The authors of this very interesting paper study the classical stability of higher-dimensional theory of Schwarzschild black holes with respect to linear perturbations in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes. The \((n+2)\)-dimensional metric has the form \[ ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\sigma_n^2, \] where \(f(r)=K-2Mr^{1-n}-\lambda r^2\), with constant parameters \(M\), \(\lambda \) and \(K=0,\pm 1\), and \(d\sigma_n^2=\gamma (z)_{ij}dz^idz^j\) denotes the metric of the \(n\)-dimensional maximally symmetric space \(\mathcal K\)\(^n\) with sectional curvature \(K=0,\pm 1\). The parameter \(M\) defines the black hole mass. In case of static background metric with a Killing parameter \(t\) in the Schwarzschild wedge, master equations have the form \[ \partial^2\Phi /\partial t^2=(\partial^2 / \partial r_{\ast }^2 -V)\Phi , \] where \(r_{\ast }\equiv \int drf^{-1}(r)\) and \(V\) is a smooth function of \(r_{\ast }\). It is shown that the differential operator \(A=-d^2/dr_{\ast }^2+V\) is a positive self-adjoint operator in \(L^2(r_{\ast }, dr_{\ast })\) (Hilbert space of square integrable functions of \(r_{\ast }\) so that the master equation possesses no negative mode normalisable solutions).
The authors show that higher-dimensional Schwarzschild black holes are stable with respect to linear perturbations. A nonexistence result for unstable solutions to the master equation with physically acceptable initial data is proved. It is also shown that there exist no well-behaved static perturbations, which are regular everywhere outside the event horizon. This is consistent with the uniqueness of a certain class of black holes in an asymptotically flat background. The stability of the maximally symmetric black holes (with a non-vanishing cosmological constant) with respect to tensor- and vector-type perturbations is discussed. A stability criterion for generalized static black holes studied by G. Gibons and Hartnoll [Phys. Rev. D (3) 66, No. 6, 064024 (2002)] is established, too.