangle. \tag{90}$$

The exact equation given above is obtained under the strict ohmic limit of the Langevin Equation in Eq. (62). The expectation values appearing above are equal time expectation values and are independent of time as the total physical system is invariant under the time translation. The first term in the above relation is zero. The rest of the equation then describe that the power (RHS) is actually proportional to the dissipation constant *γ*. Power does not vanish even at absolute zero temperature and this indicates that different parts of the physical system can and do exchange energy even at absolute zero temperature.

## **Author details**

Jishad Kumar Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel

\*Address all correspondence to: kumar.jishad@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Dissipative Quantum System and Energy Balance DOI: http://dx.doi.org/10.5772/intechopen.106474*

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## **Chapter 4**

## Centrifugal Acceleration in Relativistic Astrophysics

*Andria Rogava*

## **Abstract**

Particles moving along prescribed, relativistically rotating trajectories may exhibit quite unexpected, interesting kinematic behavior. Their dynamics may lead to a number of physical processes that could have various important consequences in a wide variety of relativistic astrophysical objects. In this chapter the author gives brief review of the theoretical ideas and results related to Machabeli and Rogava *gedankenexperiment* (1994) and their astrophysical implications. In particular three astrophysical cases—acceleration of particles by rotating magnetospheres in AGNs, centrifugal acceleration and gamma flares in Crab nebula, and self-trapping as a beaming mechanism for Fast Radio Bursts—are discussed. Conclusions and future research prospects are briefly outlined.

**Keywords:** special relativity, fundamental problems and general formalism, relativity and gravitation, centrifugal acceleration, relativistic astrophysics

## **1. Introduction**

Crisis is a Hair Toward which forces creep Past which forces retrograde … . - Emily Dickinson.

Christiaan Huygens introduced the concept of the centrifugal force in the manuscript entitled "*De vi Centrifuga*" written in 1659 [1] and published posthumously in Leiden, in 1703. Since then, the concept of the centrifugal force became an integral part of the essential vocabulary of physics and a topic of different interpretations in the philosophy of the physical science. It proved to be a useful notion for studies of a wide variety of problems involving rotation and dynamics of particles/bodies in rotating frames of reference. Rotation is omnipresent in the nature; as long as the humans advanced in the space exploration, it became increasingly evident that the physical processes related to the presence of rotation play an important, often decisive, role in the complex and still incompletely understood texture of the physical universe.

From the very beginning, there was a certain kind of dichotomy in the understanding of the centrifugal force. According to Newton and basic principles of the classic mechanics, the magnitude and the direction of the centrifugal force, acting on matter in a rotating frame of reference, are not connected with motion of the matter, but are

entirely and explicitly determined by the rotation of the reference frame. On the basis of this postulate, the centrifugal force is defined in terms of the angular velocity of the instantaneously corotating reference frame and a displacement of the moving body (particle) from the instantaneous rotation axis. Consequently, the direction of the force is always unequivocally determined by the direction of the displacement vector and it always points outward. However, it has to be remembered that the original concept of the centrifugal force, outlined by Huygens [2], emphasized another, drastically different, aspect of the force. In particular, Huygens' idea was the following: "*centrifugal force should not be connected to rotation of a reference frame, but to motion along a curved path in space.*" Within the framework of classic, nonrelativistic physics, the difference between these two postulates does not lead to a quantitative difference, because in the Newtonian gravitation theory, based on Euclidean geometry, these two definitions are equivalent. But in Relativity, when the space–time is not Euclidean, they differ from each other [2] and may lead to different results.

In recent decades, the increase of interest in the phenomenon of relativistic rotation was largely due to series of significant results by M. Abramowicz and his collaborators [2–5] related to physics of relativistic rotation in the vicinity of black holes. In particular, it was found that under certain circumstances, the centrifugal force attracts toward the rotation axis even in the case of a nonrotating Schwarzschild black hole. This outstanding fact led to a number of unusual rotational effects considered and discussed in these publications.

About 30 years ago, Machabeli and Rogava considered [6] a "*gedankenexperiment*" (viz. an experiment carried out in thought only): centrifugally driven motion of a bead inside a long, straight pipe rotating with a constant angular velocity *ω* ¼ *const* around an axis normal to its symmetry axis. The motion of the bead was studied in the frame of reference of the rotating pipe. Different classes of exact analytic solutions were found, exhibiting puzzling regimes of motion: it was discovered that centrifugally driven bead could both accelerate and decelerate, and the acceleration could change its sign; actual modes of motion strongly depended on the value of the initial velocity. The purpose of the idealized *"gedankenexperiment"* was to mimic the motion of centrifugally driven charged particles in various astronomical situations. The possible importance of this result in the context of centrifugally driven dynamics of particles in rapidly rotating pulsar magnetospheres and astrophysical jets was later repeatedly pointed out [7] and extensively studied in a number of subsequent publications [8–12]. Namely, particle acceleration by rotating magnetospheres in active galactic nuclei was investigated [13]; the role of radiation reaction forces in the dynamics of centrifugally accelerated particles was explored [14]; centrifugal acceleration was studied in isotropic photon fields [15] and in wormhole metrics [16].

In this chapter of the present book, I will try to give a brief but systematic overview of the Machabeli & Rogava *gedankenexperiment* and its astrophysical implications. The material is divided into two sections: the next one is dedicated to the theoretical ideas and results, while the final one deals with astrophysical applications of the theory. The latter also contains a very short description of unresolved issues and possible directions for further, both physical and astrophysical, research.

## **2. Theoretical background**

In this section, a brief overview of theoretical ideas and results related to the centrifugally driven motion of particles along prescribed trajectories is given. Firstly, main characteristics of the original *gedankenexperiment* [6] are derived and discussed in detail. Secondly, it is shown how the idea developed and what is the range of related physical results disclosed in the process.

#### **2.1 Original gedankenexperiment: Rotating frame study**

In Ref. [6], a straight pipe rotating around an axis normal to the pipe, and a small bead moving inside the pipe without the friction were considered; radii of the bead and the pipe being equal to one other. According to Newtonian mechanics the bead would move with ever-increasing acceleration, if at the moment *t* ¼ 0 it was located just above the axis pivot (*r*<sup>0</sup> ¼ 0) and had an initial velocity *v*<sup>0</sup> 6¼ 0, then the solution of Newton's second law of motion would imply that the radial distance *r t*ð Þ from the axis would increase in time as:

$$r(t) = (v\_0/\alpha)\sinh\left(\alpha t\right),\tag{1}$$

while the bead's radial velocity *vr*ð Þ*t* along the pipe would monotonously increase as:

$$
\upsilon\_r(t) = \upsilon\_0 \cosh\left(at\right). \tag{2}
$$

However, relativity-related common sense tells us that if the pipe is long enough and its walls are absolutely rigid, then the bead's increasing velocity sooner or later would become relativistic, *v* ≪ *c* condition sooner or later ceases to hold and the bead's motion has to be scrutinized by means of special relativity. Intuitively it seems evident that the increase of the velocity would somehow be limited since the total velocity of the bead *vtot* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *v*2 *<sup>r</sup>* <sup>þ</sup> *<sup>r</sup>*<sup>2</sup>*ω*<sup>2</sup> <sup>p</sup> cannot exceed the speed of light. Therefore, the problem has to be considered within the framework of a relativistic theory. Hereafter, we shall use *geometrical units*, in which *G* ¼ *c* ¼ 1. If the bead would reach the *light cylinder* (defined as the radial distance *<sup>r</sup>* <sup>∗</sup> � <sup>1</sup>*=ω*, that is, *<sup>v</sup>ϕ*ðÞ� *<sup>t</sup> <sup>ω</sup><sup>r</sup>* <sup>∗</sup> <sup>¼</sup> 1) its radial velocity at *<sup>r</sup>* <sup>∗</sup> should fall to zero value, implying that the radial velocity *vr*ð Þ*t* , being increasing at the initial stage of the motion, must become decreasing as the bead approaches the light cylinder.

In Ref. [6] the problem was considered in a reference frame of the rotating pipe (rotating frame, hereafter referred to as RF), where the dynamics of the bead motion appeared to be the simplest: one-dimensional, radial motion along the straight pipe. If the space–time in the laboratory frame (hereafter referred to as LF) is Minkowskian with the metric:

$$ds^2 \equiv -d\tau^2 = -dT^2 + dX^2 + dY^2,\tag{3}$$

then applying the transformation of variables: *T* ¼ *t*, *X* ¼ *rcos*ð Þ *ωt* , *Y* ¼ *rsin*ð Þ *ωt* it can be rewritten in the following form:

$$ds^2 \equiv -d\tau^2 = -\left(\mathbb{1} - a^2 r^2\right)dt^2 + dr^2,\tag{4}$$

where *τ* is a proper time of the moving bead, defined in the standard way. The Lagrangian for the derived "1+1" space–time metric (4) may be defined as [17]:

$$L \equiv \frac{\mathcal{g}\_{a\beta}}{2} \dot{\mathbf{x}}^a \dot{\mathbf{x}}^\beta = \frac{1}{2} \left[ - \left( 1 - a^2 r^2 \right) \left( \frac{dt}{d\tau} \right)^2 + \left( \frac{dr}{d\tau} \right)^2 \right]. \tag{5}$$

Components of the corresponding Lagrange equation:

$$\frac{d}{d\pi} \left( \frac{\partial L}{\partial \dot{\mathbf{x}}^a} \right) = \frac{\partial L}{\partial \mathbf{x}^a},\tag{6}$$

may be written in the following way:

$$-\left(1 - \alpha^2 r^2\right) \left(\frac{dt}{d\tau}\right) = const \equiv -E,\tag{7}$$

$$\frac{d^2r}{d\tau^2} = \alpha^2 r \left(\frac{dt}{d\tau}\right)^2. \tag{8}$$

A simpler equation for *d*<sup>2</sup> *r=dτ*<sup>2</sup> may be derived by combining (7) with the algebraic relation between 4-velocities, resulting from their normalization *<sup>g</sup>αβuαu<sup>β</sup>* ¼ �1 condition:

$$-\left(\mathbf{1} - \alpha^2 r^2\right) \left(dt/d\tau\right)^2 + \left(dr/d\tau\right)^2 = -\mathbf{1}.\tag{9}$$

The result of a simple calculation is:

$$
\left(\frac{dr}{d\tau}\right)^2 = -\mathbf{1} + \frac{E^2}{\mathbf{1} - \alpha^2 r^2}.\tag{10}
$$

On the other hand, rewriting (9) as

$$
\left(\frac{dt}{d\tau}\right)^2 = \left[1 - \alpha^2 r^2 - \left(\frac{dr}{dt}\right)^2\right]^{-1},\tag{11}
$$

and expressing from (7) *dt=dτ* by means of *E*, we can write down the following first-order differential equation for the function *r t*ð Þ:

$$drdt = \sqrt{(\mathbf{1} - \alpha^2 r^2) \left[\mathbf{1} - \frac{(\mathbf{1} - \alpha^2 r^2)}{E^2}\right]}.\tag{12}$$

A constant parameter *E* ¼ �*Ut*, coming into view in these equations, may be treated as an energy of the moving bead in the RF and may be determined through initial conditions. For the above-mentioned case, when at the moment *t* ¼ 0, the bead was at the position *r* ¼ *r*<sup>0</sup> and had a velocity *vr* ¼ *v*0, we can easily find out that

$$E = \frac{(\mathbf{1} - \alpha^2 r\_0^{-2})}{\sqrt{\mathbf{1} - \alpha^2 r\_0^{-2} - v\_0^{-2}}}.\tag{13}$$

Obviously, 0<sup>&</sup>lt; *<sup>E</sup>*<sup>&</sup>lt; <sup>∞</sup>. If, for example, *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> 0 and *<sup>v</sup>*<sup>0</sup> 6¼ 0 then *<sup>E</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>v</sup>*<sup>2</sup> 0 � ��1*=*<sup>2</sup> and is more than unity (*E* ≫ 1, if *v*<sup>0</sup> ≈1). While, if *v*<sup>0</sup> ¼ 0 and *r*<sup>0</sup> 6¼ 0 then we see that *<sup>E</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>*r*<sup>2</sup> 0 � �<sup>1</sup>*=*<sup>2</sup> , certainly, is less than unity. Note also that *E* ≈0 if *ωr*<sup>0</sup> ≈1.

It must be noted that in the nonrelativistic limit the energy of the moving bead reduces to the following expression:

*Centrifugal Acceleration in Relativistic Astrophysics DOI: http://dx.doi.org/10.5772/intechopen.109684*

$$E\_{nr} = \mathbf{1} + \mathbf{v}^2/2 - \alpha^2 r^2/2,\tag{14}$$

where a unity evidently describes rest mass energy (per unit mass) of the bead, while remaining terms have clear "nonrelativistic" physical meaning. In particular, *<sup>v</sup>*2*=*2 is a kinetic energy corresponding to radial motion in the RF, while �*ω*2*r*2*=*2 is a "centrifugal energy" [18], known from classical mechanics.

One can also derive a differential equation for a radial acceleration of the bead *d*2 *r=dt*<sup>2</sup> . The result is:

$$d^2rdt^2 = \frac{\alpha^2r}{1 - \alpha^2r^2} \left[ 1 - \alpha^2r^2 - 2\left(\frac{dr}{dt}\right)^2 \right],\tag{15}$$

at its left-hand side appears a radial acceleration of the bead as measured in the RF. Note that in nonrelativistic limit, it reduces to the conventional expression for the centrifugal force *<sup>f</sup> cf* <sup>¼</sup> *<sup>w</sup>*<sup>2</sup>*r*.

In Ref. [6] it was shown that the exact analytic solution of (12) may be expressed by means of elliptical functions. It was derived by introducing two auxiliary variables *<sup>θ</sup>* � *arccos*ð Þ *<sup>ω</sup><sup>r</sup>* and *<sup>λ</sup>* � *<sup>ω</sup>t*; and a parameter *<sup>m</sup>* � <sup>1</sup>*=E*<sup>2</sup> , reducing (12) to:

$$\frac{d\theta}{d\lambda} = -\sqrt{1 - m\sin^2\theta}.\tag{16}$$

The solution of this equation may be written as:

$$
\lambda = \int\_0^{p\_0} \frac{d\theta}{\sqrt{1 - m \sin^2 \theta}} - \int\_0^{\rho} \frac{d\theta}{\sqrt{1 - m \sin^2 \theta}},\tag{17}
$$

with *φ*<sup>0</sup> ¼ *arccos*ð Þ *ωr*<sup>0</sup> . It can be rewritten as:

$$
\rho = \mathfrak{a}m(\lambda^\* - \lambda),
\tag{18}
$$

where *am* is an amplitude of Jacobian elliptic functions [19] and we have introduced an additional parameter:

$$
\lambda^\* \equiv \int\_0^{\wp\_0} \frac{d\theta}{\sqrt{\mathbf{1} - m \sin^2 \theta}}. \tag{19}
$$

Recovering physical variables *t* and *r* one comes to the following explicit solution:

$$r(t) = \frac{1}{a} c n(\lambda^\* - a t) \tag{20}$$

where *cn* is Jckobian elliptic cosine [19].

It is the *general* solution to the problem. In Ref. [6] a particular case was considered, specified by above-mentioned initial conditions: *r* ¼ 0 and *v*<sup>0</sup> 6¼ 0 at *t* ¼ 0. For the case, *<sup>m</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>v</sup>*<sup>2</sup> <sup>0</sup> and *<sup>λ</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>K</sup>*, where *<sup>K</sup>* is a *complete elliptical integral of the first kind* [19]. It allows rewriting the general solution (20) in a different way:

$$r(t) = \frac{1}{\alpha} c n(K - \alpha t) = \left(\frac{v\_0}{\alpha}\right) \frac{\sin(\alpha t)}{\ln(\alpha t)},\tag{21}$$

where *sn* and *dn* are Jacobian elliptical sine and modulus, respectively [19]. The corresponding solution for the radial velocity *vr*ð Þ*t* is:

$$v\_r(t) = v\_0 \frac{cn(at)}{dn^2(at)}.\tag{22}$$

Both asymptotic limits—nonrelativistic and ultrarelativistic—of the solutions (21) and (22) are easy to specify. When an initial radial velocity of the bead is nonrelativistic (*v*<sup>0</sup> ≪ 1), *m* ≈1 and recalling an asymptotic behavior of Jacobi's elliptic functions [19] ð Þ *sn* ! *tanh* ,*cn* ! *sech*, *dn* ! *sech* we can see that the solutions reduce to abovementioned (1) and (2) Newtonian expressions. Whereas in the ultrarelativistic limit (*v*<sup>0</sup> ≈ 1), *m* ≈ 0, hence *sn* ! *sin* ,*cn* ! *cos* , *dn* ! 1 and consequently *r t*ð Þ≈ð Þ *v*0*=ω sin* ð Þ *ωt* , and *vr* ≈*v*<sup>0</sup> *cos*ð Þ *ωt* .

**Figure 1** is taken from Ref. [6]. It shows the temporal evolution of the radial velocity of the bead *vr* (dashed line) and the quantity *v<sup>φ</sup>* � *ωr* (solid line), which is equal to the azimuthal velocity of the bead in the laboratory frame. The curves correspond to four different values of *v*0. Namely, **Figure 1**(**a**) is drawn for the case *v*<sup>0</sup> ¼ 0*:*001. Initially, both velocities (*vr* and *vφ*) grow almost synchronously, but later radial velocity slows its increasing pace, it reaches its maximum value (*vmax* ≈0*:*5) and begins to decrease, while *v<sup>φ</sup>* continues increasing and reaches *v<sup>φ</sup>* ¼ 1 value at the moment *<sup>t</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>K</sup>=ω*, when the bead is at the distance *<sup>r</sup>* <sup>¼</sup> *<sup>r</sup>* <sup>∗</sup> from the rotation axis and its radial velocity becomes zero. This is a "turning point" since here *vr*ð Þ*t* changes its sign, and the bead begins to move toward the rotation axis with increasing speed. A modulus of the radial velocity in the time interval *t* <sup>∗</sup> <*t*<2*t* <sup>∗</sup> varies exactly in the same way as in the previous interval 0 <sup>&</sup>lt;*t*<*<sup>t</sup>* <sup>∗</sup> . At the moment *<sup>t</sup>* <sup>¼</sup> <sup>2</sup>*<sup>t</sup>* <sup>∗</sup> the bead is at its starting point *r* ¼ 0 just above the rotation axis and it has the same velocity *v*<sup>0</sup> but is directed, this time, in the opposite direction. Therefore, the bead passes over the

**Figure 1.**

*Dependence of the bead's radial velocity* vr *(dashed line) and azimuthal velocity* v*ϕ*�*ω* r *(solid line) for four different values of initial velocity: (a)* <sup>v</sup>*<sup>0</sup> = 0.001, (b)* <sup>v</sup>*<sup>0</sup> = 0.3, (c) v*<sup>0</sup> <sup>¼</sup> ffiffiffi <sup>2</sup> <sup>p</sup> *<sup>=</sup>*2*, and (d)* <sup>v</sup>*<sup>0</sup> = 0.99.*

rotation axis and, in the interval, 2*t* <sup>∗</sup> <*t*<4*t* <sup>∗</sup> repeats in the left half of the pipe the same kind of motion. **Figure 1**(**b**) is drawn for the case *v*<sup>0</sup> ¼ 0*:*3. Qualitatively it resembles the previous case. However, a maximum value of *vr* is slightly larger and an average radial velocity of the motion is apparently higher.

An interesting "threshold" case *<sup>v</sup>*<sup>0</sup> <sup>¼</sup> ffiffi 2 <sup>p</sup> *<sup>=</sup>*2 is represented in **Figure 1**(**c**). From (15) we can see that for this particular case, initial radial acceleration is equal to zero. Therefore, originally, the bead moves almost uniformly, but further on it continues to move with a decreasing radial speed. Other qualitative features of the motion remain the same as in above-specified cases. Finally, **Figure 1**(**d**) represents the case of a strongly relativistic initial speed *v*<sup>0</sup> ¼ 0*:*99. It corresponds to the asymptotic case *m* ≈ 0 discussed above and consequently the curves for *vr* and *v<sup>φ</sup>* would be well represented by usual trigonometric cosine and sine, respectively.

Thus, we see that the character of the bead motion is "oscillatory." The period of the "oscillations" *<sup>P</sup>* � <sup>4</sup>*<sup>t</sup>* <sup>∗</sup> <sup>¼</sup> <sup>4</sup>*K=ω*. The period tends to infinity when *<sup>v</sup>*<sup>0</sup> ! 0 (as it should be), while when *v*<sup>0</sup> ! 1, the period naturally tends to: *P* ! 2*π=ω*.

In Ref. [6] the authors introduced the concept of an "effective potential" *U r*ð Þ, which turns out to be quite useful for shedding more light on the qualitative characteristics of derived solutions. Substituting *dr=dt* (15) from (12) it is possible to rewrite the equation of motion as:

$$\frac{d^2r}{dt^2} = \alpha^2 r \left[ (2m - 1) - 2m\alpha^2 r^2 \right] \equiv -\frac{dU(r)}{dr},\tag{23}$$

and for our particular case (*r*<sup>0</sup> ¼ 0, *v*<sup>0</sup> 6¼ 0), taking into account (13), we can rewrite it in the following way:

$$\frac{d^2r}{dt^2} = \alpha^2 r \left[ \left( \mathbf{1} - 2v\_0^2 \right) - \mathbf{2} \left( \mathbf{1} - v\_0^2 \right) \alpha^2 r^2 \right],\tag{24}$$

while the explicit expression for *U r*ð Þ is:

$$U(r) = \frac{\alpha^2 r^2}{2} \left[ (\mathbf{1} - 2m) + m\alpha^2 r^2 \right]. \tag{25}$$

In **Figure 2**, which is also taken from [6], the function *U x*ð Þð Þ *x* � *ωr* is shown for four different values of *v*0, corresponding to the cases shown in **Figure 1**(**a**–**d**). It provides graphic illustration of the bead's motion in terms of its motion in the specific kind of potential "well." Namely, in (a) and (b) cases, the curves have "secondary" minima, ascertaining that the bead beginning motion from the point *x* ¼ 0 accelerates while "rolling down" to the "secondary" minimum point, then hampers until it reaches the point *x* ¼ 1, and stops and begins to move toward the rotation axis. Case (c) is represented by a "potential well" with an almost flat bottom, where initially, as we have seen earlier, motion happens to be almost uniform. For the fourth case, the form of the "well" naturally implies the motion with the negative radial acceleration during the whole course of the motion.

Recently the *gedankenexperiment* in the RF was reconsidered by Khomeriki and Rogava [20] and the exact solutions for the bead motion were found and analyzed for most general sets of initial conditions. The qualitative nature of the motion is largely the same, but there are interesting quantitatively new classes of solutions. We address the interested reader to this publication for more details.

**Figure 2.**

*Effective potential as a function of the quantity* v*ϕ*�*ω* r *for four different values of the initial velocity: (a)* <sup>v</sup>*<sup>0</sup> = 0.001, (b)* <sup>v</sup>*<sup>0</sup> = 0.3, (c) v*<sup>0</sup> <sup>¼</sup> ffiffiffi <sup>2</sup> <sup>p</sup> *<sup>=</sup>*2*, and (d)* <sup>v</sup>*<sup>0</sup> = 0.99.*

It must be noted that in Ref. [6], Gia Machabeli and I have interpreted the surprising behavior of the bead in terms of reversal in direction of a centrifugal force. Criticizing that interpretation, Miller and Abramowicz [21] pointed out the connection of the bead's behavior with relativistic dependence of its mass on velocity. They recommended defining the relativistic centrifugal force in the way that excludes the "reversal" of the force and argued that the deceleration of the bead has to be ascribed to the relativistic mass variation. In Ref. [22], F. de Felice considered relativistic dynamics on a rotating disk and for the case of radially constrained motion, he interpreted the vanishing of the bead's radial velocity at the light cylinder as the result of an infinite Doppler shift experienced by the inertial observer.

If we follow the standard definition and assume that the centrifugal force is an "apparent" force, appearing only in non-inertial frames of reference then one may wonder: is it possible to derive above-described dynamics of the bead inside the rotating pipe in the laboratory inertial frame (LF), relative to which the pipe rotates? Below it will be shown that all quantitative results of Ref. [6] are not related at all to peculiarities of the frame in which the problem is considered. We shall discuss also two other interesting aspects of the problem, viz. surprising and remarkable analogy of the "pipe+bead" *gedankenexperiment with*: (a) motion of classic simple gravity pendulum, and (b) the radial geodesic infall motion of a test particle in Schwarzschild geometry.

#### **2.2 The original solution: LF study**

In this section, we reconsider the *gedankenexperiment* in the LF, where the space– time is Minkowskian. Our purpose is to show that the peculiar dynamics of the system disclosed in the RF are not related at all to the arguable concept of a reversible or irreversible centrifugal force; on the contrary, it can be easily derived within the LF where no inertial forces are present.

Owing to the symmetry of the system, it is convenient to write the spatial part of the metric in polar coordinates:

$$ds^2 \equiv -d\tau^2 = -dt^2 + r^2d\rho^2 + dr^2,\tag{26}$$

The metric has three nonzero Christoffel symbols:

$$
\Gamma^{r}\_{\phi\phi} = -r, \quad \Gamma^{\phi}\_{r\phi} = \Gamma^{\phi}\_{\phi r} = \mathbf{1}/r. \tag{27}
$$

The radial velocity and angular velocity of a bead/particle are defined as:

$$v \equiv dr/dt, \quad \alpha \equiv d\phi/dt,\tag{28}$$

while the Lorentz factor is:

$$\gamma \equiv dt/d\tau = \left(\mathbf{1} - r^2 \alpha^2 - v^2\right)^{-1/2},\tag{29}$$

The motion of the bead is free in the radial direction, while the pipe wall acts on the bead with tangential force of reaction. The radial component of the equation of motion

$$
\left(\frac{d^2r}{d\tau^2} + \Gamma^r\_{a\beta}\left(\frac{d\mathbf{x}^a}{d\tau}\,\frac{d\mathbf{x}^\beta}{d\tau}\right) = \mathbf{0},\tag{30}
$$

is reduced to:

$$\frac{d^2r}{d\tau^2} = r\left(\frac{d\phi}{d\tau}\right)^2 = r\alpha^2\gamma^2. \tag{31}$$

As far as *dr=dτ* ¼ *γv* and

$$\frac{d\chi}{dt} = \chi^3 v \left(\omega^2 r + \frac{dv}{dt}\right),\tag{32}$$

it is quite easy to calculate that:

$$
\frac{d^2r}{d\tau^2} = \gamma^2 \left[ (1 + \chi^2 v^2) \frac{d^2r}{dt^2} + \alpha^2 r \gamma^2 v^2 \right],\tag{33}
$$

and it is equally straightforward to show that:

$$\mathbf{1} - \chi^2 \mathbf{v}^2 = \chi^2 \left(\mathbf{1} - r^2 \boldsymbol{\alpha}^2 - 2\boldsymbol{\nu}^2\right),\tag{34}$$

$$\mathbf{1} + \chi^2 \mathbf{v}^2 = \chi^2 \left(\mathbf{1} - r^2 \alpha^2\right),\tag{35}$$

which, in turn, after taking into account (31) and (33), allows us to derive the following equation:

$$d^2r dt^2 = \frac{\alpha^2 r}{1 - \alpha^2 r^2} \left(1 - \alpha^2 r^2 - 2r^2\right). \tag{36}$$

One can easily see that it is exactly the same Eq. (15), which has been derived in the previous subsection within the RF analyses of the same problem!

It is worthwhile to note that taking into account (34) and (35), we can rewrite this equation in the following, surprisingly beautiful, form:

$$\frac{d^2r}{dt^2} = \left(\frac{1-\chi^2v^2}{1+\chi^2v^2}\right)\alpha^2r.\tag{37}$$

Apart from its elegant form, this equation turns out to be quite informative because it distinctly shows asymptotic peculiarities of the bead dynamics. Namely, if the motion is nonrelativistic (*γv* ≪ 1), (37) reduces to the usual classical equation for centrifugal acceleration: *d*<sup>2</sup> *<sup>r</sup>=dt*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>*r*; while in the ultrarelativistic limit (*γ<sup>v</sup>* <sup>≫</sup> 1), the sign of the right-hand side of (37) changes (!) to the opposite: *d*<sup>2</sup> *<sup>r</sup>=dt*<sup>2</sup> ¼ �*ω*<sup>2</sup>*r*. Note, furthermore, that when *<sup>γ</sup>*0*v*<sup>0</sup> <sup>¼</sup> 1 (viz. *<sup>v</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* ffiffi 2 <sup>p</sup> ) above-mentioned remarkable acceleration sign reversal occurs from the very beginning of the motion!

Another mathematically elegant feature of the problem is that (36) can be rewritten as a differential equation for the function *<sup>y</sup>* � *<sup>v</sup>*<sup>2</sup>ð Þ*<sup>r</sup>* . The resulting equation is:

$$\frac{d^2y}{dt^2} + \frac{4\alpha^2r}{1 - \alpha^2r^2}y = 2\alpha^2r,\tag{38}$$

with the solution, which can be written as:

$$\mathcal{Y}(r) = \left(\mathbb{1} - \alpha^2 r^2\right) + \text{const} \cdot \left(\mathbb{1} - \alpha^2 r^2\right)^2. \tag{39}$$

In particular, for the "conventional" initial condition when initially (*t* ¼ 0) the bead is situated on top of the pivot *r*<sup>0</sup> ¼ 0 and has a velocity *v*<sup>0</sup> 6¼ 0, the latter equation implies that *const* <sup>¼</sup> *<sup>v</sup>*<sup>2</sup> <sup>0</sup> � 1. Therefore, for the function *v r*ð Þ we obtain the result that had been derived in Ref. [6]:

$$v(r) = \sqrt{(1 - \alpha^2 r^2) \left[1 - \frac{(1 - \alpha^2 r^2)}{\chi\_0^2}\right]}.\tag{40}$$

#### **2.3 An interesting pendulum analogy**

The *gedankenexperiment* reveals" surprising dynamics of relativistically rotating particles. But it happens to be remarkable also due to a couple of nontrivial and noteworthy analogies.

One of them shows when we introduce new set of variables:

$$\phi \equiv 2 \arccos \left( \alpha r \right) \tag{41}$$

$$
\lambda \equiv at\tag{42}
$$

$$
\Omega^2 \equiv \mathbf{1} - v\_0^2 \tag{43}
$$

and after, some uncomplicated calculations, find out that our equation of motion reduces to the following, remarkably familiar equation:

*Centrifugal Acceleration in Relativistic Astrophysics DOI: http://dx.doi.org/10.5772/intechopen.109684*

$$\frac{d^2\phi}{d\lambda^2} + \Omega^2 \sin\phi = 0,\tag{44}$$

we arrive at the well-known *pendulum equation*!

Note that one could easily derive (44) from (43) by rewriting it in above-

introduced notation as *dϕ=dλ* ¼ �2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>Ω</sup><sup>2</sup> *sin* <sup>2</sup> ð Þ *ϕ=*2 q , and taking one more derivative by *λ*. It is common knowledge that (44) describes nonlinear oscillations of a free "simple gravity" pendulum. In fact, it is closely related to another big achievement for Christiaan Huygens, who studied in his "Horologium Oscillatorium" in 1673. With this analogy, now being so apparent, the striking resemblance of Ref. [6] solutions with a pendulum motion becomes more transparent and understandable. If we introduce the concept of an *"analogous pendulum"*, governed by (44), then the initial conditions (*r*<sup>0</sup> <sup>¼</sup> 0, ð Þ *dr=dt <sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>v</sup>*0) are replaced by *<sup>ϕ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>π</sup>* and ð Þ *<sup>d</sup>ϕ=d<sup>λ</sup> <sup>λ</sup>*¼<sup>0</sup> ¼ �2*v*0. Therefore, it turns out that the analogous pendulum rotates in a vertical plane with the effective frequency Ω. Remarkably enough, the time interval, needed by the bead to reach *ωr* ¼ 1 ("light cylinder"), corresponds to the time needed by the analogous pendulum to reach its stable equilibrium (*ϕ* ¼ 0) point. This time interval is finite, as it, certainly should be.

### **2.4 A captivating black hole analogy**

Another striking analogy turns out to be even more unexpected. One can show that the "bead-pipe" *gedankenexperiment* features an unusual analogy with a specific kind of geodesic motion in Schwarzschild geometry!

Let us consider a radial geodesic "fall" of a test particle onto a Schwarzschild black hole with *M* mass. Let a radial velocity of the particle at infinity be *V*<sup>∞</sup> pointed inwards. If one denotes by *<sup>E</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>∞</sup> � <sup>1</sup> � *<sup>V</sup>*∞<sup>2</sup> � ��1*=*<sup>2</sup> >1 the specific energy of the particle per its rest mass, then for its radial velocity relative to the observer at infinity *V*^*<sup>r</sup>* � ffiffiffiffiffi *grr* <sup>p</sup> *dr=dt* one gets

$$\left|\mathbf{V}\_{\hat{r}}\right|^{2} = \frac{E^{2}}{4} - \left[\frac{E}{2} - \frac{1}{E}\left(\mathbf{1} - \frac{2\mathbf{M}}{r}\right)\right]^{2},\tag{45}$$

(See, e.g., Ref. [23], where this equation is written in different notation). For the quantity *dV*^*r=dt* (called by McVittie "nontenzor radial acceleration of the particle") one gets:

$$\frac{dV\_{\hat{r}}}{dt} = \frac{2M}{Er^2} \left[ \frac{E}{2} - \frac{1}{E} \left( 1 - \frac{2M}{r} \right) \right] \sqrt{1 - \frac{2M}{r}}.\tag{46}$$

Inspecting the expression it is easy to see that the acceleration of the particle is negative (that is, the modulus of the particle's infall velocity is increasing), until the particle reaches the radial distance *<sup>r</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup>*M<sup>=</sup>* <sup>2</sup> � *<sup>γ</sup>*∞<sup>2</sup> ð Þ, where the acceleration *changes* its sign, and *V*^*<sup>r</sup>* reaches its maximum velocity *Vmax* ¼ *γ*∞*=*2.

More precisely, we have the following regimes of the motion:

• *V*<sup>∞</sup> ≪ 1 (*γ*<sup>∞</sup> ≈1): the particle begins to move with increasing speed, at *r*<sup>1</sup> ≈ ¼ 4*M* the speed reaches its maximum value (*Vmax* ≈1*=*2) and further on it decelerates,

down to zero radial velocity, when the particle approaches the black hole horizon *r* ! 2*M*;


It is easy to notice that this scenario remarkably resembles (even quantitatively!) the one in Ref. [6] *gedankenexperiment*. This likeness happens due to the remarkable similarity between (45) and the corresponding equation for *dr=dt* from Ref. [6], which may be written as:

$$
\left(\frac{dr}{dt}\right)^2 = \frac{E^2}{4} - \left[\frac{E}{2} - \frac{1}{E}(1 - \alpha^2 r^2)\right]^2,\tag{47}
$$

The resemblance is not incidental, of course, but is related to the likeness of the space–time (4) in the rotating pipe:

$$ds\_p^2 = -(1 - \alpha^2 r^2)dt^2 + dr^2,\tag{48}$$

with the Schwarzschild space–time along a radial geodesic (*θ* ¼ *const*, *φ* ¼ *const*):

$$ds\_s^2 = -(\mathbf{1} - 2\mathbf{M}/r)dt^2 + \frac{dr^2}{\mathbf{1} - 2\mathbf{M}/r}.\tag{49}$$

Comparing, for example, lapse functions for these two metrics (*α<sup>p</sup>* � ffiffiffiffiffiffiffiffi �*gtt* <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>*r*<sup>2</sup> <sup>p</sup> , and *<sup>α</sup><sup>s</sup>* � ffiffiffiffiffiffiffiffi �*gtt* <sup>p</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>2</sup>*M=<sup>r</sup>* <sup>p</sup> ) one can see that *<sup>α</sup><sup>p</sup>* and *<sup>α</sup><sup>s</sup>* become infinite at the light cylinder and horizon, respectively. Obviously, the likeness is not complete: the spatial part of the rotating pipe metric is flat, while the spatial part of (49) metric is curved (*grr* 6¼ 1). The latter difference is also essential: it ensures finiteness of the time *<sup>t</sup>* <sup>∗</sup> [6], needed by the bead to reach the light cylinder *<sup>r</sup>* <sup>∗</sup> � *<sup>ω</sup>*�1. As it is well known, an analogous time interval for a test particle to reach the Schwarzschild black hole horizon, as measured by the distant observer, is infinitely long!

### **2.5 A centrifugal force "reversal"?!**

It is well known that a generalization of Newton's second law

$$
\overrightarrow{F} = d\overrightarrow{p}/dt = d\left(m\overrightarrow{v}\right)/dt\tag{50}
$$

is the most useful and convenient definition of a (three-) force *F* ! in special relativity. According to Rindler ([24], p. 88) *"This definition has no physical content until other properties of force are specified, and the suitability of the definition will depend on these other properties."*

For real applied forces, arising in relativistic dynamics, this definition is physically self-consistent and proved to be the most appropriate. But, how one should define

*Centrifugal Acceleration in Relativistic Astrophysics DOI: http://dx.doi.org/10.5772/intechopen.109684*

*inertial* forces in relativity!? Miller and Abramowicz suggested [21] using the same general method for these forces as well. The approach is fully justified but we would like to emphasize one aspect of the problem.

The definition contains a quantity *m t*ðÞ¼ *m*0*γ*ð Þ*t* , which has, clearly, the meaning of the relativistic mass in the LF. However, note that this quantity is used for the definition of another physical quantity—centrifugal force—which exists in another *non-inertial* frame of reference. A question arises: is it self-consistent to define a physical variable in one frame by means of another variable defined in another frame? Certainly, the time-variable mass may be defined also in RF, but it would not be equal to *m t*ð Þ. The important point is that *γ*ð Þ*t* , having in the LF the meaning of Lorentz factor, has not the same meaning in the RF because Lorentz factor of the moving particle is not invariant between frames [24].

This circumstance appears evident in the "1 þ 1" formulation of the same problem. In fact, for the two-dimensional curved metric (4) in the RF *V* � *v=αp*, and <sup>Γ</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>*r*<sup>2</sup> ð Þ*<sup>=</sup>* <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>*r*<sup>2</sup> � *<sup>v</sup>*<sup>2</sup> ½ � ð Þ <sup>1</sup>*=*<sup>2</sup> . Accordingly, it is possible to define relativistic mass in the RF as *M t*ðÞ� *m*0Γ and to write, instead of Eq. (50), the following definition:

$$F^\* = \frac{d}{dt}(MV) = \frac{M\alpha^2 r}{\alpha\_p}.\tag{51}$$

This definition is already made by means of the true Lorentz factor of the bead as measured in the RF. Just like (50) it, also, gives "irreversible" centrifugal force but lacks the attractive simplicity of (50). However, we should always remember that the mass, which the RF observer actually *measures*, is *M t*ð Þ and is not *m t*ð Þ. It seems, therefore, more consistent to express the physical quantity existing in the particular frame (centrifugal force *f <sup>c</sup>*, which exists in the RF) through the other physical quantity *M t*ð Þ defined and measured in the same frame.

Despite this uncertainty, we should like to note that the importance of the mass variation effect, noticed by Miller and Abramowicz, is a very remarkable and important feature of this problem. As it appears, the capability of mass to vary drastically affects the dynamics of the motion. But, is it appropriate and logically justified to describe this secondary dynamical effect as the action of some "negative self-thrust" force!? Saying "secondary" we do not mean its significance but, rather, its status in the causal order of true physical reasons. If one introduces it that would be yet another "apparent" force, like the centrifugal force. Actual dependence of the bead mass on time is governed explicitly by the concrete kind of its motion in the LF, which is entirely determined by the outer *real* force, tangential pipe reaction force, applied to this moving body.

## **3. Astrophysical applications and conclusions**

In 2003, the problem of motion of particles on relativistically rotating prescribed trajectories considered in Ref. [6] for a straight trajectory case (*gedankenexperiment*) was generalized. Rogava, Osmanov, and Dalakishvili considered the motion of rotationally driven particles along flat trajectories of arbitrarily curved shape [7]. In this paper, the problem was again studied on the level of the idealized *gedankenexperiment*, both in the laboratory (LF) and in the rotating (RF) frames of reference. For

the simple example of the Archimedes spiral, it was found that the dynamics of these particles may involve both accelerative and decelerative modes of motion. Moreover, it was also found that there are special solutions, which remain force-free during the whole course of the motion.

Additionally, basic equations were outlined and the scheme for the solution of the more general version of the same problem was given, where the angular velocity of the rotating system is *not* assumed to be constant. Instead, it is required that the full triple system—rotator+pipe+bead—is conservative and the rotator is allowed to exchange perceptible portions of energy with the bead. The latter setup is a significant step closer to real astrophysical situations where, for instance, a rapidly rotating neutron star acts as a rotator and its magnetic field lines play the role of wires (pipes) along which particles are centrifugally accelerated.

One astrophysically important difference from the linear pipe case [6] is that for curved prescribed trajectories and the motion of the accelerated particle is not any more radially bounded: there exist regimes of motion when the bead may reach infinity! This result has a simple physical explanation. For the case of the linear pipe, rotating with the constant angular velocity, the natural limit of the radial motion is given by the light cylinder radius. However, in the case of the curved pipe, even when it rotates at the constant rate, the bead moves both in radial and azimuthal directions, following the curvature of the prescribed trajectory and having a variable angular velocity Ωð Þ*t* . It means that now the role of the *"effective light cylinder* is played by the time-dependent quantity *RL*ðÞ¼ *<sup>t</sup>* <sup>Ω</sup>�<sup>1</sup> ð Þ*t* , and all those radial distances for which *r t*ð Þ<*RL*ð Þ*t* become accessible! Therefore, if both *r t*ð Þ and *RL*ð Þ*t* are monotonously increasing functions, but the former stays always smaller than the latter (e.g., that is the case for the Archimedes spiral) then the bead can reach infinity because wherever it is at any given moment of time, the light cylinder is still ahead of it and the centrifugally accelerated bead will never "overtake" the light cylinder!

## **4. Acceleration of particles by rotating magnetospheres in AGN**

It is well-known that some blazars, such as Mrk 421, Mrk 501, PKS 2155–304, 1ES 2344–514, H1426 + 428, and 1ES 1959 + 650, emit ultra-high-energy, several TeV 1 TeV � 1012 eV photons and constitute a special class of so-called *"TeV blazars"*. The standard model for a blazar assumes the presence of a supermassive black hole, surrounded by an accretion disc and ejecting twin relativistic jets, one of which is seen by a terrestrial observer almost end-on. Usually, broadband emission spectra of these objects contain two components: the low-energy (from radio to optical/UV) part associated with synchrotron radiation and the high-energy (X- and *γ*- rays) component formed by the inverse Compton scattering (ICS) of softer photons. The presence of the latter part of the spectrum is normally explained on the basis of the synchrotron self-Compton (SSC) model. However, the origin of accelerated electrons, the physical mechanism responsible for their efficient acceleration is a matter of uncertainty.

Proposed mechanisms such as the Fermi-type acceleration processes or repetitive acceleration of electron-positron pairs as a feedback mechanism could account for the observed high energy emission up to 20 TeV. However, the Fermi-type acceleration in relativistic jets is efficient only if the seed population of "pre-accelerated" electrons is present, possessing high enough (*γ*max ≥ 10<sup>2</sup> ) Lorentz factors. The provenience of the "pre-acceleration" is not well-understood.

Since pioneering work by T. Gold [25, 26] it has been proposed that centrifugally driven outflows (CDOs) consisting of centrifugally accelerated particles may acquire extremely high energies. The idea has often been discussed [8, 27] in the context of pulsar emission theory. Moreover, regarding the AGNs, it has been shown that CDOs from accretion disks may occur if the poloidal magnetic field lines are inclined at an angle ≤60<sup>∘</sup> to the equatorial plane of the disk [28].

Osmanov et al. [9] argued that the presence of the CDO would imply that despite the intense energy losses via UV radiation (when due to ICS, soft photons are upscattered on electrons and gain energy from them), electrons may still reach very high, *<sup>γ</sup>*max <sup>10</sup><sup>5</sup> , Lorentz factors. It was surmised that if robust enough, this mechanism could account not only for the "pre-acceleration" of electrons but could be considered as an independent bona fide alternative mechanism for the generation of high-energy emission (up to 20 TeV) of TeV blazars!

Gangadhara and Lesch [29] examined the role of the centrifugal force on the dynamics of electrons moving along straight magnetic field lines, situated in the equatorial plane and corotating with the spinning AGN. They showed that an upscattering of low-energy photons against accelerated electrons may readily lead to the generation of the nonthermal x-ray and *γ*-ray emission. Later on, the problem was critically re-examined by Rieger and Mannheim [13], checking whether the rotational energy gain of charged particles, moving along straight magnetic field lines, is limited not only by ICS but also by the expected breakdown of "the-bead-on-the-wire" approximation. The latter is a synonymous term for the "bead+pipe" approximation used in our *gedankenexperiment*. The breakdown happens in the vicinity of the light cylinder, where a Coriolis force acting on the particle and trying to "tear it off" the field line would exceed the Lorentz force "binding" the particle to the field line.

Bearing in mind that in real, three-dimensional astrophysical situations (e.g., jets), the magnetic field lines are not localized in the equatorial plane but are inclined with respect to it. Osmanov et al. [9] examined the same problem considering the wide range of possible inclinations. It was shown that for a wide range of AGN, the mechanism responsible for limiting the maximum Lorentz factors is ICS, but under favorable conditions, it can allow particles to reach quite high Lorentz factors *γ*max ≥10<sup>5</sup> . The breakdown of the-bead-on-the-wire approximation becomes important for the low luminosity (<sup>&</sup>lt; <sup>10</sup><sup>41</sup> erg*=*s) AGN, when *<sup>γ</sup>*max 108. For higher luminosities (> 10<sup>41</sup> erg*=*s), it can be dominant only for relatively small inclination angles of the magnetic field lines with respect to the rotation axis (≤10<sup>∘</sup> ).

Therefore, it was argued [9] that CDOs could indeed be amply efficient to account for the TeV blazar emission. For the wide range of cases, ICS remains the dominant factor in limiting the maximum Lorentz factor of accelerated particles, letting them produce very high-energy photons [1–20] TeV. Obviously, an important restriction of the [9] approach is that only straight magnetic field lines were considered, whereas, in realistic astrophysical situations, the magnetic field lines are curved. It might be especially important if one considers particle dynamics on a longer time scale and over larger length scales when the curvature of the field lines cannot be neglected. As a matter of fact, the mathematical formalism for such a study is described in [7].

### **4.1 Centrifugal acceleration and gamma flares in Crab nebula**

The Crab nebula is a source of almost steady high-energy emissions. Observations made from orbital probes (Fermi, SWIFT, and RXTE) showed evidence of its

variability in the X\x-ray range. Several years ago the Fermi and AGILE satellites detected dazzling, brief, and strong bursts of gamma radiation above 100 MeV, with its source located in the Crab Nebula. Since then, several gamma-ray flares have been reported.

The most obvious source of radiation power in the nebula is the rotational energy damping *dE=dt* <sup>¼</sup> *<sup>I</sup>*Ω*d*Ω*=dt* <sup>¼</sup> <sup>5</sup> � <sup>10</sup>38*erg<sup>=</sup> sec* of the pulsar PSR 0532. Machabeli et al. [30] demonstrated that via the centrifugal acceleration mechanism, it is possible to pump energy from the neutron star's vast rotational kinetic energy storage, � <sup>5</sup> � <sup>10</sup><sup>38</sup> erg s � 1, to proper electrostatic plasma (Langmuir) oscillations. Furthermore, it was shown that the growth rate of the perturbations is maximum in the "superluminal" area, where the phase velocity of perturbations exceeds the speed of light. That is why in this region, the condensate of plasmons is formed, which is transferred to the Crab Nebula. The transfer of the energy of the plasmon condensate from the pulsar magnetosphere to the nebula over a huge distance, 3 � <sup>10</sup><sup>17</sup>*cm*, takes place practically without any tangible losses.

But in the nebula, unlike the pulsar magnetosphere, apart from electrons and positrons, there are also protons. That is why a *modulation instability* is developed, which leads to the collapse of Langmuir waves; viz. a cavity is formed, which collapses, and on the final stage of the collapse, involved particles attain very high Lorentz factors, resulting in the powerful synchrotron emission of the nebula. The collapse stops at the scale of a few Debye radii.

It was also shown that in the course of the active phase of the collapse in the cavity due to the influence of nonlinear processes on the polarization properties of the medium, *self-trapping* [31] of the synchrotron radiation (generated within the cavity) takes place. If the conditions for the appearance of the self-trapping are fulfilled for certain values of emitted wave frequencies, for others, both higher and lower values of the frequency, they are not satisfied. It means that giant bursts of energy release have to happen in narrow energy (frequency) ranges. Therefore, it would naturally explain abrupt, burst-like increases of the radiation intensity within narrow bands, impressively explained by self-trapping in the framework of nonlinear optics (so-called "Askaryan effect" [32]). Waves propagating in the nonparallel direction to the line of sight are bent and directed to the observer due to self-trapping. Ultimately it leads to the required increase of the radiation intensity and, at the same time, unlike other mechanisms, it does not require any additional sources of energy.

### **4.2 Self-trapping as a beaming mechanism for Fast Radio Bursts!?**

Bearing in mind what we have just said about self-trapping, it is most reasonable to assume [33] that the principal reason for the beaming of electromagnetic radiation leading to Fast Radio Bursts (FRBs) could be the nonlinear self-trapping phenomenon. This mechanism implies that the part of the radiation beam directed toward the observer is augmented by its outer part, which in the absence of the self-trapping would not be focused toward the same point. As a result, the observer, while the beam is being self-trapped, sees an enhanced intensity of radiation in a very narrow frequency range. This scenario is robust and fully autonomous because unlike many other mechanisms, it does not require additional external sources of energy. Besides, self-trapping depends on quite a large number of parameters, in particular, on a specific proper value of the ratio of the wave amplitude to the amplitude of an

### *Centrifugal Acceleration in Relativistic Astrophysics DOI: http://dx.doi.org/10.5772/intechopen.109684*

incident electrostatic wave, on the temperature of the medium, and on the direction of these waves relative to the line of sight. Any, even the slightest, deviation of any of these parameters from 'favorable' values may lead to the disappearance of an observable radiation burst. This is why this is an extremely finely tuned, random, and rarely occurring event. This circumstance guarantees that an FRB event, if indeed being caused by the arrival of the self-trapped self-focused enhanced beam to the observer, is a totally random and extremely rare natural phenomenon.

If our model is correct and relevant to actual FRBs, the self-trapping condition may hold in a given direction only for a very short interval of time. Hence, it is logical to surmise that the probability of the coincidence between the line of sight and the direction of self-trapping has to be quite small. Paradoxically enough, what could be a serious drawback for a commonly occurring phenomenon that in this case 'works' just in the opposite way, it strengthens our confidence that self-trapping could be an important physical factor contributing to the appearance of this extremely rare and energetic phenomenon—fast radio bursts or FRBs.

Obviously, the self-trapping mechanism does not exclude other physically plausible, repetitive or non-repetitive and catastrophic or non-catastrophic mechanisms proposed for FRBs. We tend to believe that self-trapping may be one of a number of very efficient 'beaming' mechanisms that might be needed for interpreting FRBs as narrowly beamed radio bursts [34]. We suppose that the prime reason for any single burst appearance could be related to its generic source, located in a distant galaxy, and causing a giant outburst in the radio range. The mechanism of this primary outburst may be related to one of the previously suggested plasma mechanisms.

Presumably, these outbursts might be observed without any additional amplification if they occur on intergalactic distance scales. That was Machabeli et al. guessed in Ref. [33]. Shortly afterward first-ever FRB from within our galaxy was detected [35], confirming our expectation since it was associated with a rapidly rotating neutron star, magnetar SGR 1935 + 2154 about 30,000 light-years away in the Vulpecula constellation. The STARE2 team independently observed the burst [36], detected its fluence, and confirmed the connection between this burst (FRB 200428) and FRBs at extragalactic distances. Needless to say, a magnetar is a plausible candidate for the centrifugally driven relativistic acceleration of electrons with subsequent Langmuir collapse and self-trapping as the beaming mechanism. It only strengthened our expectation that FRB could be akin to, for instance, giant radio pulses occasionally observed from pulsars within our own Galaxy [33]. Actually, for intergalactic giant pulses, the beaming mechanism may not be even necessary for the burst to be detectable. But in order to make FRBs visible on extragalactic distance scales, some powerful additional beaming/amplification is required and this is where we believe selftrapping could play its important role.

With improved capabilities of several wide-field, broad-band surveys that recently became operational the observational situation may change dramatically. It is expected that the FRB field will change from the presently available small-number statistics up to hundreds or even thousands [37] of new FRBs per year. Therefore, in forthcoming years, one might expect that the number of detected FRBs will increase by orders of magnitude, and we could be able to see primary outbursts leading to FRBs without a need for their additional 'self-trapping-related' or any other kind of amplification. But within this large number of events, there will be a relatively small fraction of peculiar bursts undergoing secondary self-trapping amplification and having distinctly peaked structures.

## **5. Conclusions and future prospects**

Above given three examples of supposed astrophysical appearance were meant to emphasize that centrifugal acceleration may certainly be one of most efficient converters of rotational energy into the energy of directed motion and may lead to the generation of powerful observational signatures. The universe is full of rotating plasma flows containing enormous amounts of kinetic energy. There are a few physical mechanisms able to transform a considerable part of the mechanical energy into radiation energy. For instance, so-called "nonmodal" processes [38, 39], taking place in flows with nonuniform velocity fields (shear flows), are capable to infuse a part of the regular rotation-related energy in waves and instabilities appearing in these flows [40, 41]. However, none of these nonmodal mechanisms has efficiency even remotely comparable with the efficiency of the centrifugal acceleration mechanism.

Obviously, centrifugal acceleration has to be of particular importance in objects with ultra-strong magnetic fields: pulsars, magnetars, AGN, etc. In these astronomical situations, magnetic field lines provide proper "prescribed trajectories" for relativistic particles and can accelerate them up to very high Lorentz factors. That is why the issue of centrifugal acceleration occupies an important role in the domain of relativistic astrophysics.

Theoretically speaking the most principal, crucial challenge is to move from highly idealized, "gedankenexperiment" setups, and various "toy" problems up to more realistic models, replacing mechanical "bead-on-wire" approximation with real, strong magnetic fields serving as genuine accelerators of relativistic particles in related astronomical objects. We could easily envisage that with the development of observational capabilities in relativistic astrophysics possible areas of applications for the centrifugal acceleration mechanism in high-energy astrophysics will widen further. Obviously, a number of intriguing problems remain to be understood. For instance, how centrifugal acceleration operates in conjunction with space–time "rotation" nearby black holes!? In Refs. [10, 20], a theoretical formalism was developed allowing to study of centrifugally induced phenomena in general static and isotropic metrics, associated with rotating relativistic bodies. One can imagine, for example, that in a Kerr black hole ergosphere, the interlace of centrifugal acceleration with Penrose process [42] may lead to interesting physical peculiarities, some of which may turn out to be important in the astronomical-observational context.

## **Acknowledgements**

Andria Rogava's research was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) grant FR-18-14747.

*Centrifugal Acceleration in Relativistic Astrophysics DOI: http://dx.doi.org/10.5772/intechopen.109684*

## **Author details**

Andria Rogava Institute of Theoretical Physics, Ilia State University, Tbilisi, Georgia

\*Address all correspondence to: andria.rogava@iliauni.edu.ge

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## **Chapter 5**
