Supersymmetric black holes play an important role in string theory. It is natural to ask whether they are classically stable or not, i.e. does a small initial perturbation remain small with time evolution?
In supergravity from the fact that supersymmetric solution staurates BPS-bound does not imply classical stability. For example, anti-de-Sitter spacetime is a supersymmetric solution of various supergravity theories but it is classically unstable against the formation of small black holes.
In AdS classical stability was analysed under time evolution determined by the nonlinear Einstein equation. But Aretakis showed that even linear perturbations of a supersymmetric black hole can exhibit instability. He proved that derivatives of a massless scalar field grow polinomially on the horizon of extreme Reissner-Nordstrom black hole. It's interesting to investigate stability of tensor fields.
To have an analytical solution, radial infall of non-charged massive body to a ERN black hole was considered.
Einstein equations were linearized and solved in the near-horizon limit.
As the result, it was shown that all components of metric perturbation are damped exponentially, but l=0,1 spherical harmonics of rr-component oscillate and do not decay with time.