Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aretakis S.: Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations II. Ann. Henri Poincare 12, 1491 (2011)
Aretakis S.: Stability and Instability of Extreme Reissner–Nordström Black Hole Spacetimes for Linear Scalar Perturbations I. Comm. Math. Phys. 307, 17 (2011)
Bizon P., Friedrich H.: A remark about wave equations on the extreme Reissner–Nordström black hole exterior. Class. Quantum Grav. 30, 065001 (2013)
Carter, B.: Black hole equilibrium states. In: DeWitt, C., DeWitt, B. (ed.) Black holes—les astres occlus, page 61. Gordon and Breach, USA (1973)
Dafermos M.: Stability and instability of the Cauchy horizon for the spherically symmetric Einstein–Maxwell-scalar field equations. Ann. Math. 158, 875 (2003)
Dafermos, M.: The interior of charged black holes and the problem of uniqueness in general Relativity. Commun. Pure Appl. Math. LVIII:0445 (2005)
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. (2008, arXiv:0811.0354[gr-qc])
Dain, S., Dotti, G.: The wave equation on the extreme Reissner–Nordström black hole. (2012, arXiv:1209.0213)
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.: 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 spacetime: geometry, analysis, numerics. Lecture Notes in Physics, page 1. Springer, Berlin (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. Birkhausser, Basel (2004)
Friedrich H., Schmidt B.: Conformal geodesics in general relativity. Proc. R. Soc. Lond. A 414, 171 (1987)
Griffiths J.B., Podolský J.: Exact space–times in Einstein’s General Relativity. Cambridge University Press, London (2009)
Hawking S.W., Ellis G.F.R.: The Large Scale Structure of Space–Time. Cambridge University Press, London (1973)
Kruskal M.D.: Maximal extension of Schwarzschild metric. Phys. Rev. D 119, 1743 (1960)
Lawden D.F.: Elliptic Functions and Applications. Springer, Berlin (1989)
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 conformal approach for the analysis of the non-linear stability of pure radiation cosmologies. Ann. Phys. 328, 1 (2013)
Penrose, R.W. Rindler: Spinors and Space–Time. Spinor and Twistor Methods in Space–Time Geometry, Vol. 2. Cambridge University Press, London (1986)
Schmidt B.G., Walker M.: Analytic conformal extensions of asymptotically flat spacetimes. J. Phys. A Math. Gen. 16, 2187 (1983)
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)
Stewart J.: Advanced General Relativity. Cambridge University Press, London (1991)
Valiente Kroon, J.A.: Global evaluations of static black hole spacetimes (In preparation)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by James A. Isenberg.
Rights and permissions
About this article
Cite this article
Lübbe, C., Valiente Kroon, J.A. A Class of Conformal Curves in the Reissner–Nordström Spacetime. Ann. Henri Poincaré 15, 1327–1366 (2014). https://doi.org/10.1007/s00023-013-0276-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0276-2