Rigidity in vacuum under conformal symmetry
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Abstract
Motivated in part by Eardley et al. (Commun Math Phys 106(1):137–158, 1986), in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.
Keywords
Lorentzian rigidity Vacuum equations Conformal symmetryMathematics Subject Classification
53C50 83C75Notes
Acknowledgements
GJG’s research was supported in part by NSF Grants DMS-1313724 and DMS-1710808.
References
- 1.Bartnik, R.: Remarks on cosmological spacetimes and constant mean curvature surfaces. Commun. Math. Phys. 117(4), 615–624 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 2.Besse, A.: Einstein Manifolds. Classics in Mathematics. Springer, Berlin (1987)CrossRefGoogle Scholar
- 3.Eardley, D., Isenberg, J., Marsden, J., Moncrief, V.: Homothetic and conformal symmetries of solutions to Einstein’s equations. Commun. Math. Phys. 106(1), 137–158 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 4.Ehrlich, P.E., Galloway, G.J.: Timelike lines. Class. Quantum Gravity 7(3), 297 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 5.Eschenburg, J.-H.: The splitting theorem for space-times with strong energy condition. J. Differ. Geom. 27(3), 477–491 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Eschenburg, J.-H., Galloway, G.J.: Lines in space-times. Commun. Math. Phys. 148(1), 209–216 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 7.Galloway, G.J.: Splitting theorems for spatially closed space-times. Commun. Math. Phys. 96(4), 423–429 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 8.Galloway, G.J.: The Lorentzian splitting theorem without the completeness assumption. J. Differ. Geom. 29, 373–387 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Galloway, G.J.: Some rigidity results for spatially closed space-times. Mathematics of gravitation, Part I (Warsaw, 1996), Banach Center Publications, vol. 41, pp. 21–34. Polish Academy of Science, Warsaw (1997)Google Scholar
- 10.Galloway, G.J., Vega, C.: Achronal limits, Lorentzian spheres, and splitting. Ann. Henri Poincaré 15(11), 2241–2279 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 11.Galloway, G.J., Vega, C.: Hausdorff closed limits and rigidity in Lorentzian geometry. Ann. Henri Poincaré 18(10), 3399–3426 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 12.Garfinkle, D., Harris, S.G.: Ricci fall-off in static and stationary, globally hyperbolic, non-singular spacetimes. Class. Quantum Gravity 14(1), 139–151 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 13.Harris, S.G.: On maximal geodesic-diameter and causality in Lorentz manifolds. Mathematische Annalen 261(3), 307–313 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London (1973)CrossRefGoogle Scholar
- 15.Newman, R.P.A.C.: A proof of the splitting conjecture of S.-T. Yau. J. Differ. Geom. 31, 163–184 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.O’Neill, B.: Semi-Riemannian Geometry. Pure and Applied Mathematics, vol. 103. Academic Press Inc., New York (1983)Google Scholar
- 17.Yano, K.: The Theory of Lie Derivatives and Its Applications. Bibliotheca Mathematica, vol. 3. North-Holland Pub. Co., Amsterdam (1957)Google Scholar
- 18.Yau, S.-T.: Problem Section. Annals of Mathematics Studies, No. 102, pp. 669–706. Princeton University Press, Princeton (1982)Google Scholar
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