Does someone have examples of extensions of results from convergence theory for riemannian geometry to a lorentzian setting. (I am familiar with the work of M.T.Anderson and co. in CMC gauge, i would like to have other references if possible). Thanks !
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$\begingroup$ I'm far from an expert on this, but I believe articles by Johan Noldus are as good a place to start as any... $\endgroup$– RbegaCommented Aug 25, 2011 at 21:25
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$\begingroup$ oh yes, i know his work on lorentzian versions of gromov-hausdorff distances. i was in fact wondering if there exists paper in which cheeger type compactness results or for instance pinching theorems were applied to get interesting results on lorentzian geometry. $\endgroup$– michaelCommented Aug 25, 2011 at 22:02
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1$\begingroup$ I know you have said that you are familiar with Anderson's work, but in section 5 of his paper "Cheeger-Gromov theory and applications to general relativity" (math.sunysb.edu/~anderson/cargese.pdf) he seems to indicate that this is basically an open problem, and describes why using a curvature bound without choosing coordinates can be unsatisfactory. Namely that he describes the non-compact class of "plane-fronted gravitational waves" that satisfy $|R|^2 = 0$. Of course it would also be nice to hear about newer, updated work if it exists. $\endgroup$– Ken KnoxCommented Aug 25, 2011 at 22:22
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$\begingroup$ yes, in some sense due to the remark you've just noted he has to work with the "natural" 4d-riemannian metric constructed from the lorentzian metric and the time-function. It's more or less always the case in articles i know around this subject, you look at globally hyperbolic spacetimes and in some sense you reduce your problem to a problem a riemannian geometry. i was wondering if since these results advance had been made by geometers . $\endgroup$– michaelCommented Aug 25, 2011 at 22:57
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2 Answers
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I just noticed this question about Lorentzian convergence for which some results have been obtained, albeit by far not as strong as in the metric case. In case anyone is interested in having a discussion about this, getting some references or some idea as to why it is somewhat more difficult than the standard metric theory, then he or she may contact me at my homepage http://be.linkedin.com/pub/johan-noldus/1b/416/889
Let me mention at this point already that to my best knowledge, I was the first person to develop a Lorentzian theory of convergence and that no equivalence with any metric theory of convergence exists (no use of time functions and so on). There is a paper by Sormani http://arxiv.org/abs/1006.0411 which mentions my first paper; I do not understand however why she mentions that no stability theorems are known in this context since I have explained my work to her http://www.ams.org/notices/200404/lawrenceville-prog.pdf on the occasion of an AMS meeting. Certainly, the paper on this http://arxiv.org/abs/gr-qc/0402049 was alreay on the web by then.
Obviously, a link may be provided to the answers I give at my homepage, but for general reasons I prefer to limit my writings to my homepage. I notice that in spite of the fact that only I am supposed to know the password of my account and I have made maximal use of the contact setting, my homepage at Linkedin displays once in a while that I am not available for contact. The person who can solve this question for me with an appropriate proof attached, earns my reccomendation ;-) Therefore, for now, you can contact me on [email protected] or [email protected] by personal mail and I will respond on the Linkedin page.
So far, I did not receive any questions/comments by mail, something which is rather unexpected since generically, it seldomly happens that authors offer a public opportunity to provide comments about any work whatsoever. Concerning the response below, michael should define what it means to be ''nonlinearly stable''; assuming that the k_i are time functions and ''vacuum development'' is a Wick rotation, how would you define a Lorentz space in the context of general metric spaces and time functions ? For globally hyperbolic manifolds obtained by the usual Wick procedure from smooth time functions and smooth metric spaces, the answer obviously is yes since my notion of convergence does not hinge upon a chosen observer and hence, must be valid for all observers. Assuming, on the other hand, that ''vacuum development'' means solving the vacuum equations of motion, then I refer again to my first remark that he should define what it means in an observer independent context. I have never given this too much thought concerning gravity (as it is usually understood in some gauge) but there exist some pretty general results why, for the study of stability, it is sufficient to answer this question for the linearization of the field equations around some fixed point (that is, a static solution) and obviously, vacuum GR is not linearly stable in that sense (and therefore one would expect also not nonlinearly stable, albeit some submanifolds of stable perturbations certainly can be found). However, I guess one would like to broaden the notion of stability to nonstatic solutions, in that case I am not even aware of any generally accepted definition of stability since there is no God given frame of reference. So, I do not understand why Micheal thinks that Sormani would find this aspect of my work unsatisfying given that nonlinear-stability does not even hold for more conventional notions of distance. Moreover, this stability of vacuum solutions is to my knowledge interesting for (a) a prelude to the massive case in which static solutions are totally unrealistic (b) to get a semiclassical grip on a would be ''quantum theory'' in case you would believe that the canonical quantization procedure applies to gravity. Concerning (a), all physical intuition tells one that this problem is unstable and (b) is in my opinion the wrong question to ask. I refer to http://www.vixra.org/abs/1106.0029 for further explanation why.
At this point I refer to my homepage.
I just saw Christina's kind answer; one small correction though, I merely have a PhD and am not a professor by any means at this point in time ;-)
Kind regards,
Johan Noldus
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$\begingroup$ I understand the remark in Sormani's article on the fact that no stability result is known using your notion of convergence in the sense of non-linear stability in GR. Does $(\Sigma_i,g_i,k_i) \rightarrow (\Sigma,g,k)$ in the sense of convergence of riemannian manifold imply some kind of convergence of the globally hyperbolic vacuum development. It's what you could expect from a good convergence theory from lorentzian manifold i suppose ... of course such result seems completely out of reach (expect maybe in the cases for which non linear stability is known). $\endgroup$– michaelCommented Aug 28, 2011 at 20:25
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$\begingroup$ In Prof Sormani's article, the question of stability appears after mentioning the interest of the question of convergence theory w.r.t. general relativity. In mathematical relativity literature, stability results refer to my knowledge either to Klainerman-Christodoulou (resp. Klainerman-Nicolo or Lindblad-Rodnianski) type results or current work on non-linear stability of Kerr (Dafermos, Rodnianski ...). I may be wrong but i suppose that it is reasonable to give this hypothesis. $\endgroup$– michaelCommented Aug 30, 2011 at 20:39
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I just became aware of these postings and have written to Prof Noldus asking if there are particular stability results he would like me to mention in my survey article. It is still a preprint and can be updated. I do remember the excellent talk he gave in 2004 which is why I cited his work in the first place.
By stability theorem, in my survey, I was referring to a stability theorem as a theorem in which conditions are placed on a manifold and then it is proven that the manifold is close to some special manifold. In the GAFA paper cited there, for example, there is a Riemannian stability theorem in which it is shown that the spacelike Friedmann model is stable. It is open whether the spacetime Friedmann model is stable and perhaps Noldus' notion may be useful in that setting. There are also stability theorems of the form where one assumes a spacelike manifold has scalar \ge 0 and small ADM mass. I've recently shown with Dan A. Lee that this implies the manifold is close in the Intrinsic Flat sense to Euclidean space. Again, one might wish to ask this kind of question in the spacetime setting. There are also the stability theorems of Klainerman-Christodoulou which I believe are stronger and have smooth closeness between the spaces. My survey was asking for weaker notions of closeness between Lorentzian manifolds and suggesting Noldus' notion might work well for some of these questions while an even weaker notion might be useful in other settings.
I don't read math overflow often. A friend pointed this posting out to me. Please feel free to email me anytime if you have questions or suggestions for any of my articles. My email address is in the back of my articles.
Best regards,
Christina Sormani
