Non-modal linear stability of the Schwarzschild black hole    [PDF]

Gustavo Dotti
A proof is given that the space $\L$ of solutions of the linearized vacuum Einstein equations around a Schwarzschild black hole is parameterized by the first order variations $\d Q_{\pm}$ of the quadratic Weyl tensor invariants $Q_+ = C_{\a \b \g \d} C^{\a \b \g \d}$ and $Q_-= C^*_{\a \b \g \d} C^{\a \b \g \d}$. The perturbed metric $\delta g_{\a \b}$ in the Regge Wheler gauge can be reconstructed from the $\d Q_{\pm}$, the variation $\d Q_{-}$ being related to the odd Regge Wheeler modes, and $\d Q_{+}$ to the even Zerilli modes. All the information on the perturbed geometry is therefore encoded in these two scalar fields. The linearized Einstein's equation are shown to be equivalent to the scalar wave equations $(\nabla^{\a} \nabla_{\a} + \tfrac{8M}{r^3}) (r^5 \delta Q_{\pm}) =0$. This is used to prove that $|\d Q_{\pm}| < C_{\pm}/r^6$ on the exterior region $r \geq 2M$.
View original: http://arxiv.org/abs/1307.3340