- #36
Bill_K
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Normally, nonanalytic and even discontinuous extensions would be possible, but with the assumptions of vacuum and spherical symmetry I guess Birkhoff's Theorem is what prevents them here.
Bill_K said:I'm just saying that these assumptions are fine mathematically, but there is no physical justification for making them. Regions III and IV are terra incognita, and could plausibly be the source of matter and/or gravitational waves. Especially IV, where we're assuming that an exploding singularity produces nothing but vacuum.
Bill_K said:Normally, nonanalytic and even discontinuous extensions would be possible, but with the assumptions of vacuum and spherical symmetry I guess Birkhoff's Theorem is what prevents them here.
PAllen said:No additional assumption is needed, for uniqueness in the sense I gave to follow.
Bill_K said:Normally, nonanalytic and even discontinuous extensions would be possible, but with the assumptions of vacuum and spherical symmetry I guess Birkhoff's Theorem is what prevents them here.
PAllen said:Right.
TrickyDicky said:There are many ways to state Birkhoff's theorem, this is the one used in WP:"Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric." Is this one ok to you? if not please choose another one you like better.
I would say that this one doesn't imply no additional assumption is needed for uniqueness. But since I' not sure you accept this version of the theorem I await yor answer for further discussion.
PAllen said:MTW demonstrates a more general version, that any manifold or section thereof that satisfies Einstein tensor = 0 and spherical symmetry, must be part of the KS geometry. This establishes there is no way to extend other than to add more of the KS geometry; and the KS geometry is the unique maximal extension.
TrickyDicky said:I suspected you were using a variant that named the KS geometry. The original statement said Schwarzschild metric, long before the KS geometry was found. Only if one previously assumes that the KS extension is unique, ignoring the mathematical requirement that it be analytic, can state the theorem in that way. But this thread is precisely about not taking for granted that assumption unless there is a clear physical justification for analyticity tht no one has so far provided.
No need to dispute anything. If you want to claim that what is known as the "maximal analytic extensión" of the schwarzschild solution is not analytic...PAllen said:There is no requirement of analyticity. KS geometry is not assumed, it is discovered. You need only the following to arrive at full KS geometry:
(1) Einstein tensor is defined and zero everywhere on the manifold.
(2) spherical symmetry
(3) there is no other manifold meeting (1) and (2) of which this is a subset (i.e. this can't be extended further in any way meeting (1) and (2).
There is only one manifold meeting these conditions, and it is KS. Further, every manifold meeting these conditions with (3) left off, is an open subset of KS.
I am not going to dispute facts any more on this. If you have something different to say, then I might have further comment. If you re-dispute facts, this has become pointless.
TrickyDicky said:No need to dispute anything. If you want to claim that what is known as the "maximal analytic extensión" of the schwarzschild solution is not analytic...
TrickyDicky said:Ok, so for the rest of people that can read in any peer-reviewed paper on black holes or in any GR textbook that the KS metric is an analytic continuation of the Schwarzschild metric, we can go on with the argument in the OP concerning GR as a gauge theory: on the one hand diffeomorphism invariance with background independence of GR only assumes a differentiable manifold M that can be given a pseudoRiemannian metric g, mathematically it is true that all analytic real manifolds are infinitely differentiable, but the converse is not true, so any analytic extension needs the analyticity assumption.
PAllen said:Oh come on! The issue isn't whether KS is analytic, the issue is whether you have to assume this a-priori. You do not, in this special case.
TrickyDicky said:I don't mind if you assume it a-priori or a-posteriori, the issue I present is that you have to assume it.
PAllen said:Ok, I see your confusion. The key is that the statent: "The Einstein tensor is defined and zero everywhere" is sufficient to enforce the required degree of differentiability to force uniqueness with the further assumption of spherical symmetry.
Of course it is true that quite generally physicists tend to assume, for all theories, that nature is sufficiently smooth that any degree of differentiability is ok to require for a given problem.
PAllen said:There is no such thing as an a-posterior assumption. If something follows from you assumptions it is a consequence not an assumption.
TrickyDicky said:Here is an interesting discussion about analyticity in physics, one of the answers, by unknown even refers to GR (in this case it makes reference to the no-hair theorem that is also valid only for real analytic manifolds).
http://mathoverflow.net/questions/114555/does-physics-need-non-analytic-smooth-functions
PAllen said:General theorems of this type typically do required a number of technical assumptions to prove anything. Such theorems don't assume anything about the metric, thus they typically need smoothness assumptions to constrain the problem enough to accomplish the proof.
Again, in the case of uniquness of KS geometry, such additional assumption is not needed because the existence and vanishing of the Einstein tensor everywhere is already requiring a sufficient degree of smoothness.
PAllen said:I must issue a correction here. I've read too many mathematically sloppy treatments of Birkhoff. It turns out, that as strong as it is, you really do need additional assumptions to arrive uniquely at the KS geometry.