Physics/Essays/Fedosin/Heaviside vector

The Heaviside vector is a vector of energy flux density of gravitational field, which is a part of the gravitational stress-energy tensor in the Lorentz-invariant theory of gravitation. The Heaviside vector $$~ \mathbf {H} $$ can be determined by the cross product of two vectors:
 * $$ \mathbf {H} = - \frac{c^2_{g}}{4\pi G} [ \mathbf \Gamma \times \mathbf {\Omega}], $$

where $$ ~\mathbf \Gamma $$ is the vector of gravitational field strength or gravitational acceleration, $$~ G $$ is the gravitational constant, $$~ \mathbf{\Omega}$$ is the gravitational torsion field or torsion of the field, $$~ c_{g}$$ is the speed of gravity.

The Heaviside vector magnitude is equal to the amount of gravitational energy transferred through the unit area which is normal to the energy flux per unit time. The minus sign in the definition of $$~ \mathbf {H} $$ means that the energy is transferred in the direction opposite to the vector.

The momentum density of gravitational field
To determine the vector of momentum density $$~ \mathbf { P_g} $$ of gravitational field we must divide the Heaviside vector by the square of the speed of gravitation propagation:
 * $$~ \mathbf { P_g }= \frac {1}{ c^2_{g}}\mathbf {H}= - \frac{1}{4\pi G} [ \mathbf \Gamma \times \mathbf {\Omega}].$$

The vector $$~ \mathbf { P_g } c_{g} = \frac {1}{ c_{g}}\mathbf {H} =U^{0k} $$ is a part of the gravitational stress-energy tensor $$~ U^{ik} $$ in the form of three timelike components, when the indices of the tensor are i = 0, k = 1,2,3. To determine the momentum of the gravitational field, we must integrate the vector $$~ \mathbf { P_g } $$ over the moving space volume, occupied by the field, taking into account the Lorentz contraction of this volume.

The Heaviside theorem
From the law of conservation of energy and momentum of matter in a gravitational field in the Lorentz-invariant theory of gravitation should Heaviside theorem:
 * $$\nabla \cdot \mathbf {H} = - \frac{\partial {U^{00}}}{\partial {t}} - \mathbf {J} \cdot \mathbf {\Gamma }, $$

where $$~ \mathbf {J}$$ is the mass current density.

According to this theorem, the gravitational energy flowing into a certain volume in the form of the energy flux density $$~ \mathbf {H} $$ is spent to increase the energy of the field $$~ U^{00}$$ in this volume and to carry out the gravitational work as the product of field strength $$~ \mathbf {\Gamma } $$ and the mass current density $$~ \mathbf {J}$$.

Plane waves
Maxwell-like gravitational equations, in the form of which the equations of Lorentz-invariant theory of gravitation are presented, allow us to determine the properties of plane gravitational waves from any point sources of field. In a plane wave the vectors $$ ~\mathbf \Gamma $$ and $$~ \mathbf{\Omega}$$ are perpendicular to each other and to the direction of the wave propagation, and the relation $$~ \Gamma_0=c_g \Omega_0 $$ holds for the amplitudes.

If we assume that the wave propagates in one direction, for the field strengths it can be written:
 * $$~ \Gamma ( \mathbf{r}, t ) = \Gamma_0 \cos ( \omega t  -  \mathbf{k} \cdot \mathbf{r} ),  $$


 * $$~ \Omega ( \mathbf{r}, t ) = \Omega_0 \cos ( \omega t  -  \mathbf{k} \cdot \mathbf{r}), $$

where $$~ \omega$$ and $$~ \mathbf{k} $$ are the angular frequency and the wave vector.

Then for the gravitational energy flux it will be:
 * $$ H( \mathbf{r}, t ) = - \frac{c^2_{g}}{4\pi G} \Gamma_0 \Omega_0 \cos^2 ( \omega t  -  \mathbf{k} \cdot \mathbf{r} ) =- \frac{c_{g}}{4\pi G}  \Gamma^2_0  \cos^2 ( \omega t  -  \mathbf{k} \cdot \mathbf{r} ) .$$

The average value over time and space of the squared cosine is equal to ½, so:
 * $$ \left\langle H( \mathbf{r}, t ) \right\rangle = - \frac{c_{g}}{8\pi G}  \Gamma^2_0.$$

In practice, it should be noted that the pattern of waves in a gravitationally bound system of bodies has rather quadrupole than dipole character, since in case of emission we should take into account the contributions of all field sources. According to the superposition principle we must first sum up at each point of space all the existing fields $$ ~\mathbf \Gamma $$ and $$~ \mathbf{\Omega}$$, find them as functions of coordinates and time, and only then calculate with the obtained total magnitudes the energy flux in the form of the Heaviside vector.

Gravitational pressure
Suppose that there is a gravitational energy flux falling on some unit material area absorbing all the energy. The energy flux propagates at the speed $$~ c_{g}$$ and transfers the momentum density of the field
 * $$~ \mathbf { P_g }= \frac {1}{ c^2_{g}}\mathbf {H}.$$

Then the maximum possible gravitational pressure is:
 * $$~p= \mid \frac{\langle H \rangle}{ c_{g}} \mid =\frac {\Gamma^2_0}{8\pi G } ,$$

where $$ \langle H\rangle$$ is the mean Heaviside vector and $$ ~\Gamma_0$$ is the amplitude of the gravitational field strength vector of incident plane gravitational wave. The formula for the maximum pressure can be understood from the definition of pressure as force $$ ~ F $$, applied to the area $$ ~ S $$, the definition of force as the momentum of field ~ \Delta Q during the time $$ ~ \Delta t $$, provided that $$ ~ \Delta Q = Q $$; $$ ~ c_g \Delta t = \Delta x $$; volume, absorbing field momentum $$ ~ \Delta V = \Delta x S $$; average density of gravitational momentum $$~ \langle P_g \rangle = \frac {Q}{\Delta V } $$:
 * $$p=\frac {F}{S}= \frac {\Delta Q }{\Delta t S}= \frac {Q c_g }{\Delta x S}= \langle P_g \rangle c_g .$$

Since the gravitational energy flux passes through bodies with low absorption in them, to calculate the pressure it is necessary to take the difference between the incident and outgoing energy fluxes.

History
Representation of the gravitational energy flux first appeared in the works by Oliver Heaviside. Previously the Umov vector for the energy flux in substance (1874) and the Poynting vector for the electromagnetic energy flux (1884) had been determined.

The Heaviside vector is in agreement with that used by Krumm and Bedford, by Fedosin, by H. Behera and P. C. Naik.