Black holes in anti–de Sitter spacetime: Quasinormal modes, tails, and flat spacetime

Black holes (BHs) in asymptotically anti–de Sitter (AdS) spacetimes have been the subject of intense scrutiny, including detailed frequency-domain analysis and full nonlinear evolutions. Remarkably, studies of linearized perturbations in the time-domain are scarce or nonexistant. We close this gap by evolving linearized scalar wave packets in the background of rotating BHs in AdS spacetimes. Our results show a number of interesting features. Small BHs in AdS behave as asymptotically flat BHs for early/intermediate times, displaying the same ringdown modes and power-law tails. As the field bounces back and forth between the horizon and the timelike boundary it “thermalizes” and the modes of AdS settle in. Finally, we have indications that wave packets in the vicinity of fast spinning BHs grow exponentially in time, signaling a superradiant instability of the geometry previously reported through a frequency-domain analysis.

Figure 1

Very long time evolution of a spherically symmetric ($l=m=0$) wave packet in a Kerr-AdS background, with $\ell ={10}^{4}M$, contrasted against the evolution of the same initial wave packet in Schwarzschild-AdS.

Figure 2

Time evolution of a spherically symmetric ($l=m=0$) wave packet in a small Schwarzschild-AdS background, with $\ell ={10}^{4}M$, contrasted against the evolution of the same initial wave packet in asymptotically flat spacetime. The pulse shows signs of Schwarzschild ringdown and power-law tail at very early times. The complete agreement with that of asymptotically BHs is shown in the inset, which zooms-in the evolution at early times. At very late times, not shown here, the two evolutions differ appreciably.

Figure 3

Time evolution of a wave packet in a small Kerr-AdS background, with $\ell =64M$, for an initial $l=m=0$ scalar wave packet. The BH is spinning at a rate $a/M=0$ (black line) and $a/M=0.995$ (blue line). The $m=0$ pulse “thermalizes” very quickly, and at late times, we find good agreement with the Kerr-AdS QNMs computed via frequency-domain methods [12]. The inset shows an early-time zoom-in of the evolution, which shows signs of ringdown similar to that of asymptotically flat BHs; however, before ringdown “dies off,” the signal has had time to travel to and back from the boundary, significantly distorting the waveform.

Figure 4

Time evolution of a dipolar $l=1$ wave packet in a small Kerr-AdS background, with $\ell ={10}^{4}M$. Left panel shows $m=1$ waveforms in the background of a BH with spin $a=0$ and $a=0.995M$. The right panel zooms in on the corresponding signal at late times. The waveform begins with a high frequency part corresponding to a Schwarzschild ringdown. At time scales $t\sim \ell$ the effects of the cosmological constant are felt and a low frequency component eventually settles in before complete reflection at the boundary. Eventually, the low frequency component dominates the signal. The turnover to the AdS, with “thermalization” at low frequencies is more clearly seen in Fig. 3.

Figure 5

Very long time evolution of an $m=1$ wave packet in a Kerr-AdS background, with $\ell =64M$ and $a/M=0.995$, contrasted against the evolution of the same, however, with $m=0$. Instability on the scale of ${10}^{6}M$ develops in the non-axisymmetric case, as predicted by frequency-domain analysis [15, 16, 17, 18].