Four potential theory of gravitation

Fringe versus mainstream ideas
The discussion that follows has yet to enter mainstream physics and for that reason has been labeled "fringe". But a mainstream variation of this idea is well known as a linear approximation to Einstein's theory of relativity. See, for example:
 * https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll6.html

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. The basic equations are:



(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

(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}) $$ wherein $$\Phi^0$$ is the time component and $$\vec{\Phi}$$ is the spatial component of the gravitational four potential $$\Phi^\nu=(\Phi^0,\vec{\Phi})$$ and $$c\,\rho^0$$ is the time component and $$\vec{j}_0$$ is the spatial component of the rest mass four current $$j^\nu=(c\,\rho^0,\vec{j}_0)$$. $$c\,\rho^0$$ is the rest mass per unit volume and $$\vec{j}_0=\vec{v}\rho^0$$ with $$\vec{v}$$ for the velocity at which the rest mass density moves.

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



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



(4) \vec{\nabla} \times(\vec{\nabla} \times \vec{\Phi})=\frac{4\pi \,G}{c^2}\,\vec{j} $$



(5) \frac{\partial (c\,\Phi^0)}{c\,\partial t}=\vec{\nabla}\vec{\Phi} $$



(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 $$m_0$$ of the particle being subjected to the gravitation. The Dirac equation for the electron becomes under the influence of the gravitational four potential

(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



(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.

From (8) it can be seen, especially at small $$ \vec{\Phi} $$ that in order to conserve the energy $$ E $$ the rest mass $$ m_0 $$ changes under the influence of the term $$ m_0\,c^2\,E\,\Phi^0 $$. Since $$ \Phi^0 $$ is negative, a more negative $$ \Phi^0 $$, in other words a stronger gravitational potential, implies that the momentum term $$ c^2\,p^2 $$ increases correspondingly. In other words, the rest mass can be regarded to become smaller by the term $$ m_0\,c^2\,E\,\Phi^0 $$. Thus, the momentum term, i.e. the kinetic energy, increases by energy from the rest mass. When a mass falls towards a stronger gravitational potential, it does not obtain kinetic energy but converts some of its rest mass energy into its kinetic energy.

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



(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) $$ with K as a constant equal to $$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.

In (9) K is given according to equation (183) on p. 28 in by

(10) K=\frac{1}{8\pi} $$ That can be obtained from the gravitation radiation in comparison with the general theory of relativity under the assumption that the conservation of the angular momentum applies (loc. cit.).

From (9) and (10) the amplitude for $$ \Phi_0 $$ of the four potential of gravitation in a unit volume can be calculated according to formula (31) on p. 7 in



(11) \Phi_0=\sqrt{\frac{1}{E}}\sqrt{\frac{2\hbar \pi G}{c^2}} $$

wherein as a matter of course $$ E=\hbar \omega$$, with $$ \omega$$ as the frequency of the gravon. Thus, the gravitation four potential can be quantised like the electromagnetic four potential.

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