On the nature of the infinite fall toward the EH

[rjbeery]

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

Observers Alice and Bob are hovering far above the event horizon of a block hole. Alice stops hovering and enters free fall at time T_0. Bob waits an arbitrary amount of time, T_b, before reversing his hover and chasing (under rocket-propelled acceleration A_b) after Alice who continues to remain in eternal free fall.

Question: For any time T_b does there exist an acceleration A_b (however impractical yet physically possible) such that Bob can reach Alice before she crosses the event horizon, therefore rescuing her from doom?

 

[Mentz114]

I'm guessing that if Bob can survive any g-force then he could reach Alice before she crosses the EH. However, knowing how tricky relativity is, there is could be some point above the EH beyond which the rescue is impossible. This point would depend on T_b/R0 where r=R0 is the initial position.

 

[pervect]

I would say that if bob emits light flashes at regular intervals, there will be a "last flash" emitted at time T_b1 that Alice can see before she enters the horizon (and another "last flash" at T_b2 that Alice can see before she reaches the singularity).

As a consequence of this, for T>T_b1, not even light emitted by Bob could reach Alice before she reaches the horizon, and since Bob can't ever overtake a light beam, so the answer must be no.

 

[rjbeery]

I would say that if bob emits light flashes at regular intervals, there will be a "last flash" emitted at time T_b1 that Alice can see before she enters the horizon (and another "last flash" at T_b2 that Alice can see before she reaches the singularity).

As a consequence of this, for T>T_b1, not even light emitted by Bob could reach Alice before she reaches the horizon, and since Bob can't ever overtake a light beam, so the answer must be no.

Are you sure there would be a "last flash"? I'd be curious to see this analyzed mathematically. Reason being, if there were such a flash calculable by Bob then he could announce definitively "when" Alice has crossed the EH, which contradicts my understanding.

 

[PAllen]

Are you sure there would be a "last flash"? I'd be curious to see this analyzed mathematically. Reason being, if there were such a flash calculable by Bob then he could announce definitively "when" Alice has crossed the EH, which contradicts my understanding.

There is definitely such a flash. All too often, popular presentation present only half the causal structure of a BH:

- That Bob can never get a signal from Alice at or inside the horizon. Thus, horizon crossing events are never part of Bob's past light cone.

However, it is equally true that:

- Alice receives a specific last signal from Bob on crossing the horizon, and another (in the limit) on approach to the singularity. As a result, horizon crossing events are most defininitely in Bob's future light cone - just never in his past light cone.

For alice, her past light cone includes events in Bob's history until she reaches the singularity. However, once she passes the horizon, her future light cone is strictly interior to the horizon, and always includes the singularity. For a non-rotating, uncharged BH, Alice's future light cone necessarily includes less and less of the interior until reaching the singularity.FYI: Bob can definitely make such an announcement if he so desires. For example, here is one way: https://www.physicsforums.com/showpost.php?p=4165220&postcount=23

 

[tom.stoer]

Wouldn't it be much simpler to replace Bob by a photon from the very beginning? If the photon can't reach Alice before crossing the EH, Bob can't reach her, either

 

[PAllen]

Wouldn't it be much simpler to replace Bob by a photon from the very beginning? If the photon can't reach Alice before crossing the EH, Bob can't reach her, either

But the OP was all about Bob hovering after Alice started falling, and deciding at some point to try to catch Alice. Do you mean replace 'try to catch' with 'try to send a light signal'? If so, that is the essential issue; and I thought that's what Pervect was pointing out. Once a light signal would only catch Alice at or inside the horizon, it is too late for Bob to rescue Alice. Any time before this, it is possible, in principle, for Bob to rescue Alice.

 

[tom.stoer]

Do you mean replace 'try to catch' with 'try to send a light signal'?

Yes

and I thought that's what Pervect was pointing out

I overlooked that ...

Once a light signal would only catch Alice at or inside the horizon, it is too late for Bob to rescue Alice. Any time before this, it is possible, in principle, for Bob to rescue Alice.

Exactly

Shouldn't be too difficult to calculate that

 

[rjbeery]

There is definitely such a flash. All too often, popular presentation present only half the causal structure of a BH:

- That Bob can never get a signal from Alice at or inside the horizon. Thus, horizon crossing events are never part of Bob's past light cone.

However, it is equally true that:

- Alice receives a specific last signal from Bob on crossing the horizon, and another (in the limit) on approach to the singularity. As a result, horizon crossing events are most defininitely in Bob's future light cone - just never in his past light cone.

For alice, her past light cone includes events in Bob's history until she reaches the singularity. However, once she passes the horizon, her future light cone is strictly interior to the horizon, and always includes the singularity. For a non-rotating, uncharged BH, Alice's future light cone necessarily includes less and less of the interior until reaching the singularity.


FYI: Bob can definitely make such an announcement if he so desires. For example, here is one way: https://www.physicsforums.com/showpost.php?p=4165220&postcount=23

When you say "definitely", is that taking dissipative effects such as Hawking Radiation into account? If Hawking Radiation exists, in my understanding, Alice would appear to "almost" reach the EH and appear to continue to do so as the BH dissipates and the Schwarzschild radius is reduced.

 

[PAllen]

When you say "definitely", is that taking dissipative effects such as Hawking Radiation into account? If Hawking Radiation exists, in my understanding, Alice would appear to "almost" reach the EH and appear to continue to do so as the BH dissipates and the Schwarzschild radius is reduced.

Hawking radiation does not actually apply to the SC geometry. Firstly, the SC geometry never quite forms; secondly, Hawking radiation precludes an exact spherical symmetry. Further, you must distinguish classical GR (which does not include Hawking radiation), from GR + quantum corrections as an approximation to some unknown successor theory. I thought we were discussing classical GR.

If discussing GR+quantum corrections, the view that evaporation prevents matter from crossing a horizon (or from a horizon ever forming) is just one opinion. I believe it is the minority opinion, due to 2009 paper by Padmanabhan (that is, this paper refuted arguments in a 2007 paper that horizon never forms, and no major paper since has refuted Padmanabhan's arguments, that I know of).

 

[pervect]

Hawking radiation does not actually apply to the SC geometry. Firstly, the SC geometry never quite forms; secondly, Hawking radiation precludes an exact spherical symmetry. Further, you must distinguish classical GR (which does not include Hawking radiation), from GR + quantum corrections as an approximation to some unknown successor theory. I thought we were discussing classical GR.

If discussing GR+quantum corrections, the view that evaporation prevents matter from crossing a horizon (or from a horizon ever forming) is just one opinion. I believe it is the minority opinion, due to 2009 paper by Padmanabhan (that is, this paper refuted arguments in a 2007 paper that horizon never forms, and no major paper since has refuted Padmanabhan's arguments, that I know of).

Would that be http://arxiv.org/pdf/0906.1768.pdf ?

 

[pervect]

Are you sure there would be a "last flash"? I'd be curious to see this analyzed mathematically. Reason being, if there were such a flash calculable by Bob then he could announce definitively "when" Alice has crossed the EH, which contradicts my understanding.

One can find a web reference by Hamilton (and some diagrams in Eddington-Finklestein coordinates) that show the existence of a last flash.

Answer to the quiz question 5: False. You do NOT see all the future history of the world played out. Once inside the horizon, you are doomed to hit the singularity in a finite time, and you witness only a finite (in practice rather short) time pass in the outside Universe.

Eddingtion Finklstein coordinates. Yellow lines are light, white line is infalling observer. You can see there is a last flash.

http://casa.colorado.edu/~ajsh/collapse.html

For some detailed calculations:

phase 1. Show that in Schwarzschild coordinates for a black hole of mass m=2, the geodesic is given by

-\infty < \tau < 0

r = {3}^{2/3} \left( -\tau \right) ^{2/3}
t = \tau-4\,\sqrt [3]{3}\sqrt [3]{-\tau}+4\,\ln \left( \sqrt [3]{3}\sqrt <br /> [3]{-\tau}+2 \right) -4\,\ln \left( \sqrt [3]{3}\sqrt [3]{-\tau}-2<br /> \right)

By showing that it satisfies
\frac{dr}{d\tau} = \sqrt \frac {2m}{r}
\frac{dt}{d\tau} =\frac{1}{1-2m/r}

See for instance http://www.fourmilab.ch/gravitation/orbits/, or your favorite GR textbook. m=2 was chosen to make the expressions more tractable, you may choose to repeat without this attempt at simplification if you prefer

phase 2: convert to ingoing Eddington Finklestein coordinates by the transformation

v = t + r + 4\,\ln \left| \frac{r}{2m} - 1 \right|

(recall that we set m=2 in phase 1).

Phase 2a: recall, or derive, that for infalling light, v=constant. Therefore v(tau) gives you the "flash number" you are viewing at time tau.

Get
v =<br /> \tau-4\,\sqrt [3]{3}\sqrt [3]{-\tau}+4\,\ln \left( \sqrt [3]{3}\sqrt <br /> [3]{-\tau}+2 \right) -4\,\ln \left( \sqrt [3]{3}\sqrt [3]{-\tau}-2<br /> \right) +{3}^{2/3} \left( -\tau \right) ^{2/3}-8\,\ln \left( 2<br /> \right) +4\,\ln \left( \left| \left( \sqrt [3]{3}\sqrt [3]{-\tau}+<br /> 2 \right) \left( \sqrt [3]{3}\sqrt [3]{-\tau}-2 \right) \right| <br /> \right) <br />

Use the fact that ln(a*b) = ln(a)+ln(b) to rewrite this and cancel out the apparent singularity in v

<br /> v = \tau-4\,\sqrt [3]{3}\sqrt [3]{-\tau}+8\,\ln \left( \sqrt [3]{3}\sqrt <br /> [3]{-\tau}+2 \right) +{3}^{2/3} \left( -\tau \right) ^{2/3}-8\,\ln <br /> \left( 2 \right)

Observe that v is finite (zero) when tau -> 0, so that you do not in fact see the entire history of the universe before you reach the event horizon, furthermore that you don't see the entire history of the universe before you reach the singularity.

If you don't like the formal cancelllation of the divergent terms note that you can compute the limit of v as you approach the event horizon, and show that the limit exists to answer the original question. (You won't see any results inside the event horizon that way though).

Option: recompute the geodesic equations in EF coordinates and show that the modified solution satisfies them to justify the formal cancelation of the divergent terms.If this seems like waaaaay too much work, just study Hamilton's EF plot, or find the plot of an infalling observer in Eddington Finklestein coordinates in your favorite textbook.

 

[pervect]

There's a rather nice way of summarizing the above results from the infalling observer's point of view. If an observer at infinity is shining a beam of constant frequency downwards, the infalling observer can measure the doppler shift of the radially infalling light as a function of proper time, or as a function of the Schwarzschild r coordinate.

This encapsulates what one would predict an infalling observer would actually see and measure, without getting overly involved in setting up coordinate systems and such.

The doppler shift is just dv / d\tau. The equation in terms of r is particular simple compared to the rather messy equations we've seen to date:

doppler shift = sqrt(r) / [ sqrt(r) + 2]

One can see that the doppler shift starts out at 1 at infinity, and that at the event horizon at r=4, the doppler shift is 1/2, so the incoming frequency is halved.

Furthermore, the doppler shift is always < 1, there's always a redshift (assuming you are looking straight behind you), which increases as you approach the horizon, and the received frequency tends towards zero as you approach the central singularity - for a Schwarzschild black hole. This makes sense, as the only "gravity" in the free-fall frame is tidal forces, and those would tend to be of the sort to cause redshift, not blueshift.

 

[PAllen]

Would that be http://arxiv.org/pdf/0906.1768.pdf ?

Yes. I referenced it in another thread. Published in same journal as Krauss et. al.; refers to that paper explicitly as representative of the position it is refuting. And I didn't find any peer reviewed response to this Padmanabhan paper arguing the position of the 2007 paper. I also see most of the QG field just ignoring the Krauss et. al. argument and continuing discussion of how a horzion and interior behave with quantum, string, or loop corrections, and how the information paradox gets resolved; rather than accepting the view that there is no problem because a BH never forms. The whole recent debate on horizon firewalls between Polchinski and Susskind would be moot if either accepted the Krauss et. al. position.

 

[rjbeery]

Yes. I referenced it in another thread. Published in same journal as Krauss et. al.; refers to that paper explicitly as representative of the position it is refuting. And I didn't find any peer reviewed response to this Padmanabhan paper arguing the position of the 2007 paper. I also see most of the QG field just ignoring the Krauss et. al. argument and continuing discussion of how a horzion and interior behave with quantum, string, or loop corrections, and how the information paradox gets resolved; rather than accepting the view that there is no problem because a BH never forms. The whole recent debate on horizon firewalls between Polchinski and Susskind would be moot if either accepted the Krauss et. al. position.

Thanks to both of you, would you happen to have a reference to the 2007 paper as well? My only response to the text above is that BH's are interesting mathematical studies; I wouldn't personally take work in this area as evidence of their existence any more than Klein bottles.

 

[PAllen]

Thanks to both of you, would you happen to have a reference to the 2007 paper as well? My only response to the text above is that BH's are interesting mathematical studies; I wouldn't personally take work in this area as evidence of their existence any more than Klein bottles.

Here is the 2007 paper:

http://arxiv.org/abs/gr-qc/0609024 (published Phys. Rev. D 2007)

and here is a claimed refutation, link to abstract (more info) rather than PDF:

http://arxiv.org/abs/0906.1768 (published in Phys. Rev. D 2009)

 

[phinds]

... Alice who continues to remain in eternal free fall.

No, Alice only SEEMS to be in free fall to an external observer. Alice herself will know damned well that she eventually reaches the singularity and dies (actually she'll die from sphagetificaion before reaching the singularity). Don't confuse observation with reality.

 

[rjbeery]

No, Alice only SEEMS to be in free fall to an external observer. Alice herself will know damned well that she eventually reaches the singularity and dies (actually she'll die from sphagetificaion before reaching the singularity). Don't confuse observation with reality.

Actually, I'm trying to separate reality from idealized mathematical models. I understand how things would appear from Alice's perspective according to the SC analysis; I've never necessarily accepted this at face value as being what happens in reality.

 

[PeterDonis]

Actually, I'm trying to separate reality from idealized mathematical models. I understand how things would appear from Alice's perspective according to the SC analysis; I've never necessarily accepted this at face value as being what happens in reality.

There's a similar discussion going on in another thread, and I'll ask a similar question here to the one I asked there: why not? On what basis do you pick and choose which parts of the mathematical model can be "accepted at face value as being what happens in reality" and which can't? Both predictions, that Alice reaches the horizon, and later the singularity, in finite proper time, and that it takes infinite coordinate time for Alice to reach the horizon (at least, that's one way of putting it, though I think it's a misleading way), come from the same physical law: the Einstein Field Equation. If you accept its results at face value for the second prediction but not for the first, what's the difference?

 

[rjbeery]

There's a similar discussion going on in another thread, and I'll ask a similar question here to the one I asked there: why not? On what basis do you pick and choose which parts of the mathematical model can be "accepted at face value as being what happens in reality" and which can't? Both predictions, that Alice reaches the horizon, and later the singularity, in finite proper time, and that it takes infinite coordinate time for Alice to reach the horizon (at least, that's one way of putting it, though I think it's a misleading way), come from the same physical law: the Einstein Field Equation. If you accept its results at face value for the second prediction but not for the first, what's the difference?

There are a few reasons, but the simplest one is that if Bob calculates that Alice "never" crosses the EH, and can witness in a finite time the dissipation of the BH, then I'm having a problem accepting that Alice would ever get an opportunity to cross the EH regardless of what SC analysis shows her experience to be. It seems to me that she would be quickly destroyed and emitted as radiation in the Hawking radiation process from her perspective. On the other hand, if there is a point in Bob's timeline beyond which Alice is no longer able to be rescued, even at essentially the speed of light, then he could conclude that she has crossed over. Make sense?

 

[PeterDonis]

There are a few reasons, but the simplest one is that if Bob calculates that Alice "never" crosses the EH

That's not what Bob calculates. Bob calculates that he will never receive light signals from any event on Alice's worldline at or below the horizon; he does *not* calculate that Alice's worldline stops at the horizon. Bob calculates that the coordinate time he assigns to events on Alice's worldline goes to infinity as Alice approaches the horizon; he does *not* calculate that Alice can't reach the horizon. Coordinate time by itself can't tell you that; you have to look at invariants, and all the invariants are finite at the horizon. The physical law does not just include coordinate values, and does not assign direct physical meaning to coordinate values.

It seems to me that she would be quickly destroyed and emitted as radiation in the Hawking radiation process from her perspective.

But that's a different physical law than the EFE; it's a (currently unknown) law of quantum gravity. Hawking radiation is a quantum process. Your objection along these lines would be better phrased as "I know that the classical GR calculation predicts that Alice falls through the horizon and reaches the singularity; but I believe quantum corrections change that so that she never actually reaches the horizon, but instead gets turned into radiation before that." In other words, instead of questioning the validity of one physical law (the EFE), you're saying that that law doesn't really apply to Alice; some other law does. That's a different discussion than the one that appears to have been going on in this thread.

On the other hand, if there is a point in Bob's timeline beyond which Alice is no longer able to be rescued, even at essentially the speed of light, then he could conclude that she has crossed over. Make sense?

That does, yes, and by that criterion, according to classical GR, Alice does cross over, because there *is* a point in Bob's timeline beyond which he can no longer rescue Alice, even at the speed of light, before she crosses. The quantum question is more interesting, and nobody knows the answer; all we have are various speculations.

 

[pervect]

Thanks to both of you, would you happen to have a reference to the 2007 paper as well? My only response to the text above is that BH's are interesting mathematical studies; I wouldn't personally take work in this area as evidence of their existence any more than Klein bottles.

Most of the prior discussion has been oriented on trying to explain what the mathematical predictions of GR actually are. It doesn't do much good to have a theory if one gets the math wrong for what it predicts :-(.

But let's move a bit onto the observational side and away from the math for a little bit.

There's clearly something very massive and rather dark at the center of our galaxy - we can see the orbits of stars around - something.

http://arxiv.org/abs/astro-ph/0210426 "Closest Star Seen Orbiting the Supermassive Black Hole at the Centre of the Milky Way"

Furthermore, it's very black.

http://iopscience.iop.org/0004-637X/701/2/1357/

Black hole event horizons, causally separating the external universe from compact regions of spacetime, are one
of the most exotic predictions of general relativity. Until recently, their compact size has prevented efforts to study
them directly. Here we show that recent millimeter and infrared observations of Sagittarius A* (Sgr A*), the
supermassive black hole at the center of the Milky Way, all but require the existence of a horizon. Specifically, we
show that these observations limit the luminosity of any putative visible compact emitting region to below 0.4%
of Sgr A*’s accretion luminosity.

Basically, if matter falls onto the surface of a neutron star (for example), you can see the energy released and the characteristic spectrum. Astronomers have been looking very carefully at what Sag. A. has been emitting and so far it seems to be consistent with what we'd expect from a black hole and not consistent with other hypothesis.

We also have a lot of other physical evidence for GR, including terrestrial experiments.

Thus, black holes are a LOT more than a theoretical study nowadays.

 

[rjbeery]

Most of the prior discussion has been oriented on trying to explain what the mathematical predictions of GR actually are. It doesn't do much good to have a theory if one gets the math wrong for what it predicts :-(.

But let's move a bit onto the observational side and away from the math for a little bit.

There's clearly something very massive and rather dark at the center of our galaxy - we can see the orbits of stars around - something.

http://arxiv.org/abs/astro-ph/0210426 "Closest Star Seen Orbiting the Supermassive Black Hole at the Centre of the Milky Way"

Furthermore, it's very black.

http://iopscience.iop.org/0004-637X/701/2/1357/



Basically, if matter falls onto the surface of a neutron star (for example), you can see the energy released and the characteristic spectrum. Astronomers have been looking very carefully at what Sag. A. has been emitting and so far it seems to be consistent with what we'd expect from a black hole and not consistent with other hypothesis.

We also have a lot of other physical evidence for GR, including terrestrial experiments.

Thus, black holes are a LOT more than a theoretical study nowadays.

Of course this explanation is easy: we have an "almost black hole" residing in these areas. A neutron star doesn't undergo uniform and instant collapse; it would occur at the center of mass first, where pressure is greatest. If this initial point of collapse were to occur, allowing greater compacting of the remaining mass which would then itself pass the threshold for neutron collapse, we can see the progression in our mind's eye. However, if the newly infalling neutron mass takes "forever" to reach such a threshold (which some are saying it would not) OR if the rate of Hawking radiation is inversely proportional to the BH's radius (which at this point the radius would be at a lower bound), then I'm simply exploring the presumption that the EH is ever formed in the first place.

 

[harrylin]

Actually, I'm trying to separate reality from idealized mathematical models. I understand how things would appear from Alice's perspective according to the SC analysis; I've never necessarily accepted this at face value as being what happens in reality.

This is a topic that has been raised in different formulations by a number of people, and a few days ago I started a similar topic that was regretfully misunderstood, so that I intended to continue under a new header. But it isn't useful to have parallel discussions about nearly the same topic, so I'll at least for now I'll join this discussion here. I hope you won't mind that I add my own 2cts to this discussion.

The GR book of Adler, Basin and Schiffer very briefly discusses the "nature of the singularity". It suggests that Alice will cross the horizon, which they defend with the suggestion that only Alice's time reckoning is physical:

near[sic] r=2m, a finite physical time interval ds measured by a particle moving on a geodesic corresponds to an infinite time-coordinate interval. Thus the time parameter t [..] is not suited to describe the physical problem at hand.

Different from them, I see no reason to think that Alice's physics is more physical than Bob's physics (and we won't be able to decide by experiment; this discussion is very philosophical, in you case you had not realized it already). Moreover their suggestion implies that Bob's (Schwarzschild's) equally GR-based physics is wrong, while it is for me the most straightforward solution. Thus for me their argument falls flat, and what we are left with are two contradicting solutions.

Now, I'm not sufficiently familiar with Alice's predictions (and I don't mean "prediction" but a fuller view of her other predictions), and so I had in mind (and still do) to start a topic on the opposite of your topic: on the nature of the "fall through" the EH.

 

[PeterDonis]

they defend with the suggestion that only Alice's time reckoning is physical

That's not quite what they say. What they say is that Alice's time reckoning is physical along Alice's worldline. Similarly, Bob's time reckoning is physical along Bob's worldline. What you and rjbeery are trying to claim is that Bob's time reckoning should be considered "physical" along Alice's worldline, even though it conflicts with her proper time. That is what I and everyone else here are objecting to.

 

[rjbeery]

That's not quite what they say. What they say is that Alice's time reckoning is physical along Alice's worldline. Similarly, Bob's time reckoning is physical along Bob's worldline. What you and rjbeery are trying to claim is that Bob's time reckoning should be considered "physical" along Alice's worldline, even though it conflicts with her proper time. That is what I and everyone else here are objecting to.

I can tell from harrylin's single post that this is not what he's advocating. In addition, I've given my reasons for questioning what Alice experiences.

 

[PAllen]

Don't know why I bother, but to repeat yet again (assuming classical GR, no quantum effects):

Bob computes that Alice reaches the horizon in finite time on Alice's clock no matter what coordinates Bob uses (including SC exterior coordinates). Further, if Bob uses any coordinates that cover the interior (including SC internal coordinates), Bob calculates Alice reaches the singularity in finite time on Alice's clock (only a little later than when Alice reaches the horizon on Alice's clock).

The confusion all starts with asking what clock readings on Bob's clock (world line) should Bob treat as corresponding various clock readings on Alice's world line. This is where you go beyond even computed physics to pure convention. If Bob uses a convention which requires getting a signal from an event in order to assign a 'Bob' time to it, Bob cannot assign any times to a portion of Alice's world line. If, on the other hand, Bob uses a different convention, allowing assignment of Bob times to events Bob can send a signal to, then Bob can assign specific, finite times to all event's on Alice's world line. SC coordinate time just happens to be an instance of the first class of simultaneity convention.

 

[Nugatory]

near[sic] r=2m, a finite physical time interval ds measured by a particle moving on a geodesic corresponds to an infinite time-coordinate interval. Thus the time parameter t [..] is not suited to describe the physical problem at hand.

Different from them, I see no reason to think that Alice's physics is more physical than Bob's physics (and we won't be able to decide by experiment; this discussion is very philosophical, in you case you had not realized it already). Moreover their suggestion implies that Bob's (Schwarzschild's) equally GR-based physics is wrong, while it is for me the most straightforward solution.

I don't see any implication that Bob's physics is "wrong" in the quoted text. I see a statement about the limitations of using the Schwarzschild time coordinate (as opposed to the Schwarzschild metric, which of course both Bob and Alice agree about, at all points in spacetime) to describe the physics in a particular region of spacetime that is far removed from Bob.

 

[rjbeery]

Don't know why I bother, but to repeat yet again (assuming classical GR, no quantum effects):...If Bob uses a convention which requires getting a signal from an event in order to assign a 'Bob' time to it, Bob cannot assign any times to a portion of Alice's world line. If, on the other hand, Bob uses a different convention, allowing assignment of Bob times to events Bob can send a signal to, then Bob can assign specific, finite times to all event's on Alice's world line. SC coordinate time just happens to be an instance of the first class of simultaneity convention.

This is well put, PAllen. Don't lose patience, I'm not disputing anything you write. Do you have a reaction to my "almost black hole" post above?

 

[harrylin]

I don't see any implication that Bob's physics is "wrong" in the quoted text. I see a statement about the limitations of using the Schwarzschild time coordinate (as opposed to the Schwarzschild metric, which of course both Bob and Alice agree about, at all points in spacetime) to describe the physics in a particular region of spacetime that is far removed from Bob.

The authors suggests next that a test particle will cross the horizon, although that will literally never happen according to Schwarzschild. And their claim that "the time parameter t [..] is not suited to describe the physical problem at hand" implies that the solution is not suited for describing physical reality until the end of time. More complete than t->∞ is physically impossible.

Note that some people in past discussions wrongly suggested that the parameter t refers to what a distant observer literally sees - the reception of light signals at x->∞. And there were also suggestions that it is innocent like SR's relativity of simultaneity; I could not copy that thinking.

 

[PAllen]

This is well put, PAllen. Don't lose patience, I'm not disputing anything you write. Do you have a reaction to my "almost black hole" post above?

Imagine a classical theory that matches GR except as follows:

"local physics ceases to be governed by SR, and instead local physics freezes, whenever the normal progress of local physics would lead to formation of a horizon."

would be indistinguishable by a distant observer from classical GR (which has, built into its mathematical and physical foundations, that local physics is always, everywhere, governed by SR).

This modified GR, would, indeed, predict a 'frozen' star or frozen stellar cluster (for large galactic central clusters) that is externally indistinguishable from a BH.

There are some quantum approaches proposed, which rationalize this modification (which is pretty silly classically). Krauss et. al. is one; there are others. I believe the majority view remains that quantum effects do not forestall the formation of an event horizon (though its behavior is not strictly classical); nor do quantum effects prevent that matter crosses the EH. However, quantum effects are presumed to prevent any singularity and avoid the information paradox.

 

[pervect]

Of course this explanation is easy: we have an "almost black hole" residing in these areas. A neutron star doesn't undergo uniform and instant collapse; it would occur at the center of mass first, where pressure is greatest. If this initial point of collapse were to occur, allowing greater compacting of the remaining mass which would then itself pass the threshold for neutron collapse, we can see the progression in our mind's eye. However, if the newly infalling neutron mass takes "forever" to reach such a threshold (which some are saying it would not) OR if the rate of Hawking radiation is inversely proportional to the BH's radius (which at this point the radius would be at a lower bound), then I'm simply exploring the presumption that the EH is ever formed in the first place.

All you have to do now is support your "almost black hole" with some detailed math.

We've already gone over in this thread (and several others) why GR doesn't predict this "almost black hole". So , you must be discussing some other non-GR theory.

 

[harrylin]

I can tell from harrylin's single post that this is not what he's advocating. [..]

Indeed, I don't swallow the "only proper time is physical" advocacy. As a matter of fact, in view of the IMHO correct non-acceptance of Einstein's twin paradox solution I have rather the contrary view (but that is in itself a huge topic). See: http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

 

[PAllen]

The authors suggests next that a test particle will cross the horizon, although that will literally never happen according to Schwarzschild. And their claim that "the time parameter t [..] is not suited to describe the physical problem at hand" implies that the solution is not suited for describing physical reality until the end of time. More complete than t->∞ is physically impossible.

Note that some people in past discussions wrongly suggested that the parameter t refers to what a distant observer literally sees - the reception of light signals at x->∞. And there were also suggestions that it is innocent like SR's relativity of simultaneity; I could not copy that thinking.

Except ad nauseum, the t you are referring to is not a physical quantity in the theory at all. Its going to infinity has no physical meaning.

 

[harrylin]

Would that be http://arxiv.org/pdf/0906.1768.pdf ?

Interesting - thanks!

 

[harrylin]

Except ad nauseum, the t you are referring to is not a physical quantity in the theory at all. Its going to infinity has no physical meaning.

Reference please! - I wonder if this is the point where "the theory" becomes too poorly defined...

 

[PAllen]

Interesting - thanks!

And not so interesting when I posted exactly the same link for you over a week ago?

 

[PeterDonis]

in view of the IMHO correct non-acceptance of Einstein's twin paradox solution

You're going to have to elaborate on this, because I don't know what you're talking about; I'm quite familar with the Usenet Physics FAQ entry you linked to and I don't see anything like this. I suspect this is yet another issue of interpretation.

 

[harrylin]

And not so interesting when I posted exactly the same link for you over a week ago?

Sorry I did at the time not see a link to a refutation of the 2007 paper. Maybe you forgot to mention it, or maybe I overlooked your post.

 

[rjbeery]

All you have to do now is support your "almost black hole" with some detailed math.

We've already gone over in this thread (and several others) why GR doesn't predict this "almost black hole". So , you must be discussing some other non-GR theory.

Wait a minute, Pervect, you're moving goal posts. You said:

But let's move a bit onto the observational side and away from the math for a little bit.

There's clearly something very massive and rather dark at the center of our galaxy - we can see the orbits of stars around - something.

http://arxiv.org/abs/astro-ph/0210426 "Closest Star Seen Orbiting the Supermassive Black Hole at the Centre of the Milky Way"

Furthermore, it's very black.

I gave brief, qualitative description of an explanation for this. Now you're retreating back to GR? As I said before:

Actually, I'm trying to separate reality from idealized mathematical models.

Are you suggesting that GR tells the whole picture? Because on this point I'm certain you would be in the minority.

 

[PeterDonis]

Reference please!

How about every GR textbook that emphasizes, over and over, that coordinates have no physical meaning, and that you have to look at invariants to extract the physics? Look through MTW, for example, some time and count how many times they say this or something like it.

 

[stevendaryl]

The authors suggests next that a test particle will cross the horizon, although that can literally never happen according to Schwarzschild. And their claim that "the time parameter t [..] is not suited to describe the physical problem at hand" implies that the solution is not suited for describing physical reality until the end of time. More complete than t->∞ is physically impossible.

t is a label placed on a set of events in spacetime. There is no guarantee that a labeling scheme will give a label to every event. The counterexample that is easy to work with is Rindler coordinates. In two dimensional spacetime, we have coordinates X and T and a metric given by ds^2 = g^2 X^2 dT^2 - dX^2. Now, suppose at time T=0 you shine a flashlight in the negative-X direction. If I haven't screwed up, then the light signal will approach X=0 asymptotically according to:

X = X_0 e^{-gT}

So it never gets to X=0, and you might think it nonsense to ask what happens to the signal after crossing X=0. However, if we switch from Rindler coordinates back to Minkowsky coordinates, we see:

x = X cosh(gT)
t = X sinh(gT)

The path of the light signal is
x = X_0 - t

So x=0 after a finite amount of time, according to time coordinate t. The point x=0, t=X_0 in Minkowsky coordinates corresponds to the "point" X=0, T=\infty in Rindler coordinates. The point x &lt; 0, t &gt; X_0 takes place after T=\infty.

You cannot simply say that because the time coordinate running to \infty, the description of events must be complete.

 

[PeterDonis]

Are you suggesting that GR tells the whole picture? Because on this point I'm certain you would be in the minority.

Well, let's start counting votes. I'm with pervect. Anybody else?

 

[Nugatory]

The authors suggests next that a test particle will cross the horizon, although that will literally never happen according to Schwarzschild.

When you say "according to Schwarzschild", do you mean "according to the Schwarzschild metric" or do you mean "according to Schwarzschild coordinates"? They are different things.

 

[PAllen]

Reference please! - I wonder if this is the point where "the theory" becomes too poorly defined...

General covariance = all coordinates are equally good, and none are physical per se (they are conventions). Any may equally be used to make predictions = compute invariants; invariants are coordinate independent. Einstein several times regretted that relativity wasn't called the theory of invariants instead.

General covariance was a key founding principle for Einstein, along with equivalence principle. In response to a critique by Kretschmann, he admitted it has little force a 'selector of valid theories'. But he never let go of as principle of GR, and nor does modern GR.

 

[PeterDonis]

More complete than t->∞ is physically impossible.

Why?

Here's what you should have said: Once you've taken t->infinity, going further *using that coordinate chart* is mathematically impossible, because coordinates are real numbers and there are no real numbers greater than infinity.

It's a very long way from that claim to the claim that you are making.

 

[harrylin]

You're going to have to elaborate on this, because I don't know what you're talking about; I'm quite familar with the Usenet Physics FAQ entry you linked to and I don't see anything like this. I suspect this is yet another issue of interpretation.

I referred to the IMHO correct non-acceptance of Einstein's "induced real gravitational fields".

 

[PeterDonis]

if the rate of Hawking radiation is inversely proportional to the BH's radius (which at this point the radius would be at a lower bound)

Hawking radiation, whatever the correct final theory of it may be, is irrelevant for assessing whether the object at the center of our galaxy is a black hole. It's important theoretically for the study of quantum gravity theories, but practically it's irrelevant. In fact, that goes for all of the "black hole candidates" we've detected thus far.

 

[PeterDonis]

I referred to the IMHO correct non-acceptance of Einstein's "induced real gravitational fields".

I see. What does that have to do with "proper time advocacy"? Or were you just saying you are a "contrarian" in more than one area?

[Edit: I should clarify that, while I would certainly call non-acceptance of proper time as physical "contrarian", I wouldn't say the same about non-acceptance of "induced real gravitational fields". The presence of tidal gravity is a clear physical observable that distinguishes "real" gravitational fields from what the Usenet Physics FAQ calls "pseudo" ones, and that distinction is accepted in GR. So GR does not accept "induced real gravitational fields"--"induced" fields are "pseudo" fields with zero tidal gravity and would not be considered "real".]

 

[rjbeery]

Imagine a classical theory that matches GR except as follows:

"local physics ceases to be governed by SR, and instead local physics freezes, whenever the normal progress of local physics would lead to formation of a horizon."

would be indistinguishable by a distant observer from classical GR (which has, built into its mathematical and physical foundations, that local physics is always, everywhere, governed by SR).

This modified GR, would, indeed, predict a 'frozen' star or frozen stellar cluster (for large galactic central clusters) that is externally indistinguishable from a BH.

There are some quantum approaches proposed, which rationalize this modification (which is pretty silly classically). Krauss et. al. is one; there are others. I believe the majority view remains that quantum effects do not forestall the formation of an event horizon (though its behavior is not strictly classical); nor do quantum effects prevent that matter crosses the EH. However, quantum effects are presumed to prevent any singularity and avoid the information paradox.

Note the bolded portion. IMHO this concept isn't silly in any way, and I don't even need quantum approaches to accept it. All forms of measurement ultimately rely on c (or equivalently t). Change the "rate of flow of time" and a local observer would never know the difference. All of our instruments, including our physiological and cognitive processes, would also be changed accordingly. I'm not saying that is what happens, but I don't dismiss the idea prima facie.