Tensors in the Exploring of the Space-Time

[18] Lima EL. Curso de Anlise, Volume 2. Rio de Janeiro, Brasil: IMPA; 1981

*Advances on Tensor Analysis and Their Applications*

[19] Golub GH, Van Loan CF. Matrix Computations. The Johns Hopkins

University Press; 1996

**18**

**Chapter 2**

**Abstract**

Space-Time

*and Edgar Navarro*

movement of matter.

**1. Introduction**

**21**

Kinematic-Energy Measurements

We consider the relation between the twistor kinematic-energy model of the space-time and the kinematic-energy tensor as the energy-matter tensor studied in relativity theory to obtain the torsion tensor of the space-time. Measurements of the torsion tensor through their energy spectra are obtained for the movement of a particle under certain trajectories (curves whose tangent spaces twist around when they are parallel transported) when crossing an electromagnetic field. We want to give an indicium of the existence of torsion field through the electronic signals produced between the presence of electromagnetic field and the proximity of

**Keywords:** energy-matter tensor, kinematic-energy tensor, movement energy

The fundamental problem considered in this chapter is linked with the determination of energy-(space-)time variations that occur in the interaction of movement and matter-energy on a special geometry of movement or movement kinematics. However, we need a background component that permits the measure and detects under the invariance of its fields the change of matter particle spin (as could be in the torsion case [1], considering a quasi-local matter model represented through the gravitational waves of cylindrical type to measure and detect the field torsion). This last, considering only a component of geometrical torsion no vanish, along of a curve of a particle as study object that moves affected by an energy radiation that

The gauging of the torsion system using movement in an external field, which acts on a particle through the deformation space, could be the simplest way to use the dual concepts of twistor frame and spinors. The objective is to demonstrate the existence of the kinematic twistor tensor in a system that detects the torsion and obtains its image by spinors due to the duality, as demonstrated in Ref. [2].

We know the need of an intermediate gauge field to establish experimentally the relation between the kinematic twistor tensor and the energy-matter tensor (this last due to the movement in the space-time) in duality, as determined in Ref. [3].

vacuum, torsion tensor, twistor kinematic-energy model

permits the use of some physical effect like the Hall effect.

of the Torsion Tensor in

*Francisco Bulnes, Isaías Martínez, Omar Zamudio*

#### **Chapter 2**

## Kinematic-Energy Measurements of the Torsion Tensor in Space-Time

*Francisco Bulnes, Isaías Martínez, Omar Zamudio and Edgar Navarro*

#### **Abstract**

We consider the relation between the twistor kinematic-energy model of the space-time and the kinematic-energy tensor as the energy-matter tensor studied in relativity theory to obtain the torsion tensor of the space-time. Measurements of the torsion tensor through their energy spectra are obtained for the movement of a particle under certain trajectories (curves whose tangent spaces twist around when they are parallel transported) when crossing an electromagnetic field. We want to give an indicium of the existence of torsion field through the electronic signals produced between the presence of electromagnetic field and the proximity of movement of matter.

**Keywords:** energy-matter tensor, kinematic-energy tensor, movement energy vacuum, torsion tensor, twistor kinematic-energy model

#### **1. Introduction**

The fundamental problem considered in this chapter is linked with the determination of energy-(space-)time variations that occur in the interaction of movement and matter-energy on a special geometry of movement or movement kinematics. However, we need a background component that permits the measure and detects under the invariance of its fields the change of matter particle spin (as could be in the torsion case [1], considering a quasi-local matter model represented through the gravitational waves of cylindrical type to measure and detect the field torsion). This last, considering only a component of geometrical torsion no vanish, along of a curve of a particle as study object that moves affected by an energy radiation that permits the use of some physical effect like the Hall effect.

The gauging of the torsion system using movement in an external field, which acts on a particle through the deformation space, could be the simplest way to use the dual concepts of twistor frame and spinors. The objective is to demonstrate the existence of the kinematic twistor tensor in a system that detects the torsion and obtains its image by spinors due to the duality, as demonstrated in Ref. [2].

We know the need of an intermediate gauge field to establish experimentally the relation between the kinematic twistor tensor and the energy-matter tensor (this last due to the movement in the space-time) in duality, as determined in Ref. [3].

*Advances on Tensor Analysis and Their Applications*

Likewise, we consider M the space-time as the complex Minkowski model, and we define the kinematic twistor tensor as the obtained of the model in a space region Σ. Then considering the energy-matter tensor and its image in a two-dimensional surface will be two-surface twistor ð ÞS *:* The geometrical evidence of torsion is precisely through this contorted surface.

In other words, the kinematic twistor tensor A*αβ* in the radiation energy bath (electromagnetic radiation) from the energy-matter tensor T*αβ* will be defined by the interaction of two fields Z*<sup>α</sup>* <sup>1</sup> and Z*<sup>α</sup>* <sup>2</sup> that act in Σ,

$$\mathbf{A}\_{a\beta}\mathbf{Z}\_1^a\mathbf{Z}\_2^\beta = \int\_{\hat{\Sigma}} \mathbf{T}\_{a\beta}\mathbf{k}^a \mathbf{d}\sigma^\beta,\tag{1}$$

which produces an electrical total charge due to the Gauss divergence theorem on currents T*αβ*k*<sup>α</sup>* ,

$$\mathbf{Q}[\mathbf{k}] = \frac{1}{4\pi\mathbf{G}} \int\_{\Sigma} \mathbf{R}\_{a\beta\gamma\delta} \mathbf{f}^{a\beta} \mathbf{d}\sigma^{\nu\delta},\tag{2}$$

This can be identified as the source depending on the killing vector k*<sup>α</sup>* of the Minkowski space background model

$$M = \mathbb{S}^2 \otimes \mathbb{C}^2 \otimes \mathbb{M},\tag{3}$$

where M is the space-time of two components

$$\mathbf{M} = \mathbf{S}^{+} \oplus \mathbf{S}^{-}.\tag{4}$$

Likewise, by the twistor-spinor theory, and by using the duality between the tensors T*αβ*, and A*αβ*, we can determine the mechanism of measurement and characterize the geometrical context of the detection of torsion. We define the

*Supermassive mass movement field + electromagnetic energy field = torsion evidence on the surface of sensing (sphere). How can we construct a tensor whose evidence of torsion can trace the electronic signals that could come from matter and electromagnetic fields of gravitational waves? We need a tensor of invariants of movement identified by invariants in geometry. This is the kinematic twistor tensor* A*αβ. The space S is the sphere that senses the torsion and transmits its variation at the time to the surface* Σ *defined by the electromagnetic-matter movement. Two-dimensional model of spinor representation of the kinematic twistor*

<sup>¼</sup> *<sup>Z</sup><sup>α</sup>*j*Z<sup>α</sup>* <sup>¼</sup> <sup>ω</sup>A, <sup>π</sup><sup>A</sup>

for all coordinates systems A and A. We define the twistor infinity tensor I*αβ*<sup>0</sup>

which has a metric defined when *<sup>α</sup>* <sup>¼</sup> *<sup>β</sup>* and Z*<sup>β</sup>* <sup>¼</sup> <sup>Z</sup>*<sup>β</sup>* (its complex conjugate).

ð Þ! S I *αβ* Σ

*αβ*W*β:*

Then, in the infinity of the space-time, we have the sequence of mappings:

 ! I *αβ*

<sup>1</sup> <sup>ω</sup><sup>A</sup> : <sup>∗</sup> ! , with rule of correspondence on points of the space–time <sup>π</sup><sup>A</sup>0↦i*x*AA<sup>0</sup>

π<sup>A</sup><sup>0</sup> , <sup>∗</sup> <sup>¼</sup> <sup>W</sup>*<sup>α</sup>* <sup>¼</sup> <sup>π</sup>A, <sup>ω</sup><sup>A</sup><sup>0</sup>

! <sup>∗</sup> , with correspondence rule of points of the space–time <sup>ω</sup><sup>A</sup>↦‐i*x*AA<sup>0</sup>

ω<sup>A</sup> <sup>¼</sup> <sup>i</sup>*x*AA<sup>0</sup>

corresponding twistor spaces in this case are: <sup>¼</sup> <sup>Z</sup>*<sup>α</sup>* <sup>¼</sup> <sup>ω</sup>A, <sup>π</sup><sup>A</sup><sup>0</sup>

**23**

<sup>2</sup> <sup>I</sup>*αβ* : <sup>∗</sup> ! , with the correspondence rule W*α*↦Z*<sup>α</sup>*<sup>I</sup>

as the obtained directly of the all space-time whose structure obeys a Minkowski space *M*. Then the surface Σ, which is a 3-dimensional surface is obtained for the

, that is to say:

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

, (8)

<sup>Σ</sup> <sup>¼</sup> <sup>Σ</sup> <sup>Z</sup>*<sup>α</sup>*, *<sup>Z</sup><sup>β</sup>* , (9)

ð Þ Σ (10)

ω<sup>A</sup>*:* Likewise, the

ω<sup>A</sup><sup>0</sup> ¼ �i*x*AA<sup>0</sup>

,

π<sup>A</sup><sup>0</sup> *:* Also its dual π<sup>A</sup><sup>0</sup> :

πA

2

twistor space as the points set<sup>1</sup>

*tensor* A*αβ is constructed from the sphere.*

**Figure 1.**

twistor fields Z*<sup>α</sup>* and Z*<sup>α</sup>*

Then, its system has a complex set of four-dimensional solution families ffi <sup>2</sup> � �, and the family defines the two-surface twistor space ð ÞS *:*

Likewise, we can define the space of kinematic twistor tensor as the space of tensors [2]:

$$(\mathbb{T}(\mathbb{S}) \odot \mathbb{T}(\mathbb{S})) \, \ast = \left\{ \mathbf{A}\_{a\beta} \in \mathbb{T}\_2^4(\mathbb{M}) \, \big|\, \mathbf{A}\_{a\beta} Z^a Z^\beta = \mathbb{Q}[\mathbf{k}] \right\},\tag{5}$$

Though a gauge field (electromagnetic field as photons) acts on the back-ground radiation of the Minkowski space *M*, and the energy of the matter will be related to this gauge field through the equation

$$
\mathbf{j}^a = \mathbf{T}\_{a\boldsymbol{\beta}} \mathbf{k}^a,\tag{6}
$$

where k*<sup>α</sup>* can represent the density of background radiation, which establishes the curved part of the space (with spherical symmetry) together with T*αβ* (see **Figure 1**)

$$\mathbf{Q}[\mathbf{k}] = \frac{1}{4\pi\mathbf{G}} \int\_{S^2} \mathbf{T}\_{a\beta} \mathbf{k}^a \mathbf{d}\sigma^\beta \ge \int\_{S^2} \mathbf{j}^a \mathbf{d}\sigma^\beta \ge 2\pi\chi,\tag{7}$$

The corresponding electromagnetic device generates an electromagnetic radiation bath in a space region, where a movement of mass is detected inside this region, producing variations in the electromagnetic field. If we use a curvature energy sensor [3–5], we will obtain a spectrum in a twistor-spinor frame.

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time DOI: http://dx.doi.org/10.5772/intechopen.92815*

#### **Figure 1.**

Likewise, we consider M the space-time as the complex Minkowski model, and

In other words, the kinematic twistor tensor A*αβ* in the radiation energy bath (electromagnetic radiation) from the energy-matter tensor T*αβ* will be defined by

Σ

which produces an electrical total charge due to the Gauss divergence theorem

ð

R*αβγδ*f

Σ

This can be identified as the source depending on the killing vector k*<sup>α</sup>* of the

Then, its system has a complex set of four-dimensional solution families ffi <sup>2</sup> � �,

<sup>2</sup> ð Þ <sup>M</sup> �

Though a gauge field (electromagnetic field as photons) acts on the back-ground radiation of the Minkowski space *M*, and the energy of the matter will be related to

*<sup>α</sup>* <sup>¼</sup> <sup>T</sup>*αβ*k*<sup>α</sup>*

where k*<sup>α</sup>* can represent the density of background radiation, which establishes the curved part of the space (with spherical symmetry) together with T*αβ* (see

> d*σ<sup>β</sup>* ≥ ð

The corresponding electromagnetic device generates an electromagnetic radia-

S2 j

j

ð

T*αβ*k*<sup>α</sup>*

tion bath in a space region, where a movement of mass is detected inside this region, producing variations in the electromagnetic field. If we use a curvature energy sensor [3–5], we will obtain a spectrum in a twistor-spinor frame.

S2

Likewise, we can define the space of kinematic twistor tensor as the space of

<sup>2</sup> that act in Σ,

T*αβ*k*<sup>α</sup>* d*σ<sup>β</sup>*

, (1)

*αβ*d*σγδ*, (2)

*<sup>M</sup>* <sup>¼</sup> S2 <sup>⊗</sup> <sup>2</sup> <sup>⊗</sup> M, (3)

M ¼ S<sup>þ</sup> ⊕ S�*:* (4)

�A*αβZαZ<sup>β</sup>* <sup>¼</sup> Q k½ � � �, (5)

, (6)

*<sup>α</sup>*d*σ<sup>β</sup>* ≥2*πχ*, (7)

we define the kinematic twistor tensor as the obtained of the model in a space region Σ. Then considering the energy-matter tensor and its image in a two-dimensional surface will be two-surface twistor ð ÞS *:* The geometrical evidence of torsion

<sup>1</sup> and Z*<sup>α</sup>*

A*αβ*Z*<sup>α</sup>* 1Z*β* 2 ¼ ð

Q k½ �¼ <sup>1</sup>

4*π*G

is precisely through this contorted surface.

*Advances on Tensor Analysis and Their Applications*

the interaction of two fields Z*<sup>α</sup>*

,

Minkowski space background model

this gauge field through the equation

where M is the space-time of two components

and the family defines the two-surface twistor space ð ÞS *:*

ð Þ ð Þ<sup>S</sup> <sup>⊙</sup>ð Þ<sup>S</sup> <sup>∗</sup> <sup>¼</sup> <sup>A</sup>*αβ* <sup>∈</sup>T<sup>4</sup>

Q k½ �¼ <sup>1</sup>

4*π*G

on currents T*αβ*k*<sup>α</sup>*

tensors [2]:

**Figure 1**)

**22**

*Supermassive mass movement field + electromagnetic energy field = torsion evidence on the surface of sensing (sphere). How can we construct a tensor whose evidence of torsion can trace the electronic signals that could come from matter and electromagnetic fields of gravitational waves? We need a tensor of invariants of movement identified by invariants in geometry. This is the kinematic twistor tensor* A*αβ. The space S is the sphere that senses the torsion and transmits its variation at the time to the surface* Σ *defined by the electromagnetic-matter movement. Two-dimensional model of spinor representation of the kinematic twistor tensor* A*αβ is constructed from the sphere.*

Likewise, by the twistor-spinor theory, and by using the duality between the tensors T*αβ*, and A*αβ*, we can determine the mechanism of measurement and characterize the geometrical context of the detection of torsion. We define the twistor space as the points set<sup>1</sup>

$$\mathbb{T} = \left\{ Z^a | Z^a = \left( \alpha^\mathbf{A}, \pi\_\mathbf{A} \right) \right\}, \tag{8}$$

for all coordinates systems A and A. We define the twistor infinity tensor I*αβ*<sup>0</sup> 2 as the obtained directly of the all space-time whose structure obeys a Minkowski space *M*. Then the surface Σ, which is a 3-dimensional surface is obtained for the twistor fields Z*<sup>α</sup>* and Z*<sup>α</sup>* , that is to say:

$$
\Sigma = \Sigma(\mathbf{Z}^a, \mathbf{Z}^\beta),
\tag{9}
$$

which has a metric defined when *<sup>α</sup>* <sup>¼</sup> *<sup>β</sup>* and Z*<sup>β</sup>* <sup>¼</sup> <sup>Z</sup>*<sup>β</sup>* (its complex conjugate). Then, in the infinity of the space-time, we have the sequence of mappings:

$$\mathbb{T} \stackrel{\mathbf{I}^{\phi}}{\xrightarrow{\mathbf{I}^{\phi}}} \mathbb{T}(\mathbf{S}) \stackrel{\mathbf{I}^{\phi}\_{\mathbf{S}}}{\xrightarrow{\mathbf{I}}} \mathbb{T}(\mathbf{S}) \tag{10}$$

$$\mathbb{T} = \left\{ Z^a = \left( a^\Lambda, \pi\_{\Lambda'} \right) \middle| a^\Lambda = \text{i} \mathbf{x}^{\Lambda \mathsf{A} \mathsf{V}} \pi\_{\Lambda'} \right\}, \quad \mathbb{T} \,\ast = \left\{ \mathbf{W}\_a = \left( \pi\_\Lambda, a^{\Lambda \mathsf{V}} \right) \middle| a^{\Lambda \mathsf{V}} = -\text{i} \mathbf{x}^{\Lambda \mathsf{A} \mathsf{V}} \pi\_\Lambda \right\},$$
  $\mathbb{T}^2$   $\mathbf{I}\_{a\emptyset} : \mathbb{T} \ast \to \mathbb{T}$ , with the corresponding rule  $\mathbb{W}\_{a\flat} \mapsto \mathbf{Z}^a \mathbf{I}^{a\emptyset} \mathbf{W}\_{\beta}$ .

<sup>1</sup> <sup>ω</sup><sup>A</sup> : <sup>∗</sup> ! , with rule of correspondence on points of the space–time <sup>π</sup><sup>A</sup>0↦i*x*AA<sup>0</sup> π<sup>A</sup><sup>0</sup> *:* Also its dual π<sup>A</sup><sup>0</sup> : ! <sup>∗</sup> , with correspondence rule of points of the space–time <sup>ω</sup><sup>A</sup>↦‐i*x*AA<sup>0</sup> ω<sup>A</sup>*:* Likewise, the corresponding twistor spaces in this case are:

**Figure 2.**

*Kinematic twistor tensor due to the energy-matter tensor perturbation of the supermassive body, which is determined on sphere* S*:*

whose correspondence rule is given as follows:

$$\mathbf{Z}^{a} \mapsto \mathbf{I}^{a\beta} \mathbf{S}\_{\beta\beta'} \overline{\mathbf{Z}^{\beta'}} \mapsto \mathbf{I}^{a\beta} \Sigma\_{\beta\beta'} \overline{\mathbf{Z}^{\beta'}}.\tag{11}$$

<sup>A</sup>*αβ* <sup>¼</sup> <sup>1</sup>

mappings on the gauge and detection mechanism of torsion:

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

ðÞ Σ I *αβ*Σ*ββ*<sup>0</sup>

ð Þ <sup>S</sup> <sup>I</sup> *αβ* S ! ωAB

I *αβ*

**2. Torsion indicium in gravitational spin waves**

recover the most important cause of the second curvature.

S*γ αβ* <sup>¼</sup> *<sup>χ</sup>*CC<sup>0</sup>

T*γ*

where ⊙ is a symmetric tensor product.

model of torsion can be written as follows:

torsion exists [1, 2].

where it is clear that

dual fields. *Proof.* [2].

have:

**25**

<sup>16</sup>π<sup>G</sup> <sup>∮</sup> S

Σ ! A*αβ*

Finally, we can establish the following commutative diagram of twistor space

<sup>Σ</sup> ↑ ↑ <sup>T</sup>*αβ* <sup>↑</sup>A*αβ* <sup>Z</sup>*<sup>α</sup>Z<sup>β</sup>*

In this context, the use of the Einstein-Cartan-Sciama-Kibble theory is important. Likewise, this theory is convenient considering our space-time model as has been defined *M*, and the field experiments considering external fields created through the use of the spin Hall effect and movement of matter in Σ*:* We consider the curvature and twistor-spinor framework studied in Refs. [2, 4], where they

Likewise, for the curvature tensor K*αβγδ*, we start with the Riemann tensor R*αβγδ*

**Conjecture 2.1 (Bulnes F, Rabinovich I).** The curvature in the spinor-twistor framework can be perceived with the appearance of the torsion and the anti-self-

In the previous research of this conjecture [2], it was established that the spinor

<sup>B</sup><sup>0</sup> <sup>þ</sup> <sup>~</sup>*χ*CC<sup>0</sup> A0

<sup>B</sup><sup>0</sup> <sup>þ</sup> <sup>~</sup>*χ*CC<sup>0</sup> A0

> πC0 χA0 B0

Considering the spinor equation of torsion (15) in the twistor-spinor framework,

<sup>B</sup><sup>0</sup> ∈ AB, (18)

*αβ*, (19)

*<sup>α</sup>* π*<sup>β</sup>*A0, (21)

AC0, (22)

<sup>B</sup><sup>0</sup> <sup>∈</sup> *AB* , (20)

that appears in the integral (2). Likewise, considering the space-time *M*, a complex Riemannian manifold, we have the conjecture where the indicium of

AA<sup>0</sup> ∈ <sup>A</sup><sup>0</sup>

T*γ αβ* <sup>¼</sup> 2S*<sup>γ</sup>*

Then, it is obvious that the torsion tensor can be written as follows:

AA<sup>0</sup> ∈ <sup>A</sup><sup>0</sup>

<sup>I</sup>*αβ* <sup>¼</sup> <sup>π</sup><sup>A</sup><sup>0</sup>

<sup>π</sup><sup>A</sup><sup>0</sup> <sup>∇</sup>AA0π<sup>B</sup> ð Þ¼0 <sup>ξ</sup>Aπ<sup>B</sup><sup>0</sup> � <sup>2</sup>π<sup>A</sup><sup>0</sup>

and for other transformation of spinor coordinate frame (and derivative), we

*αβ* <sup>¼</sup> <sup>2</sup> *<sup>χ</sup>*CC<sup>0</sup>

we have the transformation in the infinity twistor of the space-time:

ð ÞS ⊙ð ÞS

RABω<sup>A</sup> *<sup>α</sup>* ω<sup>B</sup>

ð Þ ð ÞS ⊙ð ÞS ∗

,

*<sup>β</sup>* , (16)

(17)

We consider the symmetric part of the fields Z*<sup>α</sup>* and Z*<sup>β</sup>*, given by the spinors ωAB, which satisfy the valence-2 twistor equation:

$$\nabla\_{\mathbf{A}'}^{\mathbf{A}} \mathbf{a}^{\mathbf{B} \mathbf{C}} = -i \in \mathsf{A}^{\mathbf{A}\left(\mathbb{B} \mathbf{k}\_{\mathbf{A}'}^{\mathbf{C}}\right)},\tag{12}$$

which has a solution in a 10-dimensional space. We need limit the space region of our study to spinor waves in a four-dimensional space, that is, on a component of Eq. (3). The solution in the space of Eq. (12) is spanned by spinor fields ωAB of the form<sup>3</sup>

$$
\alpha^{\rm AB} = \alpha\_1^{\left(\rm A \alpha\_2^{\rm B}\right)} \tag{13}
$$

where each ω<sup>A</sup> <sup>i</sup> is a valence-1 twistor, satisfying the equation:

$$
\nabla\_{\mathbf{A}'}^{\mathbf{A}} \alpha^{\mathbf{B}} = \cdot i \in {}^{\mathbf{AB}} \pi\_{\mathbf{A}'},\tag{14}
$$

We need in all time, for our measurements the conservation condition, which will be given by the equation:

$$\nabla^a \mathbf{T}^{a\beta} = \mathbf{0},\tag{15}$$

that is to say, we suppose that the energy-matter is always present in the space and is constant, at least in the space region where is bounded the three-dimensional surface Σ. Likewise, when a supermassive body exists that perturbs the space-time, the energy matter of its tensor can be carried out (see **Figure 2**):

$$Z\_1 \left( \mathsf{A} Z\_1^{\mathsf{R}} \right) \dashrightarrow Z\_1 \otimes\_{\mathsf{A} \otimes \mathsf{max}} Z\_2 \oplus \mathsf{T} \otimes\_{\mathsf{A} \otimes \mathsf{max}} \mathsf{T} \dashrightarrow \mathsf{T} \otimes \mathsf{T} \dashrightarrow$$

<sup>3</sup> Here the spinors product ω<sup>1</sup> <sup>A</sup>ω<sup>2</sup> <sup>B</sup> ð Þ, comes from fields product Z1 AZ2 <sup>B</sup> ð Þ, which is a symmetric tensor product, that is to say,

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time DOI: http://dx.doi.org/10.5772/intechopen.92815*

$$\mathbf{A}\_{a\boldsymbol{\beta}} = \frac{\mathbf{1}}{\mathbf{1}\mathsf{6}\pi\mathbf{G}} \oint\_{\mathcal{S}} \mathbf{R}\_{\mathbf{AB}} \mathbf{a}\_{\boldsymbol{\alpha}}^{\mathcal{A}} \mathbf{a}\_{\boldsymbol{\beta}}^{\mathcal{B}},\tag{16}$$

Finally, we can establish the following commutative diagram of twistor space mappings on the gauge and detection mechanism of torsion:

$$\begin{array}{ll} \mathrm{T}(\boldsymbol{\Sigma}) \stackrel{\mathrm{I^{\otimes}\boldsymbol{\Sigma}\_{\beta\beta'}}}{\longleftarrow} \boldsymbol{\Sigma} \stackrel{\mathrm{A\_{q\beta}}}{\longrightarrow} (\mathbb{T}(\mathbf{S}) \boldsymbol{\odot} \boldsymbol{\Gamma}(\mathbf{S})) \; \* \\\ \mathrm{I^{\otimes}\_{\boldsymbol{\Sigma}}} \uparrow & \uparrow \mathrm{T}^{a\beta} \\\ \mathrm{T}(\mathbf{S}) \stackrel{\mathrm{I^{q\beta}}}{\longleftarrow} \boldsymbol{\mathcal{S}} \stackrel{\omega^{\mathrm{AB}}}{\longrightarrow} \boldsymbol{\mathrm{T}}(\mathbf{S}) \boldsymbol{\ominus} \boldsymbol{\Gamma}(\mathbf{S}) \end{array} \tag{17}$$

where ⊙ is a symmetric tensor product.

#### **2. Torsion indicium in gravitational spin waves**

In this context, the use of the Einstein-Cartan-Sciama-Kibble theory is important. Likewise, this theory is convenient considering our space-time model as has been defined *M*, and the field experiments considering external fields created through the use of the spin Hall effect and movement of matter in Σ*:* We consider the curvature and twistor-spinor framework studied in Refs. [2, 4], where they recover the most important cause of the second curvature.

Likewise, for the curvature tensor K*αβγδ*, we start with the Riemann tensor R*αβγδ* that appears in the integral (2). Likewise, considering the space-time *M*, a complex Riemannian manifold, we have the conjecture where the indicium of torsion exists [1, 2].

**Conjecture 2.1 (Bulnes F, Rabinovich I).** The curvature in the spinor-twistor framework can be perceived with the appearance of the torsion and the anti-selfdual fields.

*Proof.* [2].

whose correspondence rule is given as follows:

*Advances on Tensor Analysis and Their Applications*

ωAB, which satisfy the valence-2 twistor equation:

form<sup>3</sup>

**Figure 2.**

*determined on sphere* S*:*

where each ω<sup>A</sup>

will be given by the equation:

<sup>3</sup> Here the spinors product ω<sup>1</sup> <sup>A</sup>ω<sup>2</sup>

product, that is to say,

**24**

Z*<sup>α</sup>*↦I

∇<sup>A</sup>

<sup>ω</sup>AB <sup>¼</sup> <sup>ω</sup><sup>1</sup>

∇<sup>A</sup>

the energy matter of its tensor can be carried out (see **Figure 2**):

Z1 AZ2

<sup>B</sup> ð Þ,

*αβ*S*ββ*0Z*<sup>β</sup>*<sup>0</sup>

*Kinematic twistor tensor due to the energy-matter tensor perturbation of the supermassive body, which is*

↦I *αβ*Σ*ββ*´ 0Z*<sup>β</sup>*<sup>0</sup>

We consider the symmetric part of the fields Z*<sup>α</sup>* and Z*<sup>β</sup>*, given by the spinors

which has a solution in a 10-dimensional space. We need limit the space region of our study to spinor waves in a four-dimensional space, that is, on a component of Eq. (3). The solution in the space of Eq. (12) is spanned by spinor fields ωAB of the

Aω<sup>2</sup>

<sup>i</sup> is a valence-1 twistor, satisfying the equation:

We need in all time, for our measurements the conservation condition, which

that is to say, we suppose that the energy-matter is always present in the space and is constant, at least in the space region where is bounded the three-dimensional surface Σ. Likewise, when a supermassive body exists that perturbs the space-time,

comes from fields product Z1 AZ2

<sup>B</sup> ð Þ¼*<sup>Z</sup>*<sup>1</sup> <sup>⊗</sup> Symm*Z*<sup>2</sup> <sup>∈</sup> <sup>⊗</sup> Symm¼⊙*:*

*:* (11)

<sup>A</sup>0ωBC ¼ �*<sup>i</sup>* <sup>∈</sup> A BkC ð Þ <sup>A</sup><sup>0</sup> , (12)

<sup>B</sup> ð Þ¼<sup>ω</sup>AωB, (13)

<sup>A</sup>0ω<sup>B</sup> <sup>¼</sup> ‐*i*<sup>∈</sup> ABπ<sup>A</sup>0, (14)

<sup>∇</sup>*<sup>α</sup>*T*αβ* <sup>¼</sup> 0, (15)

<sup>B</sup> ð Þ,

which is a symmetric tensor

In the previous research of this conjecture [2], it was established that the spinor model of torsion can be written as follows:

$$\mathbf{S}\_{\alpha\beta}^{\prime} = \chi\_{\mathbf{A}\mathbf{A}^{\prime}}^{\prime\prime\prime} \mathbf{\tilde{e}}\_{\mathbf{A}^{\prime}\mathbf{B}^{\prime}} + \tilde{\chi}\_{\mathbf{A}^{\prime}\mathbf{B}^{\prime}}^{\prime\prime\prime} \mathbf{\tilde{e}}\_{\mathbf{A}\mathbf{B}},\tag{18}$$

where it is clear that

$$\mathbf{T}^{\mathsf{r}}\_{a\boldsymbol{\beta}} = \mathbf{2} \mathbf{S}^{\mathsf{r}}\_{a\boldsymbol{\beta}},\tag{19}$$

Then, it is obvious that the torsion tensor can be written as follows:

$$\mathbf{T}^{\mathsf{V}}\_{\mathsf{a}\boldsymbol{\beta}} = \mathfrak{Z} \big( \boldsymbol{\chi}^{\mathrm{CC}\prime}\_{\mathsf{AA'}} \in \mathsf{A'}^{\mathrm{CC}}\_{\mathsf{A'}\mathsf{B'}} + \tilde{\boldsymbol{\chi}}^{\mathrm{CC}\prime}\_{\mathsf{A'}\mathsf{B'}} \in \mathsf{A}\_{\mathsf{B}} \big), \tag{20}$$

Considering the spinor equation of torsion (15) in the twistor-spinor framework, we have the transformation in the infinity twistor of the space-time:

$$\mathbf{I}\_{a\boldsymbol{\beta}} = \pi\_a^{\mathbf{A}\prime} \pi\_{\boldsymbol{\beta}\mathbf{A}^\prime},\tag{21}$$

and for other transformation of spinor coordinate frame (and derivative), we have:

$$
\pi^{\mathbf{A}\prime}(\nabla\_{\mathbf{A}\mathbf{A}\prime}\pi\_{\mathbf{B}\prime}) = \sharp\_{\mathbf{A}}\pi\_{\mathbf{B}\prime} - 2\pi^{\mathbf{A}\prime}\pi^{\mathbf{C}\prime}\chi\_{\mathbf{A}\prime\mathbf{B}\prime\mathbf{A}\prime},\tag{22}
$$

#### **3. Curvature energy to torsion**

The following results obtained in Ref. [2] are the fundamental principles that are required to gauge and detect the torsion through the tensor A*αβ*, considering the law transformation to pass from a field Z*<sup>α</sup>* to other Z*<sup>β</sup>* through two coordinate systems *α* and *β* to transform the surface Σ:

$$
\Sigma\_{a\beta} = \mathbf{A}\_{a\beta} \mathbf{I}^{\beta \gamma} \Sigma\_{\gamma a'} \tag{23}
$$

which relates to the spinor field ωAB, with the killing vector k*<sup>α</sup>*

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

wave pulse, which can be spectrally reproduced in a function sinω<sup>L</sup>

**wave links such as spinors and wave pulses**

φ<sup>A</sup><sup>0</sup>

surface Σ, which is given as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

the strong electronic gauging study [3, 7].

given in the energy domain MN ≥ A*αβ*Z*<sup>α</sup>*I

**(A)** and **(B)**).

framework (**Figure 4**)

and

**27**

twistor equation. We use the divergence theorem when S is a 2-surface in the 3-

around the source having several censorship conditions designed through dominating energy conditions of curvature that can be used in the electronic experiments. We have a metrology [5–7] of curvature measured and detected by our curvature sensors, which permitted us to have the curvature in new units obtained under

Likewise, the energy of the kinematic twistor tensor that will be substantive energy to curvature energy measure in the case of the spinor-twistor framework is

the electronic device of electromagnetic radiation bath interacting with the proximity of supermassive object or simple mass movement (see **Figure 2**, and **Figure 3**

**4. Electronic experiment demonstration of torsion existence through**

<sup>4</sup> complies the integrals:

<sup>2</sup>π<sup>i</sup> <sup>∮</sup> <sup>Z</sup>*α*\$<sup>a</sup>

<sup>2</sup>π<sup>i</sup> <sup>∮</sup> <sup>Z</sup>*α*\$<sup>a</sup>

<sup>A</sup>*αβ* <sup>¼</sup> <sup>1</sup>

<sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>G</sup> <sup>∮</sup> S R*αβγδ*f

¼ 1 2πi i <sup>8</sup>π<sup>G</sup> <sup>∮</sup> S ωA *<sup>α</sup>* d<sup>2</sup> ω*<sup>β</sup>*<sup>A</sup>

<sup>¼</sup> <sup>1</sup> <sup>16</sup>π<sup>G</sup> <sup>∮</sup> S ωA *<sup>α</sup>* d<sup>2</sup> ω*β*A,

∇AA<sup>0</sup>

which are equivalent to □<sup>φ</sup> <sup>¼</sup> 0, for zero spin case.

<sup>4</sup> Here our electromagnetic wave equation can be characterized by the massless field equations:

φAB … <sup>L</sup> ¼ 0, ∇AA<sup>0</sup>

which for the particular case of the determination of A*αβ*, are the integrals:

<sup>16</sup>π<sup>G</sup> <sup>∮</sup> S RABω<sup>A</sup> *<sup>α</sup>* ω<sup>B</sup> *β*

<sup>B</sup><sup>0</sup> … <sup>L</sup><sup>0</sup> <sup>a</sup> ð Þ¼ <sup>1</sup>

<sup>φ</sup>AB … <sup>L</sup><sup>0</sup> <sup>a</sup> ð Þ¼ <sup>1</sup>

An electromagnetic field as detector can also be a part of establishing the perturbation in the space-time that must help us to perceive the torsion existence. Likewise, this field as a solution of the Maxwell equations in the spinor-twistor

> *∂ <sup>∂</sup>*ω<sup>A</sup> <sup>⋯</sup> *<sup>∂</sup>*

> > *αβ*dσ*γδ*,

φ<sup>A</sup><sup>0</sup>

<sup>B</sup><sup>0</sup> … <sup>L</sup><sup>0</sup> ¼ 0 ,

*βγ*Z*<sup>γ</sup>* ≥0*:* Then, the solution of the quasi-local mass is directly related to the quantity of energy-matter tensor. Likewise, this solution is a function of radius and time as

<sup>Σ</sup> <sup>¼</sup> <sup>ω</sup>AπA<sup>0</sup> <sup>þ</sup> <sup>ω</sup>AπA0, (26)

, in the valence-2

<sup>ω</sup><sup>L</sup> , under voltage of

π<sup>A</sup>0⋯π<sup>L</sup>0*f*ð Þ Z π<sup>F</sup>0dπ<sup>F</sup><sup>0</sup> (27)

*<sup>∂</sup>*ω<sup>L</sup> *<sup>f</sup>*ð Þ <sup>Z</sup> <sup>π</sup><sup>F</sup>0dπ<sup>F</sup><sup>0</sup> (28)

Then, we enunciate the following theorem.

**Theorem 3.1 (Bulnes F, Stropovsvky Y, Rabinovich I).** We consider the embedding as follows:

$$
\sigma: \Sigma \to (\mathbb{T}(\mathbb{S}) \otimes \mathbb{T}(\mathbb{S})) \*,\tag{24}
$$

The space *σ*ð Þ Σ is smoothly embedded in the twistor space ð Þ ð ÞS ⊗ ð ÞS ∗ *:* Then, their curvature energy is given in the interval MN ≥ A*αβ*Z*<sup>α</sup>*I *βγ*Z*<sup>γ</sup>* ≥0*: Proof*. [2].

We have a source to linearized gravitational field that is explained through kinematics and electrodynamics used in its construction (see **Figure 3**). The linearized Riemann tensor corresponding to the spinor frame has been constructed, considering the components

$$\mathbf{f}\_{a\emptyset} = \mathfrak{o}\_{\mathbf{AB}} \in \mathfrak{s}^{\prime\prime},\tag{25}$$

#### **Figure 3.**

*(A) Antenna with voltage feeding of length 24.5 cm. (B) Electronic device of electronic monopole to electromagnetic radiation bath.*

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time DOI: http://dx.doi.org/10.5772/intechopen.92815*

which relates to the spinor field ωAB, with the killing vector k*<sup>α</sup>* , in the valence-2 twistor equation. We use the divergence theorem when S is a 2-surface in the 3 surface Σ, which is given as follows:

$$
\Sigma = \alpha^A \overline{\pi}\_{\mathcal{A}'} + \alpha^A \overline{\pi}\_{\mathcal{A}'}, \tag{26}
$$

around the source having several censorship conditions designed through dominating energy conditions of curvature that can be used in the electronic experiments.

We have a metrology [5–7] of curvature measured and detected by our curvature sensors, which permitted us to have the curvature in new units obtained under the strong electronic gauging study [3, 7].

Likewise, the energy of the kinematic twistor tensor that will be substantive energy to curvature energy measure in the case of the spinor-twistor framework is given in the energy domain MN ≥ A*αβ*Z*<sup>α</sup>*I *βγ*Z*<sup>γ</sup>* ≥0*:*

Then, the solution of the quasi-local mass is directly related to the quantity of energy-matter tensor. Likewise, this solution is a function of radius and time as wave pulse, which can be spectrally reproduced in a function sinω<sup>L</sup> <sup>ω</sup><sup>L</sup> , under voltage of the electronic device of electromagnetic radiation bath interacting with the proximity of supermassive object or simple mass movement (see **Figure 2**, and **Figure 3 (A)** and **(B)**).

#### **4. Electronic experiment demonstration of torsion existence through wave links such as spinors and wave pulses**

An electromagnetic field as detector can also be a part of establishing the perturbation in the space-time that must help us to perceive the torsion existence. Likewise, this field as a solution of the Maxwell equations in the spinor-twistor framework (**Figure 4**) <sup>4</sup> complies the integrals:

$$\mathfrak{q}\_{\mathbf{A}'\mathbf{B}'\dots\mathbf{L}'}(\mathbb{R}^\mathbf{a}) = \frac{1}{2\pi i} \oint\_{Z^a \leftrightarrow \mathbb{R}^a} \mathfrak{q}\_{\mathbf{A}'} \dotsm \mathfrak{r}\_{\mathbf{L}} f(\mathbf{Z}) \mathfrak{r}\_{\mathbf{F}'} \mathrm{d}\pi\_{\mathbf{F}'} \tag{27}$$

and

**3. Curvature energy to torsion**

*Advances on Tensor Analysis and Their Applications*

and *β* to transform the surface Σ:

embedding as follows:

*Proof*. [2].

**Figure 3.**

**26**

*electromagnetic radiation bath.*

considering the components

Then, we enunciate the following theorem.

The following results obtained in Ref. [2] are the fundamental principles that are required to gauge and detect the torsion through the tensor A*αβ*, considering the law transformation to pass from a field Z*<sup>α</sup>* to other Z*<sup>β</sup>* through two coordinate systems *α*

*βγ*Σ*γα*<sup>0</sup> (23)

*βγ*Z*<sup>γ</sup>* ≥0*:*

<sup>B</sup>0, (25)

*σ* : Σ ! ð Þ ð ÞS ⊗ ð ÞS ∗ , (24)

Σ*αβ* ¼ A*αβ*I

**Theorem 3.1 (Bulnes F, Stropovsvky Y, Rabinovich I).** We consider the

The space *σ*ð Þ Σ is smoothly embedded in the twistor space ð Þ ð ÞS ⊗ ð ÞS ∗ *:*

We have a source to linearized gravitational field that is explained through kinematics and electrodynamics used in its construction (see **Figure 3**). The linearized Riemann tensor corresponding to the spinor frame has been constructed,

f*αβ* ¼ ωAB ∈ <sup>A</sup><sup>0</sup>

*(A) Antenna with voltage feeding of length 24.5 cm. (B) Electronic device of electronic monopole to*

Then, their curvature energy is given in the interval MN ≥ A*αβ*Z*<sup>α</sup>*I

$$\mathfrak{q}\_{\text{AB}\dots L'}(\mathbb{R}^{\mathbf{a}}) = \frac{\mathbf{1}}{2\pi \mathbf{i}\_{Z' \to \mathbb{R}^{\mathbf{a}}}} \oint\_{\mathbb{O}^{\mathbf{a}}} \frac{\partial}{\partial \mathbf{o}^{\mathbf{A}}} \cdots \frac{\partial}{\partial \mathbf{o}^{\mathbf{L}}} f(\mathbf{Z}) \pi\_{\mathbf{F}'} \mathbf{d} \pi\_{\mathbf{F}'} \tag{28}$$

which for the particular case of the determination of A*αβ*, are the integrals:

$$\begin{split} \mathbf{A}\_{\alpha\beta} &= \frac{1}{16\pi\mathbf{G}} \oint\_{\mathbf{S}} \mathbf{R}\_{\mathbf{AB}} \mathbf{o}\_{\alpha}^{\mathbf{A}} \mathbf{o}\_{\beta}^{\mathbf{B}} \\ &= \frac{1}{16\pi\mathbf{G}} \oint\_{\mathbf{S}} \mathbf{R}\_{\alpha\beta\gamma\delta} \mathbf{f}^{\alpha\beta} \mathbf{d} \mathbf{o}^{r\delta}, \\ &= \frac{1}{2\pi i} \frac{\mathbf{i}}{8\pi\mathbf{G}} \oint\_{\mathbf{S}} \mathbf{o}\_{\alpha}^{\mathbf{A}} \mathbf{d}^{2} \mathbf{o}\_{\beta\mathbf{A}} \\ &= \frac{1}{16\pi\mathbf{G}} \oint\_{\mathbf{S}} \mathbf{o}\_{\alpha}^{\mathbf{A}} \mathbf{d}^{2} \mathbf{o}\_{\beta\mathbf{A}}, \end{split}$$

$$
\nabla^{\mathsf{AA}\prime} \mathsf{q}\_{\mathsf{AB\\_L}} = \mathsf{0}, \quad \nabla^{\mathsf{AA}\prime} \mathsf{q}\_{\mathsf{A}^\prime \mathsf{B}^\prime \dots \mathsf{L}^\prime} = \mathsf{0}, \dots
$$

which are equivalent to □<sup>φ</sup> <sup>¼</sup> 0, for zero spin case.

<sup>4</sup> Here our electromagnetic wave equation can be characterized by the massless field equations:

**Figure 4.**

*(A) Two-dimensional surface of charge* Q k½ � *in monopole field. (B) Two-dimensional surface of energy-matter tensor* T*αβ in supermassive body.*

#### **Figure 5.**

*Dynamic-magnetic system defining the formula* A*αβ*Z*<sup>α</sup>*I *βγ*Z*<sup>γ</sup> :.*

where it has been applied in the field around the circle used as cycle of the displacement along the three-cylindrical spiral cycles (see **Figure 5**). As discussed in Section 2, the torsion evidence can be obtained with a good approximation (given the limitations of the electronic system) when a complete signal sinω<sup>L</sup> <sup>ω</sup><sup>L</sup> is obtained in each three cycles, where two complete spinors are produced.

The sensor device of magnetic field of Hall effect has detected the boundary whose region is an arco length of 0.045 m (see **Figure 6(A)**). Without this range, there is no detection of field, although it is evident the cyclic subsequent displacements of the magnetic dilaton. This is shown with three curves in the graph of **Figure 5**, with displacement times *t*1, *t*2, and *t*<sup>3</sup> The electric potential that is generated due to the magnetic field variation is inversely proportional to the magnetic field intensity with base in the relation of 19.4 mV/Gs (**Figure 7**).

on the gravitational field between the dilaton mass and the Earth mass. The coordi-

*(A) Magnetic sphere as magnetic dilaton of radius 0.025 m. This dilaton will be used to detect the distortion in the boundary surface* Σ, *where the interaction happens between the magnetic field of proper dilaton and the gravitational field generated for the mass of the proper dilaton and the mass of the earth. Maximum proximity of sensor is 0.010 m. (B) Rotational dynamic system of radius 0.085, with a reversible vertical displacement of 0.040 m. The sensor used is the Hall effect sensor. The device has an interface system for microcontroller and*

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

The conditioning signal is defined for the continuous variations of the electric potential, which are converted in frequency through the integrated circuit LM331 (see the **Figure 7**). The maximum response (output of frequency) of this device is 10 KHz; therefore, it is developed an electronic circuit to condition the signal and has required lectures. The digital signal obtaining each electric potential variation (0.52–0.26 V, and 0.26–0.52 V) as result of position change of the magnetic dilaton in the space is established. The intention of consider digital signal with pulse width to each respective 26 positions in the space is to do for each pulse a convolution with sinusoidal signal, this to obtain and try with periodic signals to the points study that

, … , L<sup>0</sup> are considered in our inertial reference

, B<sup>0</sup>

nate systems A, B, … , L and A<sup>0</sup>

*Frequency in the trajectory of the first cycle.*

**Figure 6.**

**Figure 7.**

**29**

*symmetric variable voltage source.*

frames used in the experiment.

In the first half of walk, the magnetic dilaton generates a decreasing potential of 0.52 V, until a minimum of 0.26 V. In the second half of walk, the magnetic dilaton generates an increasing potential of 0.26 V, until a maximum of 0.52 V, when it moves away. For the subsequent cycles, the remoteness of sensor in the trajectory obeys the spiral trajectory of the dynamic system. Both the effect of magnetic dilaton and the dynamics of system define our kinematic twistor tensor A*αβ*, which can be gauged in a more fine way with a quantum electronic device version of our electronic system used in this experimentation. The tensor of energy mass depends

#### **Figure 6.**

*(A) Magnetic sphere as magnetic dilaton of radius 0.025 m. This dilaton will be used to detect the distortion in the boundary surface* Σ, *where the interaction happens between the magnetic field of proper dilaton and the gravitational field generated for the mass of the proper dilaton and the mass of the earth. Maximum proximity of sensor is 0.010 m. (B) Rotational dynamic system of radius 0.085, with a reversible vertical displacement of 0.040 m. The sensor used is the Hall effect sensor. The device has an interface system for microcontroller and symmetric variable voltage source.*

#### **Figure 7.**

where it has been applied in the field around the circle used as cycle of the displacement along the three-cylindrical spiral cycles (see **Figure 5**). As discussed in Section 2, the torsion evidence can be obtained with a good approximation (given

*βγ*Z*<sup>γ</sup> :.*

*(A) Two-dimensional surface of charge* Q k½ � *in monopole field. (B) Two-dimensional surface of energy-matter*

The sensor device of magnetic field of Hall effect has detected the boundary whose region is an arco length of 0.045 m (see **Figure 6(A)**). Without this range, there is no detection of field, although it is evident the cyclic subsequent displacements of the magnetic dilaton. This is shown with three curves in the graph of **Figure 5**, with displacement times *t*1, *t*2, and *t*<sup>3</sup> The electric potential that is generated due to the magnetic field variation is inversely proportional to the magnetic

In the first half of walk, the magnetic dilaton generates a decreasing potential of 0.52 V, until a minimum of 0.26 V. In the second half of walk, the magnetic dilaton generates an increasing potential of 0.26 V, until a maximum of 0.52 V, when it moves away. For the subsequent cycles, the remoteness of sensor in the trajectory obeys the spiral trajectory of the dynamic system. Both the effect of magnetic dilaton and the dynamics of system define our kinematic twistor tensor A*αβ*, which can be gauged in a more fine way with a quantum electronic device version of our electronic system used in this experimentation. The tensor of energy mass depends

<sup>ω</sup><sup>L</sup> is obtained in

the limitations of the electronic system) when a complete signal sinω<sup>L</sup>

field intensity with base in the relation of 19.4 mV/Gs (**Figure 7**).

each three cycles, where two complete spinors are produced.

*Dynamic-magnetic system defining the formula* A*αβ*Z*<sup>α</sup>*I

**Figure 4.**

**Figure 5.**

**28**

*tensor* T*αβ in supermassive body.*

*Advances on Tensor Analysis and Their Applications*

*Frequency in the trajectory of the first cycle.*

on the gravitational field between the dilaton mass and the Earth mass. The coordinate systems A, B, … , L and A<sup>0</sup> , B<sup>0</sup> , … , L<sup>0</sup> are considered in our inertial reference frames used in the experiment.

The conditioning signal is defined for the continuous variations of the electric potential, which are converted in frequency through the integrated circuit LM331 (see the **Figure 7**). The maximum response (output of frequency) of this device is 10 KHz; therefore, it is developed an electronic circuit to condition the signal and has required lectures. The digital signal obtaining each electric potential variation (0.52–0.26 V, and 0.26–0.52 V) as result of position change of the magnetic dilaton in the space is established. The intention of consider digital signal with pulse width to each respective 26 positions in the space is to do for each pulse a convolution with sinusoidal signal, this to obtain and try with periodic signals to the points study that

#### *Advances on Tensor Analysis and Their Applications*

**Appendix. (A) The experimental data table to the cycles of magnetic**

**Degrees Gs** 15 30.49 14 30.62 13 30.75 12 30.87 11 30.98 10 31.08 9 31.17 8 31.25 7 31.33 6 31.39 5 31.44 4 31.49 3 31.52 2 31.54 1 31.56 0 31.57 259 31.56 258 31.54 257 31.52 256 31.49 255 31.44 254 31.39 253 31.33 252 31.25 251 31.17 250 31.08 249 30.98 248 30.87 247 30.75 246 30.62 245 30.49

**Appendix. (B) Voltage that corresponds to proximity between magnetic**

**Voltage Frequency**

0.01 26 0.02 122

**dilaton (magnetic sphere) and sensor**

**31**

**dilaton displaced along the cylindrical spiral movement**

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

**Figure 8.** *Two-dimensional model of torsion by spinors.*

determine the curve in a 3-dimensional space in field theory in terms of the signal analysis.

In the first experiment (as described in Section 3), the sphere S has not curved inside the three-dimensional surface Σ. The electromagnetic field of monopole is fixed and does not produce distortion in the space. Any matter particle complies the spherical symmetry falling in the natural gravitational Earth field.

In the two experiments (in this Section 4), the choose of a magnetic dilaton represented by the ball of certain mass, which is displaced along the cylindrical spiral trajectory, produces a distortion at least in electronic device lectures and in the space, which could be affected for the Earth magnetic field and also for the gravitational field between the dilaton mass and the Earth mass. Summarizing the above, we can consider the following two-dimensional surface model of spinors deduced directly of second experiment verifying some conclusions on the torsion existence and consistence though twistors (see **Figure 8**).

#### **5. Conclusions**

We can establish different dualities in field theory, geometry, and movement to relate the energy-matter tensor and the kinematic twistor tensor for the torsion study. The torsion is a field observable, which in geometry is a second curvature. From a point of view of the field theory, torsion is an high evidence of the birth gravity and its consequences until our days with the gravitational waves detected from astronomical observatories.

Through of electronics is designed an analogue of the measurement of torsion as evidence of gravitational waves existence. With an experiment we gave some fundamentals studied in the gravitation theories, but with a modern mathematical study on invariants as are the twistors and spinors used to microscopic and microscopic field theory.

However, the limitations of our purely electronic devices only let see and interpret using the arguments of geometry, certain traces of electronic signals of the torsion evidence considering an electromagnetic field determined in certain voltage range and a movement of cylindrical trajectory, which as we know, is the constant torsion. However, this verifies Conjecture 2.1 and Theorem 3.1 established in other studies in theoretical physics and mathematical physics. Likewise, the methods and results of the research are on parallel themes and related to the gravity (no gravity precisely), considering this method as analogous to detect gravity waves but in this case to detect waves of torsion in an indirect way.


## **Appendix. (A) The experimental data table to the cycles of magnetic dilaton displaced along the cylindrical spiral movement**

#### **Appendix. (B) Voltage that corresponds to proximity between magnetic dilaton (magnetic sphere) and sensor**


determine the curve in a 3-dimensional space in field theory in terms of the signal

In the two experiments (in this Section 4), the choose of a magnetic dilaton represented by the ball of certain mass, which is displaced along the cylindrical spiral trajectory, produces a distortion at least in electronic device lectures and in the space, which could be affected for the Earth magnetic field and also for the gravitational field between the dilaton mass and the Earth mass. Summarizing the above, we can consider the following two-dimensional surface model of spinors deduced directly of second experiment verifying some conclusions on the torsion

We can establish different dualities in field theory, geometry, and movement to relate the energy-matter tensor and the kinematic twistor tensor for the torsion study. The torsion is a field observable, which in geometry is a second curvature. From a point of view of the field theory, torsion is an high evidence of the birth gravity and its consequences until our days with the gravitational waves detected

Through of electronics is designed an analogue of the measurement of torsion as evidence of gravitational waves existence. With an experiment we gave some fundamentals studied in the gravitation theories, but with a modern mathematical study on invariants as are the twistors and spinors used to microscopic and

However, the limitations of our purely electronic devices only let see and interpret using the arguments of geometry, certain traces of electronic signals of the torsion evidence considering an electromagnetic field determined in certain voltage range and a movement of cylindrical trajectory, which as we know, is the constant torsion. However, this verifies Conjecture 2.1 and Theorem 3.1 established in other studies in theoretical physics and mathematical physics. Likewise, the methods and results of the research are on parallel themes and related to the gravity (no gravity precisely), considering this method as analogous to detect gravity waves but in this

spherical symmetry falling in the natural gravitational Earth field.

existence and consistence though twistors (see **Figure 8**).

In the first experiment (as described in Section 3), the sphere S has not curved inside the three-dimensional surface Σ. The electromagnetic field of monopole is fixed and does not produce distortion in the space. Any matter particle complies the

analysis.

**Figure 8.**

*Two-dimensional model of torsion by spinors.*

*Advances on Tensor Analysis and Their Applications*

**5. Conclusions**

from astronomical observatories.

case to detect waves of torsion in an indirect way.

microscopic field theory.

**30**


**Voltage Frequency** 0.08 220 0.07 205 0.05 174 0.04 165 0.03 155 0.02 122 `0.01 96

*Kinematic-Energy Measurements of the Torsion Tensor in Space-Time*

*DOI: http://dx.doi.org/10.5772/intechopen.92815*

**Author details**

Francisco Bulnes<sup>1</sup>

Ingeniería), Mexico

Mexico

**33**

\*, Isaías Martínez<sup>2</sup>

\*Address all correspondence to: gruposlie@yahoo.com.mx

provided the original work is properly cited.

1 IINAMEI, Research Department in Mathematics and Engineering, TESCHA,

© 2020 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,

2 IINAMEI A.C. (Investigación Internacional Avanzada en Matemáticas e

, Omar Zamudio<sup>2</sup> and Edgar Navarro<sup>2</sup>

#### *Kinematic-Energy Measurements of the Torsion Tensor in Space-Time DOI: http://dx.doi.org/10.5772/intechopen.92815*


## **Author details**

**Voltage Frequency** 0.03 155 0.04 165 0.05 174 0.07 205 0.08 220 0.09 252 0.10 275 0.11 303 0.12 324 0.13 338 0.14 344 0.15 365 0.16 380 0.17 404 0.18 422 0.19 443 0.20 457 0.21 489 0.22 495 0.23 502 0.24 530 0.25 542 0.26 559 0.25 548 0.24 530 0.23 503 0.22 495 0.21 483 0.20 457 0.19 443 0.18 422 0.17 404 0.16 380 0.15 265 0.14 344 0.13 338 0.12 324 0.11 303 0.10 275 0.09 252

*Advances on Tensor Analysis and Their Applications*

**32**

Francisco Bulnes<sup>1</sup> \*, Isaías Martínez<sup>2</sup> , Omar Zamudio<sup>2</sup> and Edgar Navarro<sup>2</sup>

1 IINAMEI, Research Department in Mathematics and Engineering, TESCHA, Mexico

2 IINAMEI A.C. (Investigación Internacional Avanzada en Matemáticas e Ingeniería), Mexico

\*Address all correspondence to: gruposlie@yahoo.com.mx

© 2020 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] Bulnes F. Detection and measurement of quantum gravity by a curvature energy sensor: H-states of curvature energy. In: Uzunov D, editor. Recent Studies in Perturbation Theory. Rijeka, Croatia: IntechOpen; 2017. Available from: https://www.intechopen. com/books/recent-studies-in-perturba tion-theory/detection-and-measureme nt-of-quantum-gravity-by-a-curvatureenergy-sensor-h-states-of-curvatureener; https://doi.org/10.5772/68026

[2] Bulnes F, Stropovsvky Y, Rabinovich I. Curvature Energy and Their Spectrum in the Spinor-Twistor Framework: Torsion as Indicium of Gravitational Waves. Journal of Modern Physics. 2017;**8**:1723-1736. DOI: 10.4236/jmp.2017.810101

[3] Bulnes F. Electromagnetic Gauges and Maxwell Lagrangians applied to the determination of curvature in the spacetime and their applications. Journal of Electromagnetic Analysis and Applications. 2012;**4**(6):252-266. DOI: 10.4236/jemaa.2012.46035

[4] Bulnes F. Gravity, curvature and energy: Gravitational field intentionality to the cohesion and union of the universe. In: Zouaghi T, editor. Gravity —Geoscience Applications, Industrial Technology and Quantum Aspect. London, UK: IntechOpen; 20 December 2017. DOI: 10.5772/intechopen.71037. Available from: https://www. intechopen.com/books/gravitygeoscience-applications-industrialtechnology-and-quantum-aspect/ gravity-curvature-and-energygravitational-field-intentionality-to-thecohesion-and-union-of-the-uni

[5] Bulnes F, Martínez I, Mendoza A, Landa M. Design and development of an electronic sensor to detect and measure curvature of spaces using curvature energy. Journal of Sensor Technology.

2012;**2**(3):116-126. DOI: 10.4236/ jst.2012.23017

[6] Bulnes F, Martínez I, Zamudio O, Negrete G. Electronic sensor prototype to detect and measure curvature through their curvature energy. Science Journal of Circuits, Systems and Signal Processing. 2015;**4**(5):41-54. DOI: 10.11648/j.cssp.20150405.12

**Chapter 3**

**Abstract**

**1. Introduction**

**35**

Spacetimes

*Pınar Kirezli Uludağ*

**Keywords:** Brans-Dicke, stationary symmetric

Brans-Dicke Solutions of

Stationary, Axially Symmetric

One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. The theory offers a new approach by taking a scalar field *ϕ* instead of Newton's gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. In this context, some solutions of Einstein general relativity, such as black hole or wormhole solutions, are discussed in BD theory. Indeed, black hole solutions in BD theory are not fully understood yet. Old and new such solutions and their analysis will be reviewed in this chapter.

Einstein's theory of general relativity (GR), which is undoubtedly one of the greatest theories of the last century, is still being tried to be understood. Recently, the theory is supported by the observations of the gravitational waves which are observed by LIGO and Virgo collaboration [1]. On the other hand, GR may have some problems regarding defining gravity accurately at all scales. One of the problems that GR faced was that it could not fully describe the accelerated expansion of the universe [2–4] without unknown materials, i.e., dark matter and dark energy. Although, in order to understand the theory and satisfy the scientific cruosity, GR is modified with higher-order Ricci scalar [5, 6] soon after the theory is published, this modifications were not paid attention. The pioneer of studies on scalar-tensor theory were done by Brans and Dicke [7] by changing Newton's gravitational constant *G* with a scalar field *ϕ* ¼ 1*=G*. In order to understand the BD theory, several experimental tests of GR are studied, and they are summarized in [8]. Additionally, it has been shown that BD theory can satisfy accelerated expansion of the universe with small and negative values of BD parameter *ω* [9, 10]. But these values of BD parameter cannot satisfy the solutions of our solar system and latest CMB datas. Extended BD theories which include a potential for the BD field are allowed to construct a number of analytic approximations [11]. Although, in the

[7] Bulnes F, Martínez I, Zamudio O. Fine curvature measurements through curvature energy and their gauging and sensoring in the space. In: Yurish SY, editor. Spain: Advances in Sensors Reviews 4, IFSA; 2016

#### **Chapter 3**

**References**

[1] Bulnes F. Detection and

[2] Bulnes F, Stropovsvky Y,

Physics. 2017;**8**:1723-1736. DOI: 10.4236/jmp.2017.810101

Electromagnetic Analysis and

10.4236/jemaa.2012.46035

Rabinovich I. Curvature Energy and Their Spectrum in the Spinor-Twistor Framework: Torsion as Indicium of Gravitational Waves. Journal of Modern

[3] Bulnes F. Electromagnetic Gauges and Maxwell Lagrangians applied to the determination of curvature in the spacetime and their applications. Journal of

Applications. 2012;**4**(6):252-266. DOI:

[4] Bulnes F. Gravity, curvature and energy: Gravitational field intentionality

to the cohesion and union of the universe. In: Zouaghi T, editor. Gravity —Geoscience Applications, Industrial Technology and Quantum Aspect. London, UK: IntechOpen; 20 December 2017. DOI: 10.5772/intechopen.71037.

Available from: https://www. intechopen.com/books/gravitygeoscience-applications-industrialtechnology-and-quantum-aspect/ gravity-curvature-and-energy-

cohesion-and-union-of-the-uni

**34**

gravitational-field-intentionality-to-the-

[5] Bulnes F, Martínez I, Mendoza A, Landa M. Design and development of an electronic sensor to detect and measure curvature of spaces using curvature energy. Journal of Sensor Technology.

measurement of quantum gravity by a curvature energy sensor: H-states of curvature energy. In: Uzunov D, editor. Recent Studies in Perturbation Theory. Rijeka, Croatia: IntechOpen; 2017. Available from: https://www.intechopen. com/books/recent-studies-in-perturba tion-theory/detection-and-measureme nt-of-quantum-gravity-by-a-curvatureenergy-sensor-h-states-of-curvatureener; https://doi.org/10.5772/68026

*Advances on Tensor Analysis and Their Applications*

2012;**2**(3):116-126. DOI: 10.4236/

[6] Bulnes F, Martínez I, Zamudio O, Negrete G. Electronic sensor prototype to detect and measure curvature through their curvature energy. Science Journal of Circuits, Systems and Signal Processing. 2015;**4**(5):41-54. DOI: 10.11648/j.cssp.20150405.12

[7] Bulnes F, Martínez I, Zamudio O. Fine curvature measurements through curvature energy and their gauging and sensoring in the space. In: Yurish SY, editor. Spain: Advances in Sensors

Reviews 4, IFSA; 2016

jst.2012.23017

## Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes

*Pınar Kirezli Uludağ*
