The question you're asking is tantamount to asking what the gravitational field of a single particle is.
There are two directions of interaction for a force or field, including gravity. The "forward reaction" is how the body responds to the given force or field. For gravity and inertia, together, the combined effect - for a given space-time metric - is given by the "geodesic equation" associated with that metric.
The Equivalence Principle, in the form originally stated by Galileo, asserts that the "forward reaction" for gravity is independent of the composition of the body: that the response to gravity is proportional to the body's response in the law of inertia. The principle, in the form re-instated by Einstein, goes one step further to assert that the law of gravity, itself, is but a warped form of the law of inertia. That statement is made in the form of the single equation - the geodesic equation - to account for them both, as one. So, it asserts the equivalence of mass and passive gravitational charge.
The other direction of interaction is "back-reaction"; which pertains to how the body's presence affects, molds or modifies the force or the field. On account of this, technically, the equivalence principle isn't quite true. A pin dropped onto the ground will fall a little bit differently than a space station as large as the moon dropped onto the ground would, because the space station is going to be pulling the ground up a lot harder. That's back-reaction at play.
The back-reaction of the body defines its active gravitational charge.
For gravity, back-reaction is determined by Einstein's field equations, which substantially modifies the pre-relativistic back-reaction law, which was Newton's inverse square force law.
If you play fast and loose with the math, you can sorta derive the geodesic law from Einstein's field equations ... namely by taking the (singular) stress tensor for a source concentrated on a world line and applying the continuity equation to the stress tensor; the continuity equation being both a pre-condition for the formulation of Einstein's field equations and derivable from it as a corollary. So, in that sense the equivalence of the active and passive gravitational charges is mandated by the theory... but not necessarily by nature. It still has to be tested, and those tests are done regularly, too.
The back-reaction described by Einstein's field equations pertains to matter such as would be described from the physics predating Quantum Theory - "classical physics". It is a back-reaction law only for classical matter. There is no known back-reaction law for matter such as would be described by contemporary theories of matter, which are all grounded in Quantum Theory. That's a gap.
A lot of things could (and do) lie in that gap which defy contemporary understanding by anyone in this world today. That's why Congress entered the survey blurb
Advanced Space Propulsion Based on Vacuum (Spacetime Metric) Engineering
into its records at the start of the subcommittee hearing on UAP's in July of 2023.
There isn't even so much as a back-reaction law for single particle, such as the photon or electron, nor any consensus (at least in the case of the photon) what such a thing would even mean, since the photon's electromagnetic field is a q-number field. The same goes for the other fundamental bosons, including the gauge bosons and Higgs boson.
In the case of the fundamental fermions, however, this much we can say: a solution to Einstein's field equations that has the same mass, same total charge (meaning to combination of all charges, for all the gauge forces, not just electro-magnetism, so you need to be talking about the Einstein-Yang-Mills-Higgs equation, not Einstein-Maxwell equation) and same angular momentum as given by their intrinsic (spin) angular momentum is a
... (wait, wait) ...
Moreover, it's not just any kind of singularity, but is a ...
specifically: a naked Kerr-Newman ring singularity; way beyond the threshold, not even close; and so much so that they had to call in the science police for its indecency.
That's semi-serious, too. It is a total violation of the...
Chronology Protection Conjecture
So, it's not a black hole at all, since by definition, these are sheathed by event horizons, but more like a mini-stargate.
That's even the case for the left neutrinos and right anti-neutrinos. They may be electrically neutral, but in case anyone forgot, they have weak nuclear force charges.
It may or may not be the case for the right neutrinos or left anti-neutrinos - if they exist; but their existence is not established since they have no electric charge, weak nuclear charge or strong nuclear charge and therefore almost nothing to see them with - except via the Higgs field and gravity, and possibly except through their "baryon minus lepton" (B-L) number, if this proves to be a charge for a heretofore unverified gauge force. If there is a B-L force, and a corresponding charge, then the right neutrino and left anti-neutrino would also fall into the naked singularity category with the other fundamental fermions. Otherwise, a Kerr-Newman solution matching the stats of a right neutrino or left anti-neutrino might not be naked, but just a regular rotating, uncharged black hole (a Kerr solution).
Being a naked singularity means there is no event horizon, no ergosphere, and the singularity is directly exposed to and seen by everyone. That also means no Hawking evaporation, since that's predicated on the sheathing by an event horizon. It also means there is causality violation near the singularity and indeterminacy - something that (in retrospect) looks suspiciously like it could serve as a classical underpinning to quantum indeterminacy - an Einstein's Revenge Scenario.
In that vein, though I don't know enough about
ER - EPR
in my mind, it raises the question: could such singularities also serve as an underpinning to this?
All of this, of course, raises the question: is that what the fermion actually is? Here are the two cases:
(1) The actual fermions we observe, measure, track in bubble chambers, irradiate CRT screens with, are totally different than these solutions. But if so, that raises the questions: (a) what else would it be if not this, and (b) shouldn't we already be seeing the discrepancies between the two models soon (something we could measure or experimentally test for), since we're talking about the Compton scale?
... or ...
(2) Fundamental fermions are indeed Kerr-Newman ring singularities. In that case, there should already exist some kind of deep correspondence between the Dirac equation and its solutions versus the Kerr-Newman solution.
So, does such a correspondence exist?
The answer is yes - very much so!
And by "very much so", I mean not just at the level of the semi-classical theory of first quantization, but even broaching the level of second quantization, also being able to account for some of the results therein.
The Dirac - Kerr-Newman Electron
It's one of those things that's been sitting in the attic that nobody's really been paying attention to, on account of the attention having been drawn away from this (the question of single-particle back-reaction / gravity fields) by the distraction of other things, such as Loop Quantum Gravity or String Theory, that may (and probably will) prove to be red herrings.