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 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.
Kind regards,
Johan Noldus