# On the nature of the "infinite" fall toward the EH

by rjbeery
Tags: fall, infinite, nature
Mentor
P: 16,283
 Quote by PeterDonis Adding Hawking radiation and evaporation of the hole changes the spacetime, so that light rays from the infalling observer as he gets closer and closer to the horizon no longer take a time approaching infinity to get out; instead, they take a time approaching the time it takes for the distant observer to see the hole's final evaporation.
This seems intuitively like the right conclusion. Do you have a supporting reference?
PF Gold
P: 4,777
 Quote by Mike Holland Sorry, PAllen, but I still have a problem with this. We observe time dilation at the surface of the Sun by studying the effect on spectral lines and such. But what we measure is affected by the dilation we experience here on Earth, so there is a coordinate factor present. However, we can calculate the effect of Earth's gravity field and also the effect of our orbital movements, etc, and derive an ideal reference frame, like that of the remote observer used in the O-S calculations. Once we have eliminated local effects, which are all calculable, we arrive at an "absolute" time dilation value for the surface of the Sun. Any other obserrver in the galaxy can do the same compensating calculations and arrive at the same value for time dilation at the Sun's surface. So this value is coordinate-independant.
False. What you are doing is isolating a family emitter world lines that encounter a nearly static metric in the vicinity of the sun, and using these to define coordinates near the sun. This is not even possible in general. See next comment.
 Quote by Mike Holland Admittedly this coordinate frame we use in our calculations is an unattainable ideal, just as absolute zero is in temperature measurements. But using this ideal we can ascertain time dilation values which depend only on the mass present and the distance from it, just as O-S did in their calculations. Edit: OK, so my ideal coordinate frame is still a coordinate frame, but is is one that would be used by astronomers everywhere, and it is a "special" one, much as you might dislike that idea.
It is possible to do even remotley as you suggest because there is only one major gravitating body in some large region. Replace the sun with pair tightly co-orbiting neutron stars and you are SOL (hint: time varying metric perturbations not centered on either body would be significant). Meanwhile, the coordinate independent definition of GR Doppler is unaffected, and defines how any emitter, on any world line, anywhere in the vicinity of the binary would be shifted for any given receiver world line further away. Both (dynamic) curvature and different states of emitter and receive motion would have an impact. But you would be unable to define something you call gravitational time dilation. The latter is not a general GR feature; it is something you can define in sufficiently simple spacetimes to simplify calculation. It never necessary. It is not manifested in perfectly good coordinates for simple spacetimes (e.g. the Fermi-Normal coordinates of a free fall observer).

So, I still claim, no exceptions in SR or GR:

- time dilation is a coordinate feature, not an observable.
- Doppler between a chosen emitter and a chosen receiver is an invariant observation.
- Differential aging between different space time paths between given events is an invariant observation.
PF Gold
P: 4,777
 Quote by DaleSpam Point well taken. However, what is coordinate independent is that the later light paths from Lucky do not reach Unlucky before Unlucky hits the singularity.
Correct, indisputable.
Physics
PF Gold
P: 5,321
 Quote by DaleSpam This seems intuitively like the right conclusion. Do you have a supporting reference?
There's a Penrose diagram of an evaporating black hole on p. 200 of Carroll's lecture notes. Also, the thread in the astrophysics area that was linked to earlier has some good discussion and links; see in particular this post by George Jones:

http://www.physicsforums.com/showpos...9&postcount=23
PF Gold
P: 4,777
 Quote by stevendaryl Well, the original poster went on to talk about Hawking radiation, and the question of reconciling two points of view: From the point of view of Schwarzschild coordinates (modified suitably to allow for a slow time-dependence in the M parameter), the black hole evaporates BEFORE the infalling observer reaches the event horizon. From the point of view of the infalling observer, the infalling observer reaches the singularity in a finite amount of proper time, presumably long before Hawking radiation would be relevant. There really is no definitive way to resolve this without a quantum theory of gravity, although it seems that there should be a qualitative way of understanding how these are not contradictory. For someone falling into a black hole, it's all over in a short amount of time--you pass through the event horizon and hit the singularity pretty quickly (for small black holes, anyway). It wouldn't seem that Hawking radiation would change this picture very drastically, because Hawking radiation is pretty puny; it shouldn't make a big change to the geometry of the black hole, except after long, long, long periods of time. On the other hand, from the point of view of a distant observer, the black hole evaporates in a finite amount of time. What happens to the infalling observer, then? This puzzle is really not about classical General Relativity, since it involves quantum corrections. But if there are any real black holes in the universe, then they're going to be quantum black holes, not classical black holes. So it would be nice to have a qualitative understanding of quantum black holes, even if a definitive understanding is years away (if ever). It would be nice to have a feel for which features of the classical description of a black hole are likely to be present (approximately, anyway) in a more realistic black hole, and which features are likely to be completely tossed out in a quantum theory of black holes.
Yes it would. Unfortunately, the correct answer is not known. Somewhere in this thread I posted links to a 2007 paper by Krauss et.al. that argues one position; and a paper by Padmanabhan et.al. from 2009 that claims to refute the former. My belief is that the 2009 paper represents the 'majority view' (and I can't find any response to it from the 2007 authors), but it is far from 'settled physics'. Without responding to the 2009 paper, there are certainly new papers written in the framework of the 2007 paper. It appears to me that both string theory and LQG are more consistent with the framework of the 2009 paper, as is Hawking's proposal for resolving the information paradox.
PF Gold
P: 4,777
 Quote by PeterDonis I'm not sure I agree; I think in this case the same reasoning would apply that I gave before, at least in part: there will at least be a subset of surfaces of constant time in such a chart can be mapped one-to-one to the ingoing null rays that Lucky emits from e1 to e2, and must be ordered the same way. It won't be *all* of the surfaces of constant time now, because the region covered by the chart is bounded by two ingoing null rays, so any spacelike surfaces in this region will "exit" the chart on at least one side. But I think that having even a subset of spacelike surfaces mapped one-to-one to ingoing null rays is sufficient.
I don't agree. I think this will be forced only if e1 and e2 are too far apart (or if you try to include too much outside the region defined by two ingoing null paths, Lucky world line, and the singularity). That is, there is a global prohibition, but not quasi-local problem.

Think of the Kruskal chart, and singularity region in the right half (that matches the GP singularity ordering). Specifically, to make it easy, think of a two singularity arrival events that are nearly horizontal in the chart, and close together, and connect them out with light paths to some static, radial, external world line. Now, within this sliver, we just change all simultaneity surfaces by one degree from horizontal, counterclockwise.

[Edit: Actually, K-S shows a stronger result. It is global, and reverses GP singluarity arrival ordering for 'half' the singularity arrival events. It seems to me, that distortions of K-S can reverse GP ordering in a consistent global chart for all singularity arrival events before (in GP sense) any chosen one. What you can't do is accomplish this for the whole singularity arrival sequence.]
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PF Gold
P: 5,321
 Quote by PAllen think of a two singularity arrival events that are nearly horizontal in the chart, and close together, and connect them out with light paths to some static, radial, external world line. Now, within this sliver, we just change all simultaneity surfaces by one degree from horizontal, counterclockwise.
I see what you mean, but I'm not sure the time ordering on the singularity will be monotonic if you do this. I don't really trust my powers of visualization for this, so I'll have to think about it some more to see if I can come up with a mathematical way to tackle it.
PF Gold
P: 4,777
 Quote by PeterDonis I see what you mean, but I'm not sure the time ordering on the singularity will be monotonic if you do this. I don't really trust my powers of visualization for this, so I'll have to think about it some more to see if I can come up with a mathematical way to tackle it.
Just consider the simple transform (producing ugly metric):

V' = V - k U
U' = U

using the conventions where V is the K-S time coordinate, -1 < k < 1. Lines of constant V' are spacelike everywhere; lines of constant U' are the same as lines of constant U. While the metric gets ugly, it is not hard to see that increasing k towards 1 shifts the inflection in singularity ordering as far to the right as desired; decreasing k towards -1 shifts the inlection to the left. So, for any two events on the singularity, you can get an ordering where the left is first for some k close to -1, and where the right is first for some k close to 1. Each of these charts is global, with the same time ordering for causal curves as the original K-S chart.

This fully justifies (better late than never) my original statement that for any two light signals reaching the singularity from Lucky, Lucky can consider the arrival events to be the reverse of the emission events. The only thing Lucky can't do is achieve such an inversion over the whole history of a static world line. It can be achieved for any segment of interest, but not for the whole past/future eternal history.

Also, note that none of this contradicts Dalespam's improved wording: If light from e1 reaches Unlucky as Unlucky reaches the singularity, light from any event e2, later on Lucky's world line, will not reach Unlucky at all. This wording is coordinate independent. Wording on the order of the two singularity arrival events is coordinate dependent (as expected by the spacelike relation between them).

A final observation is that Lucky can achieve total time order on the singularity consistent with their world line time order using a variety of coordinates (Lemaitre, GP, EF, etc.). A mirror Lucky in region III would use a mirror version of each these coordinate systems to achieve a total singularity ordering consistent with their world line.
Physics
PF Gold
P: 5,321
 Quote by PAllen increasing k towards 1 shifts the inflection in singularity ordering as far to the right as desired; decreasing k towards -1 shifts the inlection to the left.
Yes, but there will always *be* an inflection point; you can never produce a completely monotonic ordering on the singularity this way. That's all I am saying; that a *monotonic* ordering on the singularity can't be reversed without also reversing the time ordering of events on timelike curves (unless you restrict attention only to portions of timelike curves inside the horizon). No K-S style chart gives a monotonic ordering.

 Quote by PAllen This fully justifies (better late than never) my original statement that for any two light signals reaching the singularity from Lucky, Lucky can consider the arrival events to be the reverse of the emission events.
Only if he's willing to accept a non-monotonic ordering of events on the singularity. The emission events are outside the horizon, so there's no way to obtain a reversed monotonic ordering of all events on the singularity that keeps the ordering of emission events the same. If you only want to reverse the arrival events, but allow the complete ordering to be non-monotonic, then yes, you can always do that, as you have shown.

 Quote by PAllen Also, note that none of this contradicts Dalespam's improved wording: If light from e1 reaches Unlucky as Unlucky reaches the singularity, light from any event e2, later on Lucky's world line, will not reach Unlucky at all. This wording is coordinate independent.
Yes, agreed.

 Quote by PAllen A final observation is that Lucky can achieve total time order on the singularity consistent with their world line time order using a variety of coordinates (Lemaitre, GP, EF, etc.).
Yes, and once he's done this, he can't reverse that order while still keeping the ordering the same on his own worldline. (In fact, he can't even reverse it and still *cover* his own worldline; see below.)

 Quote by PAllen A mirror Lucky in region III would use a mirror version of each these coordinate systems to achieve a total singularity ordering consistent with their world line.
Yes, but any such coordinate chart won't cover region I at all. So Lucky and mirror Lucky can never have a common chart that (1) covers both of their worldlines, and (2) agrees on a monotonic ordering of events on the singularity.
PF Gold
P: 4,777
 Quote by PeterDonis Only if he's willing to accept a non-monotonic ordering of events on the singularity. The emission events are outside the horizon, so there's no way to obtain a reversed monotonic ordering of all events on the singularity that keeps the ordering of emission events the same. If you only want to reverse the arrival events, but allow the complete ordering to be non-monotonic, then yes, you can always do that, as you have shown.
Except possibly as a brief, initial speculation, corrected almost immediately, I never claimed monotonic was possible. After backing off from that, everything else I thought turned out to be justified; much more than just a chart bounded e1 to e2 on Lucky's world line, light rays to the singularity, and the singularity arrival events - that reverses arrival order relative to transmission order. Instead, the whole of the K-S manifold can be covered, reversing e1 and e2 arrival; all that can't be done is to reverse the entire singularity arrival ordering.
PF Gold
P: 4,777
 Quote by PeterDonis Yes, but any such coordinate chart won't cover region I at all. So Lucky and mirror Lucky can never have a common chart that (1) covers both of their worldlines, and (2) agrees on a monotonic ordering of events on the singularity.
Yes, I completely understand this. I don't think I suggested otherwise.
Physics
PF Gold
P: 5,321
 Quote by PAllen Except possibly as a brief, initial speculation, corrected almost immediately, I never claimed monotonic was possible.
Yes, I agree, you didn't. I was only trying to make the point that, even though the singularity is spacelike, there *is* a possible monotonic "time ordering" of events on the singularity, which matches the time ordering of events on Lucky's worldline. That's kind of counterintuitive for a spacelike surface.
P: 97
 Quote by PeterDonis Mike, I think what PAllen was referring to is that this "ideal coordinate frame" of yours is only valid if all the objects involved are at rest relative to one another, since gravitational time dilation can only be defined in a system that is static.
OK, but I was thinking of observers taking into account any motion relative to the black hole or any local gravity fields, and using SR and GR to calculate its effects on their observations. I thought they would be left with a time dilation which would be the same for all distant observers observing the same clock near the same supermassive object. They should see what O-S calculated for their ideal case.

PAllen, I obviously need to read up on close orbiting neutron stars, but what does SOL mean?
Physics
PF Gold
P: 5,321
 Quote by Mike Holland OK, but I was thinking of observers taking into account any motion relative to the black hole or any local gravity fields, and using SR and GR to calculate its effects on their observations.
But any such "effects" will be frame dependent. There are no invariants corresponding to "gravitational time dilation" for objects that are falling into the black hole.

 Quote by Mike Holland I thought they would be left with a time dilation which would be the same for all distant observers observing the same clock near the same supermassive object.
For a static clock, yes, you can meaningfully define a "time dilation" relative to distant observers. But you can't for an infalling clock.

 Quote by Mike Holland They should see what O-S calculated for their ideal case.
O-S calculated the *proper* time along an infalling worldline. That's not the same as calculating a time dilation; they didn't do that for an infalling object, because it can't be done. There is no invariant relationship between the proper time O-S calculated for an infalling object and any sort of "time dilation".
P: 3,178
 Quote by PAllen Yes it would. Unfortunately, the correct answer is not known. Somewhere in this thread I posted links to a 2007 paper by Krauss et.al. that argues one position; and a paper by Padmanabhan et.al. from 2009 that claims to refute the former. My belief is that the 2009 paper represents the 'majority view' (and I can't find any response to it from the 2007 authors), but it is far from 'settled physics'. Without responding to the 2009 paper, there are certainly new papers written in the framework of the 2007 paper. It appears to me that both string theory and LQG are more consistent with the framework of the 2009 paper, as is Hawking's proposal for resolving the information paradox.
The 2007 paper gives a clear conclusion about pure GR in a separate GR discussion; however the 2009 paper doesn't as clearly separate GR from GR+QM, and so I did not spot or understand what error in the "classical" GR calculation was supposedly demonstrated in the 2009 paper - or even if they claim that they did.
Please clarify what the claimed error is according to you - I'm sure that you understand it much better than I do.
PF Gold
P: 4,777
 Quote by harrylin The 2007 paper gives a clear conclusion about pure GR in a separate GR discussion; however the 2009 paper doesn't as clearly separate GR from GR+QM, and so I did not spot or understand what error in the "classical" GR calculation was supposedly demonstrated in the 2009 paper - or even if they claim that they did. Please clarify what the claimed error is according to you - I'm sure that you understand it much better than I do.
Except that your interpretation of an alleged classical result is at odds with how every expert here reads the paper, how every expert here reads the press releases, and how other authors refer to the 2007 paper (it is considered new only insofar as the quantum result).
You are entitled to your interpretation, but it is important to note that it is considered incorrect by every expert here.