Help Understanding Black Hole Theories

[sciencegem]

Hi,
I think I have several misconceptions about the theoretical framework of black holes, I'm just not sure where my intuition (or, lack of) goes wrong. So sorry if this sounds really stupid, any help is appreciated.
The scenario that I find confusing is what Alice sees when she throws Bob into a black hole, and what happens to Bob as he passes the event horizon (other than the extreme tidal forces he experiences). As far as I understand, Bob passes the event horizon fine using proper time, but Alice is in hyperbolic coordinates (don't know if that's the right name) so she sees Bob asymptotically approach the horizon for infinity but never cross it. The signal every tick of Bob's clock sends back to Alice is increasingly redshifted so she sees Bob's time slow down. Now, assuming I got all that right, here's what I don't get. If Bob brought everything but the kitchen sink with him into the black hole, than theoretically Alice would see Bob and everything but the kitchen sink outside the black hole forever. Wouldn't black holes be kinda luminous than? Would Alice never see the black hole grow as the result of Bob and everything but the kitchen sink? Don't we "see" black holes grow, and call them black for a reason? Does this have anything to do with the Unruh effect (which I've heard of but know nothing about) or the radiation jets I see in pictures but also know nothing about? Also, the equation I know for the Schwarzschild metric is dτ^2=[1-(2MG/r)]dt^2 -[1-(2MG/r)]^-1dr^2 -dΩ^2 . So after passing the horizon (r=2MG), the t and r components of the metric switch signs as if time and space have swapped roles. What am I missing there? Finally, does anyone have any good references on Rindler coordinates? When I was trying to read up on this there was something about the singularity lying on the time axis therefore being time-like rather than spatial--which I'm sure I just stupidly misunderstood but I'd like som intuition about that too.
A huge thanks to anybody who read all that! And once again, I apologize for my chaotically incorrect reasoning. Any hints are appreciated!Sent from my iPhone using Physics Forums

 

[PeterDonis]

Hi, sciencegem! Black holes are certainly counterintuitive, and it takes everyone some time when they first come to grips with how they work.

Bob passes the event horizon fine using proper time

Yes, although a better way of saying it is that the proper time for Bob to reach the horizon ("proper time" is just the time elapsed on a clock that Bob carries with him) is finite. Proper time is not something you "use"; it's just the mathematical representation of the elapsed time on a clock following a particular path through spacetime.

Alice is in hyperbolic coordinates (don't know if that's the right name) so she sees Bob asymptotically approach the horizon for infinity but never cross it.

Yes, although Alice is not "in" a particular set of coordinates, and what she sees doesn't depend on what coordinates she uses. As you say, she sees Bob asymptotically approach the horizon but never sees him reach it.

The signal every tick of Bob's clock sends back to Alice is increasingly redshifted so she sees Bob's time slow down.

Yes, each signal is increasingly redshifted; but that also means that signals that Bob emits at constant intervals by his clock (for example, once every second by his clock) reach Alice further and further apart by her clock. That is why Bob's time appears to "slow down"; the images of Bob's clock that Alice sees appear to tick slower and slower, because they are coming to Alice further and further apart by her clock.

Wouldn't black holes be kinda luminous than?

No, because of the redshift of the light emitted by objects falling in. The description above of what Alice sees is highly idealized: it assumes that Alice can detect light of arbitrarily low frequency/arbitrarily long wavelength. No real detector can do that; in any real case, the light from Bob, or anything Bob is carrying with him such as kitchen sinks, would fairly quickly become so redshifted that Alice would no longer be able to detect it.

Would Alice never see the black hole grow as the result of Bob and everything but the kitchen sink?

No, because the measurements Alice makes to determine the mass of the hole don't depend on what happens to Bob as he gets close to the horizon, or what happens after he falls through the horizon. After Bob passes Alice, if she makes repeated measurements of the hole's mass, she will measure it to increase to its new value, including Bob's mass, even though she never sees him reach the horizon. Basically, the mass she measures doesn't just include the mass inside the horizon; it includes any mass that's closer to the horizon than she is. (Technically, that's only true as I stated it for the idealized case of a perfectly spherically symmetrical black hole and a perfectly spherically symmetrical shell of matter falling in; but the basic conclusion remains the same for cases like Bob, who certainly isn't a spherically symmetric shell of matter, falling in; all that changes are some of the mathematical details.)

Does this have anything to do with the Unruh effect

Not as far as objects falling in, no. There are analogies between the Unruh effect and Hawking radiation, which is a quantum phenomenon that, as far as we know, doesn't affect the basic picture of objects falling into the horizon that I described above.

the radiation jets I see in pictures but also know nothing about?

Those are generated by rotating black holes, which are more complicated. I would not try to tackle those until you're clear about the simpler case of non-rotating holes, which is what I was describing above.

after passing the horizon (r=2MG), the t and r components of the metric switch signs as if time and space have swapped roles.

This is only true of the particular coordinates you mention, which are called Schwarzschild coordinates. Since these coordinates are singular at the horizon, a more correct way of saying what you are saying here is that there are two Schwarzschild coordinate patches: the "exterior" patch, covering the region outside the horizon, and the "interior" patch, covering the region inside the horizon. With the standard coordinate labels being held the same between the two patches, the $t$ coordinate is timelike in the exterior and spacelike in the interior, while the $r$ coordinate is spacelike in the exterior and timelike in the interior. This is sometimes described as time and space "swapping roles", but that can be misleading, because $t$ and $r$ are just coordinates; neither one has any necessary connection to "time" or "space". (Also, there are other coordinate charts in which $r$, although it is defined the same way, is spacelike everywhere, so the "role" a coordinate plays depends on the chart.)

does anyone have any good references on Rindler coordinates?

The Wikipedia page is a reasonably good start:

http://en.wikipedia.org/wiki/Rindler_coordinates

When I was trying to read up on this there was something about the singularity lying on the time axis therefore being time-like rather than spatial

This seems garbled, so either you misread something or the reference you were getting this from didn't do a good job of explaining. The "singularity" in Rindler coordinates is only a coordinate singularity, and it corresponds to a pair of null lines, so it isn't either timelike or spacelike. The Wikipedia page talks about this (it uses the term "Rindler horizon" for what you're calling the "singularity").

 

[sciencegem]

That was a truly awesome explanation PeterDonis, thank you!