Abstract
We establish a complete classification theorem for the topology and for the orbital type of the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum \(3+1\)-spacetimes. We show that one and only one of the following must hold: (i) all generators are closed, (ii) only two generators are closed and any other densely fills a two-torus, (iii) every generator densely fills a two-torus, or (iv) every generator densely fills the horizon. We then show that, respectively to (i)–(iv), the horizon’s manifold is either: (i’) a Seifert manifold, (ii’) a lens space, (iii’) a two-torus bundle over a circle, or, (iv’) a three-torus. All the four possibilities are known to arise in examples. In the last case, (iv), (iv’), we show in addition that the spacetime is indeed flat Kasner, thus settling a problem posed by Isenberg and Moncrief for ergodic horizons. The results of this article open the door for a full parameterization of the metrics of all vacuum spacetimes with a compact Cauchy horizon. The method of proof permits direct generalizations to higher dimensions.
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Notes
That is, the quotient of \({\mathbb {R}}^{n}\) by a lattice.
References
Alexandrino, M.M., Bettiol, R.G.: Lie groups and geometric aspects of isometric actions, vol. 8. Springer, Cham (2015)
Chrusciel, P.T., Rendall, A.D.: Strong cosmic censorship in vacuum space-times with compact, locally homogeneous Cauchy surfaces. Ann. Phys. 242(2), 349–385 (1995)
Chrusciel, Piotr T.: Lake, Kayll: cauchy horizons in gowdy space-times. Class. Quant. Grav. 21, S153–S170 (2004)
Fukaya, Kenji: A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. J. Differ. Geom. 28(1), 1–21 (1988)
Isenberg, J., Moncrief, V.: On spacetimes containing Killing vector fields with non-closed orbits. Class Quant Gravity 9(7), 1683 (1992)
Larsson, Eric.: Smoothness of compact horizons. In: Annales Henri Poincaré, volume 16, pages 2163–2214. Springer, Cham (2015)
Lee, John M.: Introduction to Smooth Manifolds. Springer, New York (2012)
Minguzzi, E.: Completeness of cauchy horizon generators. J. Math. Phys. 55(8), 082503 (2014)
Minguzzi, E.: Area theorem and smoothness of compact Cauchy horizons. Commun. Math. Phys. 339(1), 57–98 (2015)
Misner, C.W.: Taub-Nut Space as a Counterexample to almost anything. Relativity Theory and Astrophysics: Relativity and Cosmology. American Mathematical Society and Cornell University. J. Ehlers Ed., volume 8, page 160 (1967)
Moncrief, Vincent: Isenberg, James: Symmetries of cosmological Cauchy horizons with non-closed orbits. Commun. Math. Phys. 374(1), 145–186 (2020)
Myers, Sumner.B., Steenrod, Norman.Earl.: The group of isometries of a Riemannian manifold. Annals of Mathematics, pages 400–416 (1939)
Oliver Lindblad Petersen. Wave equations with initial data on compact Cauchy horizons. arXiv preprint arXiv:1802.10057 (2018)
Oliver Lindblad Petersen and István Rácz. Symmetries of vacuum spacetimes with a compact Cauchy horizon of constant non-zero surface gravity. arXiv:1809.02580 (2018)
Reiris, Martin: Bustamante, Ignacio: On the existence of Killing fields in smooth spacetimes with a compact Cauchy horizon. Classical and Quantum Gravity, Accepted manuscript - (2020)
Rendall, Alan.D.: Compact null hypersurfaces and collapsing Riemannian manifolds. eprint arXiv:dg-ga/9510002, pages dg–ga/9510002, October (1995)
Scott, Peter: The geometries of \(3\)-manifolds. Bull. London Math. Soc. 15(5), 401–487 (1983)
Acknowledgements
The second author is greatly indebted to Oliver Petersen for letting him know about the important Riemannian metric \(\sigma \) on \({\mathcal {C}}\) invariant under the flow of the vector field V normalizing the surface gravity to a constant.
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Bustamante, I., Reiris, M. A classification theorem for compact Cauchy horizons in vacuum spacetimes. Gen Relativ Gravit 53, 36 (2021). https://doi.org/10.1007/s10714-021-02809-z
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DOI: https://doi.org/10.1007/s10714-021-02809-z