In my old paper i stuffed:
and out popped:
In the paper it is seen that this gives the expected results as far as the case of a circular orbit, no velocity at all, the anomalous perihelion shift and the Schwarzschild radius. From this site it seems like we also get the right value for the radius of the ”innermost stable circular orbit”.
In the paper we saw that the ”post-Newtonian expansion” at the 1PN level (that expression is for instance used by JPL-Horizons when calculating ephemerides) gives strange results for the case of a circular orbit and the case of no velocity at all. From this site we see that in the strong field limit we get very strange results in the form of ”bouncing”. This is because NASA/JPL approximates GR by using a repulsive ”inverse r cube” term.
I managed to get to peer review at the paper ”General Relativity and Gravitation” back in 2012 but instead of appreciating that my expression seems to give results closer to what is expected from the Schwarzschild solution then what the post-Newtonian expansion does I got the response:
”The post-Newtonian expansion is well known to correctly describe all observable quantities for orbits in the Solar System and in the binary pulsar systems, for instance. The author should be demonstrating that his model reproduces post-Newtonian observables, not that it gives different results”
he also wrote:
”If the author were to write a paper with a rigorous derivation of a model that correctly reproduced post-Newtonian observables then this would stand a much greater chance of publication.”
I tried to get published in around seven different jornals before I gave up.
Anyhow, moving forward to 2019, the energy of a test-body moving in a spherically symmetric gravitational field can, according to GR, be written as:
This can be re-written as:
If we write:
we could write ”E=m(r,v)(c(r))^2”, so we sort of have a ”general relativistic relativistic mass”. Now, this is ad hoc but can maybe be be motivated by some symmetry assumption, we need to throw in an extra factor to get the orbits right. (The symmetry assumption would be that the mass of the central mass must increase (in the sense that we are dealing with here) by the same amount as the mass of the orbiting object, because they are both getting closer to each other. Sort of the same argument can be made as regards of why the speed of light slows down extra much in the radial direction.) We write:
and throw this into:
For the case of pure non-radial motion we get:
For the case of pure radial motion we instead get:
If we take a superposition of these two orthogonal solutions we get:
(Maybe it should be proven that you can take the superposition of the two solutions to get the general solution)
This final expression that is an attempt to generalize the Newtonian gravitational acceleration to get to the accelerations expected from the Schwarzschild solution has several features in common with what is expected from the Schwarzschild solution:
- Acceleration whenever v=0. For the case of no velocity the extra terms vanishes and we retain the Newtonian acceleration, as is expected.
- Acceleration for pure non-radial motion. We see that the extra terms vanishes for pure non-radial motion, which means that we get the classical velocity for a body in circular orbit, just as expected.
- Weak field perihelion shift. Although not proven here, in the weak field limit we get precisely the anomalous perihelion shift we expect. (I do not know if there is a strong field perihelion shift to test the expression against)
- The Schwarzschild radius. As seen from simulations, (these are made using the expression in my old paper ), the expression reproduces the Schwarzschild radius.
- The radius of the innermost stable circular orbit. As seen from these simulations, made by using the expression in my old paper, we get a spinning around in a circle behaviour just inside the ISCO-radius before the particle falls into the black hole, which is the expected behaviour. It is hard to say if the results in the simulations are exactly right.
- Acceleration of pure radial infall. According to this answer on astronomy.stackexchange the expected acceleration for pure radial infall is exactly what we got (se two expressions up in this post).
Given that we have all of these effects more or less precisely as expected from the Schwarzschild solution we must at least be very close to have generalized the classical Newtonian gravitational acceleration to give results that exactly coincides with what is expected from the Schwarzschild solution.
I have started a few threads on the stackexchange network related to this issue:
What is the coordinate acceleration for pure radial motion?
Why are JPL using this expression to simulate Schwarzschild orbits?
3PN and higher order post-Newtonian Schwarzschild approximation
Is there a general relativistic mass in general relativity?
If you find out that the last expression above somehow give orbits that differs from what is expected from the Schwarzschild solution, please tell me. Also, if you have any idea of what kind of testing I can do to check the orbits against what is expected from the Schwarzschild solution, please let me know.