[MrEldritch]

Bingo.

Well, not quite - black holes would be respective of charge conservation! A black hole only has three properties, in our current understanding of general relativity - but "electrical charge" is one of those properties.

But there's nothing that stops it, say, eating protons and spitting out positrons later.

[madrocketsci]

Do you mind if I ask you some questions about this?

I always have trouble picturing how, dynamically, a charge inside an event horizon is supposed to be able to propagate an electric field outside the event horizon of a black-hole. (retarded vector potential travels from a charge along light-paths to another point in space-time. There isn't any way for the influence of a point charge to get out?)

Perhaps some other related questions too: Charge and current density is a 4-vector in SR, which transforms along with all the other 4-vectors (momentum-energy 4-vector, etc). In a situation where the effective mass of an object reversibly lowered to the event-horizon (slowly moved relative to the event horizon with small velocity) goes to zero (all the mass energy ends up somewhere else) - wouldn't the effective charge density from a non-infalling external observer's perspective also be going to zero?

If we're just drawing a box around a black-hole and declaring that charge is conserved, we would have as much/little reason to declare any other conservation also holds?

[raattgift]

These are good questions! You might consider taking them to a forum where you'll get a more rigorous answer though. :D

Fully classically, the field lines point to the sources; they get "stuck" to the horizon as the source crosses. To a naive outside distant observer for whom the horizon subtends a small angle of the sky, so does each source. When thinking about collapsing charged matter forming a new black hole, substitute a spherically symmetric shell and shrink its area, while keeping the charge and mass constant and uniformly distributed on the shell -- the electric field and gravitational field outside the shell then both follow gauss-laws, so even without a horizon, observers outside the shell at a large distance (such that the shell looks virtually pointlike) cannot get the full information about the shell using only local measurements, including whether the shell is inside or outside a horizon.

Semiclassically, one can use virtual photons which aren't as restricted as real matter, especially in that the black hole horizon is not necessarily a trapping surface for them. Typically one sets up the black hole as a background that has already determined the relevant quantum fields, and then introduces a test particle onto that background. If the test particle radiates a photon, the black hole will only react once the photon enters the horizon; unless and until that happens, the background is kept constant. (Hawking introduces negative energy particles in his formalism precisely to keep the background always constant.) Changing the background is tricky, but never involves real particles crossing from the interior of the horizon to the exterior.

> we would have as much/little reason to declare any other conservation also holds

Sure, no-hair as a theorem (rather than as a principle) only tells you that given classical vacuum, Maxwell's electromagnetism (in tensor form), and an eternal black hole metric, spacetime and all its contents are totally determined everywhere by a small handful of parameters. As a principle it suggests that perturbing that setup (e.g. by adding a source outside the horizon) does not lead to wildly inaccurate results.

I'm sorry that I don't understand what it is that you're asking in your second-last paragraph. There is a body of literature on black hole "mining" (it's a common thought-experiment when trying to distinguish between general relativity an alternative theory, especially a quantum one) that maybe touches on what you're curious about.

[raattgift]

> A black hole only has three properties, in our current understanding of general relativity - but "electrical charge" is one of those properties

An isolated black hole, at a suitable coordinate time, has mass, electric charge, angular momentum (three components), linear momentum (three components) and spatial position (three components). Holding the BH at the spatial origin drops the last six.

Also, a slight wrinkle: this state is asymptotic -- at timescales less than light-crossing there can be substantial additional hair. At longer timescales, some configurations can persist on much longer than light-crossing scales -- one example is the magnetic field at the newly formed horizon of a isolated collapsed rotating magnetar.

"isolated" here can get tricky in practice as well.

But in the usual case, no, you can't look at a black hole and tell whether someone much earlier threw (classical picture) in one shell of matter of mass M vs two concentric shells of matter at 1/2 M each or three concentric shells of matter of 1/4 M, 1/2 M and 1/4 M (or 1/2 M, 1/4 M and 1/4 M, etc.). But what's this shells picture for non-negligible charge? (Switching to a dust doesn't help, fwiw).

More formally, the no-hair theorem says that in a stationary electrovacuum, a black hole solution takes on a specific form. That mostly means that we should be able to perturb a Kerr-Newman BH solution and get the right results for an astrophysical BH.

[MrEldritch]

I would argue that "position" and "linear momentum" are more properties of one's particular choice of inertial reference frame than properties of the black hole itself specifically. And yes, angular momentum has three scalar components, but it's also one single vector. "Mass, charge, and angular momentum" makes three properties.

>at timescales less than light-crossing there can be substantial additional hair.

Ahh, thank you - that clears up some misconceptions of mine that have always confused me, like "Wait, so if black holes have no hair, how can they wobble and ring-down and produce gravitational waves after a black hole merger?"

[raattgift]

> inertial reference frame

Ditch that when thinking about black holes.

A BH's position in a general curved spacetime can be described by many arbitrary coordinate systems, but a black hole spacetime is not flat spacetime (by definition!) so special relativistic ideas involving frames of reference tend to fail pretty spectacularly.

As to linear and angular momentum and balding, I rather like these four sentences from Hawking & Penrose, "What the no-hair theorems show is that a large amount of information is lost when a body collapses to form a black hole. The collapsing body is described by a very large number of parameters. These are the types of matter and the multipole moments of the mass distribution. Yet the black hole that forms is completely independent of the type of matter and rapidly loses all the multipole moments except the first two: the monopole moment, which is the mass, and the dipole moment, which is the angular momentum." [1]

Merging black holes, from sufficient distance that resolving them individually is difficult, look very much like a collapsing body.

- --

[1] https://books.google.co.uk/books?id=6a-agBFWuyQC&lpg=PA39&pg... at the bottom of the page.

[raattgift]

> But there's nothing that stops it, say, eating protons and spitting out positrons later

Oh, I see what you mean, but it's not clearly because of symmetries breaking inside the horizon, rather than high-energy pairs taking energy from the background. Semiclassically: collapse an isolated star (here we depart from Hawking's formalism), and observe nothing but photons (with wavelengths comparable to the curvature radius) forming an atmosphere densest around ~ 3m \lt r \lessapprox 4m until m is very small, at which point you'll observe pair production in the atmosphere densest around ~ 4m. \lessapprox 4m is the back reaction mess of the inner atmosphere on chaotic and mostly plunging trajectories, and the highly dynamical part of the spacetime. [cf. Unruh https://link.aps.org/doi/10.1103/PhysRevD.15.365 nb 3rd paragraph, "It must be remembered that talk about particles is a very crude and metaphorical way of talking about the physics near the horizon of the black hole", and Giddings https://arxiv.org/abs/1511.08221 ] In the Hawking formalism the background is static, and that forces the use of an infalling negative energy; that's not a real symmetry being broken gravitationally, and so I'm wary about the idea of breaking baryon (and lepton) symmetry with black holes.

So really this is mostly a wordy objection to "spitting out".