In a recent paper https://arxiv.org/abs/2007.00082, two authors propose a new model (RMOND) which seems succesful to explain many cosmological data. Obviously, the GR theory must have as classical limit the modified equations of gravity of RMOND. Is the Schwarzschild metric modified by this new theory? And if the answer is 'yes', how?
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Of course the Schwarzschild metric is modified by this new RMOND theory.
Newton's gravity is the weak field limit of Schwarzschild metric. And MOND is a modification of Newton's gravity. Therefore any relativistic extension of MOND theory (including RMOND) has to modify the Schwarzschild metric at least in the weak field limit.
The paper https://arxiv.org/abs/2007.00082 you cited focuses on the FLRW metric, since it discuses the RMOND's cosmological consequences.
If you are interest in the modified Schwarzschild metric formula, you can refer to equation (16) of another paper by the same author (Tom Złosnik): https://arxiv.org/abs/0707.3519
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$\begingroup$ I haven't looked into the paper, but it's not automatically necessary that a new theory modify a particular solution (for instance, the modifying terms could have a zero solution for the case of spherical symmetry, or the Einstein-Maxwell equations having a Q=0 black hole solution.) $\endgroup$ Commented Jan 27, 2021 at 22:54
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$\begingroup$ Let's be specific, by "the modifying terms could have a zero solution", do you mean the new theory recovers Newton gravity at weak field limit in your "instance"? $\endgroup$– MadMaxCommented Jan 27, 2021 at 23:04
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$\begingroup$ I'm saying (not for this theory), that just because the underlying equation is different than Einstein's equation, that it is still possible to have "the swarzschild metric" as a solution to this other equation. For a trivial example, like how Einstein-Maxwell theory has the exact schwarzschild solution as a solution, or how a freely falling radially infalling observer in Schwarzschild has the same equation for $r(\tau)$ that you would get for $r(t)$ in Newtonian theory. $\endgroup$ Commented Jan 28, 2021 at 17:28
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$\begingroup$ like, if the general RMond solution for spherically symmetric matter is something like schwarzschild + (iintegration constant) * (parity violating metric), then exact schwarzschild is a solution. $\endgroup$ Commented Jan 28, 2021 at 17:29
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$\begingroup$ Is the general RMond solution for spherically symmetric matter something like schwarzschild + (iintegration constant) * (parity violating metric)? $\endgroup$– MadMaxCommented Jan 28, 2021 at 18:12