Instability of the Cauchy horizon of Reissner-Nordström black holes

Abstract

The stability of the inner Reissner-Nordstr\"om geometry is studied with test massless integer-spin fields. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate most of the results. Monochromatic waves which are of small amplitude and ingoing near the outer horizon develop infinite energy densities near the inner Cauchy horizon (as measured by a freely falling observer). Previous work has shown that certain derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon ($r={r}_{\ensuremath{-}}$) if energy densities are to remain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past null infinity (${\mathcal{I}}^{\ensuremath{-}}$), cannot be localized near ${r}_{+}$, the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the black hole. Price, Sibgatullin, and Alekseev have shown that such moments are radiated away as "tails" which travel outward and are rescattered inward yielding a wave field with a time dependence ${t}^{\ensuremath{-}p}$, $p>0$. This decay in time is sufficiently slow that the tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with the time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. G\"ursel et al. have further shown that even perturbations which are localized as they cross the outer horizon produce singularities at the inner horizon. By a ray-tracing scheme we are able to show that this singularity arises when the incoming radiation is first scattered for $r\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{<}{r}_{+}$ (i.e., just inside the outer horizon), whence the exponentially small scattered radiation is efficiently rescattered when the potential becomes strong. The exponentially small first scattering near the outer horizon is translated by the second scattering into exponentially decaying waves near the inner horizon. Their exponential decay is, however, so slow that the resultant energy density is singular on the horizon.