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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: [1]

{\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 or torsion of the field, ~c  is the 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 the definition of  ~ \mathbf {H}   means that the energy is transferred in the direction opposite to the vector.

Contents

  • 1 Energy flux density of gravitational field
  • 2 Heaviside theorem
  • 3 Plane waves
  • 4 Gravitational pressure
  • 5 History
  • 6 References
  • 7 See also
  • 8 External links

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 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 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 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 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. [2] 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, [3] by Fedosin, [4] by H. Behera and P. C. Naik.[5]

References

  1. Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
  2. Oliver Heaviside. A Gravitational and Electromagnetic Analogy, Part I, The Electrician, 31, 281-282 (1893).
  3. P. Krumm and D. Bedford, Am. J. Phys. 55 (4), 362 (1987).
  4. Fedosin S.G. (1999), written at Perm, pages 544, Fizika i filosofiia podobiia ot preonov do metagalaktik, ISBN 5-8131-0012-1.
  5. Harihar Behera and P. C. Naik. Gravitomagnetic Moments and Dynamics of Dirac (Spin 1/2 ) Fermions in Flat Space-Time Maxwellian Gravity. International Journal of Modern Physics A, Vol. 19, No. 25 (2004), P. 4207-4229.

See also

External links

 

Source: http://sergf.ru/vhen.htm

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