Energy multipliers for perturbations of the Schwarzschild metric
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 \(\Phi\) of \(g\) and \(\nu=0\) on \(\Phi\). Furthermore, the sufficient conditions are also proved. On the contrary, as to the Kerr metrics, the situation is much worse; there exists a continuous family of partial photon spheres, accumulating to \(\Phi\).
scientific article; zbMATH DE number 5607329