Abstract
The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild–de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild–de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild–de Sitter spacetime in the asymptotic region. We show that small enough perturbations of asymptotic initial data for the Schwarzschild–de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild–de Sitter spacetime.
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Andersson, L., Galloway, G.J.: dS/CFT and space–time topology. Adv. Theor. Math. Phys. 6, 307 (2003)
Bażański, S.L., Ferrari, V.: Analytic extension of the Schwarzschild–de Sitter metric. Il Nuovo Cimento B 91, 126 (1986)
Beig, R., O’Murchadha, N.: The momentum constraints of General Relativity and spatial conformal isometries. Commun. Math. Phys. 176, 723 (1996)
Beyer, F.: Asymptotics and singularities in cosmological models with positive cosmological constant. Ph.D. thesis, University of Potsdam (2007)
Beyer, F.: Non-genericity of the Nariai solutions: I. Asymptotics and spatially homogeneous perturbations. Class. Quantum Gravit. 26, 235015 (2009)
Beyer, F.: Non-genericity of the Nariai solutions: II. Investigations within the Gowdy class. Class. Quantum Gravit. 26, 235016 (2009)
Bičák, J., Podolský, J.: Cosmic no-hair conjecture and black-hole formation: an exact model with gravitational radiation. Phys. Rev. D 52, 887–895 (1995)
Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton University Press, Princeton (1993)
Dain, S., Friedrich, H.: Asymptotically flat initial data with prescribed regularity at infinity. Commun. Math. Phys. 222, 569 (2001)
Friedrich, H.: The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system. Proc. R. Soc. Lond. A 378, 401 (1981)
Friedrich, H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. R. Soc. Lond. A 375, 169 (1981)
Friedrich, H.: On the existence of analytic null asymptotically flat solutions of Einstein’s vacuum field equations. Proc. R. Soc. Lond. A 381, 361 (1982)
Friedrich, H.: Cauchy problems for the conformal vacuum field equations in General Relativity. Commun. Math. Phys. 91, 445 (1983)
Friedrich, H.: Some (con-)formal properties of Einstein’s field equations and consequences. In: Flaherty, F.J. (ed.) Asymptotic Behaviour of Mass and Space–Time Geometry. Lecture Notes in Physics, vol. 202. Springer, Berlin (1984)
Friedrich, H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys. 100, 525 (1985)
Friedrich, H.: Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant. J. Geom. Phys. 3, 101 (1986)
Friedrich, H.: On purely radiative space-times. Commun. Math. Phys. 103, 35 (1986)
Friedrich, H.: On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure. Commun. Math. Phys. 107, 587 (1986)
Friedrich, H.: On the global existence and the asymptotic behaviour of solutions to the Einstein–Maxwell–Yang–Mills equations. J. Diff. Geom. 34, 275 (1991)
Friedrich, H.: Einstein equations and conformal structure: existence of anti-de Sitter-type space–times. J. Geom. Phys. 17, 125 (1995)
Friedrich, H.: Evolution equations for gravitating ideal fluid bodies in general relativity. Phys. Rev. D 57, 2317 (1998)
Friedrich, H.: Gravitational fields near space-like and null infinity. J. Geom. Phys. 24, 83 (1998)
Friedrich, H.: Conformal Einstein evolution. In: Frauendiener, J., Friedrich, H. (eds.)The Conformal Structure of Space–Time: Geometry, Analysis, Numerics, Lecture Notes in Physics. Springer, Berlin, p. 1 (2002)
Friedrich, H.: Conformal geodesics on vacuum spacetimes. Commun. Math. Phys. 235, 513 (2003)
Friedrich, H.: Smoothness at null infinity and the structure of initial data. In: Chruściel, P.T., Friedrich, H. (eds.) 50 Years of the Cauchy Problem in General Relativity. Birkhauser, Basel (2004)
Friedrich, H.: Geometric asymptotics and beyond. In: Bieri, L., Yau, T.-S. (eds.) One Hundred Years of General Relativity. Surveys in Differential Geometry, vol. 37. International Press, Vienna (2015)
Friedrich, H., Kánnár, J.: Bondi-type systems near space-like infinity and the calculation of the NP-constants. J. Math. Phys. 41, 2195 (2000)
Friedrich, H., Schmidt, B.: Conformal geodesics in General Relativity. Proc. R. Soc. Lond. A 414, 171 (1987)
Galloway, G.J.: Cosmological spacetimes with \(\Lambda >0\). In: Dostoglou, S., Ehrlich, P. (eds.) Advances in Differential Geometry and General Relativity, Contemporary Mathematics. AMS, Providence (2004)
García-Parrado, A., Gasperín, E., Valiente Kroon, J.: Conformal Geodesics in the Schwarzshild–de Sitter and Schwarzschild–anti-de Sitter Spacetimes (in preparation, 2015)
García-Parrado, A., Martín-García, J.M.: Spinors: a Mathematica package for doing spinor calculus in General Relativity. Comp. Phys. Commun. 183, 2214 (2012)
Geyer, K.H.: Geometrie der Raum-Zeit der Maßbestimmung von Kottler, Weyl und Trefftz. Astr. Nach. 301, 135 (1980)
Griffiths, J.B., Podolský, J.: Exact Space–Times in Einstein’s General Relativity. Cambridge University Press, Cambridge (2009)
Hackmann, E., Lämmerzahl, C.: Geodesic equation in Schwarzschild-(anti)-de Sitter spacetimes: analytical solutions and applications. Phys. Rev. D 78, 024035 (2008)
Jaklitsch, M.J., Hellaby, C., Matravers, D.R.: Particle motion in the spherically symmetric vacuum solution with positive cosmological constant. Gen. Rel. Gravit. 21, 941 (1989)
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time. Cambridge University Press, Cambridge (1973)
Kato, T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181 (1975)
Kreiss, H.-O., Lorenz, J.: Stability for time-dependent differential equations. Acta Numer. 7, 203 (1998)
Lübbe, C., Tod, P.: An extension theorem for conformal gauge singularities. J. Math. Phys. 50, 112501 (2009)
Lübbe, C., Valiente, J.A.: Kroon, On de Sitter-like and Minkowski-like spacetimes. Class. Quantum Grav. 26, 145012 (2009)
Lübbe, C., Valiente Kroon, J.A.: The extended Conformal Einstein field equations with matter: the Einstein–Maxwell system. J. Geom. Phys. 62, 1548 (2012)
Lübbe, C., Valiente Kroon, J.A.: A class of conformal curves in the Reissner–Nordström spacetime. Ann. Henri Poincaré 15, 1327 (2013)
Lübbe, C., Valiente Kroon, J.A.: Spherically symmetric anti-de Sitter-like Einstein–Yang–Mills spacetimes. Phys. Rev. D 90, 024021 (2014)
Paetz, T.-T.: Killing initial data on spacelike conformal boundaries. J. Geom. Phys. 106, 51 (2016)
Penrose, R., Rindler, W.: Spinors and Space–Time, Two-Spinor Calculus and Telativistic Fields, vol. 1. Cambridge University Press, Cambridge (1984)
Penrose, R., Rindler, W.: Spinors and Space–Time, Spinor and Twistor Methods in Space–Time Geometry, vol. 2. Cambridge University Press, Cambridge (1986)
Podolký, J.: The structure of the extreme Schwarzschild–de Sitter space–time. Gen. Relat. Gravit. 31, 1703 (1999)
Schlue, V.: Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies. Commun. Math. Phys. 334(2), 977 (2015)
Schmidt, B.G.: Conformal geodesics. Lect. Notes. Phys. 261, 135 (1986)
Stanciulescu, C.: Spherically symmetric solutions of the vacuum Einstein field equations with positive cosmological constant. Master thesis, University of Vienna (1998)
Stewart, J.: Advanced General Relativity. Cambridge University Press, Cambridge (1991)
Tod, K.P.: Isotropic cosmological singularities. In: Frauendiener, J., Friedrich, H. (eds.) The Conformal Structure of Space–Time. Geometry, Analysis, Numerics. Lect. Notes. Phys., vol. 604. p. 123 (2002)
Valiente Kroon, J.A.: Conformal Methods in General Relativity. Cambridge University Press, Cambridge (2016)
Valiente Kroon, J.A.: Global evaluations of static black hole spacetimes (in preparation, 2016)
Wald, R.M.: General Relativity. The University of Chicago Press, Chicago, IL (1984)
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Communicated by James A. Isenberg.
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Gasperín, E., Valiente Kroon, J.A. Perturbations of the Asymptotic Region of the Schwarzschild–de Sitter Spacetime. Ann. Henri Poincaré 18, 1519–1591 (2017). https://doi.org/10.1007/s00023-016-0544-z
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DOI: https://doi.org/10.1007/s00023-016-0544-z