Black Holes and Charged Particles

[DuckAmuck]

What happens when charged particles fall into a black hole?
Say like N electrons fall in, giving the black hole a net charge of -N.
Since light cannot escape the event horizon, I imagine electric fields cannot either, since they are mediated by photons.
So is that charge effectively lost until the black hole decays?

Or is there some kind of compensation mechanism with the virtual particles near the event horizon such that N positrons are pulled out of the vacuum, into the black hole to neutralize the net charge, and N electrons are ejected into space? This would conserve charge outside the black hole. Thanks.

 

[Nugatory]

So is that charge effectively lost until the black hole decays?
...
Or is there some kind of compensation mechanism with the virtual particles near the event horizon such that N positrons are pulled out of the vacuum, into the black hole to neutralize the net charge, and N electrons are ejected into space? This would conserve charge outside the black hole.

Neither. You end up with a charged black hole, described in the simplest case by the Reissner-Nordstrom solution to the Einstein Field Equations. Google will find many references, although much of it is (unavoidably) fairly math-intensive.

Since light cannot escape the event horizon, I imagine electric fields cannot either, since they are mediated by photons.

Although electric fields are mediated by photons, that does not mean that the electric field is carried by photons moving through space or crossing the horizon. The charge is still there and there's still an electric field outside the event horizons (note the plural! R-N black holes have two event horizons).

 

[BiGyElLoWhAt]

Well, you can't ever observe something cross the event horizon, you would just see it approach the event horizon and sit there, creating a static field.

 

[PeterDonis]

is that charge effectively lost until the black hole decays?

No. One key point to grasp that helps to resolve this apparent difficulty is that "charge" in GR is not a localized property of a particle; it's a property of the spacetime as a whole. A quick and dirty heuristic way to think of this is to imagine a charged particle sending out electric field lines; these lines go all the way out to infinity, and the presence and configuration of the field lines at infinity tells you about the presence of charge somewhere in the spacetime. Those field lines are still there even if there is a black hole present and the source of the charge falls into it.

Another important point to keep in mind is that, at any event in spacetime, the electromagnetic field at that event is entirely determined by the sources (charges and currents) that are present in the past light cone of the event. (The formal way to demonstrate this is to write everything in terms of retarded potentials, known as the Lienard-Wiechert potentials.) At any event outside the horizon, the past light cone lies entirely outside the horizon; so even if there are "now" no sources outside the hole, the electromagnetic field that you detect at that event is not actually coming from inside the hole. It's coming from the charges and currents that fell into the hole in the past.

 

[MeJennifer]

Well, you can't ever observe something cross the event horizon, you would just see it approach the event horizon and sit there, creating a static field.

Not true.

For the sake of argument we could all right now pass some event horizon.

 

[BiGyElLoWhAt]

I could see me pass an event horizon, but you couldn't see me pass an event horizon. Once I hit the event horizon, I will stop and sit there, as you see me.
https://en.wikipedia.org/wiki/Event_horizon

Light emitted from inside the event horizon can never reach the outside observer. Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon.

 

[PeterDonis]

Once I hit the event horizon, I will stop and sit there, as you see me.

No, that is not what an outside observer will see. An outside observer will see you get closer and closer to the horizon, moving more and more slowly, but never actually reach it.

Also, Wikipedia is not a good source to be using for these kinds of discussions, certainly not in an "A" level thread.

 

[BiGyElLoWhAt]

Technically, yes, but for all intents and purposes I'll eventually "stop".

 

[PeterDonis]

for all intents and purposes I'll eventually "stop".

Only if "eventually" means "never as seen by the outside observer". In the idealized model we are talking about here, the outside observer literally never sees you stop, even if he waits for an infinite amount of time by his clock.

Remember that this is an "A" level thread, and approximate statements that would be OK in a "B" or possibly even an "I" level thread are not appropriate here. We should be giving the most exact, precise, detailed statement of our best current theory, as applied to the situation under discussion, that we can.

 

[PAllen]

Also, putting together two of Peter's answers, for any event outside the horizon, the EM field originates from charges in the past positioned just outside the horizon; these are the events in the past light cone of some external event. On the other hand, EM fields exist inside the horizon(s) as well, and these fields may originate from charges both outside and inside the horizon(s) - the past light cone of an interior event includes much of the exterior.

 

[MeJennifer]

Light emitted from inside the event horizon can never reach the outside observer. Likewise, any object approaching the horizon from the observer's side appears to slow down and never quite pass through the horizon.

It all depends on how you define "the outside observer".

For instance an inertial observer B trailing a person A falling through the event horizon could remain in visual contact provided A's light signal reaches B after B falls through the event horizon.

 

[PeterDonis]

It all depends on how you define "the outside observer".

An observer that is outside the event horizon.

For instance an inertial observer B trailing a person A falling through the event horizon could remain in visual contact provided A's light signal reaches B after B falls through the event horizon.

And once B falls through the horizon, B is no longer an outside observer.

 

[MeJennifer]

And once B falls through the horizon, B is no longer an outside observer.

True, but B was an outside observer before and when A inside emitted light B cached up to it when he fell through it himself.

 

[PeterDonis]

B was an outside observer before

He isn't when he receives the light from A that you are talking about. That is the key point. We have been talking about what the outside observer sees; he only sees things when the light reaches him, not when it is emitted.

 

[MeJennifer]

He isn't when he receives the light from A that you are talking about. That is the key point. We have been talking about what the outside observer sees; he only sees things when the light reaches him, not when it is emitted.

We are actually agreeing...

 

[PeterDonis]

We are actually agreeing...

If so, that's good. I couldn't tell.

 

[Markus Hanke]

Is the diverging coordinate in-fall time for an external observer also true in R-N spacetime, which is different in many respects from the Schwarzschild case ?
Also, if you let a charged particle fall towards the black hole, then that charge should be affected by its very own electromagnetic field, meaning that in principle it wouldn't trace out a geodesic at all. How does that affect the total coordinate in-fall time as seen by an external observer ?

 

[TheMeInTeam]

Only if "eventually" means "never as seen by the outside observer". In the idealized model we are talking about here, the outside observer literally never sees you stop, even if he waits for an infinite amount of time by his clock.

Remember that this is an "A" level thread, and approximate statements that would be OK in a "B" or possibly even an "I" level thread are not appropriate here. We should be giving the most exact, precise, detailed statement of our best current theory, as applied to the situation under discussion, that we can.

With that being said, what would that look like? When I read up on it my understanding was that you'd never see someone cross, as you say. However, they'd still stop having light reach outside observers, due to crossing the event horizon. Rather than seeing a "stop", wouldn't you see a fading approach until you stopped seeing anything (IE it just fades enough that you can't detect it)? This is pretty difficult to visualize for me.

 

[PeterDonis]

Is the diverging coordinate in-fall time for an external observer also true in R-N spacetime, which is different in many respects from the Schwarzschild case ?

It is for an uncharged object falling into a R-N hole. The case of a charged object falling into an R-N hole is different, though; see below.

if you let a charged particle fall towards the black hole, then that charge should be affected by its very own electromagnetic field, meaning that in principle it wouldn't trace out a geodesic at all.

Yes, that's correct. A particle with the same charge as the hole will, heuristically, feel a force pushing it away from the hole, so it will fall in more slowly than an uncharged object, and will eventually decelerate, stop, and turn around and accelerate away. If this happens before the object reaches the horizon, then of course it will never fall in at all. If it happens after the object crosses the horizon--more precisely the outer horizon (there is also an inner one), things get more complicated, and I won't discuss that case further here; the only thing we would need to know is that the qualitative behavior will be the same as for the Schwarzschild case, but possibly with different details, as for the case of a particle with opposite charge, discussed below.

A particle with opposite charge from the hole will, heuristically, feel a force pulling it towards the hole, so it will fall in more quickly than an uncharged object. I actually have not seen a computation of how that affects what an observer who remains far away from the hole sees. My quick and dirty intuitive guess is that the faraway observer still sees the object appear to slow down more and more as it gets closer to the horizon, and never quite reach it. That's based on the fact that the horizon (more precisely the outer horizon) is still an outgoing null surface, where radially outgoing light remains at the same radial coordinate forever. The details of exactly how the slowdown is seen, as a function of the faraway observer's clock time, I would expect to be different from the Schwarzschild case; but I would expect the qualitative behavior to be the same.

 

[dllahr]

Thank you all for the discussions and explanations, @PeterDonis thank you for taking the time to explain in detail and with great heuristics. I have a follow up question - based on your discussion of the light cone of the event, assume for simplicity a single charged particle has fallen into a black hole. Based on measurements of the static field strength, would the charge appear to be at the center of the black hole, or would it be at the point where it crossed the event horizon?

 

[PeterDonis]

Based on measurements of the static field strength, would the charge appear to be at the center of the black hole, or would it be at the point where it crossed the event horizon?

What do you mean by "measurements of static field strength"? The particle isn't static, it's falling into the hole.

 

[dllahr]

Assume time long enough that there is no measurable change - e.g. it's finished falling.

 

[PeterDonis]

Assume time long enough that there is no measurable change - e.g. it's finished falling.

There is no such time. The particle continues to fall until it reaches the singularity, at which point it ceases to exist.

 

[PeterDonis]

The particle continues to fall until it reaches the singularity, at which point it ceases to exist.

Actually, there are complications here which are perhaps worth mentioning.

First, the answer in the quote above assumes a Schwarzschild black hole (i.e., an uncharged hole). But strictly speaking, if there is charge present anywhere, you can't have a Schwarzschild black hole. The hole should be a Reissner-Nordstrom black hole, i.e., a charged hole.

Second, in the Reissner-Nordstrom case, the inner horizon is a Cauchy horizon, which means, in the idealized case, that we can't actually predict what happens once the particle passes it. But in any real case, we expect that classical GR would break down prior to reaching the inner horizon. (Similarly, in the Schwarzschild case, we expect that classical GR would break down prior to reaching the singularity.) We would need a theory of quantum gravity to go beyond that point, and we don't have one.

These complications don't change the basic answer that there is no point where the particle is "finished falling" and there is "no measurable change". But it's good to recognize that in this domain, questions that might look simple really aren't.

 

[dllahr]

Does the Reissner-Nordstrom metric apply when a single charged particle falling past an event horizon? I don't think it does - it

is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. [https://en.wikipedia.org/wiki/Reissner–Nordström_metric]​

So I don't think we can say there is an inner / Cauchy horizon.

I don't think we need a theory of quantum gravity, because I'm not concerned about the small distances around/near/in the singularity. I'm asking if the static field appears to originate at the event horizon, or at the singularity (that's a substantial distance for stellar or supermassive black holes)

I really have to disagree with this statement - every instrument has a measurement error, hence there will at some point be no measurable change - the subsequent change will be smaller than can be measured by any instrument.

These complications don't change the basic answer that there is no point where the particle is "finished falling" and there is "no measurable change".​

Regardless of whether it's a Schwartzchild or Reissner-Nordstrum, once it crosses the outer event horizon, I would think we should not be receiving any "updates" - photons / radiation - indicating its movement toward the singularity or inner horizon. Hence it seems that the static field would appear as if the charge were permanently implanted at the event horizon where the particle crossed.

 

[PAllen]

I wonder if the gist of the question is that as the charged body is falling, from some distance away, the electric field around the system would be asymmetric. Then, what happens later? The answer is that it rapidly evolves toward symmetry, just as for gravity becoming symmetrical. These facts are consequences of the BH no hair theorems.

 

[PeterDonis]

Does the Reissner-Nordstrom metric apply when a single charged particle falling past an event horizon?

I don't think this case has been explicitly studied in the literature. My "yes" answer is based on the fact that charge is a conserved quantity, so either it's present in the 4-dimensional spacetime geometry or it's not. If it is, then the Reissner-Nordstrom metric should apply.

it
is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.

The exterior portion (i.e., outside the outer horizon) of the Reissner-Nordstrom geometry can be interpreted this way, yes. But if we have a black hole present, we can't just restrict ourselves to the exterior portion of whatever spacetime geometry the hole has. We have to consider the event horizon (which is the outer horizon in the R-N geometry) and the region below it.

I don't think we need a theory of quantum gravity, because I'm not concerned about the small distances around/near/in the singularity. I'm asking if the static field appears to originate at the event horizon, or at the singularity.

You're contradicting yourself. To even ask the question of whether the static field "appears to originate" at the singularity, you have to consider regions of spacetime arbitrarily close to the singularity.

every instrument has a measurement error, hence there will at some point be no measurable change - the subsequent change will be smaller than can be measured by any instrument.

Sorry, this is not valid reasoning. You really need to get more familiar with black hole spacetime geometries. You are using intuitive reasoning that is based on intuitions which are not valid at or below the event horizon of a black hole.

Regardless of whether it's a Schwartzchild or Reissner-Nordstrum, once it crosses the outer event horizon, I would think we should not be receiving any "updates" - photons / radiation - indicating its movement toward the singularity or inner horizon.

That's irrelevant to the question of where the static field "appears to originate". What your statement here really means is, once the particle crosses the outer event horizon, the geometry is indistinguishable, to an observer outside that horizon, from the R-N geometry. But where does the static field "appear to originate" from in the R-N geometry? There is no well-defined answer to this question. So there is no well-defined answer to your question either.

 

[dllahr]

I don't think you're being reasonable and I think this discussion has become unproductive, so I'm going to stop. I think I've hit a nerve because this isn't something with a well defined answer.

I don't think this case has been explicitly studied in the literature. My "yes" answer is based on the fact that charge is a conserved quantity, so either it's present in the 4-dimensional spacetime geometry or it's not. If it is, then the Reissner-Nordstrom metric should apply.
The exterior portion (i.e., outside the outer horizon) of the Reissner-Nordstrom geometry can be interpreted this way, yes. But if we have a black hole present, we can't just restrict ourselves to the exterior portion of whatever spacetime geometry the hole has. We have to consider the event horizon (which is the outer horizon in the R-N geometry) and the region below it.
You're contradicting yourself. To even ask the question of whether the static field "appears to originate" at the singularity, you have to consider regions of spacetime arbitrarily close to the singularity.
Sorry, this is not valid reasoning. You really need to get more familiar with black hole spacetime geometries. You are using intuitive reasoning that is based on intuitions which are not valid at or below the event horizon of a black hole.
That's irrelevant to the question of where the static field "appears to originate". What your statement here really means is, once the particle crosses the outer event horizon, the geometry is indistinguishable, to an observer outside that horizon, from the R-N geometry. But where does the static field "appear to originate" from in the R-N geometry? There is no well-defined answer to this question. So there is no well-defined answer to your question either.

 

[PeterDonis]

I think I've hit a nerve because this isn't something with a well defined answer.

You haven't "hit a nerve", but yes, the question you are asking does not have a well-defined answer. In such a case I'd rather just say so, instead of trying to cobble together an answer that doesn't really apply.

Perhaps it might clarify things somewhat to go back to your original formulation of the question:

would the charge appear to be at the center of the black hole, or would it be at the point where it crossed the event horizon?

This question does not have a well-defined answer, because it's not clear what "where the charge appears to be" means. Intuitively, if we have an ordinary object that is charged, we would say the charge "appears to be" at the center of the object, if we have to pick a single point. But there is no "center" of a black hole in this sense. That is to say, there is no "point in space" that marks the center of the hole. "Space" in a black hole spacetime simply does not work the way it does in our ordinary intuitions.

 

[nikkkom]

What your statement here really means is, once the particle crosses the outer event horizon, the geometry is indistinguishable, to an observer outside that horizon, from the R-N geometry.

For the outside observer, this never happens, right? It takes infinite time, as observed from outside, to cross the horizon?

 

[PeterDonis]

For the outside observer, this never happens, right? It takes infinite time, as observed from outside, to cross the horizon?

No. The outside observer will never see the particle cross the horizon. But the outside observer cannot deduce from that that the horizon crossing never happens or that it takes infinite time.

 

[nikkkom]

No. The outside observer will never see the particle cross the horizon. But the outside observer cannot deduce from that that the horizon crossing never happens or that it takes infinite time.

Outside observer can bounce a radar pulse off the falling object.

There are practical problems both with trying to do that with the "object" being a single electron, with the long time to get the pulse come back, and with red/blueshifting ("radio" waves may be seen as gamma ray by the object, and even if they do bounce off it, they will be red-shifted on the way back), but theoretically existence of the falling object is not unobservable.

 

[PAllen]

Outside observer can bounce a radar pulse off the falling object.

There are practical problems both with trying to do that with the "object" being a single electron, with the long time to get the pulse come back, and with red/blueshifting ("radio" waves may be seen as gamma ray by the object, and even if they do bounce off it, they will be red-shifted on the way back), but theoretically existence of the falling object is not unobservable.

No they can’t. There is a well defined time after which any light pulse sent sent by outside observer towards infaller will reach infaller only inside the horizon or not at all

 

[dllahr]

So this is much simpler than @PeterDonis is making it. The charge in the Reissner–Nordström metric is located at the singularity:
https://en.wikipedia.org/wiki/Reissner–Nordström_metric#Charged_black_holes
In the section above, the electromagnetic potential is defined:

So yes, we know effectively where the charge is. We don't need to invoke a lot of obfuscating mumbo-jumbo.

 

[PeterDonis]

The charge in the Reissner–Nordström metric is located at the singularity

No, it isn't. The "singularity" isn't even a single point, and it's certainly not a spatial point at the "center" of the black hole. Spacetime geometry in this case does not work the way you are assuming it does.