# Gravitational tensor

The gravitational tensor or gravitational field tensor, (sometimes called the gravitational field strength tensor) is an antisymmetric tensor, combining two components of gravitational field – the gravitational field strength and the gravitational torsion field – into one. It is used to describe the gravitational field of an arbitrary physical system and for invariant formulation of gravitational equations in the covariant theory of gravitation. The gravitational field is a component of general field.

## Definition

Tensor of gravitational field is defined by the gravitational four-potential of gravitational field ${\displaystyle ~D_{\mu }}$ as follows: [1] [2]

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

Due to the antisymmetry of this formula the difference of two covariant derivatives is equal to the difference between the two partial derivatives with respect to the 4-coordinates.

## Expression for the components

If we consider the definition of the 4-potential of gravitational field:

${\displaystyle ~D_{\mu }=\left({\frac {\psi }{c_{g}}},-\mathbf {D} \right),}$

where ${\displaystyle ~\psi }$ is the scalar potential, ${\displaystyle ~\mathbf {D} }$ is the vector potential of the gravitational field, ${\displaystyle ~c_{g}}$ is the propagation speed of gravitational effects (speed of gravity),

and if we introduce for Cartesian coordinates ${\displaystyle ~(c_{g}t,x,y,z)}$ the gravitational field strengths by the rules:

${\displaystyle ~\mathbf {\Gamma } =-\nabla \psi -{\frac {\partial \mathbf {D} }{\partial t}},}$
${\displaystyle ~\mathbf {\Omega } =\nabla \times \mathbf {D} ,}$

where ${\displaystyle ~\mathbf {\Gamma } }$ is the gravitational field strength or gravitational acceleration, ${\displaystyle ~\mathbf {\Omega } }$ is the torsion field,

then the covariant components of the gravitational field tensor according to (1) will be:

${\displaystyle ~\Phi _{\mu \nu }={\begin{vmatrix}0&{\frac {\Gamma _{x}}{c_{g}}}&{\frac {\Gamma _{y}}{c_{g}}}&{\frac {\Gamma _{z}}{c_{g}}}\\-{\frac {\Gamma _{x}}{c_{g}}}&0&-\Omega _{z}&\Omega _{y}\\-{\frac {\Gamma _{y}}{c_{g}}}&\Omega _{z}&0&-\Omega _{x}\\-{\frac {\Gamma _{z}}{c_{g}}}&-\Omega _{y}&\Omega _{x}&0\end{vmatrix}}.}$

According to the rules of tensor algebra, raising (lowering) of the tensors’ indices, that is the transition from the covariant components to the mixed and contravariant components of tensors and vice versa, is done by means of the metric tensor ${\displaystyle ~g_{\mu \nu }}$. In particular ${\displaystyle \Phi _{\alpha }^{\mu }=g^{\mu \nu }\Phi _{\nu \alpha }}$, as well as ${\displaystyle ~\Phi ^{\alpha \beta }=g^{\alpha \nu }g^{\mu \beta }\Phi _{\mu \nu }.}$

In Minkowski space the metric tensor turns into the tensor ${\displaystyle ~\eta _{\mu \nu }}$, which does not depend on the coordinates and time. In this space, which is used in the special relativity, the contravariant components of the gravitational field tensor are as follows:

${\displaystyle ~\Phi ^{\mu \nu }={\begin{vmatrix}0&-{\frac {\Gamma _{x}}{c_{g}}}&-{\frac {\Gamma _{y}}{c_{g}}}&-{\frac {\Gamma _{z}}{c_{g}}}\\{\frac {\Gamma _{x}}{c_{g}}}&0&-\Omega _{z}&\Omega _{y}\\{\frac {\Gamma _{y}}{c_{g}}}&\Omega _{z}&0&-\Omega _{x}\\{\frac {\Gamma _{z}}{c_{g}}}&-\Omega _{y}&\Omega _{x}&0\end{vmatrix}}.}$

Since the vectors of gravitational field strength and torsion field are the components of the gravitational field tensor, they are transformed not as vectors, but as the components of the tensor of the type (0,2). The law of transformation of these vectors in the transition from the fixed reference frame K into the reference frame K', moving at the velocity V along the axis X, has the following form:

${\displaystyle \Gamma _{x}^{\prime }=\Gamma _{x},~~~\Gamma _{y}^{\prime }={\frac {\Gamma _{y}-V\Omega _{z}}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}},~~~\Gamma _{z}^{\prime }={\frac {\Gamma _{z}+V\Omega _{y}}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}},}$
${\displaystyle \Omega _{x}^{\prime }=\Omega _{x},~~~\Omega _{y}^{\prime }={\frac {\Omega _{y}+V\Gamma _{z}/c_{g}^{2}}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}},~~~\Omega _{z}^{\prime }={\frac {\Omega _{z}-V\Gamma _{y}/c_{g}^{2}}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}}.}$

In the more general case where the velocity ${\displaystyle ~\mathbf {V} }$ of the reference frame K’ relative to the frame K is aimed in any direction, and the axis of the coordinate systems parallel to each other, the gravitational field strength and the torsion field are converted as follows:

${\displaystyle \mathbf {\Gamma } ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {\Gamma } )+{\frac {1}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}}\left(\mathbf {\Gamma } -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {\Gamma } )+[\mathbf {V} \times \mathbf {\Omega } ]\right),}$
${\displaystyle \mathbf {\Omega } ^{\prime }={\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {\Omega } )+{\frac {1}{\sqrt {1-{V^{2} \over c_{g}^{2}}}}}\left(\mathbf {\Omega } -{\frac {\mathbf {V} }{V^{2}}}(\mathbf {V} \cdot \mathbf {\Omega } )-{\frac {1}{c_{g}^{2}}}[\mathbf {V} \times \mathbf {\Gamma } ]\right).}$

## Properties

• ${\displaystyle ~\Phi _{\mu \nu }}$ is an antisymmetric tensor of rank 2, for it ${\displaystyle ~\Phi _{\mu \nu }=-\Phi _{\nu \mu }}$. The tensor has six independent components, three of which are related to the components of the vector of gravitational field strength ${\displaystyle ~\mathbf {\Gamma } }$, and the other three are related to the components of the torsion field vector ${\displaystyle ~\mathbf {\Omega } }$.
• The Lorentz transformations of the coordinates preserve two invariants arising from the tensor properties of the field:
${\displaystyle \Phi _{\mu \nu }\Phi ^{\nu \mu }={\frac {2}{c_{g}^{2}}}(\Gamma ^{2}-c_{g}^{2}\Omega ^{2})=inv,}$
${\displaystyle {\frac {1}{4}}\varepsilon ^{\mu \nu \sigma \rho }\Phi _{\mu \nu }\Phi _{\sigma \rho }=-{\frac {2}{c_{g}}}\left(\mathbf {\Gamma } \cdot \mathbf {\Omega } \right)=inv.}$

The first expression is the contraction of the tensor, and the second is defined as the pseudoscalar invariant. In the latter expression the Levi-Civita symbol ${\displaystyle \varepsilon ^{\mu \nu \sigma \rho }}$ is used for the four-dimensional space, which is a completely antisymmetric unit tensor, with its gauge ${\displaystyle \varepsilon ^{0123}=1.}$

• Tensor determinant is also the Lorentz invariant:
${\displaystyle \det \left(\Phi _{\mu \nu }\right)={\frac {4}{c_{g}^{2}}}\left(\mathbf {\Gamma } \cdot \mathbf {\Omega } \right)^{2}.}$

## Application

Let us consider the following expression:

${\displaystyle {\frac {\partial \Phi _{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial \Phi _{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial \Phi _{\sigma \mu }}{\partial x^{\nu }}}=0.\qquad \qquad (2)}$

Equation (2) is satisfied identically, which is proved by substituting into it the definition for the gravitational field tensor according to (1). If in (2) we use nonrecurring combinations 012, 013, 023 and 123 as the indices ${\displaystyle ~\mu \nu \sigma }$, and if we pass from the field potentials to the strengths, this leads to two vector equations:

${\displaystyle ~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},\qquad \qquad (3)}$
${\displaystyle ~\nabla \cdot \mathbf {\Omega } =0.\qquad \qquad (4)}$

Equations (3) and (4) are two of the four Heaviside's equations for the gravitational field strengths in the Lorentz-invariant theory of gravitation. According to (3), the change in time of the torsion field creates circular gravitational field strength, which leads to the effect of gravitational induction, and equation (4) states that the torsion field, as well as the magnetic field, has no sources. Equations (3) and (4) can also be obtained from equality to zero of the 4-vector, which is found by the formula:

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

Another couple of gravitational field equations is also expressed in terms of the gravitational field tensor:

${\displaystyle ~\nabla _{\nu }\Phi ^{\mu \nu }={\frac {4\pi G}{c_{g}^{2}}}J^{\mu },\qquad \qquad (5)}$

where ${\displaystyle J^{\mu }=\rho _{0}u^{\mu }=\left({\frac {c_{g}\rho _{0}}{\sqrt {1-V^{2}/c_{g}^{2}}}},{\frac {\mathbf {V} \rho _{0}}{\sqrt {1-V^{2}/c_{g}^{2}}}}\right)=(c_{g}\rho ,\mathbf {J} )}$ is the 4-vector of mass current density, ${\displaystyle \rho _{0}}$ is the matter density in the comoving reference frame, ${\displaystyle \mathbf {V} }$ is the velocity of the matter unit, ${\displaystyle ~G}$ is the gravitational constant.

In the expanded form the equation for the field strengths with field sources are as follows:

${\displaystyle ~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho ,}$
${\displaystyle ~\nabla \times \mathbf {\Omega } ={\frac {1}{c_{g}^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),}$

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

According to the first of these equations, the gravitational field strength is generated by the matter density, and according to the second equation the circular torsion field is always accompanied by the mass current, or emerges due to the change in time of the gravitational field strength vector.

Gravitational four-force acting on the mass ${\displaystyle ~M}$ of a body can be expressed in terms of the gravitational field tensor and the 4-velocity of the body:

${\displaystyle ~F_{\mu }=M\Phi _{\mu \nu }u^{\nu }.}$

This expression can be derived, in particular, as the consequence of the axiomatic construction of the covariant theory of gravitation in the language of 4-vectors and tensors. [3]

If we take the covariant divergence of both sides of (5), and taking into account (1) we obtain the continuity equation for the mass 4-current: [4]

${\displaystyle ~\nabla _{\alpha }\nabla _{\beta }\Phi ^{\alpha \beta }=\nabla _{\alpha }\nabla _{\beta }\nabla ^{\alpha }D^{\beta }-\nabla _{\alpha }\nabla _{\beta }\nabla ^{\beta }D^{\alpha }=-R_{\mu \alpha }\Phi ^{\mu \alpha }={\frac {4\pi G}{c_{g}^{2}}}\nabla _{\alpha }J^{\alpha }.}$

In Minkowski space the Ricci tensor ${\displaystyle ~R_{\mu \alpha }}$ equal to zero, the covariant derivative becomes the partial derivative, and the continuity equation becomes as follows:

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

## Action and Lagrangian

Total Lagrangian for the matter in gravitational and electromagnetic fields includes the gravitational field tensor and is contained in the action function: [4] [5]

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

where ${\displaystyle ~L}$ is Lagrangian, ${\displaystyle ~dt}$ is differential of 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, ${\displaystyle ~c=c_{g}}$ is the speed of light as a measure of the propagation speed of electromagnetic and gravitational interactions, ${\displaystyle ~A_{\mu }=\left({\frac {\varphi }{c}},-\mathbf {A} \right)}$ is the electromagnetic 4-potential, where ${\displaystyle ~\varphi }$ is the electric scalar potential, and ${\displaystyle ~\mathbf {A} }$ is the electromagnetic vector 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 the constants of acceleration field and pressure field, respectively, ${\displaystyle ~u_{\mu \nu }}$ is the acceleration tensor, ${\displaystyle ~\pi _{\mu }}$ is the 4-potential of 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 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 matter unit in gravitational and electromagnetic fields and pressure field: [6]

${\displaystyle ~-u_{\beta \sigma }\rho _{0}u^{\sigma }=\rho _{0}{\frac {dU_{\beta }}{d\tau }}-\rho _{0}u^{\sigma }\partial _{\beta }U_{\sigma }=\Phi _{\beta \sigma }\rho _{0}u^{\sigma }+F_{\beta \sigma }\rho _{0q}u^{\sigma }+f_{\beta \sigma }\rho _{0}u^{\sigma },}$

where the first term on the right is the gravitational force density, expressed with the help of the gravitational field tensor, 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.

If we vary the action function by the gravitational four-potential, we obtain the equation of gravitational field (5).

## Gravitational stress-energy tensor

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

${\displaystyle ~U^{ik}={\frac {c_{g}^{2}}{4piG}}\left(-g^{im}\Phi _{mr}\Phi ^{rk}+{\frac {1}{4}}g^{ik}\Phi _{rm}\Phi ^{mr}\right)}$.

The covariant derivative of the gravitational stress-energy tensor determines the 4-vector of gravitational force density:

${\displaystyle ~f^{\alpha }=-\nabla _{\beta }U^{\alpha \beta }={\Phi ^{\alpha }}_{k}J^{k}.}$

## Generalized momentum and Hamiltonian mechanics

By definition, the generalized momentum ${\displaystyle \mathbf {P} }$ characterizes the total momentum of the matter unit taking into account the momenta from the gravitational and electromagnetic fields. In the covariant theory of gravitation the generalized force, as the rate of change of the generalized momentum by the coordinate time, depends also on the gradient of the energy of gravitational field associated with the matter unit and determined by the gravitational field tensor. [7]

In the weak-field approximation Hamiltonian as the relativistic energy of a body with the mass ${\displaystyle ~m}$ and the charge ${\displaystyle ~q}$ with ${\displaystyle ~c=c_{g}}$ equals:

${\displaystyle ~H=c{\sqrt {m^{2}c^{2}+(\mathbf {P} -m\mathbf {D} -q\mathbf {A} )^{2}}}+m\psi +q\varphi -}$
${\displaystyle ~-\int {({\frac {c^{2}}{16piG}}\Phi _{\mu \nu }\Phi ^{\mu \nu }-{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu })}dx^{1}dx^{2}dx^{3}+const.}$

If we use the covariant 4-vector of generalized velocity

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

then in the general case the Hamiltonian has the form: [4]

${\displaystyle ~H=\int {(s_{0}J^{0}-{\frac {c^{2}}{16piG}}\Phi _{\mu \nu }\Phi ^{\mu \nu }+{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }+{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }+{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},}$

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

If we move to the reference frame that is fixed relative to the center of mass of system, Hamiltonian will determine the invariant energy of the system.