Are Finkelstein/Kruskal Black Hole Solutions Compatible with Einstein's GR?

[stevendaryl]

There is no 't' in Schwarzshild.

...because the metric is independent of t.

 

[stevendaryl]

According to GR there is matter in the universe

I would say that it is an empirical question. Look around, you see matter, so the vacuum solutions are not relevant. But GR can describe both a universe with matter and a universe without matter.

 

[PeterDonis]

I would say that it is an empirical question. Look around, you see matter, so the vacuum solutions are not relevant.

There is matter, but not everywhere; there are certainly regions of the actual universe which are, at least to a very good approximation, vacuum. Vacuum solutions are certainly relevant for describing such regions. We use the Schwarzschild metric to describe spacetime around the Earth; we just don't use the entire global manifold, we use a portion of it. The EFE is local, so this is perfectly valid.

 

[harrylin]

[..] I do realize that the title of this thread is unclear, so I will continue this with a clearer title.

By chance (or perhaps not?) just now a new topic has been started that is very close to the topic title that I had in mind to continue with. In order not to duplicate threads I joined the discussion there:
https://www.physicsforums.com/showthread.php?p=4185579

 

[harrylin]

I would say that it is an empirical question. Look around, you see matter, so the vacuum solutions are not relevant. But GR can describe both a universe with matter and a universe without matter.

Concerning the second part, that's assuming that GR was intended for other universes than our own. I don't think so. Why would other universes have the same laws of nature as ours? I'm afraid that this really gets too philosophical and speculative...

There is matter, but not everywhere; there are certainly regions of the actual universe which are, at least to a very good approximation, vacuum. [..]

Quite so; but wasn't the white hole solution intended to start near a black hole?

 

[Mentz114]

...because the metric is independent of t.

But there are 2 'c's, one of which I missed out. Irony.

 

[harrylin]

I apologise for my tetchy attitude. But when it comes to GR there is only one brand. The one invented by Einstein - so why put the soubriquet on. You seem to think there are many brands.

Anyhow, I'll stay out of this now.

In the literature and discussions I found different flavours of GR, and for me it is an unanswered question if that matters or not for the metaphysics. But thanks for your apology, such little things make PF a nice place to be in. :!)

It would depend on what the discussion was about. Yes, I noticed "mislabeling", in the sense that, as I said, the "Einstein Equivalence Principle" as it is currently used in GR (Pepsi) is not precisely the same principle that Einstein himself stated (Coke).

If the discussion is about what Einstein said, then yes, asking for Coke is perfectly reasonable. But if the discussion is about what's currently used in GR, then a client who keeps asking for Coke even after everybody has pointed out repeatedly that the discussion is really about Pepsi would seem a little weird.

This thread was intended to have the exact taste of Coke, in order to reduce the mutual misunderstandings that were experienced earlier (but it didn't work because I didn't explain the topic well enough).

 

[harrylin]

In other words, you meant "Adam' thinks he is in inertial motion, but then he discovers that he really isn't". But on what basis would Adam' even *think* he was in inertial motion? On the standard definition of "inertial motion", Adam' could measure directly that he was in inertial motion, by using an accelerometer, as I said. But on your definition, inertial motion doesn't mean free fall, it means motion in a straight line with respect to the gravitating body. On what basis would Adam' first think he is moving in a straight line, but then be forced to conclude otherwise?

He would only be forced to conclude otherwise if he could discern the existence of a gravitating body in the vicinity. Else, he simply wouldn't know. Similarly, a bee flying towards a window has no reason to expect the existence of the window - until hitting the glass.

 

[stevendaryl]

Concerning the second part, that's assuming that GR was intended for other universes than our own

I think you're being a little silly. Laws of physics typically describe more than what is actually the case. Newtonian physics will tell you what would happen in a perfectly elastic collision of two perfect spheres the size of the Earth. There are no perfect spheres the size of the Earth that are perfectly elastic.

Typically, theories of physics tell you what follows from hypothesized initial conditions. Usually, the theories don't tell you what the initial conditions are, you have to find those out empirically.

It would be pretty weird if GR only applied to our universe.

 

[PeterDonis]

Quite so; but wasn't the white hole solution intended to start near a black hole?

The short answer is "no", but perhaps it's worth expanding on this.

(First, a quick note: "near the black hole" is still vacuum. The black hole region is vacuum at the horizon, and all the way down to r = 0. But I think that's a minor point compared to what I'm going to say below.)

Suppose we want to solve the Einstein Field Equation subject to the following conditions:

(1) The spacetime is spherically symmetric.
(2) The spacetime is vacuum everywhere--i.e., there is no matter *anywhere*, ever.

The complete solution to the EFE under these conditions includes an exterior region (which I'll call region I), a black hole region (region II), a second exterior region (region III), and a white hole region (region IV). The solution doesn't "start near a black hole"; it doesn't "start near" anywhere. It's just the complete solution we get when we impose those conditions ("complete" meaning "including all possible regions which are indicated by the math, whether they are physically reasonable or not").

Suppose we want to solve the Einstein Field Equation subject to the following somewhat different conditions:

(1') The spacetime is spherically symmetric.
(2') On some spacelike slice, the spacetime is vacuum for radius > R_0 (where R_0 is some positive value), but is *not* vacuum for radius <= R_0; instead, the region r <= R_0 on this spacelike slice is filled with dust (where "dust" means "a perfect fluid with positive energy density and zero pressure") which is momentarily at rest.
(3') We are only interested in the spacetime to the future of the spacelike slice given in #3.

The complete solution we get when we impose these conditions is what I'll call the "modernized Oppenheimer-Snyder model" ("modernized" to avoid any concerns about whether or not it was the model O-S originally proposed; this model is described, for example, in MTW). This spacetime has three regions: an exterior vacuum region (which I'll call region I'), a black hole interior vacuum region (region II'), and a non-vacuum collapsing region (region C'). There is no white hole region, and no second exterior region, in this spacetime.

Now, in the vacuum regions I' and II', the solution of the EFE is the vacuum solution: that is, it is *exactly the same* as the solution in the corresponding portions of regions I and II. Another way of saying this: if I describe regions I and II in a suitable coordinate chart, and regions I' and II' in a suitable coordinate chart, I can identify an open set of coordinate values in regions I and II that meet the following conditions:

(A) The coordinate values are exactly the same as the ones in regions I' and II'; and
(B) The invariant quantities at each corresponding set of coordinate values (I <-> I', and II <-> II') are identical.

A fairly common shorthand, I believe, for what I've said above is that region I' is isometric to a portion of region I, and region II' is isometric to a portion of region II. Or, speaking loosely, regions I' and II' can be thought of as "pieces" of regions I and II that have been "cut and glued" to region C'.

Hopefully all this makes somewhat clearer how the term "solution" is being used, and what it means to say that "the same solution" appears in different models.

 

[PeterDonis]

This thread was intended to have the exact taste of Coke

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.

 

[PeterDonis]

He would only be forced to conclude otherwise if he could discern the existence of a gravitating body in the vicinity.

We've already stipulated that he can, because he can detect tidal gravity (as can Eve'). But given that, why would he ever assume he was moving in a straight line in the first place?

Maybe I should expound a bit more on what I'm looking for here. The standard view of this scenario is that the two cases are exactly parallel: in both cases, the accelerated observer (Eve, Eve'), because of her proper acceleration, is unable to observe or explore a region of spacetime that the free-falling observer (Adam, Adam') can. The physical criterion that distinguishes them is clear, and is the same in both cases (zero vs. nonzero proper acceleration).

You are claiming that, contrary to the above, the cases are different: Adam is "privileged" in the first case, but Eve' is in the second. So I'm looking for some criterion that picks out Adam in the first case, but picks out Eve' in the second; in other words, something that applies to Adam but not Eve, and applies to Eve' but not Adam'. The only criterion I have so far is "moves in a straight line according to my chosen coordinates", but that only pushes the problem back a step: 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.

 

[DrGreg]

But there are 2 'c's, one of which I missed out. Irony.

OK if you assume c=1.

 

[harrylin]

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.

 

[PeterDonis]

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.

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.

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.

 

[Dale]

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.

 

[PeterDonis]

OK if you assume c=1.

Of course, in quantum gravity we would set c = h = 1, to get...Swarzsild?

 

[harrylin]

[..] 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.

 

[PeterDonis]

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.

 

[Dale]

I will insist on discussing numbers - as I also tried (but without insisting on it) in this thread.

What numbers are you interested in?

 

[harrylin]

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.

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.

 

[harrylin]

[..] 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.

 

[PeterDonis]

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".]

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.

 

[Dale]

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.

 

[harrylin]

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.

 

[PAllen]

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.

 

[PeterDonis]

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.

 

[Dale]

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.

 

[PeterDonis]

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).

 

[PeterDonis]

And just to add some quick numbers based on the formula in my last post (which isn't strictly correct for an object that starts from rest at finite r, but which will be *less* than the time for falling from rest any finite r, and the error gets smaller as r gets larger): if we plug in 2M for the Sun (about 3 km), and start from the radius of the actual Sun (about 700,000 km), we have:

\frac{r}{2M} = \frac{700000}{3} = 233333

\frac{\tau}{2M} = ( 233333 )^{\frac{3}{2}} = 112710467

Multiplying by 2M (i.e., 3), and converting from km to seconds by dividing by c (299792), we get 1128 seconds. The time to the horizon is only 10 microseconds smaller (since that's 3 km divided by c). Again, this is a lower bound (since a Lemaitre observer is moving inward at "escape velocity" at any finite r); the actual proper time to fall from rest at r = 233333M will be larger.

If we run the same calculation for the million solar mass black hole at the center of the Milky Way, and start from the same value of r / 2M (which will equate to 233 billion km or about 1556 AU, about 13 times the distance to the Voyager spacecraft but still only about 10^-5 light year, so very close by interstellar standards), the result for \tau / 2M remains the same, and we just scale \tau up by a factor of a million; so the time to the singularity would be 1128 million seconds or about 36.4 years, and the time to the horizon would be about 10 seconds shorter.

Finally, for a billion solar mass black hole, such as the ones that are thought to be at the centers of quasars, if we start from the same value of r / 2M, we will be about 10^-2 light year away when we start; it will take 36,400 years to fall to the singularity, and the time to the horizon will be about 2 hours 47 minutes shorter.

Again, all of these times are lower bounds; I suspect the actual numbers for a fall from rest at finite r will be significantly higher even for such a high r / 2M.