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Lyapunov functional approach to radiative metrics. (English) Zbl 0536.53035

Summary: 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 surface S being topologically a two-sphere, where S is invariantly defined by the intersection of the hypersurfaces \(u=const\) and \(r=const\). It is shown that \(\int_{S}K^ 2 d\sigma\) is a Lyapunov functional, where K is the Gaussian curvature and \(d\sigma\) is the invariant measure on S. The critical point occurs at \(K=0\) or, equivalently, at \(\partial^ 2K=0\), which condition is shown to characterize Schwarzschild space-time.

MSC:

53B50 Applications of local differential geometry to the sciences
83C30 Asymptotic procedures (radiation, news functions, \(\mathcal{H} \)-spaces, etc.) in general relativity and gravitational theory
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References:

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