Abstract
We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled byclosed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the “surface gravity” of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy horizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions).
For the special case of a null surface diffeomorphic toT 3 we characterize the degenerate vacuum solutions completely. These consist of an infinite dimensional family of “plane wave” spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic toT 3 and prove the generality of the previously constructed non-degenerate solutions.
We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry.
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References
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Moncrief, V.: Paper in preparation
See for example the articles by Geroch, R., Horowitz, G.T., Penrose, R.: In: General relativity: an Einstein centenary survey. Hawking, S., Israel, W. (eds.). Cambridge: Cambridge University Press 1979
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See, for example: Sperb, R.: Maximum principles and their applications. New York: Academic Press 1981
To treat smooth functions on S1, one can apply the ordinary maximum principle for functions on R1 to the special case of periodic functions. For the analytic case treated in this paper one can avoid the use of the maximum principle altogether, replacing it by an inductive argument applied to the successive derivatives of Raychaudhuri's equation
Hawking, S., Ellis, G.: The large scale structure of space-time. Cambridge: Cambridge University Press 1973. See Proposition 6.4.4 on p. 191
Fischer, A., Marsden, J., Moncrief, V.: The structure of solutions of Einstein's equations. I. One Killing field. Ann. Inst. H. Poincaré33, 147 (1980). See especially Sect. (2)
See, for example: Carter, B.: In: Black Holes, DeWitt, C., DeWitt, B.S. (eds.). New York: Gordon and Breach 1973. See in particular Eq. (5.9) on p. 147
See Sect. (8.5) of Ref. [7],, especially the discussion on pp. 297–298
The examples which suggest this “rigidity conjecture” are the polarized Gowdy models discussed in Ref. [1]
Moncrief, V.: Global properties of Gowdy spacetimes with T3×R topology. Ann. Phys. (N.Y.)132, 87 (1981). See especially Sect. IV
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Communicated by S.-T. Yau
Research supported in part by NSF grant No. PHY79-16482 at Yale and No. PHY79-13146 at Berkeley
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Moncrief, V., Isenberg, J. Symmetries of cosmological Cauchy horizons. Commun.Math. Phys. 89, 387–413 (1983). https://doi.org/10.1007/BF01214662
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DOI: https://doi.org/10.1007/BF01214662