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

[martinbn]

It sounds as if what you consider "orthodox GR" is opposed to Einstein's GR, which I consider "orthodox GR"; and the answer to your question appears to be in Einstein's 1935 paper and partly in Finkelstein's 1958 paper. But I will not discuss modified EFE equations here as that distracts from the purpose of this thread, which is to discuss and enhance understanding of Finkelstein/Kruskal solutions - and those have nothing to do with modified EFE's as Dalespam kindly showed.

Wait! Are you saying that, according to Einstein, GR does _not_ include the field equations.

 

[harrylin]

Wait! Are you saying that, according to Einstein, GR does _not_ include the field equations.

No, I did not say that and I did not see him say that; apparently slightly different solutions are possible. If you like I can send you the paper, but please start a new thread on that topic which distracts from Finkelstein/Kruskal.

 

[martinbn]

Then can you clarify what you mean by "In Einstein's GR the equations are not the foundation of GR." ?

 

[harrylin]

Then can you clarify what you mean by "In Einstein's GR the equations are not the foundation of GR." ?

Sigh - I already did -one last try with my comparison: the LT are not the foundation of SR, they are the result; equations don't come falling from heaven. This was my last reply here for a few days. And then the discussion will be about Kruskal.

 

[Dale]

In Einstein's GR the equations are not the foundation of GR, just as SR's equations are not the foundation of SR either - that's putting things upside down. Equations without theory are meaningless, just as theory without equations is useless. And I think that I have been very clear in posts #22 and #25

I read, re-read, and re-re-read your 22 and 25, but I don't see any information in either on what you consider to be the foundation of GR or SR. Do you perhaps mean that the foundation is the way that the equations are mapped to experimental outcomes? Or perhaps you mean that the foundations are the postulates from which the equations are derived?

In either case, the basic equations (Lorentz transform for SR or EFE for GR) are certainly at least compatible with the foundations of their respective theory. So proving that something is a solution to the basic equations also proves that it is compatible with the foundations. Therefore, I still insist that the question is clearly and directly answered by the homework-style problem I outlined above.

As for any confusion you may have between Schwarzschild and KS, I think that we can and should certainly discuss that, but it should be with the clear understanding that KS is demonstrably a legitimate part of GR. Can you accept that now? If not, please identify what doubt can possibly remain on that topic.

 

[PeterDonis]

Einstein and Finkelstein considered modifying the EFE; apparently neither thought that GR is based on the EFE.

References, please? I have no idea what you are talking about.

I can't follow you and I 'm pretty sure that it is due to different meaning of words; please stick for this discussion to Einstein's definitions of words.

If we do that, then Adam' is not in "inertial motion" in the first place; by the definition you give, it is Eve' who is in "inertial motion", because she is moving in a "straight line" in the coordinate system that you consider privileged (the one which is fixed with reference to the gravitating body). Your statement about Adam' having to conclude that his "inertial motion" is an illusion only makes sense on the standard definition of "inertial motion", the one I was using. If you're not going to stick to Einstein's definitions, how do you expect the rest of us to do so?

Adam' can detect the same tidal gravity as Eve', from which he will infer that he is falling towards a black hole.

Yes, he can. So what? If anything, this adds to the things that Adam' *cannot* conclude are "illusions". He can directly measure that he is in free fall (I'll use that term to avoid any confusion about "inertial motion"), so that's not an "illusion". He can also directly measure tidal gravity in his vicinity, so he can conclude that he is freely falling towards a BH; so that's not an "illusion" either. And given that he is falling towards the black hole, he has no reason to think he is in "inertial" motion by your definition in the first place; as I said above, it's Eve' who is in "inertial" motion by that definition, not Adam'. What, exactly, is Adam' supposed to conclude is an "illusion"?

 

[PeterDonis]

Check my first post again: what you propose is exactly what this thread is meant for! Only, I'm a bit confused as to which one to take... Is Finkelstein's equation the simplest?

Either the Eddington-Finkelstein chart or the Painleve chart work fine; which one is more suitable depends on what you want to calculate. E-F is better suited for describing the worldlines of ingoing light rays that cross the horizon; Painleve is better suited for describing the worldlines of ingoing timelike objects that cross the horizon.

 

[martinbn]

Sigh - I already did -one last try with my comparison: the LT are not the foundation of SR, they are the result; equations don't come falling from heaven. This was my last reply here for a few days. And then the discussion will be about Kruskal.

I think you are confused about what EFE are. LT in SR are not analogous
to EFE in GR. The EFE are not a consequence of GR, you need a postulate for them, either the equations themselves or something equivalent, so they are part of the theory not a consequence.

 

[PAllen]

I think you are confused about what EFE are. LT in SR are not analogous
to EFE in GR. The EFE are not a consequence of GR, you need a postulate for them, either the equations themselves or something equivalent, so they are part of the theory not a consequence.

To add to this point:

- various sets of postulates can lead to SR formalized as theory of invariants and transformations.

- On this framework, there are many theories of matter and fields consistent with SR: Maxwell's EM; QED; modifications of Newtonian dynamics.

For GR:

- The analog of SR invariants and transformations is the specification that geometry is pseudo-riemannian; coordinate transforms are general, and objects transform as per differential geometry (tensor calculus, in Einstein's day).

- The EFE are a specific theory gravity on the above foundations (others are possible, most rejected by experimental evidence). There are many sets of assumptions which lead to the EFE as the theory of gravity. The EFE are much closer in spirit to Maxwell's equations than to the Lorentz transform or the definition of proper time = clock time.

 

[zonde]

I think you are confused about what EFE are. LT in SR are not analogous
to EFE in GR. The EFE are not a consequence of GR, you need a postulate for them, either the equations themselves or something equivalent, so they are part of the theory not a consequence.

EFE are not empirical law. So they give quantitative expression of some qualitative idea.

 

[zonde]

Then it seems to me that the question is poorly-defined. In my opinion the foundation of GR is the EFE, so demonstrating that a metric is a solution to the EFE is the same as demonstrating that it is consistent with the foundations of GR.

Even so I might disagree that the foundation of GR is the EFE demonstrating that particular solution is a solution to the EFE is very good argument for it's compatibility with foundations of GR.

In other words, I feel that the question has been taken "by the horns" and answered clearly and unambiguously: KS is consistent with GR. To me there seems absolutely no room whatsoever for doubt on the matter, so I do not understand exactly what you think remains.

What it takes for the claim that particular solution is solution to EFE?
Is it enough to demonstrate that particular patch of spacetime is OK? Or do we have to demonstrate that particular solution is globally consistent?
If global consistency is required then BH type coordinates can't be solution to EFE while "frozen star" coordinates can be solution to EFE.

 

[PeterDonis]

If global consistency is required then BH type coordinates can't be solution to EFE

Why not? What does "global consistency" mean?

 

[Dale]

Even so I might disagree that the foundation of GR is the EFE demonstrating that particular solution is a solution to the EFE is very good argument for it's compatibility with foundations of GR.

Sorry, I cannot tell if you agree or disagree that demonstrating that a metric is a solution to the EFE also demonstrates it is compatible with the "foundations" of GR. Can you clarify, and if you disagree provide some rationale for your disagreement?

What it takes for the claim that particular solution is solution to EFE?

Plug the solution into the EFE and check that the RHS is the same as the LHS. There is nothing unique to the EFE for proving something is a solution.

 

[stevendaryl]

Plug the solution into the EFE and check that the RHS is the same as the LHS. There is nothing unique to the EFE for proving something is a solution.

There is a little bit more to it than that. In the general case, you may need to split spacetime up into pieces, find metrics for each piece that solves the EFE as a differential equation, and then show that the pieces are compatible (in the boundary between neighboring regions, the metric in the two regions are compatible).

 

[martinbn]

There is a little bit more to it than that. In the general case, you may need to split spacetime up into pieces, find metrics for each piece that solves the EFE as a differential equation, and then show that the pieces are compatible (in the boundary between neighboring regions, the metric in the two regions are compatible).

Of course, but when he says "the solution" he means the metric, which is an object independent of coordinates.

 

[stevendaryl]

Of course, but when he says "the solution" he means the metric, which is an object independent of coordinates.

Yes, but to demonstrate mathematically that the metric is a solution to the field equations, you have to express the metric as a function of some coordinates, and show that the differential equations are satisfied. At least, I don't know how to specify a metric and prove that it satisfies the EFE without using coordinates.

 

[martinbn]

Yes, but to demonstrate mathematically that the metric is a solution to the field equations, you have to express the metric as a function of some coordinates, and show that the differential equations are satisfied. At least, I don't know how to specify a metric and prove that it satisfies the EFE without using coordinates.

Yes, of course, no one disagrees with this.

 

[Dale]

There is a little bit more to it than that. In the general case, you may need to split spacetime up into pieces, find metrics for each piece that solves the EFE as a differential equation, and then show that the pieces are compatible (in the boundary between neighboring regions, the metric in the two regions are compatible).

Sure, but even if it doesn't cover the entire manifold each piece individually is still a valid solution of the EFE and hence compatible with GR. So the extra work you mentioned is nice, but not necessary for the purpose of this thread.

 

[pervect]

There is a little bit more to it than that. In the general case, you may need to split spacetime up into pieces, find metrics for each piece that solves the EFE as a differential equation, and then show that the pieces are compatible (in the boundary between neighboring regions, the metric in the two regions are compatible).

That's an interesting observation, though it's not necessary in this case since the Kruskal coordinates cover the whole space-time. But you could use it to show that the Kruskal coordinates "glue together" the interior and exterior Schwarzschid coordinates.

The only problem is that I very strongly suspect all the people (2, I think) who are still arguing (against the position of the 3-4 Science Advisors which seems pretty uniform, so it's fair to divide it up into sides) in this thread don't follow the math even for the simpler case of showing that the Kruskal coordinates satisfy the EFE. (The main reason for this suspicion is that I don't see how they could continue to argue if they did follow the math.)

So I doubt demonstrating how Kruskal glues together Schwarzschild would actually accomplish anything as fa as the argument goes. But it might generate an interesting disussion (the neverending argument here begins to pall for me).

 

[Mentz114]

The only problem is that I very strongly suspect all the people (2, I think) who are still arguing (against the position of the 3-4 Science Advisors which seems pretty uniform, so it's fair to divide it up into sides) in this thread don't follow the math even for the simpler case of showing that the Kruskal coordinates satisfy the EFE. (The main reason for this suspicion is that I don't see how they could continue to argue if they did follow the math.)

I endorse this wholeheartedly. Not only do the 2 seem to not follow the maths, but they don't seem to know what 'field equations' are, never mind solutions to same. People who cannot refer to GR (or GTR if you prefer) without tacking 'Einsteins' in front of it also warrant suspicion.

The OP's original question should have been answered with one word - 'Yes' and the thread closed.

 

[harrylin]

Then it seems to me that the question is poorly-defined. In my opinion the foundation of GR is the EFE, so demonstrating that a metric is a solution to the EFE is the same as demonstrating that it is consistent with the foundations of GR.

In other words, I feel that the question has been taken "by the horns" and answered clearly and unambiguously: KS is consistent with GR. To me there seems absolutely no room whatsoever for doubt on the matter, so I do not understand exactly what you think remains.

OK, I'm reading it now and the room for doubt immediately strikes the eye: Kruskal includes the white hole solution, which is held to be not realistic and according to PeterDonis not consistent with GR. And I must say, while I have great difficulty understanding anything of that article, the Wikipedia article on so-called(?!) Finkelstein coordinates looks much clearer and much more useful. Moreover, that article mentions a disadvantage of Kruskal coordinates that sounds very weird: "in those coordinates the metric depends on both the time and space coordinates."
- http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates

I am thus going to read that one instead, and will ask more about it next. The whole point of this topic is to do a reality check by means of a worked out example such as the one pervect gave but for the Earth falling towards a black hole.

BTW Perhaps I should have given this thread the title "Finkelstein vs Schwartzschild in the light of Einstein's GR; if that is clearer this discussion can continue with that title. It is the logical continuation of the Oppenheimer thread.

 

[PeterDonis]

Kruskal includes the white hole solution, which is held to be not realistic and according to PeterDonis not consistent with GR.

Please don't misquote me. Here's what I said (not direct quotes since I've said this in multiple places): The full K-S solution, including all four regions, is a consistent mathematical solution of the EFE. Nobody believes it is physically reasonable because of the presence of the white hole and the second exterior region (and the fact that it is vacuum everywhere, which is why the white hole and the second exterior region are there); but that doesn't change the fact that it's a consistent mathematical solution of the EFE. That makes it "consistent with GR".

Any physical theory is going to contain "unphysical" solutions--solutions which are consistent with the math of the theory but which make predictions that aren't physically reasonable. That doesn't mean those solutions are "incompatible" with the theory. It means that the theory isn't a mindless machine that cranks out physical predictions; it's a tool for humans to use in making physical predictions, and like any sophisticated tool, it requires care in its use.

If you are going to insist that only "physically reasonable" solutions be considered "consistent with GR" or "compatible with GR", then this needs to be a completely different discussion. It needs to be about what makes a solution physically reasonable or unreasonable. But we can't even have that discussion until we have agreement on what the space of solutions is, so we can sort them into the "reasonable" and "unreasonable" buckets. Everybody else appears to agree (I think--but I'm sure I'll be corrected if I'm wrong ) that the "space of solutions" with which to start sorting is the set of mathematical solutions to the EFE. The K-S solution is indisputably one such mathematical solution; DaleSpam even explicitly showed that in this thread.

So, as with previous threads on similar topics, I'm confused about what your position is.

 

[harrylin]

[..] If we do that, then Adam' is not in "inertial motion" in the first place; by the definition you give, it is Eve' who is in "inertial motion", because she is moving in a "straight line" in the coordinate system that you consider privileged (the one which is fixed with reference to the gravitating body). Your statement about Adam' having to conclude that his "inertial motion" is an illusion only makes sense on the standard definition of "inertial motion", the one I was using. If you're not going to stick to Einstein's definitions, how do you expect the rest of us to do so?

I checked and could not find an inconsistency in my description as based on the definitions... Please cite the "guilty" phrase and point it out exactly, thanks!

[..] If anything, this adds to the things that Adam' *cannot* conclude are "illusions". He can [..] directly measure tidal gravity in his vicinity, so he can conclude that he is freely falling towards a BH; so that's not an "illusion" either. And given that he is falling towards the black hole, he has no reason to think he is in "inertial" motion by your definition in the first place; as I said above, it's Eve' who is in "inertial" motion by that definition, not Adam'. What, exactly, is Adam' supposed to conclude is an "illusion"?

In out 1D example, it is not immediately clear for them (especially if only using accelerometers and not looking too far around) what physical reality is; and as we discussed earlier, the way they reckon such things as distant time and light propagation depends on their assessment of physical reality.

How does this work out with for example Finkelstein's coordinate system? (this reminds me a bit on the FAQ on the Twin paradox, with Finkelstein and Kruskal competing with Schwarzschild there are almost "too many solutions"!).

 

[Mentz114]

OK, I'm reading it now and the room for doubt immediately strikes the eye: Kruskal includes the white hole solution, which is held to be not realistic and according to PeterDonis not consistent with GR. And I must say, while I have great difficulty understanding anything of that article, the Wikipedia article on so-called(?!) Finkelstein coordinates looks much clearer and much more useful. Moreover, that article mentions a disadvantage of Kruskal coordinates that sounds very weird: "in those coordinates the metric depends on both the time and space coordinates."
- http://en.wikipedia.org/wiki/Eddington-Finkelstein_coordinates

Why is it weird if " the metric depends on both the time and space coordinates." What else could it depend on ?

I am thus going to read that one instead, and will ask more about it next. The whole point of this topic is to do a reality check by means of a worked out example such as the one pervect gave but for the Earth falling towards a black hole.

There is no 'worked out example' for the Earth falling towards a black hole.

BTW Perhaps I should have given this thread the title "Finkelstein vs Schwartzschild in the light of Einstein's GR; if that is clearer this discussion can continue with that title. It is the logical continuation of the Oppenheimer thread.

There is no 't' in Schwarzshild.

Kruskal includes the white hole solution, which is held to be not realistic and according to PeterDonis not consistent with GR.

I'll bet he said no such thing. He may have said that it is non-physical - but nearly all solutions of the EFE are unphysical.

 

[Dale]

OK, I'm reading it now and the room for doubt immediately strikes the eye: Kruskal includes the white hole solution, which is held to be not realistic

I agree that it isn't realistic, but I wasn't addressing a question about its realism. I was addressing the question of its consistency with GR, and it IS clearly and unambiguously consistent with GR. Do you have any doubt about the answer to that specific question?

BTW Perhaps I should have given this thread the title "Finkelstein vs Schwartzschild in the light of Einstein's GR; if that is clearer this discussion can continue with that title. It is the logical continuation of the Oppenheimer thread.

OK, so then why don't you make a new thread with that title and ask the question that you are actually interested in. This question seems resolved.

 

[harrylin]

Please don't misquote me.

You said:

' "Modern GR" does not consider white holes to be physically reasonable. They are valid mathematical solutions of the EFE only if the spacetime is vacuum everywhere. Nobody believes that this mathematical solution describes any actual, physical spacetime. Any actual, physical spacetime contains matter somewhere;" '

I did not quote you but indicated the essence of your stated opinion. In case you tried to say that GR allows for white holes because according to GR there is vacuum everywhere, then my paraphrase was wrong.
[..]

So, as with previous threads on similar topics, I'm confused about what your position is.

My position is that I must make up for myself if Finkelstein's solution is better and more realistic (firstly according to theory and secondly according to my own philosophy) than that of Schwarzschild. And for that I gladly accept the help of you and others to come to an understanding of that solution.

 

[Dale]

Everybody else appears to agree (I think--but I'm sure I'll be corrected if I'm wrong ) that the "space of solutions" with which to start sorting is the set of mathematical solutions to the EFE. The K-S solution is indisputably one such mathematical solution; DaleSpam even explicitly showed that in this thread.

I don't know about everyone else, but I agree.

 

[Dale]

How does this work out with for example Finkelstein's coordinate system? (this reminds me a bit on the FAQ on the Twin paradox, with Finkelstein and Kruskal competing with Schwarzschild there are almost "too many solutions"!).

There are, in fact, an infinite number of spherically symmetric vacuum solutions, all related to each other via some coordinate transform.

 

[harrylin]

I agree that it isn't realistic, but I wasn't addressing a question about its realism. I was addressing the question of its consistency with GR, and it IS clearly and unambiguously consistent with GR. Do you have any doubt about the answer to that specific question?

OK, so then why don't you make a new thread with that title and ask the question that you are actually interested in. This question seems resolved.

According to GR there is matter in the universe - the difference between a theory of physics and mathematics can hardly be better clarified.
However, I do realize that the title of this thread is unclear, so I will continue this with a clearer title.

 

[Dale]

According to GR there is matter in the universe

This is false, GR does not assert that there is matter in the universe. In fact, many spacetimes which are clearly part of GR and widely discussed in the GR literature are vacuum solutions, including Schwarzschild.

In the sense in which there is no matter in KS there is also no matter in Schwarzschild (and Finkelstein). If you wish to exclude KS on that ground then you must also exclude Schwarzschild and all other vacuum solutions.

 

[Mentz114]

According to GR there is matter in the universe ...

No. There are plenty of solutions of the EFE where no matter is present. According to our experience, there is matter in the universe and this enables us to declare as unphysical such solutions of the EFE.

[Edit]I didn't see Dalespam's post ...

 

[PAllen]

A few thoughts of mine on some of the recent discussion:

- There is exactly one complete (maximal) spherically symmetric solution of GR with asymptotically flat boundary conditions and vacuum throughout.

- Any open subset of this geometry is also a solution of GR (albeit, with different boundary conditions). For example, the subset that is covered by exterior SC coordinates has a boundary condition not present in the maximal geometry.

- The unique maximal geometry can be covered with uncountably infinite different coordinate systems or collections of coordinate patches. This is also true of any subset of the maximal solution. These different coordinate coverings are not different solutions of GR, they are different descriptions of the same solution. Kruskal coordinates, SC coordinates, Eddington-Fikelstein, Panlieve, Lemaitre, etc. are all just relabellings of any parts they cover in common.

The way to relate this to the EFE is that the EFE are insufficient by themselves to specify coordinate expression of the metric. To get a specific metric expression, you need to impose coordinate conditions in one of several common ways (e.g. Harmonic; De Donder; or use a metric 'ansatz').

 

[PeterDonis]

I checked and could not find an inconsistency in my description as based on the definitions... Please cite the "guilty" phrase and point it out exactly, thanks!

I already have. Are you reading my posts?

You said that Adam' must conclude that his inertial motion is an illusion. That claim makes no sense unless you believe that Adam' is in inertial motion. But inertial motion, according to the definition you cited, means motion in a straight line in a coordinate system which is fixed with reference to the gravitating body. By that definition, Adam' is not in inertial motion; Eve' is.

In out 1D example, it is not immediately clear for them (especially if only using accelerometers and not looking too far around) what physical reality is

I'm not sure I understand what this means, and I'm not sure it's worth trying to disentangle it. The physical observations that Adam' and Eve' can make are perfectly clear.

How does this work out with for example Finkelstein's coordinate system?

Since that coordinate system is physically equivalent to Schwarzschild coordinates in the region outside the horizon--both have exactly the same geometric invariants--everything I said applies in Finkelstein coordinates just as it does in Schwarzschild coordinates. The only difference is that Finkelstein coordinates can be used to continuously describe the motion of Adam' at and below the horizon, while Schwarzschild coordinates cannot. This is exactly parallel to the way that Minkowski coordinates can be used to describe the motion of Adam at and beyond the Rindler horizon, while Rindler coordinates cannot.

I'm beginning to wonder if you understand what a coordinate chart is and what two charts both covering the same region of a spacetime means.

 

[PeterDonis]

You said:

' "Modern GR" does not consider white holes to be physically reasonable.

Yes. That is not the same as saying "the K-S solution is not consistent with GR". At least, not the way I would use words. If that's the way you want to use words, then you should at least have said something like "PeterDonis says the K-S solution isn't physically reasonable; IMO that means it's not consistent with GR." Then at least people would know that it was you who were using the word "consistent" in a totally unusual way, not me.

I did not quote you but indicated the essence of your stated opinion.

No, you didn't. You indicated the essence of your additional claim based on my stated opinion. There's a difference. See above.

My position is that I must make up for myself if Finkelstein's solution is better and more realistic (firstly according to theory and secondly according to my own philosophy) than that of Schwarzschild. And for that I gladly accept the help of you and others to come to an understanding of that solution.

First we need to come to a common understanding of what a "solution" is. See, for example, PAllen's post #67 and my #68.

 

[PeterDonis]

the subset that is covered by exterior SC coordinates has a boundary condition not present in the maximal geometry.

Can you be more specific about the boundary condition you have in mind? I assume it's at the horizon, because the boundary condition at spatial infinity is the same (asymptotic flatness). But I'm not sure I would characterize the presence of a coordinate singularity at the horizon in SC coordinates as a "boundary condition", and I'm not sure what else you could be referring to.