Four potential theory of gravitation

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Four potential theory of gravitation

In the following four potential theory of gravitation a field theory of gravitation is set out in which the gravitational potential is represented by a four potential like the electromagnetic four potential.

The Lorentz-invariant theory of gravitation Lorentz-invariant theory of gravitation can be considered as such a theory.

An alternative theory has been published in 2014 by Poth [1]. The basic equations are:

${\displaystyle (1)~~~{\frac {1}{c^{2}}}{\frac {\partial ^{2}(c\,\Phi ^{0})}{\partial t^{2}}}-{\vec {\nabla }}^{2}(c\,\Phi ^{0})={\frac {4\pi \,G}{c^{2}}}\,(c\,\rho ^{0})}$

and

${\displaystyle (2)~~~{\frac {1}{c^{2}}}{\frac {\partial ^{2}({\vec {\Phi }})}{\partial t^{2}}}-{\vec {\nabla }}^{2}\,({\vec {\Phi }})={\frac {4\pi \,G}{c^{2}}}\,{\vec {j}}_{0}({\vec {r}})}$

wherein ${\displaystyle \Phi ^{0}}$  is the time component and ${\displaystyle {\vec {\Phi }}}$  is the spatial component of the gravitational four potential ${\displaystyle \Phi ^{\nu }=(\Phi ^{0},{\vec {\Phi }})}$  and ${\displaystyle c\,\rho ^{0}}$  is the time component and ${\displaystyle {\vec {j}}_{0}}$  is the spatial component of the rest mass four current ${\displaystyle j^{\nu }=(c\,\rho ^{0},{\vec {j}}_{0})}$ . ${\displaystyle c\,\rho ^{0}}$  is the rest mass per unit volume and ${\displaystyle {\vec {j}}_{0}={\vec {v}}\rho ^{0}}$  with ${\displaystyle {\vec {v}}}$  for the velocity at which the rest mass density moves.

These potential equations are obviously Lorentz invariant. There are further field equations:

${\displaystyle (3)~~~{\vec {\nabla }}^{2}\Phi ^{0}=-{\frac {4\pi \,G}{c^{2}}}\,\rho ^{0}+{\frac {\partial ({\vec {\nabla }}{\vec {\Phi }})}{c^{2}\,\partial t}}}$
${\displaystyle (4)~~~{\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {\Phi }})={\frac {4\pi \,G}{c^{2}}}\,{\vec {j}}}$
${\displaystyle (5)~~~{\frac {\partial (c\,\Phi ^{0})}{c\,\partial t}}={\vec {\nabla }}{\vec {\Phi }}}$
${\displaystyle (6)~~~{\frac {\partial ^{2}\,{\vec {\Phi }}}{c^{2}\,\partial t^{2}}}={\frac {\partial {\vec {\nabla }}\Phi ^{0}}{\partial t}}}$

which lead to the potential equations (1) and (2).

That four potential interacts similarly to the electromagnetic four potential, however with the rest mass ${\displaystyle m_{0}}$  of the particle being subjected to the gravitation. The Dirac equation for the electron becomes under the influence of the gravitational four potential

${\displaystyle (7)~~~i\,\hbar {\frac {\partial \psi }{\partial t}}=\left[c\,{\vec {\alpha }}\,\left({\vec {p}}-m_{0}\,{\vec {\Phi }}\,\right)+m_{0}\,c^{2}\,\Phi ^{0}+\,c^{2}\beta \,m_{0}\right]\psi \quad }$

It appears to yield an interaction with the orbital momentum of a planet and thus the known perihelion precession. Further gravitational effects can be derived, also gravitational radiation. (7) leads eventually to

${\displaystyle (8)~~~E^{2}-c^{2}\,p^{2}=m_{0}^{2}\,c^{4}+m_{0}\,c^{2}\,E\,\Phi ^{0}-m_{0}\,c^{2}\,{\vec {p}}\,{\vec {\Phi }}}$

for each one of a pair of particles interacting through gravitation. Thus we can understand the influence of gravitation without having to change the time space metric as a function of the gravitation as in the general theory of relativity, more specifically that the four length of the energy momentum four vector of a mass particle depends Lorentz invariantly on the gravitation what seems to be equivalent to a time space metric being dependent on gravitation. The difference between the model of general relativity and the present model is that in the present model the time space metric remains unchanged under gravitation, and only the rest mass interacts with gravitation.

The Lagrangian density ${\displaystyle l_{\Phi }}$  of the field could be

${\displaystyle (9)~~~l_{\Phi }=K\,{\frac {c^{4}}{G}}\,\sum _{\lambda }\left({\frac {\partial \Phi ^{\lambda }}{\partial x^{\mu }}}\,g^{\mu \nu }\,{\frac {\partial \Phi ^{\lambda }}{\partial x^{\nu }}}\right)}$

with K as a constant equal to ${\displaystyle 3/(8\pi )}$  if the gravitational radiation should equal that according to the general theory of relativity Gravitational wave.

The field equations describe a longitudinally polarized with zero spin.

The scalar theory of gravitation Gravitation/Scalar gravitation/Poth can be considered as a first approximation for ${\displaystyle \Phi ^{0}}$  of the present theory being valid for most non relativistic applications.

References

1. 'Four Potential Gravitation and its Quantization', from Hartwig Poth on 05.04.2014 under ISBN 978-3-8442-9165-0