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

In summary: Adam-like object) on a Schwarzschild spacetime, then what she observes changes depending on her frame of reference. But that's not what happens on a proper acceleration spacetime.In summary, on a proper acceleration spacetime, what Eve observes changes depending on her frame of reference, but on a Schwarzschild spacetime, she remains in the same place and observes the same thing.
  • #1
harrylin
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Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's GR?

This topic is a spin-off from a number of recent discussions:
"Are "flowing space" models compatible with GR?"
"Schwartzchild and Synge once again"
"Oppenheimer-Snyder model of star collapse"
"Notions of simultaneity in strongly curved spacetime"
"limit of Rindler coordinates"
As well as my post with more links in How do black holes grow?:
https://www.physicsforums.com/showthread.php?p=4165405

According to Einstein, the "Schwarzschild singularities" do not exist in physical reality. And it seems that most1 black hole specialists think that Einstein was simply wrong. But if the models of Finkelstein (Physical Review 1958) and Kruskal resemble in any way those of Hamilton, then I wonder how they can be compatible with Einstein's GR - with which I mean Einstein's theory of gravitation as expounded in the period 1916-1920.
Due to different meanings of words by different schools of thought it is difficult to estimate if their theory merely sounds different or really is different from his. I would like to get to the bottom of this, hopefully with the help of GR experts on this forum.

Before elaborating that question I will start with a short historical introduction including some disambiguation and definitions:

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"General relativity" started out as a theory about a "general PoR", which gave it its name; that was not directly a theory about gravitation but about relative motion. However, as Einstein put it:
the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can "create" a gravitational field by a simple variation of the co-ordinate system. -E. 1916
Most people (and as I now found, even Einstein) - abandoned this 1916 flavour of GR. I think that it just cannot work, and one of the reasons will come up here below. Compare the physics FAQ:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

Einstein's concept of transforming acceleration into a "real" gravitational field had to be dropped; and typical for him, he did not shout it from the rooftops . What remained was his theory about the effects of real gravitational fields, based on a subtly rephrased EEP. In this discussion I will label that version of his theory as "Einstein's GR", and accordingly with "EEP" I will mean his final version.

Terminology that is incompatible with that of Einstein obfuscates a correct understanding of his theory. In this discussion I will therefore stick to his definitions, and kindly request anyone participating to do the same. I found that the following terms require precision:

relative motion: the difference of the motion of two entities, as measured with a reference system
- §3 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

inertial: in uniform motion (same as in classical mechanics: not geodesic)
- https://en.wikisource.org/wiki/A_Brief_Outline_of_the_Development_of_the_Theory_of_Relativity

Special Relativity (term created by Einstein): Theory based on the PoR and the light principle, relating to coordinate systems relative to which isolated, material points move uniformly in straight lines. Does not include the EEP.
- https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

Einstein Equivalence Principle (EEP, 1935 definition):
Principle of Equivalence: If in a space free from gravitation a reference system is uniformly accelerated, the reference system can be treated as being "at rest," provided one interprets the condition of the space with respect to it as a homogeneous gravitational field. - Einstein et al, Physical Review 1935
It has a footnote that is relevant for this discussion, see further.

gravitational field (simplified definition from already2 ca.1919): effects due to the neighbourhood of matter (the equivalent effect due to acceleration is an apparent field).
- "The geometrical states of bodies and the rates of clocks depend in the first place on their gravitational fields, which again are produced by the material systems concerned - https://en.wikisource.org/wiki/Time,_Space,_and_Gravitation
- "we shall have to take account of the fact that the ponderable masses will be the determining factor in producing the field, or, according to the fundamental result of the special theory of relativity, the energy density" - https://en.wikisource.org/wiki/A_Brief_Outline_of_the_Development_of_the_Theory_of_Relativity

proper acceleration: not found:rolleyes:. And https://en.wikipedia.org/wiki/Proper_acceleration is incompatible with the EEP. :bugeye:

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And now (finally!) a retake of earlier discussions, some of which are still going on.

pervect, DrGreg and Atyy suggested that we take a look at Rindler coordinates and there has been quite some discussion on this.
In referral to http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html, it is obvious that Adam and Eve have different views of physical reality: for example when Adam "falls away" according to Eve, she ascribes the frequency difference from her two clocks fully to the effect of a gravitational field which makes her clocks go at different rates; while in contrast, Adam ascribes the frequency difference that Eve observes to the Doppler effect from her acceleration. But when Eve discovers that in reality she is not in a gravitational field, then she will agree with Adam; her changed conception of reality also changes her conception of what really happens with Adam.
PeterDonis said:
Then what's the difference in Schwarzschild spacetime? That's the whole point of the Rindler horizon analogy: that if Eve does not think Adam never crosses the horizon, Eve' who is hovering above a black hole horizon should not think that Adam', who drops off her spaceship and falls into the hole, never crosses the horizon either.

If you think there is a difference, what's the difference? Why can't Eve' reason the same way that Eve does, to conclude that Adam' does cross the horizon?
The difference started with the realisation (perhaps already in 1918, as fall-out of Einstein's Twin paradox paper) that "induced gravitational fields" are an illusion that do not really work. Thus the equivalence principle cannot swap physical reality as is the case with SR's "relativity of simultaneity". The moment that Eve realizes that her "gravitational field" is an illusion, she will agree with Adam that she is looking at the Doppler effect instead of clocks running at different rates in a gravitational field.

Inversely, if Adam realizes that his "inertial motion" is an illusion and that in fact he is falling towards a black hole while Eve is staying put, then he should agree with Eve, which also means that his wristwatch is quickly slowing down compared to her clocks.

pervect came with a similar example, even with numbers:
pervect said:
[..] The experiment involves launching a spaceship that accelerates at 1g for a year shiptime - or .1g for 10 years shiptime - or .001 g for 1000 years shiptime.

The spaceship observes the Earth through a telescope. The prediction of SR in this case [...] [note: this is SR as long as the EEP is not used] that the Earth appears to fall behind an event horizon. There will be some last event that the spaceship sees - say year 2100 exactly on the new years day celebration in Grenwich.

The metric from the accelerating spaceship looks like this, assuming the spaceship accelerates in the z direction. (There are some variant forms of the metric, this version is normalized so that g_uv = diag(-1,1,1,1) at the origin.

ds^2 = -(1+ gz)^2 dt^2 + dx^2 + dy^2 + dz^2
It looks to me that Rindler's metric is standard EEP metric; and as some people think that Einstein was ignorant about those and would have changed his position if he had heard of this argument, it will be useful to elaborate on this. From Einstein's 1935 paper:

It is worth pointing out that this metric field does not represent the whole Minkowski space but only part of it.
Thus, the transformation that converts ds2=-dε12-dε22-dε32+dε42
into (1) is
ε1=x1cosh αx4, ε2=x2, ε3=x3, ε4=x1sinh αx4.
It follows that only those points for which ε12≥ε42 correspond to points for which (1) is the metric.


Thus the wikipedia article http://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=522511984 describes EEP metric. :smile:
As the observer on the spaceship watches the Earth approach New Years 2100,, the spaceship sees the image grow dimmer and dimmer, and the Earth's clocks appear to slow down. Just as it would if the Earth were falling through the event horizon of a very large black hole, as g_00 falls towards zero at the critical value z = -1/g. (In non-geometric units, that's z = c^2/g). This is the critical value because g_00 goes to zero. I believe you call it something like "time stopping?" I forget how you referred to this condition.
Now, if we apply your argument, the Earth ceases to exist in the year 2100 at new Years in some philosophically meaningful sense. At the very least, something dramatic happens on that date, as "time stops".

My position is that it's pretty obvious the Earth won't cease to exist at New Years day on the year 2100 in any sort of meaningful sense. And that the people on Earth won't even notice this, or notice anything about "time stopping" or anythign like that. In fact, they'll find New Years day 2100 quite unremarkable.
Thanks - and for completeness I will give a counter argument which I guess to be similar to Einstein's interpretation. Of course the Earth won't cease to exist at New years day 2100; your prediction is a misinterpretation of the idea that the Earth (and the whole universe with it) is falling into a gigantic black hole. If we model that our whole universe is falling into a giant black hole then according to Einstein's GR the Earth will not reach a local time of New Years day 2100; nothing exists forever and it will literally take forever to reach that day. In other words, that event will not happen.3
However, most of us believe that the idea that our universe is falling into a gigantic black hole does not conform to reality; we think that that black hole does not exist. And just as in the earlier example of Adam and Eve, our physical reinterpretation implies changing the redshift interpretation, with as consequence that now the Earth's predicted local time is simply that of SR.

Thus you used the same "map" as Einstein, but you interpreted it differently. Note that in both examples the correct prediction according to both entrenched positions is assumed to be based on physical reality, and not on mere appearances based on fictive fields that lead to contradicting predictions.

Anyway, this was just the introduction, the purpose of this topic is to get an understanding of Finkelstein's or Kruskal's reference systems, hopefully with the help of such examples. :smile:

Can someone explain this in a simple, qualitative way, if possible by means of an extension of pervect's numerical example? Of course that solution will only be 1 spatial dimension, but it should be feasible to next imagine it in 2D/3D. Then everyone can judge for themselves how compatible such solutions are with Einstein's GR.

1 Reading between the lines I think that Kraus and colleagues share Einstein's opinion

2 As early as 1919 - that was new to me! That may explain why no English version of Einstein's 1918 paper appeared at the time.
3. Even if infinite time could happen, people on Earth could not notice it; to think that they could is a mix-up of frames.
 
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  • #2


First, despite your attempted explanation I don't understand what you think distinguishes "Einstein's GR" from "standard GR".

IMO, GR is the EFE, which were developed by Einstein. Therefore, there are no flavors of GR, if you are using the EFE then you are doing GR, and GR is Einstein's theory. To me it is as simple as that, and I don't understand where you are trying to draw the line.

Perhaps more importantly, I don't know why you are trying either. It seems pointless, a historical exercise unrelated to the science.
harrylin said:
But when Eve discovers that in reality she is not in a gravitational field, then she will agree with Adam; her changed conception of reality also changes her conception of what really happens with Adam.
Do you have a reference for this idea? It seems to run expressly counter to the equivalence principle. If not, then I think it is not a topic for PF.
 
  • #3


DaleSpam said:
First, despite your attempted explanation I don't understand what you think distinguishes "Einstein's GR" from "standard GR". [..]
Discussions in the other threads demonstrated differing formulations of GR, even different opinions about what is the EP. For example:
PeterDonis said:
You and pervect are using "EEP" to refer to two different things. You mean "the EEP that Einstein stated." He means "the EEP that is actually used, today, in GR." They're not necessarily the same [..]
However, also for me is Einstein's GR standard GR; I made sure that we all talk about the same thing here to prevent confusions. The question that has remained unanswered in too many threads is if Finkelstein/Kruskal is conform with it.
[..] It seems pointless, a historical exercise unrelated to the science.
Can you do science without comparing models to precisely defined theory? :bugeye: I have no idea what you try to say, or what it has to do with the topic.
Do you have a reference for this idea? It seems to run expressly counter to the equivalence principle. If not, then I think it is not a topic for PF.
Some references are already in the first post: Einstein's 1921 summary of GR and his 1935 comments on exactly that subtopic. Similarly the physics FAQ explains that "uniform gravitational fields" are pseudo fields. Your answer gives the impression that you did not really read my post (and you are amazing if you had the time to read all the references!). [addition: It has been suggested by others here that Einstein did not accept the internal black hole solution but they did not give a reference; you are the first one to doubt that.] Anyway, neither perfect's speculations about physical reality nor mine are important for the topic at hand.

What I hope for is that someone will provide an elaboration of the "extended Schwartzschild solution" by means of a simple example, based on Rindler coordinates. :smile:
 
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  • #4


harrylin said:
Discussions in the other threads demonstrated differing formulations of GR, even different opinions about what is the EP.
Yes, this I agree with. The Wikipedia article describes Einstein's original phrasing of the EP, as well as three different modern flavors of the EP (weak, strong, and Einstein's).
http://en.wikipedia.org/wiki/Equivalence_principle

harrylin said:
However, also for me is Einstein's GR standard GR; I made sure that we all talk about the same thing here to prevent confusions. The question that has remained unanswered in too many threads is if Finkelstein/Kruskal is conform with it.
OK, then let me clearly answer the question: Yes, the Finkelstein/Kruskal metrics are solutions to the EFE and therefor conform to GR.
 
  • #5


DaleSpam said:
Yes, this I agree with. The Wikipedia article describes Einstein's original phrasing of the EP, as well as three different modern flavors of the EP (weak, strong, and Einstein's).
http://en.wikipedia.org/wiki/Equivalence_principle

OK, then let me clearly answer the question: Yes, the Finkelstein/Kruskal metrics are solutions to the EFE and therefor conform to GR.
Thanks! I don't understand how that can be possible: it is a paradox for me as well as for probably about half of the people who participated in these discussions. Once more: the purpose of this thread is to get to the bottom of this - hopefully by means of an extension of the Rindler coordinates in the simple example as presented by pervect. How does that work?
 
  • #6


harrylin said:
Thanks! I don't understand how that can be possible: it is a paradox for me as well as for probably about half of the people who participated in these discussions. Once more: the purpose of this thread is to get to the bottom of this - hopefully by means of an extension of the Rindler coordinates in the simple example as presented by pervect. How does that work?
Going through the Rindler coordinates is not the most straightforward way to do this. The easiest and most direct way is to simply plug the metrics in question into the EFE and verify that they are solutions. I can do that later this afternoon.
 
  • #7


harrylin said:
... it is a paradox for me as well as for probably about half of the people who participated in these discussions.
Do you mean a 'causal' paradox ? Or just that an event is postponed for a long time in some coordinates ?

Once more: the purpose of this thread is to get to the bottom of this - hopefully by means of an extension of the Rindler coordinates in the simple example as presented by pervect. How does that work?
The Rindler horizon is like the SC horizon but not exactly the same thing. In the Rindler case, the position of the horizon changes if the acceleration changes, so things can pop out or in.
 
  • #8


harrylin said:
But if the models of Finkelstein (Physical Review 1958) and Kruskal resemble in any way those of Hamilton, then I wonder how they can be compatible with Einstein's GR - with which I mean Einstein's theory of gravitation as expounded in the period 1916-1920.

I'll have to take some time to read through the details and the links you posted, but I have one quick comment about this: the Einstein Field Equation, as published by Einstein in November 1915, is still used in exactly the same form today. Since that equation is the basis of "Einstein's theory of gravitation as expounded in the period 1916-1920", the theory has not changed since then. What has changed is that we now understand much better all the implications of the EFE and of various well-known solutions to it.

In other words, we are much better today at computing the *consequences* of the theory. But the theory itself is unchanged: we use exactly the same EFE today that Einstein, Schwarzschild, etc. used. We're just better at solving it, in no small part because we have learned from the experience of Einstein, Schwarzschild, etc. what works and what doesn't.
 
  • #9


A general comment after a bit more reading: there's a big difference between making judgments about a theory based on knowledge of how it's actually used to make physical predictions, and making judgments about a theory based on reading popular presentations of what it says. The way physicists actually use GR to make physical predictions has always been by solving the EFE and then looking at the properties of the solutions. Physicists have never used GR to make predictions by looking at Einstein's popular books to figure out what "gravitational field" means.

So to answer the title question of this thread as it stands, much of what is cited in the OP is simply irrelevant. The question can be answered very simply: solve the EFE for the case of a vacuum, spherically symmetric spacetime, and figure out whether the solution includes a black hole interior region. It does. Question answered. This is not a matter of "interpretation", or of what terms like "gravitational field" or "inertial" mean. It's a simple question of mathematics, with an unambiguous answer. The fact that the answer was not properly understood by physicists until the 1960's does not change that fact.

A better title question for this thread, IMO, would have been: "How well do various ways of describing GR in layman's terms work? Do they aid understanding, or do they just cause confusion?"

harrylin said:
Most people (and as I now found, even Einstein) - abandoned this 1916 flavour of GR.

A key thing to note, as mentioned in my last post in this thread: this "1916 flavour of GR", that has been "abandoned", used the same EFE that we do today. The only thing that was "abandoned" was a particular way of talking about the theory.

Further comments to follow.
 
  • #10


Mentz114 said:
Do you mean a 'causal' paradox ? Or just that an event is postponed for a long time in some coordinates ?
Perhaps I should have provided more summary of earlier discussions. I mean that events that are predicted not to happen before the end of time (which I can only interpret as meaning that they cannot happen) according to Shwarzschild's solution, are predicted to happen according to Finkelstein/Kruskal; and both solutions should be valid. That was shortly discussed in the Oppenheimer thread, but it appears that Oppenheimer's solution doesn't really get beyond that of Schwarzschild on this matter: Oppenheimer includes gravitational time dilation, which has dτ/dt->0 for r->r0. This thread is the logical continuation of that discussion and related ones.
The Rindler horizon is like the SC horizon but not exactly the same thing. In the Rindler case, the position of the horizon changes if the acceleration changes, so things can pop out or in.
Is that a problem? The purpose here is to understand how it is possible to extend the solution for a constant gravitational field.
 
  • #11


DaleSpam said:
Going through the Rindler coordinates is not the most straightforward way to do this. The easiest and most direct way is to simply plug the metrics in question into the EFE and verify that they are solutions. I can do that later this afternoon.
Thanks but ... isn't that reinventing the wheel? Probably that has been done, it's a matter of making it understandable and to explain how in your opinion it prevents creating a paradox.
 
  • #12


PeterDonis said:
[..] The way physicists actually use GR to make physical predictions has always been by solving the EFE and then looking at the properties of the solutions. Physicists have never used GR to make predictions by looking at Einstein's popular books to figure out what "gravitational field" means. [..]
I hope that you understand that that is similar to using equipment without reading the operation manual. Equations are a tool of physics and not the other way round; they have a field of applicability that is not inherent in themselves. Consequently my disambiguation may turn out to be relevant (I really don't know; we'll see!).
solve the EFE for the case of a vacuum, spherically symmetric spacetime, and figure out whether the solution includes a black hole interior region. It does. Question answered. [..]
That's really great! Please give the equation, or a link to it (why didn't you give it before?). Then we can plug in numbers and evaluate it. :smile:
 
  • #13


harrylin said:
Thanks but ... isn't that reinventing the wheel? Probably that has been done, it's a matter of making it understandable and to explain how in your opinion it prevents creating a paradox.
Yes, it definitely has been done. It is a homework-type problem, completely straightforward and mechanical.

I don't know what paradox you are concerned about. As we previously agreed, Einstein's GR is the EFE. So if a metric is a solution to the EFE then it is part of GR. Where is the paradox in that?
 
  • #14


harrylin said:
Perhaps I should have provided more summary of earlier discussions. I mean that events that are predicted not to happen before the end of time (which I can only interpret as meaning that they cannot happen) according to Shwarzschild's solution, are predicted to happen according to Finkelstein/Kruskal; and both solutions should be valid.
That is not paradoxical. Just because the 'observer at infinity' will never see a body cross the EH means nothing. After all, they are an infinite distance from the event ! We can easily show that infalling radial geodesics do continue through the EH. As any textbook will state.

The purpose here is to understand how it is possible to extend the solution for a constant gravitational field.
OK, that's something else. I'm not sure what you mean by 'constant'. As far as I know the tangent space coords of the Rindler frame do not cover the region beyond the horizon and I don't know if a chart can be found that includes the whole spacetime.
 
  • #15


Mentz114 said:
That is not paradoxical. Just because the 'observer at infinity' will never see a body cross the EH means nothing. [..]
Sorry, that is a misinterpretation; it was elaborated at length in earlier discussions that Schwartzschild does not describe what observers at infinity literally see. Also, "observer at infinity" is merely for simplification of the math, it is an exact solution, and qualitatively it is the same near the black hole. But we are discussing things that cannot be verified by Earth physics; it's just theoretical interpretation.
OK, that's something else. I'm not sure what you mean by 'constant'. As far as I know the tangent space coords of the Rindler frame do not cover the region beyond the horizon and I don't know if a chart can be found that includes the whole spacetime.
As you can see in my first post, Einstein admitted that his standard EEP frame (equivalent to a "Rindler frame") does not cover the whole of Minkowski space.
 
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  • #16


harrylin said:
Sorry, that is a misinterpretation; it was elaborated at length in earlier discussions that Schwarzschild does not describe what observers at infinity literally see. Also, "observer at infinity" is merely for simplification of the math, it is an exact solution, and qualitatively it is the same near the black hole.
Do you mean that no static observer outside the EH can see anything crossing the EH ?

But we are discussing things that cannot be verified by Earth physics; it's just theoretical interpretation.
True.

As you can see in my first post, Einstein admitted that his standard EEP frame (equivalent to a "Rindler frame") does not cover the whole of Minkowski space.
OK. Is this what bothers you ?

(Please note that there is no 't' in 'Schwarzschild').
 
  • #17


harrylin said:
I hope that you understand that that is similar to using equipment without reading the operation manual.

In so far as there is an "operation manual" other than the EFE itself, it is pretty simple: all physical observables must be expressible as invariants, i.e., scalars that are formed by contracting one or more geometric objects like vectors, tensors, etc. The stuff you are linking to is not the "operation manual"; it's various people's attempts to translate the theory's technical terminology into layman's English (or actually German, in the case of most of the Einstein stuff you've been quoting--with a second translation step from German into English after that).

harrylin said:
That's really great! Please give the equation, or a link to it (why didn't you give it before?). Then we can plug in numbers and evaluate it. :smile:

Oh, for goodness' sake. What do you think all the talk about physical invariants being finite at the horizon, and how that means the solution can be extended below the horizon, was about? The easiest way to express it so you can "plug in numbers" is to write the solution in a chart that's not singular at r = 2m, such as Painleve or Eddington-Finkelstein or Kruskal. All of those are solutions of the EFE; did you not realize that? You can then plug in numbers to one of those metrics and calculate anything that strikes your fancy. Or, if you insist on using SC coordinates, just take limits as r -> 2m whenever you want to calculate anything at the horizon, as PAllen pointed out somewhere in one of these threads.

In short, many of us have already given you what you're asking for here, many times. You just haven't wanted to accept it.
 
  • #18


OK, even if harrylin is not interested in the proof, I think that it is worth posting since it directly confirms that the KS metric is, in fact, part of GR.

The KS metric (see http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates ) is given by:
[tex]ds^2=\frac{32 M^3}{r}e^{-r/2M}(-dV^2+dU^2)+r^2 (d\theta^2+sin^2 \theta \; d\phi^2)[/tex]
where
[tex]r=2M\left( 1+ W\left( \frac{U^2-V^2}{e} \right) \right)[/tex]

The non-zero elements of the Riemann curvature tensor are:
[tex]
\begin{array}{c}
R ^V{}_{\theta ,\theta ,V}=R ^U{}_{\theta ,\theta ,U}=-R ^V{}_{\theta ,V,\theta }=-R ^U{}_{\theta ,U,\theta }=\frac{m}{r} \\
R ^V{}_{U,U,V}=R ^U{}_{V,U,V}=-R ^V{}_{U,V,U}=-R ^U{}_{V,V,U}=\frac{32 m^3 (2 m - r)}{r^4 (U-V) (U+V)} \\
R ^{\theta }{}_{V,V,\theta }=R ^{\theta }{}_{U,\theta ,U}=R ^{\phi
}{}_{V,V,\phi }=R ^{\phi }{}_{U,\phi ,U}=-R ^{\theta }{}_{V,\theta ,V}=-R ^{\theta }{}_{U,U,\theta }=-R ^{\phi
}{}_{V,\phi ,V}=-R ^{\phi }{}_{U,U,\phi }=\frac{16 m^3 (2 m-r)}{r^4 (U-V)
(U+V)} \\
R ^{\phi }{}_{\theta ,\phi ,\theta }=-R ^{\phi }{}_{\theta ,\theta ,\phi }=\frac{2 m}{r} \\
R ^{\theta }{}_{\phi ,\theta ,\phi }= -R ^{\theta }{}_{\phi ,\phi ,\theta }=\frac{2 m \sin ^2(\theta )}{r} \\
R ^V{}_{\phi ,\phi ,V}=R ^U{}_{\phi ,\phi ,U}=-R ^V{}_{\phi ,V,\phi }=-R ^U{}_{\phi ,U,\phi }=\frac{m \sin ^2(\theta)}{r} \\
\end{array}
[/tex]

From that there are no non-zero elements of the Ricci curvature tensor and the curvature scalar is also zero. So we have
[tex]R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=0[/tex]

Therefore the KS metric is a solution to the EFE for vacuum. So, the answer to the posted question is that the KS metric is definitely, clearly, and unambiguosly compatible with GR.
 
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  • #19


harrylin said:
The moment that Eve realizes that her "gravitational field" is an illusion, she will agree with Adam that she is looking at the Doppler effect instead of clocks running at different rates in a gravitational field.

But what is it, physically, that makes Eve realize that her "gravitational field" is an illusion? The only difference that I can see between Eve and Eve' is that Eve' could measure nonzero tidal gravity in her vicinity, while Eve could not; Eve is in flat spacetime, and tidal gravity there is zero.

(Note that the proper definition of "tidal gravity" here requires some care; for example, if Eve's rocket is tall enough, the acceleration measured by an accelerometer at the top of her rocket will be less than that measured by an accelerometer at the bottom. So the "acceleration due to gravity" varies with spatial position in Rindler coordinates, in which Eve's rocket is at rest, but the tidal gravity is still zero. This is a good illustration of the fact that the common heuristic that tidal gravity = spatial variation in "acceleration due to gravity" is not quite correct.)

So basically, you are saying that the absence of tidal gravity is what makes Eve realize that there is no "true" gravitational field in her spacetime. Conversely, then, the *presence* of tidal gravity would signal the presence of a "true" gravitational field.

This is actually consistent, as far as it goes, with the modern understanding in GR; in the modern understanding, tidal gravity is spacetime curvature, and it is spacetime curvature that signals the presence of a "true" gravitational field.

However, this doesn't go far enough to do the work you would like it to do. See further comments below.

harrylin said:
Inversely, if Adam realizes that his "inertial motion" is an illusion

Here we have the same question as before: what is it, physically, that makes Adam' (I put the prime on his name because we're talking about the free-faller in the black hole case, not the Rindler coordinates case) realize that his "inertial motion" is an illusion? Is it that tidal gravity is present?

Here's one problem with that: tidal gravity is finite at the horizon, and gets smaller as the mass of the black hole gets bigger. With a sufficiently massive hole, neither Eve' nor Adam' would be able to detect *any* tidal gravity, so their situation would be physically indistinguishable from that of Eve and Adam. But this can be finessed by allowing Eve' and Adam' to have more and more accurate instruments.

Here's another, bigger problem: "inertial motion" is a direct physical observable. You can measure it with an accelerometer, and you can do so whether or not tidal gravity is present. So the claim that Adam' must conclude that his "inertial motion" is an illusion is a far more drastic claim than the claim that Eve must conclude that the "gravitational field" she thinks she detects is an illusion.

In fact, if we focus on physical invariants, we find that they *support* Eve concluding that her "gravitational field" is an illusion, because tidal gravity is an invariant. But the invariants do *not* support Adam' concluding that his inertial motion is an illusion, because inertial motion is also an invariant.

So where does that leave Eve'? She can conclude that a "true" gravitational field is present, unlike in the case of Eve, because tidal gravity is present. But she *cannot* conclude that Adam's inertial motion is an "illusion", any more than Adam can, because the reading of zero on the accelerometer that Adam' carries is just as invariant as the tidal gravity that Eve' detects. So Eve' has to conclude that the presence or absence of tidal gravity is simply irrelevant to the rest of what Eve concludes about the motion of Adam and whether or not it continues into a region behind the Rindler horizon; and therefore it's irrelevant to what Eve' concludes about the motion of Adam' and whether or not it continues into a region behind the BH horizon.
 
  • #20


DaleSpam said:
OK, even if harrylin is not interested in the proof, I think that it is worth posting since it directly confirms that the KS metric is, in fact, part of GR.
This was never in any doubt ( except perhaps in the mind of the OP)
The KS coordinates are made by transforming SC with a regular holonomic transformation where the differentials transform like this
[tex]
{dx'}^\mu=\frac{\partial x^\mu}{\partial {x'}^\mu}d{x'}^\mu
[/tex]
and the components of any covariant tensor written in SC can be transformed into KC by
[tex]
A^{x}_{x'}=\frac{\partial x^\mu}{\partial {x'}^\mu}
[/tex]
So a nil Ricci tensor in SC is still nil in KC.
 
  • #21


harrylin said:
If we model that our whole universe is falling into a giant black hole then according to Einstein's GR the Earth will not reach a local time of New Years day 2100; nothing exists forever and it will literally take forever to reach that day. In other words, that event will not happen.3

This is not what the Einstein Field Equation predicts. And since "Einstein's GR" uses the EFE, it is not what Einstein's GR predicts. It may be what your misinterpreted version of Einstein's GR predicts, but that's not what the actual theory predicts.

(It may also very well have been what Einstein himself thought his theory, or rather the Schwarzschild solution, predicted. If he thought that, he was wrong.)
 
  • #22


Mentz114 said:
Do you mean that no static observer outside the EH can see anything crossing the EH ?
That is not what I mean, it sounded as if it is what you think that it means. :rolleyes:
Of course, if nothing ever crosses the event horizon according to us, then we should not see it happen either. Everyone agrees about events that we can see on Earth; the issue is not about Earth physics but about what might be called metaphysics: the interpretation.
The Schwarzschild solution says that nothing will ever cross the EH according to a static observer; from that I infer that according to that solution it also can literally never happen as determined with our ECI frame, so that nothing will ever cross the horizon - and probably no black holes will fully form according to us. However it was suggested that according to Finkelstein and Kruskal fully formed black holes exist with matter crossing the horizon before the end of the universe, based on their "extended solutions". Note that this is very unlike SR's relativity of simultaneity, where different distant times are assigned to events but nobody disagrees about what events take place - such contradictions would make an inconsistent theory.
OK. Is this what bothers you ?
In fact I'm not bothered at all: for me a "frozen star" solution is fine as an interpretation of "black hole". Consequently I had no intention to discuss more about that topic. However, some people here suggest that Schwartzschild's solution is wrong about that (they call it "incomplete") because half a century ago people were bothered by that, so much that they searched for and found another solution than Schwarzschild, claiming that it is a "logical extension" of what they seem to contradict. This thread is meant to solve that paradox by means of a discussion of that extended solution.
 
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  • #23


PeterDonis said:
This is not what the Einstein Field Equation predicts. And since "Einstein's GR" uses the EFE, it is not what Einstein's GR predicts. It may be what your misinterpreted version of Einstein's GR predicts, but that's not what the actual theory predicts.

(It may also very well have been what Einstein himself thought his theory, or rather the Schwarzschild solution, predicted. If he thought that, he was wrong.)
Einstein and Finkelstein considered modifying the EFE; apparently neither thought that GR is based on the EFE. That looks logical to me as the EFE are a solution of the foundations from which they were derived. We should not let this discussion be derailed by that: we can test if the Finkelstein/Kruskal solutions are consistent with the foundations.
 
  • #24


PeterDonis said:
[..] So basically, you are saying that the absence of tidal gravity is what makes Eve realize that there is no "true" gravitational field in her spacetime. Conversely, then, the *presence* of tidal gravity would signal the presence of a "true" gravitational field. This is actually consistent, as far as it goes, with the modern understanding in GR; in the modern understanding, tidal gravity is spacetime curvature, and it is spacetime curvature that signals the presence of a "true" gravitational field.
I do not pretend that humans are necessarily able to measure everything that is important for them. However, Eve may be so lucky situation that she can distinguish the presence of a massive object, or she simply doesn't consider it reasonable to think that she is hovering above a black hole that is of a size comparable to the visible universe or bigger.
[..] Here's another, bigger problem: "inertial motion" is a direct physical observable. You can measure it with an accelerometer, and you can do so whether or not tidal gravity is present. So the claim that Adam' must conclude that his "inertial motion" is an illusion is a far more drastic claim than the claim that Eve must conclude that the "gravitational field" she thinks she detects is an illusion. [..]
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. Adam' can detect the same tidal gravity as Eve', from which he will infer that he is falling towards a black hole.

[addendum]
PeterDonis said:
[..] The easiest way to express it so you can "plug in numbers" is to write the solution in a chart that's not singular at r = 2m, such as Painleve or Eddington-Finkelstein or Kruskal. All of those are solutions of the EFE; did you not realize that? You can then plug in numbers to one of those metrics and calculate anything that strikes your fancy. [..] In short, many of us have already given you what you're asking for here, many times. You just haven't wanted to accept it.
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?
 
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  • #25


Mentz114 said:
This was never in any doubt ( except perhaps in the mind of the OP)
The KS coordinates are made by transforming SC with a regular holonomic transformation [..]
Hopefully you now understand that that this isn't really the issue (and I didn't question that it is a mathematical solution); it is claimed that that solution is an "extension" of Schwarzschild's, so that it doesn't contradict it (and nobody doubts that Schwarzschild's solution is consistent with GR). In order to make it concrete I would like to know what Finkelstein's or Kruskal's solution predicts for pervect's example, assuming that the Earth is falling into a gigantic black hole. I'm sure that many people would like to "get" that. :smile:
 
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  • #26


Just an amusing aside. One creationist hypothesized that early Earth hovered close to a black hole for a long period, and that is why the Earth is only 7000 years old while the universe is millions of years old. He never explained how it escaped from the BH.

Mike
 
  • #27


harrylin said:
Einstein and Finkelstein considered modifying the EFE; apparently neither thought that GR is based on the EFE. That looks logical to me as the EFE are a solution of the foundations from which they were derived. We should not let this discussion be derailed by that: we can test if the Finkelstein/Kruskal solutions are consistent with the foundations.
I have already shown that the KS metric is consistent weth the EFE. Precisely which "foundations" are you concerned about and do you have any actual evidence that the EFE are not consistent with said "foundations". If not, then it seems to me that the EFE embody GR and therefore any solution to the EFE must logically be consistent with all foundations of GR.
 
  • #28


harrylin said:
Hopefully you now understand that that this isn't really the issue (and I didn't question that it is a mathematical solution); it is claimed that that solution is an "extension" of Schwarzschild's, so that it doesn't contradict it (and nobody doubts that Schwarzschild's solution is consistent with GR). In order to make it concrete I would like to know what Finkelstein's or Kruskal's solution predicts for pervect's example, assuming that the Earth is falling into a gigantic black hole.
Again, this is not the most straightforward way to show that KS is an extension of Schwarzschild. The most direct way is to show:
1) that there exists a coordinate transform from S to KS
2) that the S coordinates cover only a portion of the KS coordinates

Both of these are shown on the Wiki page I linked to above. So there should be no doubt that KS is indeed an extension of S.
 
  • #29


DaleSpam said:
I have already shown that the KS metric is consistent weth the EFE. Precisely which "foundations" are you concerned about and do you have any actual evidence that the EFE are not consistent with said "foundations". If not, then it seems to me that the EFE embody GR and therefore any solution to the EFE must logically be consistent with all foundations of GR.
I have no idea about the answers to those questions; they are part of the question. The paradox is that apparently contradictory solutions are accepted. Instead of making empty hypotheses the purpose here is to take them by the horns.
DaleSpam said:
Again, this is not the most straightforward way to show that KS is an extension of Schwarzschild. The most direct way is to show:
1) that there exists a coordinate transform from S to KS
2) that the S coordinates cover only a portion of the KS coordinates

Both of these are shown on the Wiki page I linked to above. So there should be no doubt that KS is indeed an extension of S.
I had not seen that. I'll study it (and that may take some time) - thanks!
 
  • #30


harrylin said:
I have no idea about the answers to those questions; they are part of the question. The paradox is that apparently contradictory solutions are accepted. Instead of making empty hypotheses the purpose here is to take them by the horns.
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.
 
  • #31


Mike Holland said:
Just an amusing aside. One creationist hypothesized that early Earth hovered close to a black hole for a long period, and that is why the Earth is only 7000 years old while the universe is millions of years old. He never explained how it escaped from the BH.
Mike
Wow that's smart thinking. :biggrin: But for now I have no idea if Kruskal's solution gives the same result; time-out to do some more reading. :-p
 
  • #32


DaleSpam said:
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.
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, which is merely my summary of what I and others have been bringing up in a number of threads.
If still not clear enough, perhaps some onlookers can clarify it to you in their words while I spend time on the wiki page (that will need a few free evenings!).
 
  • #33


harrylin said:
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, which is merely my summary of what I and others have been bringing up in a number of threads.
If still not clear enough, perhaps some onlookers can clarify it to you in their words while I spend time on the wiki page (that will need a few free evenings!).

To me it is still unclear what the point of this thread is. It seems that you have an unorthodox view about what GR is. For example you keep repeating that the equations are not the foundation, but what do you mean by that!? If you change EFE to some other equations you will (in general) get a completely different theory, so how can they not be part of the foundations!?
 
  • #34


harrylin said:
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, which is merely my summary of what I and others have been bringing up in a number of threads.
If still not clear enough, perhaps some onlookers can clarify it to you in their words while I spend time on the wiki page (that will need a few free evenings!).

The equations are the theory. A theory must provide a means of calculating outcomes so they can be compared with observations. It is the equations that are tested.

I agree with martinbn's remark above
To me it is still unclear what the point of this thread is. It seems that you have an unorthodox view about what GR is. For example you keep repeating that the equations are not the foundation, but what do you mean by that!? If you change EFE to some other equations you will (in general) get a completely different theory, so how can they not be part of the foundations!?
 
  • #35


martinbn said:
To me it is still unclear what the point of this thread is. It seems that you have an unorthodox view about what GR is. For example you keep repeating that the equations are not the foundation, but what do you mean by that!? If you change EFE to some other equations you will (in general) get a completely different theory, so how can they not be part of the foundations!?
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.
Mentz114 said:
The equations are the theory. A theory must provide a means of calculating outcomes so they can be compared with observations. It is the equations that are tested. [..]
Equations without theory are meaningless, for example applying Lorentz transformations to accelerating reference systems leads to paradoxes (commonly known as "twin paradox"). And this is where I must take leave from this thread until I can give my feedback about the wiki explanation.
 

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