Draft:Alena Tensor

The Alena Tensor is a class of energy-momentum tensors that allows for equivalent description and analysis of physical systems in flat spacetime (with fields and forces) and in curved spacetime (using Einstein Field Equations). This approach assumes that the metric tensor is not a feature of spacetime, but only a method of its mathematical description.

The use of the Alena Tensor allows for transformation of equations in such a way that they can be used in curvilinear (General Relativity), classical (fields and forces) and quantum (Quantum Mechanics and Quantum Field Theory) descriptions as well as in Cosmology, Continuum Mechanics and Thermodynamics. Due to this property, the Alena Tensor is a useful tool for studying unification problems, quantum gravity and many other applications in physics.

General description of the concept of Alena Tensor

To understand the construction of the Alena Tensor, it is easiest to recreate the reasoning that led to its creation using the example of the electromagnetic field. Let us consider the energy-momentum tensor in flat spacetime for a physical system with an electromagnetic field in the following form:

$T^{\alpha \beta }=\varrho U^{\alpha }{U}^{\beta }-\frac{1}{{\mu }_r}{\mathrm{{\rm Y}}}^{\alpha \beta }$

where:

$T^{\alpha \beta }$ - energy-momentum tensor for a physical system

$\varrho$ - density of matter

$U^{\alpha }$ - four-velocity

${\mu }_r$ - relative permeability

${\mathrm{{\rm Y}}}^{\alpha \beta }$ - energy-momentum tensor for the electromagnetic field

The density of four-forces acting in a physical system can be considered as a four-divergence. Let us therefore denote the four-force densities occurring in the system:

${\partial }_{\alpha }\varrho U^{\alpha }{U}^{\beta }=f^{\beta }$

is the density of the total four-force acting on matter.

Forces related to electromagnetic field energy-momentum tensor are:

${\partial }_{\alpha }\frac{1}{{\mu }_r}{\mathrm{{\rm Y}}}^{\alpha \beta }=\frac{1}{{\mu }_r}{f}_{em}^{\beta }+f_u^{\beta }$

where:

$\frac{1}{{\mu }_r}{f}_{em}^{\beta }=\frac{1}{{\mu }_r}{\partial }_{\alpha }{\mathrm{{\rm Y}}}^{\alpha \beta }$ - density of the electromagnetic four-force, corrected by a factor $\frac{1}{{\mu }_r}$

$f_u^{\beta }={\mathrm{{\rm Y}}}^{\alpha \beta }{\partial }_{\alpha }\frac{1}{{\mu }_r}$ - the density of some yet unknown four-force

Let us further assume that the forces balance, which will provide a vanishing four-divergence of the energy-momentum tensor for the entire system:

$0={\partial }_{\alpha }{T}^{\alpha \beta }=f^{\beta }-\frac{1}{{\mu }_r}{f}_{em}^{\beta }-f_u^{\beta }$

Now let's do the following reasoning. If we wanted to use the above tensor for a curvilinear description, which would describe the same physical system but curvilinearly, then in curved spacetime the forces $\frac{1}{{\mu }_r}{f}_{em}^{\beta }$ and $f_u^{\beta }$ can be replaced by Christoffel symbols of the second kind. The vanishing four-divergence in the curvilinear system makes these forces unnecessary in the equation.

This means that the entire field term can simply disappear from the equation, because instead of a field and the forces associated with it, there will be curvature. This would mean, although it is not currently the dominant view in physics, that in curved spacetime $\frac{1}{{\mu }_r}{\mathrm{{\rm Y}}}^{\alpha \beta }=0$.

Let us take a closer look at the energy-momentum tensor of the electromagnetic field ${\mathrm{{\rm Y}}}^{\alpha \beta }$, generalizing its classical form and making the following substitution

${\mathrm{{\rm Y}}}^{\alpha \beta }={\mathrm{\Lambda }}_{\rho }{g}^{\alpha \beta }-\frac{1}{{\mu }_o}{F}^{\alpha \delta }{g}_{\delta \gamma }{F}^{\beta \gamma }={\mathrm{\Lambda }}_{\rho }(g^{\alpha \beta }-\xi h^{\alpha \beta })$

where:

$F^{\alpha \delta }$ - electromagnetic field strength tensor

${\mu }_o$ - vacuum magnetic permeability

$g^{\alpha \beta }$ - metric tensor of the considered spacetime

${\mathrm{\Lambda }}_{\rho }=\frac{1}{4{\mu }_o}{F}^{\alpha \mu }{g}_{\mu \gamma }{F}^{\beta \gamma }{g}_{\alpha \beta }$ - invariant of the electromagnetic field tensor

$\xi =\frac{4}{g_{\mu \nu }{h}^{\mu \nu }}$ - parameter that ensures the disappearing trace of the tensor ${\mathrm{{\rm Y}}}^{\alpha \beta }$

$h^{\alpha \beta }=2\frac{F^{\alpha \gamma }{F}_{\gamma }^{\beta }}{\sqrt{F^{\alpha \gamma }{F}_{\gamma }^{\mu }{F}_{\alpha \xi }{F}_{\mu }^{\xi }}}$ - metric tensor of spacetime for which ${\mathrm{{\rm Y}}}^{\alpha \beta }$ vanishes.

In this way we obtain a generalized description of the tensor ${\mathrm{{\rm Y}}}^{\alpha \beta }$, which has the following properties:

• in flat spacetime ${\mathrm{{\rm Y}}}^{\alpha \beta }$ is the usual, classical energy-momentum tensor of the electromagnetic field

• its trace vanishes in any spacetime, regardless of the considered metric tensor $g^{\alpha \beta }$

• for spacetimes for which $g^{\alpha \beta }=h^{\alpha \beta }$ the entire tensor ${\mathrm{{\rm Y}}}^{\alpha \beta }$ vanishes

• $h^{\alpha \beta }{h}_{\alpha \beta }=4$ which is expected property of the metric tensor

The original definition of $h^{\alpha \beta }$ is more complex (it uses the metric tensor $g^{\alpha \beta }$) but it can be simplified to the example given here, following the author's original reasoning. Assuming that there is a curved spacetime with the metric tensor $h^{\alpha \beta }$, it cannot depend on the $g^{\alpha \beta }$ adopted for analysis. For this reason, the value of $h^{\alpha \beta }$ can be defined in flat spacetime and does not change, irrespective of the $g^{\alpha \beta }$ adopted.

In the above manner we obtained the Alena Tensor $T^{\alpha \beta }$ in the form:

$T^{\alpha \beta }=\varrho U^{\alpha }{U}^{\beta }-\frac{1}{{\mu }_r}{\mathrm{\Lambda }}_{\rho }(g^{\alpha \beta }-\xi h^{\alpha \beta })$

with the yet unknown $\frac{1}{{\mu }_r}$ for which in curved spacetime the energy-momentum tensor of the field ${\mathrm{{\rm Y}}}^{\alpha \beta }$ vanishes. This is because by substituting $g^{\alpha \beta }=h^{\alpha \beta }$ we simply get $T^{\alpha \beta }=\varrho U^{\alpha }{U}^{\beta }$ in curved spacetime.

The reasoning carried out above for electromagnetism is universal and allows to consider the Alena Tensor also for energy-momentum tensors associated with other fields. This leads to obtaining an energy-momentum tensor that can be considered both in flat spacetime and in curved spacetime.

Connection with Continuum Mechanics and General Relativity

To make the Alena Tensor consistent with Continuum Mechanics in flat spacetime, it is enough to adopt the following substitution

$\frac{1}{{\mu }_r}=\frac{p}{{\mathrm{\Lambda }}_{\rho }}$

where

p is the pressure in the system and is equal to

$p=\varrho c^2+{\mathrm{\Lambda }}_{\rho }$

and where c is the speed of light in a vacuum.

We can now introduce an additional tensor

${\mathrm{\Pi }}^{\alpha \beta }=-c^2\varrho \xi h^{\alpha \beta }$

which will play the role of deviatoric stress tensor. This allows us to write the Alena Tensor in flat spacetime as follows:

$T^{\alpha \beta }=\varrho U^{\alpha }{U}^{\beta }-p{\eta }^{\alpha \beta }-{\mathrm{\Pi }}^{\alpha \beta }+{\mathrm{\Lambda }}_{\rho }\xi h^{\alpha \beta }$

where ${\eta }^{\alpha \beta }$ is the metric tensor of flat Minkowski spacetime.

The vanishing four-divergence of $T^{\alpha \beta }$ leads to the equation:

$f^{\alpha }={\partial }^{\alpha }p+{\partial }_{\beta }{\mathrm{\Pi }}^{\alpha \beta }+f_{em}^{\alpha }$

which expresses the relativistic equivalence of Cauchy momentum equation (convective form) in which only $f_{em}$ appears as a body force.

Importantly, the above substitution also provides a connection to General Relativity in curved spacetime. For this purpose, let us introduce the following tensors, which can be analyzed in both flat and curved spacetime:

$R^{\alpha \beta }=2\varrho U^{\alpha }{U}^{\beta }-p{g}^{\alpha \beta }$

$R=R^{\alpha \beta }{g}_{\alpha \beta }=-2p-2{\mathrm{\Lambda }}_{\rho }$

$G^{\alpha \beta }=R^{\alpha \beta }-\frac{1}{2}R\xi h^{\alpha \beta }$

This allows us to rewrite the Alena Tensor as follows:

$G^{\alpha \beta }-{\mathrm{\Lambda }}_{\rho }{g}^{\alpha \beta }=2T^{\alpha \beta }+\varrho c^2(g^{\alpha \beta }-\xi h^{\alpha \beta })$

Analyzing the above equation in curved spacetime ($g^{\alpha \beta }=h^{\alpha \beta }$), we obtain its simplification to the form:

$G^{\alpha \beta }-{\mathrm{\Lambda }}_{\rho }{g}^{\alpha \beta }=2T^{\alpha \beta }$

which can be interpreted as the main equation of General Relativity up to the constant $\frac{4\pi G}{c^4}$

Because in curved spacetime we get

$G^{\alpha \beta }=R^{\alpha \beta }-\frac{1}{2}R{g}^{\alpha \beta }$

this means that $G^{\alpha \beta }$ and $R^{\alpha \beta }$ can be interpreted, respectively, as Einstein curvature tensor and Ricci tensor both with an accuracy of $\frac{4\pi G}{c^4}$ constant.

Analyzing the $G^{\alpha \beta }$ tensor in flat spacetime one can see that it is related to the non-body forces seen in the description of the Cauchy momentum equation:

${\partial }_{\beta }{G}^{\alpha \beta }={\partial }^{\alpha }p+{\partial }_{\beta }{\mathrm{\Pi }}^{\alpha \beta }$

This means that in the Alena Tensor analysis method, gravity is not a body force.

Description of gravity in the Alena Tensor

In flat spacetime, based on the conclusions from , the four-force densities associated with the tensor $G^{\alpha \beta }$ can be also represented as follows:

${\partial }_{\beta }{G}^{\alpha \beta }=f_{gr}^{\alpha }+f_{rr}^{\alpha }$

where these four-force densities can be represented as follows:

$f_{rr}^{\alpha }=(\frac{1}{{\mu }_r}-1)f_{em}^{\alpha }$ - density of the radiation reaction four-force

$f_{gr}^{\alpha }=\varrho (\frac{d\varphi }{d\tau }{U}^{\alpha }-c^2{\partial }^{\alpha }\varphi )$ - density of the four-force related to gravity,

where:

$\varphi =-ln({\mu }_r)$ - is related to the effective potential in the system with gravity

$\tau$ - proper time

It can be calculated that $f_{gr}^{\alpha }$ vanishes in two cases:

$-\overrightarrow{u}=c\frac{\mathrm{\nabla }\varphi }{{\partial }^0\varphi }$ - which turns out to be the case of free fall

${\partial }^{\alpha }\varphi =0$ - which occurs in the case of circular orbits

Neglecting the electromagnetic force and the radiation reaction force, using the above equation one can reproduce the motion of bodies in the effective potential obtained from the solutions of General Relativity. Such a description has already been done for the Schwarzschild metric.

In the above description, gravity is not a force, because the above description is based on an effective potential. However, one can see a similarity to Newton's classical equations for the case of a stationary observer who is at a constant distance from the source of gravity.

For such an observer $f_{gr}^{\alpha }=-\varrho c^2{\partial }^{\alpha }\varphi$ which can be approximated by the gradient of Newtonian potential. However, even in this case this force has the opposite sign than in the classical description of gravity and represents a force that must exist for a stationary observer suspended above the source of gravity to stay in one place (e.g. a rocket suspended above the ground in one place stays in one place thanks to the thrust of its engines).

The description of gravity obtained in this way is surprisingly consistent with current knowledge, despite the fact that gravity itself in this description is not a force, and the resulting force $f_{gr}^{\alpha }$ is not a body force.

Quantum image resulting from the use of the Alena Tensor

The biggest surprise resulting from the use of the Alena Tensor is the quantum image obtained from it. The very fact of obtaining quantum equations is surprising in light of the common belief in the world of physics about the extraordinary difficulty of including gravity in the quantum description and even, as some authors claim, about the impossibility of such an option.

Even more surprisingly, the quantum equations obtained from the Alena Tensor lead to the conclusion that gravity and the radiation reaction force have always been present in Quantum Mechanics and Quantum Field Theory. This conclusion follows from the fact that the quantum equations obtained from the Alena Tensor are the three main quantum equations currently used:

1. Simplified Dirac equation for Quantum Electrodynamics (QED):

${ℒ}_{QED}=\frac{1}{4{\mu }_o}{𝔽}^{\alpha \beta }{𝔽}_{\alpha \beta }=\frac{1}{2{\mu }_o}{𝔽}^{0\gamma }{\partial }^0{A}_{\gamma }=\frac{1}{2}\overline{\psi }(i\hslash cD-m{c}^2)\psi$

2. Klein-Gordon equation (fully consistent)

3. The relativistic equivalent of the Schrödinger equation

$ic\hslash {\partial }^0\psi =-\frac{{\hslash }^2}{m(\gamma +\frac{1}{\gamma })}{\mathrm{\nabla }}^2\psi +cq{\hat{A}}^0\psi$

which in the limit of small energies and particle speeds u < 0.4c (Lorentz factor $\gamma \approx 1$) turns into the classical Schrödinger equation considered for charged particles.

Other properties of Alena Tensor and further research

The use of the Alena Tensor leads to many further conclusions and opens up many new research possibilities. These are, for example:

• The relationship of the energy of charged particles with the magnetic moment has been obtained

• The presence of the energy density of the magnetic moment (related to particles) in the Poynting four-vector has been obtained as an additional component complementing the energy density related to light

• The relationship of the field tensor invariant with the cosmological constant has been obtained

• Exclusion of black hole singularities resulting from the presence of a radiation reaction force that prevents non-physical phenomena

• A new interpretation of the free particle in the Dirac equation is possible

• It is possible to study issues related to Dark Matter based on a new mathematical apparatus, which may lead to an explanation of its nature

• It is possible to simplify the description of cosmological phenomena

• and many other new areas of further research.

Despite the very promising results and conclusions from the use of the Alena Tensor, it is currently impossible to say whether the interpretation of physical phenomena resulting from it is fully correct. The Alena Tensor itself seems to be a mature tool, leading to compatibility or the possibility of compatibility with currently used physical descriptions and allowing for many further studies.

Paradoxically, the agreement of Alena Tensor's results with current knowledge makes it difficult to verify both theoretically and experimentally. The decisive argument, as in the case of any physical theory, will be the experimental results.