zbMATH — the first resource for mathematics

A class of conformal curves in the Reissner-Nordström spacetime. (English) Zbl 1295.83042
Summary: A class of curves with special conformal properties (conformal curves) is studied on the Reissner-Nordström spacetime. It is shown that initial data for the conformal curves can be prescribed so that the resulting congruence of curves extends smoothly to future and past null infinity. The formation of conjugate points on these congruences is examined. The results of this analysis are expected to be of relevance for the discussion of the Reissner-Nordström spacetime as a solution to the conformal field equations and for the global numerical evaluation of static black hole spacetimes.

83C57 Black holes
83C15 Exact solutions to problems in general relativity and gravitational theory
83C10 Equations of motion in general relativity and gravitational theory
53Z05 Applications of differential geometry to physics
Full Text: DOI
[1] Aretakis, S., Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II, Ann. Henri Poincare, 12, 1491, (2011) · Zbl 1242.83049
[2] Aretakis, S., Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I, Comm. Math. Phys., 307, 17, (2011) · Zbl 1229.85002
[3] Bizon, P.; Friedrich, H., A remark about wave equations on the extreme Reissner-Nordström black hole exterior, Class. Quantum Grav., 30, 065001, (2013) · Zbl 1267.83049
[4] Carter, B.: Black hole equilibrium states. In: DeWitt, C., DeWitt, B. (ed.) Black holes—les astres occlus, page 61. Gordon and Breach, USA (1973)
[5] Dafermos, M., Stability and instability of the Cauchy horizon for the spherically symmetric Einstein-Maxwell-scalar field equations, Ann. Math., 158, 875, (2003) · Zbl 1055.83002
[6] Dafermos, M.: The interior of charged black holes and the problem of uniqueness in general Relativity. Commun. Pure Appl. Math. LVIII:0445 (2005) · Zbl 1071.83037
[7] Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. (2008, arXiv:0811.0354[gr-qc]) · Zbl 1263.83188
[8] Dain, S., Dotti, G.: The wave equation on the extreme Reissner-Nordström black hole. (2012, arXiv:1209.0213) · Zbl 1263.83087
[9] 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) · Zbl 0737.53070
[10] Friedrich, H., Einstein equations and conformal structure: existence of anti-de Sitter-type space-times, J. Geom. Phys., 17, 125, (1995) · Zbl 0840.53055
[11] Friedrich, H., Gravitational fields near space-like and null infinity, J. Geom. Phys., 24, 83, (1998) · Zbl 0896.53053
[12] Friedrich, H.: Conformal Einstein evolution. In: Frauendiener, J., Friedrich, H. (eds.). The conformal structure of spacetime: geometry, analysis, numerics. Lecture Notes in Physics, page 1. Springer, Berlin (2002)
[13] Friedrich, H., Conformal geodesics on vacuum spacetimes, Commun. Math. Phys., 235, 513, (2003) · Zbl 1040.53079
[14] 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. Birkhausser, Basel (2004) · Zbl 1072.83003
[15] Friedrich, H.; Schmidt, B., Conformal geodesics in general relativity, Proc. R. Soc. Lond. A, 414, 171, (1987) · Zbl 0629.53063
[16] Griffiths J.B., Podolský J.: Exact space-times in Einstein’s General Relativity. Cambridge University Press, London (2009) · Zbl 1184.83003
[17] Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, London (1973) · Zbl 0265.53054
[18] Kruskal, M.D., Maximal extension of Schwarzschild metric, Phys. Rev. D, 119, 1743, (1960) · Zbl 0098.19001
[19] Lawden D.F.: Elliptic Functions and Applications. Springer, Berlin (1989) · Zbl 0689.33001
[20] Lübbe, C.; Valiente Kroon, J.A., The extended conformal Einstein field equations with matter: the Einstein-Maxwell system, J. Geom. Phys., 62, 1548, (2012) · Zbl 1239.53018
[21] Lübbe, C.; Valiente Kroon, J.A., A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies, Ann. Phys., 328, 1, (2013) · Zbl 1263.83188
[22] Penrose, R.W. Rindler: Spinors and Space-Time. Spinor and Twistor Methods in Space-Time Geometry, Vol. 2. Cambridge University Press, London (1986) · Zbl 0591.53002
[23] Schmidt, B.G.; Walker, M., Analytic conformal extensions of asymptotically flat spacetimes, J. Phys. A Math. Gen., 16, 2187, (1983) · Zbl 0533.53053
[24] Stephani H., Kramer D., MacCallum M.A.H., Hoenselaers C., Herlt E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, London (2003) · Zbl 1057.83004
[25] Stewart J.: Advanced General Relativity. Cambridge University Press, London (1991) · Zbl 0752.53048
[26] Valiente Kroon, J.A.: Global evaluations of static black hole spacetimes (In preparation) · Zbl 1073.83015
[27] Zenginoglu, A.: A conformal approach to numerical calculations of asymptotically flat spacetimes. PhD thesis, Max-Planck Institute for Gravitational Physics (AEI) and University of Potsdam (2006) · Zbl 1239.53018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.