Testing the membrane paradigm with holography

One version of the membrane paradigm states that, as far as outside observers are concerned, black holes can be replaced by a dissipative membrane with simple physical properties located at the stretched horizon. We demonstrate that such a membrane paradigm is incomplete in several aspects. We argue that it generically fails to capture the massive quasinormal modes, unless we replace the stretched horizon by the exact event horizon, and illustrate this with a scalar field in a Banados-Teitelboim-Zanelli (BTZ) black hole background. We also consider as a concrete example linearized metric perturbations of a five-dimensional AdS-Schwarzschild black brane and show that a spurious excitation appears in the long-wavelength response that is only removed from the spectrum when the membrane paradigm is replaced by ingoing boundary conditions at the event horizon. We interpret this excitation in terms of an additional Goldstone boson that appears due to symmetry breaking by the classical solution ending on the stretched horizon rather than the event horizon.

Figure 1

Absolute value of the inverse of the retarded/advanced Green’s function (green/red) and the membrane paradigm approximations of the former at $u_{\delta }=0.9$ (blue) and $u_{\delta }=0.999$ (magenta) as a function of $\Im (\tilde{\omega })$ for $\Re (\tilde{\omega })=0$ (left) and $\Re (\tilde{\omega })=5$ (right). One can clearly see that the stretched horizon approximation works for $\Im (\tilde{\omega })>-1$, whereas for $\Im (\tilde{\omega })<-1$ it leads to the advanced Green’s function, in line with the approximation in Eq. (5). Zeros of the green curve correspond to the locations of the quasinormal modes, as given by Eq. (8), and lie beyond the range of applicability of the membrane paradigm.