Energy multipliers for perturbations of the Schwarzschild metric
type: Item
instance of: scholarly article
title: Energy multipliers for perturbations of the Schwarzschild metric
zbMATH Open document ID: 1196.53053
published in: Communications in Mathematical Physics
publication date: 2009-09-25T00:00:00Z
review text: Consider a space-time homeomorphic to \({\mathbb R}^2\times S^2\), and let \((t,r,\theta,\phi)\) be the standard coordinates there; \(t\) is the time variable, \(r\) is the radial coordinate, \(\theta\) and \(\phi\) are longitude and colatitude coordinates, respectively. In the region \(0<2m<r_0 \leq r \leq r_1<\infty\), the present paper reveals several aspects of the wave equations \(\square_g\psi=0\) associated to two types of Lorenzian metrics \(g\) close to the Schwarzschild metric in the theory of relativity: {\parindent6mm \begin{itemize}\item[i)] Rotationally invariant metrics: Let \(d\sigma^2\) be the metric on the 2-dimensional sphere \(S^2\), i.e., \(d\sigma^2=d\theta^2+\sin^2\theta d\phi^2\). Then \(g\) is said to be a rotationally invariant metric if there are coordinates \((u,v)\) such that \(g\) is of the form \[ g=4\Omega^2dudv+f^2d\sigma^2 \] where \(\Omega\) and \(f\) are non-negative and smooth functions of \((u,v)\). \item[ii)] Kerr metrics: Let \((r,\theta,\phi)\) be the spherical coordinate in \({\mathbb R}^3\). Then \(ds^2\) is said to be a Kerr metric if \[ ds^2=-((\Delta-a^2\sin^2\theta)/\Sigma)dt^2-4amr(\sin^2\theta/\Sigma)dtd\phi+(\Sigma/\Delta)dr^2 +[.]/\Sigma \sin^2\theta d\phi^2+\Sigma d\theta^2, \] where \[ \Sigma=r^2+a^2\cos^2\theta, \quad \Delta=r^2+a^2-2mr, \quad [.]=(r^2+a^2)^2-\Delta a^2\sin^2\theta. \] Moreover, \(\Phi\) is said to be a photon sphere if null bicharacteristics starting tangentially to \(\Phi\) must stay on it. \end{itemize}} Then, a necessary condition for the existence of non-negative multipliers is proved: Theorem. Let \(g\) be a rotationally invariant metric close enough to the Schwarzschild metric to have a photon sphere \(\Phi\), and \(X=a(u,v)\partial_u +b(u,v)\partial_v\) be a smooth space-like field such that \(a(u,v)\) and \(b(u,v)\) are close to \(1\). If there exists a function \(\nu\) with \(d\nu \neq0\) such that \(\nu X\) is non-negative for \(g\), then \(X\) is orthogonal to the photon sphere \(\Phi…
reviewed by: Tokio Matsuyama
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 83C20
zbMATH DE Number: 5607329
zbMATH Keywords: Kerr metrics
zbMATH Keywords: rotationally invariant metric
zbMATH Keywords: Schwarzschild metric
zbMATH Keywords: partial photon spheres
author: S. Alinhac
MaRDI profile type: Publication
full work available at URL: s00220-009-0770-z
OpenAlex ID: W1990341883
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DOI: 10.1007/S00220-009-0770-Z
description: scientific article; zbMATH DE number 5607329