Physics/Essays/Fedosin/Heaviside vector

Heaviside vector is a vector of energy flux density of gravitational field, which is a part of gravitational stress-energy tensor in Lorentz-invariant theory of gravitation. The Heaviside vector $~\mathbf {H}$ can be determined by cross product of two vectors:

$\mathbf {H} =-{\frac {c^{2}}{4\pi G}}[\mathbf {\Gamma } \times \mathbf {\Omega } ],$

where $~\mathbf {\Gamma }$ is vector of gravitational field strength or gravitational acceleration, $~G$ is gravitational constant, $~\mathbf {\Omega }$ is gravitational torsion field, $~c$ is speed of light.

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 definition of $~\mathbf {H}$ means that the energy is transferred in direction opposite to the vector.

Energy flux of gravitational field

The vector quantity $~{\frac {1}{c}}\mathbf {H} =U^{0k}$ represents the time components of gravitational stress-energy tensor $~U^{ik}$, with the tensor indices i = 0, k = 1,2,3.

To determine the flux of gravitational energy through a surface, it is necessary to integrate the vector $~\mathbf {H}$ over the area of this surface, taking into account its motion in reference frame under consideration. Such integration takes into account mutual orientation of vector $~\mathbf {H}$ and normal vector of the surface, where orientation of normal vector and the surface area depend on the speed and direction of motion of the surface due to effects of special theory of relativity. In general theory of relativity, additional effects arise from curvature of space-time.

Heaviside theorem

From the law of conservation of energy and energy flux of 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 mass current density.

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

Plane waves

Maxwell-like gravitational equations, in the form of which 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 direction of wave propagation, and the relation $~\Gamma _{0}=c\Omega _{0}$ 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 gravitational energy flux it will be:

$H(\mathbf {r} ,t)=-{\frac {c^{2}}{4\pi G}}\Gamma _{0}\Omega _{0}\cos ^{2}(\omega t-\mathbf {k} \cdot \mathbf {r} )=-{\frac {c}{4\pi G}}\Gamma _{0}^{2}\cos ^{2}(\omega t-\mathbf {k} \cdot \mathbf {r} ).$

The average value over time and space of the squared cosine is equal to ½, so:

$\left\langle H(\mathbf {r} ,t)\right\rangle =-{\frac {c}{8\pi G}}\Gamma _{0}^{2}.$

In practice, it should be taken into account that the wave pattern in a gravitationally bound system of bodies is more likely to be quadrupole than dipole in nature, since during radiation it is necessary to take into account the contributions of all field sources, some of which move in opposite directions. 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. Then the maximum possible gravitational pressure is:

$~p=\mid {\frac {\langle H\rangle }{c}}\mid ={\frac {\Gamma _{0}^{2}}{8\pi G}},$

where $\langle H\rangle$ is a mean Heaviside vector and $~\Gamma _{0}$ is amplitude of gravitational field strength vector of incident plane gravitational wave.

The formula for maximum pressure can be understood from definition of pressure as a force $~F$ applied to an area $~S$, definition of force as a change in the field energy $~\Delta E$ along the path of the wave $~\Delta x$ over time $~\Delta t$, provided that $~c\Delta t=\Delta x$:

$p={\frac {F}{S}}={\frac {\Delta E}{\Delta xS}}={\frac {\Delta E}{c\Delta tS}}=\mid {\frac {\langle H\rangle }{c}}\mid .$

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 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 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.

See also

• Umov vector
• Poynting vector
• Gravitoelectromagnetism
• Gravitational field strength
• Gravitational torsion field
• Lorentz-invariant theory of gravitation
• Maxwell-like gravitational equations
• Covariant theory of gravitation

External links

• Heaviside vector in Russian