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## Are Finkelstein/Kruskal "interior black hole solution" compatible with Einstein's GR?

 Quote by Mentz114 But there are 2 'c's, one of which I missed out. Irony.
OK if you assume c=1.

 Quote by PeterDonis But in so far as Coke is different from Pepsi here, nobody actually uses Coke as a physical theory today. Everybody uses Pepsi (i.e., "modern GR", not "Einstein's GR", to whatever extent they are different, which I'm not even taking a position on right now). So if you're really interested in whether the Finkelstein or Kruskal metrics are consistent with Einstein's GR, as opposed to the GR that is actually used as a scientific theory today, you're interested in a question that only matters for history, not physics. If that's really your intent, you should make it crystal clear in the OP of a new thread that you're interested in the history, not the physics.
A number of people who participated in these threads hold that the GR that is actually used is effectively that theory; I don't know, perhaps it only sounds different. But physics is concerned with predictions based on established theory that has not been invalidated by experiment - else it would be religion. Thus the question concerns not just history but correct current presentation of physics theory.

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 Quote by harrylin A number of people who participated in these threads hold that the GR that is actually used is effectively that theory
Yes, I think I'm one of them. But it does depend on what you consider to be "effectively that theory", and that, to me, is a matter of history (and perhaps terminology), not physics.

 Quote by harrylin physics is concerned with predictions based on theory that has not been invalidated by experiment - else it would be religion.
I agree. My point about history vs. physics is simply that if you're interested in our best current theory that hasn't been invalidated by experiment, whether or not it's "the same theory that Einstein used" is irrelevant. You're not going to read Einstein to learn it anyway; you're going to read the most up to date textbooks and literature you can find.

 Quote by harrylin Thus the question concerns not just history but correct current presentation of physics theory.
To me these are two different questions, and I'm trying to figure out which one we should be talking about: the history question or the current physics question. I don't see how "correct presentation" of the current theory has to even take any position on the historical question. Of course the historical question is interesting, but the current theory stands or falls on its own merits regardless of how, historically, it has gotten to this point.

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 Quote by PeterDonis No argument with that; I just don't like using the term "boundary condition" to refer to this, since it's not something you impose before you derive the solution, it's something you discover in the course of doing the solution. But as I said, that's a matter of terminology, not physics or mathematics.
Agreed. To me a condition on the boundary is a boundary condition regardless of whether you found it by solving the differential equation and then specifying the resulting constants or if you inserted in the condition before solving the differential equation. The math doesn't care about the order, but as you say, this is just terminology.

If you want to distinguish the two then I would suggest "constant of integration" for the post-hoc constants and "boundary condition" for the a-priori constants. Under that categorization (which I wouldn't use) I would agree that the curvature at the horizon arises from a constant of integration rather than a boundary condition.

You can always change a constant of integration into a boundary condition by changing the order of operations.

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 Quote by DrGreg OK if you assume c=1.
Of course, in quantum gravity we would set c = h = 1, to get...Swarzsild?

 Quote by PeterDonis [..] My point about history vs. physics is simply that if you're interested in our best current theory that hasn't been invalidated by experiment, whether or not it's "the same theory that Einstein used" is irrelevant. You're not going to read Einstein to learn it anyway; you're going to read the most up to date textbooks and literature you can find. [..]
To my knowledge Einstein's GR as I defined here is our best current theory that hasn't been invalidated by experiment. It is always possible to reformulate a theory in such a way that the interpretation changes but the verifiable predictions remain the same. And I agree with the mentors that differing philosophies should not be debated on this forum, as that is useless. Tempting as it is to continue with discussing philosophy (which would deteriorate into debating it), I will insist on discussing numbers - as I also tried (but without insisting on it) in this thread.

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 Quote by harrylin To my knowledge Einstein's GR as I defined here is our best current theory that hasn't been invalidated by experiment.
As far as I can tell, you are defining "Einstein's GR" in such a way that your claim that nothing can ever actually reach a BH horizon is part of the theory. That means what you are calling "Einstein's GR" is *not* the best current theory that hasn't been invalidated by experiment.

If we take GR as it has been validated by experiment, and use that theory, without any changes, to make physical predictions about black holes, we find that it predicts that horizons and singularities form, and objects can fall in past the horizons and be destroyed in the singularities. That's not a matter of "interpretation"; it's a matter of using the theory as it's been validated, with the same math and the same rules for translating the math into physical observables, and extending it into a regime where there is no direct experimental validation.

When you make the claim that "Einstein's GR says that nothing can ever reach the horizon", you are taking the theory, GR, as it has been validated by experiment, and *changing the rules* for how it is used to make physical predictions in a regime where there is no direct experimental data. The theory, as it has been validated by experiment, uses proper time and other invariants, not coordinate time and other coordinate-dependent quantities, to make physical predictions. Proper time and all other invariants are finite at and below the horizon; the fact that coordinate time goes to infinity at the horizon is irrelevant, because the theory as it's been validated by experiment does not assign any physical meaning to coordinate time. By making coordinate time privileged for a particular scenario, black holes, you are changing the theory; the theory you are using is no longer GR, but "GR with a special patch for this situation".

It's true that, since we have no direct experimental evidence in this situation, there is no way to experimentally distinguish GR from your "GR with a patch". But that doesn't mean your "GR with a patch" is the same theory as GR. It isn't. All it means is that there is no experimental test we currently know of that distinguishes your theory, "GR with a patch", from GR.

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 Quote by harrylin I will insist on discussing numbers - as I also tried (but without insisting on it) in this thread.
What numbers are you interested in?

 Quote by PeterDonis When you make the claim that "Einstein's GR says that nothing can ever reach the horizon", [..]
That's surprising as I'm not aware of having made such a claim; however I asked questions on that topic (I checked the quoted part with Google, but only found Peter's remark here).
 you are taking the theory, GR, as it has been validated by experiment, and *changing the rules* for how it is used to make physical predictions [...] your "GR with a patch" [..]
To my knowledge it is Einstein theory as formulated by him that has been put to the test, and that without any patch; but that is a different topic, not belonging to this discussion. Note also that, obviously, his theory is fully his and certainly not mine.
 Quote by DaleSpam What numbers are you interested in?
A simple example of a rocket with a clock in the front and in the back that is falling into a black hole with full description incl. distant time t1 according to Schwarzschild and Finkelstein (r,τ,t,t1) would probably be interesting for many people; I supposed that such examples are available in the literature, but perhaps that isn't the case. So, that's for a next discussion.

 Quote by PeterDonis [..] what is it that applies to the coordinates of Adam but not Eve, *and* to those of Eve' but not Adam'? I haven't seen an answer yet.
As the discussion continues in the other thread I replied there although I don't suppose to have all the answers; I'm among those who ask questions about black holes. Anyway, thanks for your participation.

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 Quote by harrylin To my knowledge it is Einstein theory as formulated by him that has been put to the test
Actually it is the theory of GR as originally formulated by Einstein and refined by physicists for almost a century now, that has been put to the test. The Einstein Field Equation, which is what was originally published by Einstein, is unchanged, yes, but Einstein obviously did not know a lot of things about the consequences of the EFE that we know today, and some of the things he apparently believed about those consequences have turned out not to be true. [Edit: perhaps "solutions and their properties" would be a better word than "consequences".]

 Quote by harrylin A simple example of a rocket with a clock in the front and in the back that is falling into a black hole with full description incl. distant time t1 according to Schwarzschild and Finkelstein (r,τ,t,t1) would probably be interesting for many people; I supposed that such examples are available in the literature, but perhaps that isn't the case. So, that's for a next discussion.
I'll await another thread on this specific topic.

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 Quote by harrylin A simple example of a rocket with a clock in the front and in the back that is falling into a black hole with full description incl. distant time t1 according to Schwarzschild and Finkelstein (r,τ,t,t1) would probably be interesting for many people; I supposed that such examples are available in the literature, but perhaps that isn't the case. So, that's for a next discussion.
The easiest way I know of for this is to use the generalized Schwarzschild coordinates as presented here: http://arxiv.org/abs/gr-qc/0311038

The form of the metric in the generalized SC is given by their eq 2. The coordinate time as a function of r for a radial free-falling object is given by eq 12. The proper time as a function of r is given by eq 18. They also give explicit expressions for the free function B for standard Schwarzschild coordinates, Eddington-Finkelstein coordinates, and also for Painleve-Gullstrand coordinates.

 Quote by DaleSpam The easiest way I know of for this is to use the generalized Schwarzschild coordinates as presented here: http://arxiv.org/abs/gr-qc/0311038 The form of the metric in the generalized SC is given by their eq 2. The coordinate time as a function of r for a radial free-falling object is given by eq 12. The proper time as a function of r is given by eq 18. They also give explicit expressions for the free function B for standard Schwarzschild coordinates, Eddington-Finkelstein coordinates, and also for Painleve-Gullstrand coordinates.
Nice - that's constructive. Thanks.

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 Quote by DaleSpam The easiest way I know of for this is to use the generalized Schwarzschild coordinates as presented here: http://arxiv.org/abs/gr-qc/0311038 The form of the metric in the generalized SC is given by their eq 2. The coordinate time as a function of r for a radial free-falling object is given by eq 12. The proper time as a function of r is given by eq 18. They also give explicit expressions for the free function B for standard Schwarzschild coordinates, Eddington-Finkelstein coordinates, and also for Painleve-Gullstrand coordinates.
One observation about this paper is the authors suggest you can 'hide' the white hole issue by using this family of coordinates, and avoiding the corresponding Kruskal family. Not really, IMO, because you can easily show there exist timelike paths beginning and ending on the SC radius, encompassing finite proper time (in standard SC coordinates, the beginning and end t coordinates would be -∞ and +∞, despite finite clock time along the path). The existence of such a timelike path leads immediately to the question of what happened before the beginning of the total path of finite proper time. This leads directly into the white hole region.

It then becomes necessary to posit a physically plausible origin, e.g. O-S collapse, that really does remove the white hole region.

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 Quote by PAllen One observation about this paper is the authors suggest you can 'hide' the white hole issue by using this family of coordinates, and avoiding the corresponding Kruskal family. Not really, IMO
I agree, and on a quick reading the easiest way to show this would be to construct a similar generalized coordinate chart that, instead of covering regions I and II (exterior and black hole interior) would cover regions IV and I (white hole interior and exterior). I think that can be done just by changing the sign of the du dr term in their generalized line element.

 Mentor I agree, it does not allow you to cover the maximally extended spacetime using their equations. In that sense it is not truly "generalized", but it is generalized enough to easily calculate the quantities of interest by harrylin using a wide variety of coordinates over regions I and II.

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 Quote by DaleSpam The proper time as a function of r is given by eq 18.
A key thing to note about this equation is that, when you combine it with equation 12 (since the first term on the RHS of equation 18 is the coordinate time u(r), which is given by equation 12), B cancels out. In other words, the proper time for a radially infalling geodesic, as a function of r, is *independent* of B. That means it's the *same* for *all* of the charts that are included in the family described by this generalized line element.

As a quick check, I computed the explicit formula from equation 18 for the proper time to fall for a Lemaitre observer (who falls "from rest at infinity"), from radius r to the singularity at r = 0:

$$\tau ( r ) = \frac{1}{\sqrt{2M}} r^{\frac{3}{2}}$$

This matches what is given in MTW, although they write it in normalized form, which actually looks neater:

$$\frac{\tau}{2M} = \left( \frac{r}{2M} \right)^{\frac{3}{2}}$$

To get the proper time to the horizon, just subtract 2M from the RHS in the first formula, or 1 from the RHS in the second (to get $\tau / 2M$ to the horizon).

 Tags eep, finkelstein, kruskal, rindler coordinates