User:Hapoth/Gravitation theory

Four potential theory of gravitation
A Four potential theory of gravitation is a field theory of gravitation in which the gravitational field is described using 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 below can be considered as a first approximation for $$\Phi^0$$ of the present theory being valid for most non relativistic applications.

Scalar theory of gravitation
The basic field equation of a scalar theory of gravitation from Poth published in 2012/2013 is


 * $$ (1)  \frac{1}{c^2} \frac{\partial^2 \Phi}{\partial t^2}  -\nabla^2 \Phi = \kappa \cdot \rho, $$

wherein $$\Phi$$ is the scalar gravitational potential and $$\rho$$ the proper time density of the mass being the source of the gravitational potential and $$\kappa$$ is a constant incorporating the gravitational constant G. $$\kappa$$ is given by
 * $$ (2) \kappa = 4\pi\frac{h}{c^2}G   , $$

wherein h is Planck's constant and c is the veleocity of light.

The proper time density $$\rho$$ is defined by
 * $$ (3) \rho = \frac{\nu_M}{\Delta V}   , $$

wherein $$\nu_M $$ is the frequency of mass M being regarded as an oscillator according to de Broglie and $$\Delta V$$ is the volume in which that mass is situated. Thus $$\rho$$ is a scalar. Hence, (3) is Lorentz invariant.

Thus, the gravitational interaction becomes essentially an interaction between oscillators or between clocks representing masses. That yields directly the dependency of clock rates on the gravitational potential. And as the frequencies for the masses are velocity dependent, the gravitational interaction becomes dependent on the velocities of the interacting masses.

Thus, the heavy mass $$m_h $$ of an inertial rest mass $$m_0 $$ is given by
 * $$ (4) m_h=m_0 \sqrt{1-\frac{v^2}{c^2}} $$

Usually, that can not be noticed, in particular not under common laboratory conditions for determining the gravitational constant G.

The relativistic action integral for a mass $$m_0 $$ moving at velocity v in the gravitational potential of an other mass M at rest becomes
 * $$ (5) A = -\int \left(m_0\,c^2 - \frac{GM}{r}m_0 \, \sqrt{1-\frac{v^2}{c^2}}\right)\, \sqrt{1-\frac{v^2}{c^2}}\cdot dt, $$

That results eventually in the energy H given by
 * $$ (6) H= \frac {m_0c^2} {\sqrt{1-\displaystyle \frac{v^2}{c^2}}} -\frac{G M m_0}{r} \left(1+\frac{v^2}{c^2}\right), $$

Hence, the gravitational potential has become „velocity dependent“. And that eventually results amongst others in the perihelion precession. The geodesic precision and the frame dragging effect as observed by the Gravity B Probe follow also and more.

If two stars of equal mass M orbit each other at the radius r from their center of mass as in a binary star under essentially nonrelativistic conditions, a gravitational wave is created from their retardation difference as seen from a distant observer. That is not a dipole radiation but corresponds to a quadrupole radiation. The radiant power P of that wave is given up to a factor by
 * $$ (5) P \propto \frac{G}{c^5}\, \omega ^6 \,M^2\,r^4, $$

as according to the general theory of relativity. The factor appears to be 3/2 times of that obtained from the general theory of gravitation cf. e.g..

The basic field equation (1) can be easily derived from the correponding Lagrangian and, hence, also the energy density of the gravitational waves which can be directly quantized. As they are scalar, their spin is zero.

Comments
Thus there are in fact two theories, a scalar one and a four potential one. Both theories seem to be interesting in themselves - apart from being used for the phenomenon of gravitation.

The more advanced theory is of course the four potential theory. Its field equations are different from Maxwell's field equations, but they yield also the common wave equation for the four potential. Thus it should be also of interest as a four potential as such, not necessarily being restricted to its application to gravitation.

The more simple theory seems to be the scalar theory, because its field equation simply coincides with the common wave equation. In fact, when being applied it is also not so simple. It appears that Feynman had such a scalar model in mind but eventually dismissing it; at that time he had apparently no physical entity in mind providing for the gravitation interaction and being porportional to $$\sqrt{1-v^2/c^2} $$. In particular he didn't consider the proper time of a mass particle $$m $$ as a possible entity, albeit it had been commonly know that the Lagrangian $$L_m $$ of a mass particle is given by

L_m=-mc^2\sqrt{1-v^2/c^2} $$ All the less some kind of spatial density of such an entity had he considered. Nevertheless, the scalar theory can be drafted, at least as such, abeit not every mathematical model has to be implemented by nature.