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

  1. Dec 2, 2012 #1
    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:

    -----------------------------------------------------------------
    "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 :grumpy:. 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:uhh:. And https://en.wikipedia.org/wiki/Proper_acceleration is incompatible with the EEP. :bugeye:

    -------------------------------------------------------------------
    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.
    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 realises 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 realises 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:
    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:
    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.
     
    Last edited: Dec 2, 2012
  2. jcsd
  3. Dec 2, 2012 #2

    Dale

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
    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.
     
  4. Dec 2, 2012 #3
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    Discussions in the other threads demonstrated differing formulations of GR, even different opinions about what is the EP. For example:
    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.
    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.
    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:
     
    Last edited: Dec 2, 2012
  5. Dec 2, 2012 #4

    Dale

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
     
  6. Dec 2, 2012 #5
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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?
     
  7. Dec 2, 2012 #6

    Dale

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
     
  8. Dec 2, 2012 #7

    Mentz114

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    Do you mean a 'causal' paradox ? Or just that an event is postponed for a long time in some coordinates ?

    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.
     
  9. Dec 2, 2012 #8

    PeterDonis

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
     
  10. Dec 2, 2012 #9

    PeterDonis

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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?"

    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.
     
  11. Dec 2, 2012 #10
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
    Is that a problem? The purpose here is to understand how it is possible to extend the solution for a constant gravitational field.
     
  12. Dec 2, 2012 #11
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
     
  13. Dec 2, 2012 #12
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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!).
    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:
     
  14. Dec 2, 2012 #13

    Dale

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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?
     
  15. Dec 2, 2012 #14

    Mentz114

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.

    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.
     
  16. Dec 2, 2012 #15
    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.
    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.
     
    Last edited: Dec 2, 2012
  17. Dec 2, 2012 #16

    Mentz114

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    Do you mean that no static observer outside the EH can see anything crossing the EH ?

    True.

    OK. Is this what bothers you ?

    (Please note that there is no 't' in 'Schwarzschild').
     
  18. Dec 2, 2012 #17

    PeterDonis

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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).

    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.
     
  19. Dec 2, 2012 #18

    Dale

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    Re: Are Finkelstein/Kruskal "interior black hole solution" compatible with Einstein's

    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.
     
    Last edited: Dec 2, 2012
  20. Dec 2, 2012 #19

    PeterDonis

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    Re: Are Finkelstein/Kruskal interior black hole solutions compatible with Einstein's

    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.

    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.
     
  21. Dec 2, 2012 #20

    Mentz114

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    Re: Are Finkelstein/Kruskal "interior black hole solution" compatible with Einstein's

    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.
     
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