# Acceleration tensor

The acceleration tensor is an antisymmetric tensor describing the four-acceleration of particles and consisting of six components. The tensor components are at the same time the components of two three-dimensional vectors – the acceleration field strength and the solenoidal acceleration vector. The acceleration field stress-energy tensor, the acceleration field equations and the four-force density are determined with the help of the acceleration tensor. Acceleration field in matter is a component of general field.

## Definition

The expression for the acceleration tensor can be found in papers by Sergey Fedosin, [1] where the tensor is defined using the 4-curl:

${\displaystyle u_{\mu \nu }=\nabla _{\mu }u_{\nu }-\nabla _{\nu }u_{\mu }={\frac {\partial u_{\nu }}{\partial x^{\mu }}}-{\frac {\partial u_{\mu }}{\partial x^{\nu }}}.\qquad \qquad (1)}$

Here the acceleration 4-potential ${\displaystyle ~u_{\mu }}$ is given by:

${\displaystyle ~u_{\mu }=\left({\frac {\vartheta }{c}},-\mathbf {U} \right),}$

where ${\displaystyle ~\vartheta }$ is the scalar potential, ${\displaystyle ~\mathbf {U} }$ is the vector potential of acceleration field, ${\displaystyle ~c}$ is the speed of light.

## Expression for the components

The acceleration field strength and the solenoidal acceleration vector are found with the help of (1):

${\displaystyle ~S_{i}=c(\partial _{0}u_{i}-\partial _{i}u_{0}),}$
${\displaystyle ~N_{k}=\partial _{i}u_{j}-\partial _{j}u_{i},}$ and in the second expression three numbers ${\displaystyle ~i{,}j{,}k}$ are composed of non-recurring sets 1,2,3; or 2,1,3; or 3,2,1 etc.

In vector notation we can write:

${\displaystyle ~\mathbf {S} =-\nabla \vartheta -{\frac {\partial \mathbf {U} }{\partial t}},}$
${\displaystyle ~\mathbf {N} =\nabla \times \mathbf {U} .}$

The acceleration tensor consists of the components of these vectors:

${\displaystyle ~u_{\mu \nu }={\begin{vmatrix}0&{\frac {S_{x}}{c}}&{\frac {S_{y}}{c}}&{\frac {S_{z}}{c}}\\-{\frac {S_{x}}{c}}&0&-N_{z}&N_{y}\\-{\frac {S_{y}}{c}}&N_{z}&0&-N_{x}\\-{\frac {S_{z}}{c}}&-N_{y}&N_{x}&0\end{vmatrix}}.}$

The transition to the acceleration tensor with contravariant indices is carried out by multiplying by double metric tensor:

${\displaystyle ~u^{\alpha \beta }=g^{\alpha \nu }g^{\mu \beta }u_{\mu \nu }.}$

In the special relativity, this tensor has the form:

${\displaystyle ~u^{\alpha \beta }={\begin{vmatrix}0&-{\frac {S_{x}}{c}}&-{\frac {S_{y}}{c}}&-{\frac {S_{z}}{c}}\\{\frac {S_{x}}{c}}&0&-N_{z}&N_{y}\\{\frac {S_{y}}{c}}&N_{z}&0&-N_{x}\\{\frac {S_{z}}{c}}&-N_{y}&N_{x}&0\end{vmatrix}}.}$

For the vectors, related to the specific point particle, we can write:

${\displaystyle ~\mathbf {S} =-c^{2}\nabla \gamma -{\frac {\partial (\gamma \mathbf {v} )}{\partial t}},}$
${\displaystyle ~\mathbf {N} =\nabla \times (\gamma \mathbf {v} ),}$

where ${\displaystyle ~\gamma ={\frac {1}{\sqrt {1-{v^{2} \over c^{2}}}}}}$, ${\displaystyle ~\mathbf {v} }$ is the velocity of the particle.

To transform the components of the acceleration tensor from one inertial system to another we must take into account the transformation rule for tensors. If the reference frame K' moves with an arbitrary constant velocity ${\displaystyle ~\mathbf {V} }$ with respect to the fixed reference frame K, and the axes of the coordinate systems are parallel to each other, the acceleration field strength and the solenoidal acceleration vector are transformed as follows:

${\displaystyle \mathbf {S} ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {S} )+{\frac {1}{\sqrt {1-{V^{2} \over c^{2}}}}}\left(\mathbf {S} -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {S} )+[\mathbf {V} \times \mathbf {N} ]\right),}$
${\displaystyle \mathbf {N} ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {N} )+{\frac {1}{\sqrt {1-{V^{2} \over c^{2}}}}}\left(\mathbf {N} -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {N} )-{\frac {1}{c^{2}}}[\mathbf {V} \times \mathbf {S} ]\right).}$

## Properties of the tensor

• ${\displaystyle ~u_{\mu \nu }}$ is an antisymmetric tensor of rank 2, from this the condition follows: ${\displaystyle ~u_{\mu \nu }=-u_{\nu \mu }}$. Three of the six independent components of the acceleration tensor are associated with the components of the acceleration field strength ${\displaystyle ~\mathbf {S} }$, and the other three – with the components of the solenoidal acceleration vector ${\displaystyle ~\mathbf {N} }$. Due to antisymmetry such an invariant as the contraction of the tensor with the metric tensor vanishes: ${\displaystyle ~g^{\mu \nu }u_{\mu \nu }=u_{\mu }^{\mu }=0}$.
• Contraction of the tensor with itself ${\displaystyle u_{\mu \nu }u^{\mu \nu }}$ is an invariant, and contraction of the tensor product with Levi-Civita symbol as ${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }u_{\mu \nu }u_{\sigma \rho }}$ is a pseudoscalar invariant. These invariants in the special theory of relativity can be expressed as follows:
${\displaystyle u_{\mu \nu }u^{\mu \nu }=-{\frac {2}{c^{2}}}(S^{2}-c^{2}N^{2})=inv,}$
${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }u_{\mu \nu }u_{\sigma \rho }=-{\frac {2}{c}}\left(\mathbf {S} \cdot \mathbf {N} \right)=inv.}$
• Determinant of the tensor is also a Lorentz invariant:
${\displaystyle \det \left(u_{\mu \nu }\right)={\frac {4}{c^{2}}}\left(\mathbf {S} \cdot \mathbf {N} \right)^{2}.}$

## Acceleration field

The acceleration field equations are written with the acceleration tensor:

${\displaystyle \nabla _{\sigma }u_{\mu \nu }+\nabla _{\mu }u_{\nu \sigma }+\nabla _{\nu }u_{\sigma \mu }={\frac {\partial u_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial u_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial u_{\sigma \mu }}{\partial x^{\nu }}}=0.\qquad \qquad (2)}$
${\displaystyle ~\nabla _{\nu }u^{\mu \nu }=-{\frac {4\pi \eta }{c^{2}}}J^{\mu },\qquad \qquad (3)}$

where ${\displaystyle J^{\mu }=\rho _{0}u^{\mu }}$ is the mass 4-current, ${\displaystyle \rho _{0}}$ is the mass density in the comoving reference frame, ${\displaystyle u^{\mu }}$ is the 4-velocity, ${\displaystyle ~\eta }$ is a constant determined in each task.

Instead of (2) it is possible to use the expression:

${\displaystyle ~\varepsilon ^{\mu \nu \sigma \rho }{\frac {\partial u_{\mu \nu }}{\partial x^{\sigma }}}=0.}$

Equation (2) is satisfied identically, which is proved by substituting into it the definition for the acceleration tensor according to (1). If in (2) we insert the tensor components ${\displaystyle u_{\mu \nu }}$, this leads to two vector equations:

${\displaystyle ~\nabla \times \mathbf {S} =-{\frac {\partial \mathbf {N} }{\partial t}},\qquad \qquad (4)}$
${\displaystyle ~\nabla \cdot \mathbf {N} =0.\qquad \qquad (5)}$

According to (5), the solenoidal acceleration vector has no sources since its divergence vanishes. From (4) it follows that the time variation of the solenoidal acceleration vector leads to emerging of the curl of the acceleration field strength.

Equation (3) relates the acceleration field to its source in the form of the mass 4-current. In Minkowski space of the special theory of relativity the equation form is simplified and becomes:

${\displaystyle ~\nabla \cdot \mathbf {S} =4\pi \eta \rho ,}$
${\displaystyle ~\nabla \times \mathbf {N} ={\frac {1}{c^{2}}}\left(4\pi \eta \mathbf {J} +{\frac {\partial \mathbf {S} }{\partial t}}\right),}$

where ${\displaystyle ~\rho }$ is the density of moving mass, ${\displaystyle ~\mathbf {J} }$ is the density of mass current.

According to the first of these equations, the acceleration field strength is generated by the mass density, and according to the second equation the mass current or the change in time of the acceleration field strength generate the circular field of the solenoidal acceleration vector.

From (3) and (1) can be obtained continuity equation:

${\displaystyle ~R_{\mu \alpha }u^{\mu \alpha }={\frac {4\pi \eta }{c^{2}}}\nabla _{\alpha }J^{\alpha }.}$

This equation means that thanks to the curvature of space-time when the Ricci tensor ${\displaystyle ~R_{\mu \alpha }}$ is non-zero, the acceleration tensor ${\displaystyle ~u^{\mu \alpha }}$ is a possible source of divergence of mass 4-current. If space-time is flat, as in Minkowski space, the left side of the equation is set to zero, the covariant derivative becomes the 4-gradient and remains the following:

${\displaystyle ~\partial _{\alpha }J^{\alpha }={\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}$

## Covariant theory of gravitation

### Action and Lagrangian

The total Lagrangian for the substance in the gravitational and electromagnetic fields includes the acceleration tensor and is part of the action function: [1]

${\displaystyle ~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}D_{\mu }J^{\mu }+{\frac {c}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }-{\frac {1}{c}}A_{\mu }j^{\mu }-{\frac {c\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }-}$
${\displaystyle ~-{\frac {1}{c}}u_{\mu }J^{\mu }-{\frac {c}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }-{\frac {1}{c}}\pi _{\mu }J^{\mu }-{\frac {c}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}d\Sigma ,}$

where ${\displaystyle ~L}$ is the Lagrangian, ${\displaystyle ~dt}$ is the differential of the coordinate time, ${\displaystyle ~k}$ is a certain coefficient, ${\displaystyle ~R}$ is the scalar curvature, ${\displaystyle ~\Lambda }$ is the cosmological constant, which is a function of the system, ${\displaystyle ~c}$ is the speed of light as a measure of the propagation speed of electromagnetic and gravitational interactions, ${\displaystyle ~D_{\mu }}$ is the gravitational four-potential, ${\displaystyle ~G}$ is the gravitational constant, ${\displaystyle ~\Phi _{\mu \nu }}$ is the gravitational tensor, ${\displaystyle ~A_{\mu }}$ is the electromagnetic 4-potential, ${\displaystyle ~j^{\mu }}$ is the electromagnetic 4-current, ${\displaystyle ~\varepsilon _{0}}$ is the electric constant, ${\displaystyle ~F_{\mu \nu }}$ is the electromagnetic tensor, ${\displaystyle ~u_{\mu }}$ is the 4-potential of acceleration field, ${\displaystyle ~\eta }$ and ${\displaystyle ~\sigma }$ are some constants, ${\displaystyle ~u_{\mu \nu }}$ is the acceleration tensor, ${\displaystyle ~\pi _{\mu }}$ is the 4-potential of the pressure field, ${\displaystyle ~f_{\mu \nu }}$ is the pressure field tensor, ${\displaystyle ~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}}$ is the invariant 4-volume, ${\displaystyle ~{\sqrt {-g}}}$ is the square root of the determinant ${\displaystyle ~g}$ of the metric tensor, taken with a negative sign, ${\displaystyle ~dx^{1}dx^{2}dx^{3}}$ is the product of differentials of the spatial coordinates.

The variation of the action function by 4-coordinates leads to the equation of motion of the substance unit in the gravitational and electromagnetic fields and the pressure field:

${\displaystyle ~\rho _{0}a_{\beta }=\rho _{0}{\frac {Du_{\beta }}{D\tau }}=\rho _{0}u^{k}\nabla _{k}u_{\beta }=\rho _{0}{\frac {du_{\beta }}{d\tau }}-\rho _{0}\Gamma _{k\beta }^{s}u^{k}u_{s}=\Phi _{\beta \sigma }\rho _{0}u^{\sigma }+F_{\beta \sigma }\rho _{0q}u^{\sigma }+f_{\beta \sigma }\rho _{0}u^{\sigma },}$

where ${\displaystyle ~a_{\beta }}$ is the four-acceleration with the covariant index, the operator of proper-time-derivative with respect to the proper time ${\displaystyle ~\tau }$ is used, the first term on the right is the gravitational force density, expressed with the help of the gravitational field tensor, the second term is the Lorentz electromagnetic force density for the charge density ${\displaystyle ~\rho _{0q}}$ measured in the comoving reference frame, and the last term sets the pressure force density, and the relation holds:

${\displaystyle ~\rho _{0}a_{\beta }=\nabla ^{k}B_{\beta k}=-u_{\beta k}J^{k}.}$

In the special theory of relativity, this relation is simplified and can be written in the form of two expressions:

${\displaystyle ~\rho _{0}a_{0}=\nabla ^{k}B_{0k}=-u_{0k}J^{k}=-{\frac {\gamma \rho _{0}}{c}}(\mathbf {S} \cdot \mathbf {v} ),}$
${\displaystyle ~\rho _{0}a_{i}=\nabla ^{k}B_{ik}=-u_{ik}J^{k}=\gamma \rho _{0}(\mathbf {S} +[\mathbf {v} \times \mathbf {N} ]),}$

where ${\displaystyle ~i=1{,}2{,}3.}$

If we vary the action function by the acceleration 4-potential, we obtain the acceleration field equation (3).

### Acceleration field stress-energy tensor

With the help of the acceleration tensor in the covariant theory of gravitation the acceleration stress-energy tensor is constructed:

${\displaystyle ~B^{ik}={\frac {c^{2}}{4\pi \eta }}\left(-g^{im}u_{nm}u^{nk}+{\frac {1}{4}}g^{ik}u_{mr}u^{mr}\right)}$.

The covariant derivative of the acceleration field stress-energy tensor determines the four-force density:

${\displaystyle ~f^{\alpha }=\nabla _{\beta }B^{\alpha \beta }=-u_{k}^{\alpha }J^{k}=-\rho _{0}u_{k}^{\alpha }u^{k}=\rho _{0}a^{\alpha }=\rho _{0}{\frac {Du^{\alpha }}{D\tau }}.\qquad \qquad (6)}$

### Generalized velocity and Hamiltonian

The covariant 4-vector of generalized velocity is given by:

${\displaystyle ~s_{\mu }=u_{\mu }+D_{\mu }+{\frac {\rho _{0q}}{\rho _{0}}}A_{\mu }+\pi _{\mu }.}$

With regard to the generalized 4-velocity, the Hamiltonian contains the acceleration tensor and has the form: ${\displaystyle ~H=\int {(s_{0}J^{0}-{\frac {c^{2}}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }+{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }+{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},}$

where ${\displaystyle ~s_{0}}$ and ${\displaystyle ~J^{0}}$ are the time components of the 4-vectors ${\displaystyle ~s_{\mu }}$ and ${\displaystyle ~J^{\mu }}$.

In the reference frame that is fixed relative to the system's center of mass, the Hamiltonian will determine the invariant energy of the system.

## Special theory of relativity

Studying Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form ${\displaystyle ~F^{\mu }={\frac {dp^{\mu }}{d\tau }}.}$ [2]

This led them to conclude that the four-acceleration must be expressed with the help of some antisymmetric tensor ${\displaystyle ~A_{\nu }^{\mu }}$:

${\displaystyle ~c{\frac {du^{\mu }}{d\tau }}=A_{\nu }^{\mu }u^{\nu }.}$

Based on the analysis of different types of motion, they estimated the required values of the components of the acceleration tensor, thereby giving this tensor indirect definition.

From comparison with (6) it follows that the tensor ${\displaystyle ~A_{\nu }^{\mu }}$ coincides with the acceleration tensor ${\displaystyle ~u_{k}^{\alpha }}$ up to a sign and a constant factor.

Mashhoon and Muench considered transformation of inertial reference frames, associated with the accelerated reference frame, and came to the relation: [3]

${\displaystyle ~c{\frac {d\lambda _{\alpha }}{d\tau }}=\Phi _{\alpha }^{\beta }\lambda _{\beta }.}$

The tensor ${\displaystyle ~\Phi _{\alpha }^{\beta }}$ has the same properties as the acceleration tensor ${\displaystyle ~u_{\alpha }^{\beta }.}$

## Other theories

In the articles [4] [5] [6] devoted to the modified Newtonian dynamics (MOND), in the tensor-vector-scalar gravity appear scalar function ${\displaystyle ~\psi }$ or ${\displaystyle ~\phi }$, that defines a scalar field, and 4-vector ${\displaystyle {\mathfrak {U}}_{\mu }}$ or ${\displaystyle A_{\mu }}$, and 4-tensor ${\displaystyle {\mathfrak {U}}_{[\mu \nu ]}}$ or ${\displaystyle F_{ab}={\frac {\partial A_{b}}{\partial x^{a}}}-{\frac {\partial A_{a}}{\partial x^{b}}}.}$

The analysis of these values in the corresponding Lagrangian demonstrates that scalar function ${\displaystyle ~\psi }$ or ${\displaystyle ~\phi }$ correspond to scalar potential ${\displaystyle ~\vartheta }$ of the acceleration field; 4-vector ${\displaystyle {\mathfrak {U}}_{\mu }}$ or ${\displaystyle A_{\mu }}$ correspond to 4-potential ${\displaystyle ~u_{\mu }}$ of the acceleration field; 4-tensor ${\displaystyle {\mathfrak {U}}_{[\mu \nu ]}}$ or ${\displaystyle F_{ab}}$ correspond to acceleration tensor ${\displaystyle u_{\mu \nu }}$.

As it is known, the acceleration field is not intended to explain the accelerated motion, but for its accurate description. In this case, it can be assumed that the tensor-vector-scalar theories cannot pretend to explain the rotation curves of galaxies. At best, they can only serve to describe the motion, for example to describe the rotation of stars in galaxies and the rotation of galaxies in clusters of galaxies.