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.
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 by bogdanb
· score: 0 (0 votes) · LW
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.)