UPDATE March 25, 2023: Our claim about a very fast collapse of a shell and Cherenkov radiation is probably false. See our new blog post.
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Newtonian gravity looks much like the Coulomb interaction between static electric charges.
Comparison of lagrangian densities
Their formulae are quite different, though. We should analyze where the analogy is hiding.
In the first formula, the Ricci scalar somehow plays the role of the inner product of the electromagnetic tensor F in the second formula.
Analogous Einstein equations for electromagnetism
The Einstein equations calculate the elliptic orbits of planets correctly. They calculate the newtonian gravity field. They calculate the precession of the perihelion of Mercury.
Let us have a system with no gravity and such that the electric charge of each object is linearly proportional to its mass. The system behaves much like gravity. The precession of the perihelion of "electric Mercury" is caused by the extra inertia which Mercury obtains when it is close to the electric Sun. Mercury moves electric field energy around, and gets extra inertia from that.
If we write the Einstein equations for our electromagnetic system, they describe the system to some degree. However, there is no electric black hole. With extremely strong fields we get a different behavior.
The Einstein equations define a "metric" for our electric system.
We have two descriptions for our system:
1. the classical one with the Coulomb force, magnetic fields, and the Poynting vector which tells how much field energy we move around;
2. the Einstein equations.
These descriptions are (almost) equivalent at least in certain situations.
They are not equivalent for quadrupole radiation. We noted in an earlier blog post that the radiative power of a gravity quadrupole is four times the power of the analogous electric quadrupole. That is a major difference between the electromagnetic field and the gravity field.
A fundamental difference of gravity and electromagnetics is that the field energy of gravity carries a gravity charge, while the field energy of electromagnetism has no electric charge. A graviton possesses a gravity charge while a photon has no electric charge. This fundamental distinction may explain all the differences between gravity and electromagnetism.
Electromagnetism differs prominently from general relativity for a collapsing/expanding symmetric shell; Cherenkov radiation
In this blog we have argued that a collapsing shell in our Minkowski & newtonian gravity does not gain extra inertia. It does not cause field energy to move around.
In the case of electromagnetics, the magnetic field B is zero, and the Poynting vector ~ E × B is zero, which means that no energy flows.
Robert Oppenheimer and Hartland Snyder (1939) calculated the collapse of a spherically symmetric dust ball in general relativity. Its surface falls according to the Schwarzschild metric. That is a collapsing shell falls just like a point particle. A point particle a acquires huge extra inertia when it approaches the horizon. We conclude that the shell does acquire extra inertia in general relativity.
In our Minkowski & newtonian gravity, a shell can fall faster than what is the speed of an individual photon. If our model is correct, we except to see Cherenkov type radiation from matter falling into a black hole.
Reza Mansouri's proof of the nonexistence for a uniform collapse solution with a pressure p = p(ρ)
We have already discussed Reza Mansouri's (1977) result in our blog.
Mansouri shows that if the pressure in a uniformly collapsing symmetric ball of perfect fluid only depends on the density ρ of the fluid, then there is no solution in general relativity. The sole case where a solution exists is when p = 0 everywhere, that is, a collapsing dust ball.
Do we have any reason to assume that the collapse should be uniform? That is, the spatial metric at each point (t, x) has contracted as much in the radial direction as in the tangential direction.
Conclusions
Einstein's equations can describe the movement of electric charges fairly well in certain cases. There is a big difference in the power of quadrupole radiation, however.
Another difference is in the collapse of a spherical shell of charge. General relativity claims that it acquires inertia which slows down the collapse.
Reza Mansouri's result about a collapse of a perfect fluid ball raises the question: under what circumstances does a theory of gravity have solutions? An iterative method to obtain a solution may fail if a new iteration of the metric changes the pressure so much that the next iteration of the metric differs a lot from the previous one.
Proving the existence of solutions for nonlinear differential equations is mathematically too challenging. Few results exist. We need to study the problem heuristically. Does our Minkowski & newtonian gravity have solutions for the Mansouri equation of state p = p(ρ)?
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