**2. Numerical mathematical models: field torsion, torsion energy, spinors**

The torsion is a double curvature of a space or body, which is result of interaction of two fields, where one field is the electromagnetic field and other a field relative to the matter, which is the gravitational field in a wide sense (even considering the quantum gravity).

Likewise, as mentioned in the introduction, the torsion born from the matter spins which under the action of the electromagnetic field produce spin-waves whose geometrical invariants are spinors in the invariant theory [6, 7] (see **Figure 1**).

This has been formulated in a first conjecture (**Figure 2**).

**Conjecture 2.1.** The curvature in spinor-twistor framework can be perceived with the appearing of the torsion and the anti-self-dual fields [7].

#### **Figure 1.**

*Spinor image of spin-waves in torsion. This is a 2-dimensional model of spin-waves generated for a magnetic dilaton in a cylindrical-spiral trajectory of movement [8, 9]. In this 2-Dimensional surface* sð Þ ω1, ω<sup>2</sup> *have been considered a mesh of 100 points of density and the corresponding torsion model is obtained as cylindrical solution* Z = exp(0.5x)BesselY(0.1x, 2y)sin(4x) + BesselJ(0.6x, 6)*, of the corresponding field equation* ð Þ *P* þ χT *φ* ¼ 0*:*

*Numerical Simulations of Detections, Experiments and Magnetic Field Hall Effect Analysis… DOI: http://dx.doi.org/10.5772/intechopen.96779*

**Figure 2.**

causes and origin of these and the evolution of the Universe, being the curvature energy κ ωð Þ 1, ω<sup>2</sup> , an theoretical and practical element to consider in all modern Universe studies, and also the technological applications derived of these as corol-

As has been mentioned, the microscopic phenomena in the universe are the causes and are the things that give origin to the gravity and thus to the matter such as is known, through a large process in the Universe development. Likewise, an field observable that result of the microscopic interactions from particles spin level is the field torsion, which let in evidence much other space–time phenomena as the inflation, possibly the role of neutrinos, the baryongenesis, the proliferation of **H**particles and the form of evolution of all sidereal bodies, which in their different steps show mechanism where curvature and torsion are geometrical invariants that give meaning to the evolution of these sidereal objects and possibly to explain the

The present chapter will explain and expose some theoretical models, numerically obtained in 2D and 3D-dimensions of torsion and its meaning in field theory and the intrinsic study of torsion as field observable, as geometrical invariant and possible central concept to the new and future technologies that will be required to the human survival.

The torsion is a double curvature of a space or body, which is result of interaction of two fields, where one field is the electromagnetic field and other a field relative to the matter, which is the gravitational field in a wide sense (even consid-

Likewise, as mentioned in the introduction, the torsion born from the matter spins which under the action of the electromagnetic field produce spin-waves whose geo-

**Conjecture 2.1.** The curvature in spinor-twistor framework can be perceived

*Spinor image of spin-waves in torsion. This is a 2-dimensional model of spin-waves generated for a magnetic dilaton in a cylindrical-spiral trajectory of movement [8, 9]. In this 2-Dimensional surface* sð Þ ω1, ω<sup>2</sup> *have been considered a mesh of 100 points of density and the corresponding torsion model is obtained as cylindrical solution* Z = exp(0.5x)BesselY(0.1x, 2y)sin(4x) + BesselJ(0.6x, 6)*, of the corresponding field equation*

**2. Numerical mathematical models: field torsion, torsion energy,**

metrical invariants are spinors in the invariant theory [6, 7] (see **Figure 1**).

This has been formulated in a first conjecture (**Figure 2**).

with the appearing of the torsion and the anti-self-dual fields [7].

role of the Higgs boson, the meaning of the dark matter and more.

laries of the great curvature energy theory.

*Recent Advances in Numerical Simulations*

**spinors**

**Figure 1.**

**30**

ð Þ *P* þ χT *φ* ¼ 0*:*

ering the quantum gravity).

*(a). The corresponding solution to the field equation in cylindrical regime is Z = exp(*�*0.5(x + 1))BesselY (0.5x, 8y)sin(7x)BesselJ(0.3y,20). The rotation of cylinder was realized b). (b) Corresponding signals detected in the torsion detection and showed in the wave generator, under electronics measuring conditions: frequency of 235 kHz, wave sample of 13 seconds, voltage of 4 Volts (Figure 3a)).*

From a point of view of an electronics study [10] (using a magnetic Hall sensor<sup>1</sup> ) was obtained an evidence of the existence of torsion as field observable, where was used a magnetic particle as dilaton [5, 6] moving through of a trajectory whose torsion is constant in all space [4]. Likewise, was obtained the signal<sup>2</sup> :

$$\pi(\boldsymbol{\omega}) = \frac{\mathbf{1}}{2\pi} \frac{\mathbf{b}}{\mathbf{l}^3} \int\_{-0.5\mathbf{s}}^{+0.5\mathbf{s}} \Pi\left(\frac{\mathbf{t}}{\mathbf{l}}\right) \mathbf{e}^{-j\omega t} d\mathbf{t} = \frac{\mathbf{1}}{2\pi} \left[\frac{(2.5\mathbf{V} - \ddot{\boldsymbol{\alpha}})\mathbf{b}}{\mathbf{l}^3}\right] \frac{\sin\left(0.5\boldsymbol{\alpha}\right)}{0.5\boldsymbol{\alpha}},\tag{1}$$

which belongs to the signals set that evidence the torsion under permanent torsion conditions:

$$\left\{\frac{\sin a\mathcal{L}}{a\mathcal{L}}, \frac{\sinh a\mathcal{L}}{a\mathcal{L}}, \frac{\cos a\mathcal{L}}{a\mathcal{L}}, \frac{\cosh a\mathcal{L}}{a\mathcal{L}}\right\}, \forall \mathcal{L} = \frac{\mathbf{n}2\pi\mathbf{t}}{a\mathcal{T}}, \mathbf{T} > \mathbf{0} \tag{2}$$

$$\boldsymbol{\pi} = \frac{\mathbf{V}}{2\pi} \frac{\mathbf{b}}{\left(\mathbf{a}^2 + \mathbf{b}^2\right)} \frac{1}{\mathbf{l}} \left(= \frac{\text{Volts}}{\left(\text{meter}\right)^3}\right).$$

<sup>2</sup> The torsion is detected with conditions of movement. Likewise, by the lemma 3. 1 [10], the produced magnetic field in the dilaton must be corresponding to

$$H = \frac{\mathbf{l} \cdot \mathbf{a}^2}{2\pi \, \mathbf{l}^3},$$

whose magnetic field of the dilaton must be decreasing in the cycles for seconds of the turns for that these are detectable by the Hall type sensor (with low velocity). Then is conditioned the signal the initial constant voltage signal

$$V\_0 = \Pi \left(\frac{t}{13}\right) = \begin{cases} 2.5V - a & |t| \le 0.5s \\ 0 & |t| \ge 0.5s \end{cases},$$

which is a rectangular signal of conditioning. Here *α*~, is an amplifier factor of the voltage for be detectable.

<sup>1</sup> **Lemma** [10]. We consider a sensor Hall device <sup>L</sup>*<sup>H</sup>* Hall. The current deflection detected for the magnetic field change in the sensor produces per volume unit a torsion energy:

In our experiments and for our case the spin/waves detection are very little, and need signals filters to obtain a signal in a time range of the 10 seconds, at least. The corresponding screen and 2-dimensional spinor surface sð Þ ω1, ω<sup>2</sup> can be viewed in the **Figures 3a** and **4b**. The oscillations are little and the dilaton or particle reveals the torsion in the space–time [2, 4] under action of an electromagnetic field considering the kinematic invariance. Then chooses a trajectory in our electronic device of constant torsion (**Figures 3c** and **4**) [11].

**Conjecture 2.2 (F. Bulnes)** [9, 10]**.** The torsion is the geometrical invariant of the interaction between energy and space.

The appearing of waves in the space–time agitation is due the energy agitation of the particles that from their spins, are interchanging by the screw effect generated by the force *δ*ð Þ t1, t2 (see the **Figure 4**).

**3. Other cosmological phenomena re-interpretations related to 2D and 3D-torsion geometrical models and the signal analysis realized**

*(a). The screw effect in the space–time. (b). Mono-pole of electromagnetic field. This represents only* δð Þ t1, t2 *, of a field source in the space–time, whose torsion is established in the Z-axis. (c). 2-dimensional surface shows the*

*Numerical Simulations of Detections, Experiments and Magnetic Field Hall Effect Analysis…*

*DOI: http://dx.doi.org/10.5772/intechopen.96779*

In the studies of the Universe the creation of geometrical models or numerical simulation of the space–time phenomena is more complicated and obeys to a reinterpretation that obeys a modern field theories and cohomology theory of topological spaces, even the incorporation of microscopic theories of the Universe,

However, as has been said the most important geometrical invariant is the

For example, a re-interpretation that obeys a modern field theories and cohomology theory of topological spaces, with the incorporation of microscopic theories

We can create a 2D-numerical model of the screw effect and re-interpretation in the cosmology objects as black holes or sources as stars or behavior intersidereal magnetic alignment of galaxies, using 2-dimensional complex surfaces considering the Morera's and Cauchy-Goursat's theorems to be evaluate them and can be applied in an numerical program. For example, on singularities or poles in the space–time, considering the space–time a complex Riemannian manifold with singularities. This

C

*;* <sup>Ω</sup><sup>r</sup> ð Þ¼ <sup>C</sup>*:* This element is a contour around the singularity

g zð Þdz ¼ ∮

C 0‐ 1 z‐z1

*<sup>z</sup>*�<sup>1</sup> *:* The moduli

dz <sup>¼</sup> <sup>0</sup>‐2*π*<sup>i</sup> <sup>¼</sup>

because here are cause of all phenomena in the Universe.

*field source with torsion effect waving the 2-dimensional surface around the source.*

could represent the surface of the real part of the function *g z*ð Þ¼ *<sup>z</sup>*<sup>2</sup>

space of this point is less than 2 and thus lie inside one contour. Likewise, the contour integral can be split into two smaller integrals using the Cauchy-Goursat

‐2*π*i, (see the **Figure 5a**)). Likewise this value is a traditional cohomological func-

Form a cosmological evolution, the torsion play a fundamental role in the forming of all interstellar objects, even the Universe itself. The macroscopic density

curvature, being the torsion a second curvature.

of the Universe can be established as follows.

theorem having finally the contour integral [12] ∮

tional element of H<sup>f</sup> <sup>Π</sup>‐ℓ<sup>0</sup>

**33**

**Figure 4.**

as can be viewed in the **Figure 5a**).

#### **Figure 3.**

*(a) Energy spectra obtained for the conditioning of voltage signals. (b) This represents a quasi-periodic signal where in each 14 seconds to risk a pico-voltage* Vp, *when the dilaton spins to a velocity of 12 rpm, where is observed that their amplitude is very major than the subsequent amplitudes until to risk a new maximum* Vp*: (c). The frequency ω*, *is maintained in a 227 to 236 kHz range. In the sensing process was realized a closer of the energy spectra 1 sec/div (figure a)) to 5 sec/div (figures b, c) and 10sc/div (figure d)) to 50 mV for the three cases. We consider a conditioning of the signal through the operational amplifier of instrumentation AD620, which permit us view an output voltage proportional to 1.4 mV per Gauss detected.*

*Numerical Simulations of Detections, Experiments and Magnetic Field Hall Effect Analysis… DOI: http://dx.doi.org/10.5772/intechopen.96779*

**Figure 4.**

In our experiments and for our case the spin/waves detection are very little, and need signals filters to obtain a signal in a time range of the 10 seconds, at least. The corresponding screen and 2-dimensional spinor surface sð Þ ω1, ω<sup>2</sup> can be viewed in the **Figures 3a** and **4b**. The oscillations are little and the dilaton or particle reveals the torsion in the space–time [2, 4] under action of an electromagnetic field considering the kinematic invariance. Then chooses a trajectory in our electronic device

**Conjecture 2.2 (F. Bulnes)** [9, 10]**.** The torsion is the geometrical invariant of

The appearing of waves in the space–time agitation is due the energy agitation of the particles that from their spins, are interchanging by the screw effect generated

*(a) Energy spectra obtained for the conditioning of voltage signals. (b) This represents a quasi-periodic signal where in each 14 seconds to risk a pico-voltage* Vp, *when the dilaton spins to a velocity of 12 rpm, where is observed that their amplitude is very major than the subsequent amplitudes until to risk a new maximum* Vp*: (c). The frequency ω*, *is maintained in a 227 to 236 kHz range. In the sensing process was realized a closer of the energy spectra 1 sec/div (figure a)) to 5 sec/div (figures b, c) and 10sc/div (figure d)) to 50 mV for the three cases. We consider a conditioning of the signal through the operational amplifier of instrumentation AD620, which permit us view an output voltage proportional to 1.4 mV per*

of constant torsion (**Figures 3c** and **4**) [11].

*Recent Advances in Numerical Simulations*

the interaction between energy and space.

by the force *δ*ð Þ t1, t2 (see the **Figure 4**).

**Figure 3.**

*Gauss detected.*

**32**

*(a). The screw effect in the space–time. (b). Mono-pole of electromagnetic field. This represents only* δð Þ t1, t2 *, of a field source in the space–time, whose torsion is established in the Z-axis. (c). 2-dimensional surface shows the field source with torsion effect waving the 2-dimensional surface around the source.*
