34 08 June 2012 11:43PM

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Comment author: 03 July 2012 09:10:48PM 1 point [-]

That is a good question. There is more than one way to formulate the answer in nonmathematical terms, but I'm not sure which would be the most illuminating.

One is that the electromagnetic force (as opposed to electromagnetic radiation) is transmitted by virtual photons, not real photons. No real, detectable photons escape a charged black hole, but the exchange of virtual photons between a charge inside and one outside results in an electric force. Virtual particles are not restricted by the rules of real particles and can go "faster than light". (Same for virtual gravitons, which transmit the gravitational force.) The whole talk of virtual particles is rather heuristic and can be misleading, but if you are familiar with Feynman diagrams you might buy this explanation.

A different explanation that does not involve quantum theory: Charge and mass (in the senses relevant here) are similar in that they are defined through measurements done in the asymptotic boundary of a region. You draw a large sphere at large distance from your black hole or other object, define a particular integral of (respectively) the gravitational or the electromagnetic field there, and its result is defined as the total mass/charge enclosed. So saying a black hole has charge is just equivalent to saying that it is a particular solution of the coupled Einstein-Maxwell equations in which the electromagnetic field at large distances takes such-and-such form.

Notice that whichever explanation you pick, the same explanation works for charge and mass, so the peculiarity of gravity not being part of the energy-momentum tensor that I mentioned above is not really relevant for why the black hole attracts you. Where have you read this?

Comment author: 05 August 2012 11:42:46AM *  0 points [-]

Hi Alejandro, I just remembered I hadn’t thanked you for the answer. So, thanks! :-)

I don’t remember where I’ve seen the explanation (that gravity works through event horizons because gravitons themselves are not affected), it seemed wrong so I didn’t actually give a lot of attention to it. I’m pretty sure it wasn’t a book or anything official, probably just answers on “physics forums” or the like.

For some reason, I’m not quite satisfied with the two views you propose. (I mean in the “I really get it now” way, intellectually I’m quite satisfied that the equations do give those results.)

For the former, I never really grokked virtual particles, so it’s kind of a non-explanatory explanation. (I.e., I understand that virtual particles can break many rules, but I don’t understand them enough to figure out more-or-less intuitively their behavior, e.g. I can’t predict whether a rule would be broken or not in a particular situation. It would basically be a curiosity stopper, except that I’m still curious.)

For the latter, it’s simply that retreating to the definition that quickly seems unsatisfying. (Definitions are of course useful, but less so for “why?” questions.)

The only explanation I could think of that does make (some) intuitive sense and is somewhat satisfactory to me is that we can never actually observe particles crossing the event horizon, they just get “smeared”* around its circumference while approaching it asymptotically. So we’re not interacting with mass inside the horizon, but simply with all the particles that fell (and are still falling) towards it.

( * : Since we can observe with basically unlimited precision that their height above the EH and vertical speed is very close to zero, I can sort of get that uncertainty in where they are around the hole becomes arbitrarily high, i.e. pretty much every particle becomes a shell, kind of like a huge but very tight electronic orbital. IMO this also “explains” the no-hair theorem more satisfyingly than the EH blocking interactions. Although it does get very weird if I think about why should they seem to rise as the black hole grows, which I just dismiss with “the EH doesn’t rise, the space above it shrinks because there are more particles pulling on it”, which is probably not much more wrong than any other “layman” explanation.)

Of course, all this opens a different** can of worms, because it’s very unintuitive that particles should be eternally suspended above an immaterial border that is pretty much defined as no-matter-how-hard-you-try-you'll-still-fall-through-it. But you can’t win them all, and anyway it’s already weird that falling particles see something completely different, and for some reason relativity always seemed to me more intuitive than quantum physics, no matter how hairy it gets.

(**: Though a more accurate metaphor would probably be that it opens the same can of worms, just on a different side of the can...)

Comment author: 08 August 2012 03:11:24AM *  0 points [-]

OK, here is another attempt at explanation; it is a variation of the second one I proposed above, but in a way that does not rely on arguing by definition.

Imagine the (charged, if you want) star before collapsing into a black hole. If you have taken some basic physics courses, you must know that the total mass and charge can be determined by measurements at infinity: the integral of the normal component of the electric field over a sphere enclosing the star gives you the charge, up to a proportionality constant (Gauss's Law), and the same thing happens for the gravitational field and mass in Newton's theory, with a mathematically more complicated but conceptually equivalent statement holding in Einstein's.

Now, as the star begins to collapse, the mass and charge results that you get applying Gauss's Law at infinity cannot change (because they are conserved quantities). So the gravitational and electromagnetic fields that you measure at infinity do not change either. All this keeps applying when the black hole forms, so you keep feeling the same gravitational and electric forces as you did before.

Comment author: 10 August 2012 07:23:55PM *  0 points [-]

Yeah, you’re right, putting it this way at least seems more satisfactory, it certainly doesn’t trigger the by-definition alarm bells. (The bit about mass and charge being conserved quantities almost says the same thing, but I think the fact that conservation laws stem from observation rather than just labeling things makes the difference.)

However, by switching the point of view to sphere integrals at infinity it sort of side-steps addressing the original question, i.e. exactly what happens at the event horizon such that masses (or charges) inside it can still maintain the field outside it in such a state that the integral at infinity doesn’t change. Basically, after switching the point of view the question should be how come those integrals are conserved, after the source of the field is hidden behind an event horizon?

(After all, it takes arbitrarily longer to pass a photon between you and something approaching an EH the closer it gets, which is sort of similar to it being thrown away to infinity the way distant objects “fall away” from the observable universe in a Big Rip, it doesn’t seem like there is a mechanism for mass and charge to be conserved in those cases.)

Comment author: 10 August 2012 10:36:43PM 0 points [-]

how come those integrals are conserved, after the source of the field is hidden behind an event horizon?

First, note that there are no sources of gravity or of electromagnetism inside a black hole. Contrary to popular belief, black holes, like wormholes, have no center. In fact, there is no way to tell them apart from outside.

Second, electric field lines are lines in space, not spacetime, so they are not sensitive to horizons or other causal structures.

it takes arbitrarily longer to pass a photon between you and something approaching an EH

This is wrong as stated, it only works in the opposite direction. It takes progressively longer to receive a photon emitted at regular intervals from someone approaching a black hole. Again, this has nothing to do with an already present static electric field.

Comment author: 11 August 2012 01:11:06PM *  0 points [-]

how come those integrals are conserved, after the source of the field is hidden behind an event horizon?

First, note that there are no sources of gravity or of electromagnetism inside a black hole. Contrary to popular belief, black holes, like wormholes, have no center.

For your second sentence, I sort of get that; there’s no point one can travel to that satisfies any “center” property; the various symmetries would have a center on finitely-curved spacetime, but for a black hole that area gets stretched enough that you can only define the “center” as a sort of limit (as far as I can tell, you can define the direction to it, it’s just infinitely far away no matter where you start from—technically, the direction to it becomes “in the future” once the EH forms, right?). However, I didn’t say “center”, I said just “behind the EH”. “Once” a particle “crosses” it already seems as it should no longer have an influence to the outside.

Basically, intuition says that we should see the mass (or charge, to disentangle the generated field from the spacetime) sort of disappear once it crosses. Time slowing near the EH would help intuition because it suggests we’d never see the particle cross (thus, we always see a charge generating the field we’re measuring), but we’d see it redshift (signals about it moving take longer to arrive, thus the field becomes closer to static), it’s just that I’m not sure I’m measuring that time from the right reference frame.

it takes arbitrarily longer to pass a photon between you and something approaching an EH

This is wrong as stated, it only works in the opposite direction. It takes progressively longer to receive a photon emitted at regular intervals from someone approaching a black hole.

OK, wait a minute. Are you saying that if a probe falls to a BH, a laser on the probe sends pulses every 1s (by its clock), and a laser on my orbiting Science Vessel shines a light on it ever 1s (by my clock), I’ll see the probe’s pulses slow down, but my reflected pulses will return with 1Hz, just redshifted further (closer to a static field) the closer the probe falls? That seems weird, but it might be so, my intuition kind of groans for these setups.

But there must be some formulation around those lines that works, I’m just too in love with my “smearing” intuition. And I really feel a local explanation is needed, the integral at infinity basically only explains the mass of the black hole (how strongly it pulls), not its position (where it pulls towards).

I’m having a bit of trouble to explain my conflicting intuition, because stretching space affects both distance and redshift. If I understand correctly, the closer something is to an EH (as measured in our external-but-finite-distance-away reference frame), the further it is redshifted. So we can’t see it crossing pretty much because it’s too dark. But, in our reference frame, does it seem to be still approaching the EH, or did it also seem to stop above it before disappearing due to redshift? Another formulation: denoting d the distance between the two masses, D the Schwarzschild radius of the combined masses, and R the redshift of signals sent by the probe mass, all measured in our reference frame, outside but at a finite distance from the experiment, my understanding is that R(d) goes to infinity as d nears D from above; however, what is d(t) doing in that vicinity, is it nearing zero or going to (negative) infinity as well? Remember, d and t are measured in our reference frame. If we get ridiculously better instruments from Omega, can we observe the “impact” further into the future, or just closer to a fixed point. E.g., say our old instruments can measure until 11:59, afterwards the redshift is too much. If we get arbitrarily better ones, do we get to see until, say, 12:36, or is there a limit like 12:00 that you can approach but can’t cross no matter how good your telescopes are, and we get to see, say, 11:59.9 with d'(t) = 0.9 c, 11:59.99 with d'(t) = 0.99c and so on?

Consider the following mental experiment: A test particle of mass m falls towards a black hole of mass M. (For extra points, the test particle could be a small black hole, itself.)

At t_0, the two masses are a distance away, and the gravitational field (or electric one, if the BH and test mass are charged), when tested closely enough but not “touching” any of the two masses, looks like that generated by two point masses at a certain distance.

At t_h, the test particle is very close to the EH of the big black hole. (This is somewhat easier to define for a mini-blackhole as a test particle, just say that the distance between their event horizons is smaller than an epsilon.) So, at t_h, by probing closely but not too close, we should see a field looking like one generated by two point particles, at a distance a bit larger than the Schwarzschild radius of their combined masses.

At some point t_H > t_h, we should see just a black hole of mass (M+m), and the no hair theorem says that (if we don’t go too near to the EH, but definitely closer than infinity) it should look like the field is generated by a point mass of (M+m) at the center of gravity of the original system. Looking carefully a bit closer, we should see a slightly larger EH around that (more-or-less imaginary) point.

But what happens between t_h and t_H? I see no reason for discontinuity, so that means that we should be able to see two point masses getting closer and touching. (Or, more precisely, by measuring the field at a finite distance during that interval, we should see a (non-static) field that looks like one generated by two point masses getting closer and closer until they touch.)

Within the “smearing” intuitive view, that works out quite nicely: we see two point masses going nearer and nearer. But the closer the test mass is to the EH, the more its position is “smeared” around the BH. Basically the field we measure continuously keeps looking like two point masses approaching, but we can no longer tell where each is (the orientation of the line connecting the two points becomes unspecified as the line’s extremities rotate at lightspeed), so rotational momentum pretty much becomes pure spin.

In other words, between t_0 and t_h the test mass spins faster and faster, such that at t_crossing it rotates at lightspeed, one of the space dimensions becomes time, and we (already) no longer can tell where it is around the BH “center”. Which means that at “t_crossing” the gravitational field already has only mass and spin.

(Side-note 1: I sort of see two individually non-rotating black holes as having spherical EHs, and a rotating one as having an ellipsoid one, further away from spherical the faster it rotates. The above thought experiment asks more-or-less how do two spheres become an ellipsoid (or two ellipsoids become a bigger, less spherical one) when two BH merge. In my intuition, the two spheres simply rotate so far around one another we can no longer tell which is which, sort of blurring together. This seems very intuitive, although it gets a bit complicated to explain what happens when the total angular momentum happens to be zero, and I get nothing trying to explain what happens when a two–non-rotating–BHs system with zero total angular momentum collapses. I.e., if a non-rotating test black-hole falls straight into a non-rotating black hole, resulting in a bigger non-rotating black-hole, I don’t get at all how we get from observing a field that looks like it’s generated by two point masses to one that looks generated by just one, with no rotation. Which does suggest that what my intuition matches the rotating case just by accident.)

(Side-note 2: I was under the impression that time slowing down near the horizon, such that we never see anything cross, meant we “see” things hovering just above it, redshift making it just harder to see. But given that we see big black holes and astronomers say they became bigger in time, and your comment, it might have meant just that they disappear through red-shift before we can see them cross. Is that so? Is the positional uncertainty semi-explanation my intuition feeds me total poppy-cock, with just the “infinite” rotation speed being the only reason for the “hairs” disappearing? If so, that sort of kills my hope that it would make evaporation easier to swallow—tunneling from being frozen and red-shifted just above the horizon, and more importantly carrying back information, seems more intuitive than virtual particles becoming real just because of space flowing fast enough around them, which AFAIK is the usual explanation and doesn’t explain at all what’s going on with entropy.)

Comment author: 11 August 2012 06:37:07PM 0 points [-]

TL:DR :)

I recommend learning the Penrose space-time diagrams, they make things intuitive.

Comment author: 06 August 2012 03:26:28AM 0 points [-]

I'm sorry that my explanations didn't work for you; I'll try to think of something better :).

Meanwhile, I don't think it is good to think in terms of matter "suspended" above the event horizon without crossing it. It is mathematically true that the null geodesics (lightray trajectories) coming from an infalling trajectory, leaving from it over the finite proper time period that it takes for it to get to the event horizon, will reach you (as a far-away observer) over an infinite range of your proper time. But I don't think much of physical significance follows from this. There is a good discussion of the issue in Misner, Thorne and Wheeler's textbook: IIRC, a calculation is outlined showing that, if we treat the light coming from the falling chunk of matter classically, its intensity is exponentially suppressed for the far-away observer over a relatively short period of time, and if we treat it in a quantum way, there are only a finite expected amount of photons received, again over a relatively short time. So the "hovering matter" picture is a kind of mathematical illusion: if you are far away looking at falling matter, you actually do see it disappear when it reaches the event horizon.