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/Four potential gravitation From user User:Hapoth

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 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. 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.

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



(7) 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.

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 $$\Phi^0$$ of the present theory being valid for most non relativistic applications.