Radiative falloff in Schwarzschild–de Sitter spacetime

Abstract

We consider the evolution of a scalar field propagating in Schwarzschild--de Sitter spacetime. The field is non-minimally coupled to curvature through a coupling constant $\ensuremath{\xi}$. The spacetime has two distinct time scales, ${t}_{e}{=r}_{e}/c$ and ${t}_{c}{=r}_{c}/c$, where ${r}_{e}$ is the radius of the black-hole horizon, ${r}_{c}$ the radius of the cosmological horizon, and c the speed of light. When ${r}_{c}\ensuremath{\gg}{r}_{e},$ the field's time evolution can be separated into three epochs. At times $t\ensuremath{\ll}{t}_{c},$ the field behaves as if it were in pure Schwarzschild spacetime; the structure of spacetime far from the black hole has no influence on the evolution. In this early epoch, the field's initial outburst is followed by quasi-normal oscillations, and then by an inverse power-law decay. At times $t\ensuremath{\lesssim}{t}_{c},$ the power-law behavior gives way to a faster, exponential decay. In this intermediate epoch, the conditions at radii $r\ensuremath{\gtrsim}{r}_{e}$ and $r\ensuremath{\lesssim}{r}_{c}$ both play an important role. Finally, at times $t\ensuremath{\gg}{t}_{c},$ the field behaves as if it were in pure de Sitter spacetime; the structure of spacetime near the black hole no longer influences the evolution in a significant way. In this late epoch, the field's behavior depends on the value of the curvature-coupling constant $\ensuremath{\xi}$. If $\ensuremath{\xi}$ is less than a critical value ${\ensuremath{\xi}}_{c}=3/16$, the field decays exponentially, with a decay constant that increases with increasing $\ensuremath{\xi}$. If $\ensuremath{\xi}>{\ensuremath{\xi}}_{c},$ the field oscillates with a frequency that increases with increasing $\ensuremath{\xi}$; the amplitude of the field still decays exponentially, but the decay constant is independent of $\ensuremath{\xi}$. We establish these properties using a combination of numerical and analytical methods.