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Lyapunov functional approach to raiative metrics

Lyapunov's second method is applied to the spherical radiative Robinson-Trautman vacuum space-times to prove that they asymptotically settle down to Schwarzschild space-time. This class of Robinson-Trautman metrics is characterized by the surfaceS being topologically a two-sphere, whereS is invariantly defined by the intersection of the hypersurfacesu=const andr=const. It is shown that ∝ _S K ² dσ is a Lyapunov functional, whereK is the Gaussian curvature anddσ is the invariant measure onS. The critical point occurs atK=0 or, equivalently, at ð² K=0, which condition is shown to characterize Schwarzschild space-time.

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1. Central Research Institute for Physics, P.O.B. 49, H-1525, Budapest 114, Hungary

   B. Lukács & Z. Perjés

2. Department of Mathematics, University of Pittsburgh, 15260, Pittsburgh, Pennsylvania

   J. Porter

3. Central Research Institute for Physics, P.O.B. 49, H-1525, Budapest 114, Hungary

   Á. Sebestyén

1. B. Lukács
2. Z. Perjés
3. J. Porter
4. Á. Sebestyén

A product of a collaboration supported by the NSF and the Hungarian Academy of Sciences.

Lukács, B., Perjés, Z., Porter, J. et al. Lyapunov functional approach to raiative metrics. Gen Relat Gravit 16, 691–701 (1984). https://doi.org/10.1007/BF00767861

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• Received: 21 October 1983

• Issue Date: July 1984

• DOI: https://doi.org/10.1007/BF00767861

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