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Hidden symmetries and decay for the wave equation on the Kerr spacetime. (English) Zbl 1373.35307

In this paper the authors consider the covariant wave equation \(\nabla^{a}\nabla_{a}\psi =0\) in the exterior region of the slowly rotating Kerr spacetime. They prove boundedness and integrated energy decay for solutions of the above stated equation. Introduce \(r_{+}=M+\sqrt{M^2-a^2}\), \(\Sigma = r^2+a^2\cos^2\theta \); \(M\) is the mass. Then in Boyer-Lindquist coordinates \((x^{\alpha }) = (t, r, \theta , \phi )\), the exterior region is given by the manifold \(\mathbb{R}\times (r_{+},\infty ) \times S^2\) with the Lorentz metric \(g_{\mu\nu }\). It is known that for \(0\leq |a| \leq M\), the Kerr family of metrics describes an asymptotically flat, stationary and axi-symmetric solution of the vacuum Einstein equation, containing a rotating black hole with mass \(M\) and angular momentum \(M a\), and with the horizon located at \(r = r_{+} \). The Schwarzschild spacetime is the subcase with \(a = 0\). Here \(M>0\) is fixed and the authors study the slowly rotating case, that is, \(|a|\ll M\). The exterior region is globally hyperbolic, with the surfaces \(\Sigma_t \) of constant \(t\) as Cauchy surfaces. Then the wave equation is well posed in the exterior region even though the Kerr spacetime can be extended. Here it is considered some initial data on the hypersurface \(\Sigma_0 \).
The authors establish new original methods to prove two known and at the same time very important results. The first is the following statement known as Uniformly bounded positive energy, that is: “Given \(M>0\), then there are positive constants \(C\) and \(\bar{a}\) such that if \(|a|\leq \bar{a}\) and \(\psi : \mathbb{R}\times (r_{+},\infty )\times S^2\to \mathbb{R}\) is a solution of the wave equation \(\nabla^{a}\nabla_{a}\psi =0\), then for all \(t\), \(E_{\text{model,3}}(\Sigma_t)\leq C E_{\text{model,3}}(\Sigma_0)\)”. Here \(E_{\text{model,3}}(\Sigma_t)\) is the model energy.
The second result is known as Morawetz estimate, that is: “Given \(M > 0\), there are positive constants \(\bar{a}\), \(\bar{r}\), \(C\) and a function that is identically one for \(|r - 3 M | > \bar{r}\) and zero otherwise, such that for all \(|a|\leq\bar{a}\) and all smooth \(\psi \) solving the above stated wave equation, there exists an integral inequality in which takes place the angular gradient in Boyer-Lindquist coordinates in the measure \(d^3\mu dt \) and the model energy”.

MSC:

35Q76 Einstein equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57 Black holes
83F05 Relativistic cosmology
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
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