Accelerating black holes: Quasinormal modes and late-time tails

Black holes found in binaries move at very high velocities relative to our own reference frame and can accelerate due to the emission of gravitational radiation. Here, we investigate the numerical stability and late-time behavior of linear scalar perturbations in accelerating black holes described by the $C$-metric. We identify a family of quasinormal modes associated with the photon surface and a brand new family of purely imaginary modes associated with the boost parameter of the accelerating black hole spacetime. When the accelerating black hole is charged, we find a third family of modes which dominates the ringdown waveform near extremality. Our frequency- and time-domain analysis indicates that such spacetimes are stable under scalar fluctuations, while the late-time behavior follows an exponential decay law, dominated by quasinormal modes. This result is in contrast with the common belief that such perturbations, for black holes without a cosmological constant, always have a power-law cutoff. In this sense, our results suggest that the asymptotic structure of black hole backgrounds does not always dictate how radiative fields behave at late times.

Figure 1

The Penrose diagram, adapted from [50], of the charged $C$-metric, neglecting the axis $\theta =0$. The dashed lines are identified, while the wiggled lines correspond to curvature singularities at $r=0$. In turn, ${\mathcal{H}}^{\pm }$, ${\mathcal{H}}_{\alpha }^{\pm }$, ${\mathcal{CH}}^{\pm }$, and ${\mathcal{J}}^{\pm }$ correspond to the future $(+)$ and past $(-)$ event, acceleration, Cauchy horizon, and null infinity.

Figure 2

Real (left) and imaginary (right) parts of $n=0$ (solid curves) and $n=1$ (dashed curves) scalar QNMs with $m_0=1$ ($\ell =1$) vs the boost parameter $\alpha M$ of an accelerating Schwarzschild black hole. The blue curves indicate the photon surface QNMs, while the red ones indicate the acceleration modes. The circles at $\alpha =0$ designate the scalar $l=1$ QNMs of the static Schwarzschild black hole.

Figure 3

Real (left) and imaginary (right) parts of $n=0$ (solid curves) and $n=1$ (dashed curves) scalar QNMs with $m_0=0$ ($\ell =0$) vs the electric charge $Q/M$ of an accelerating Reissner-Nordström black hole with boost parameter $\alpha M=0.13$. The blue curves indicate the photon surface QNMs, the red ones indicate the acceleration modes, while the green curves indicate the near-extremal modes.

Figure 4

Real (left) and imaginary (right) parts of $n=0$ photon surface QNMs with $m_0=0$ ($\ell =0$) (top panel) and $m_0=1$ ($\ell =1$) (bottom panel) vs the electric charge $Q/M$ of an accelerating Reissner-Nordström black hole, for various boost parameters $\alpha M$.

Figure 5

Imaginary part of the fundamental photon surface modes of the $C$-metric (left) and the charged $C$-metric with $Q/M=0.5$ (right) as a function of $m_0$ and $\alpha M$. Recall that when $\alpha =0$, then $m_0=\ell$.

Figure 6

Imaginary parts of $n=0$ acceleration modes with $m_0=0$ ($\ell =0$) (solid lines) and $m_0=1$ (dashed lines) vs the electric charge $Q/M$ of an accelerating Reissner-Nordström black hole for various boost parameters $\alpha M$.

Figure 7

Left: $n=0$ (solid curves) and $n=1$ (dashed curves) near-extremal modes with $m_0=0$ ($\ell =0$) vs the boost parameter $\alpha M$ of an accelerating Reissner-Nordström black hole with electric charge $Q/M=0.999$. Right: fundamental near-extremal modes with $m_0=0$ ($\ell =0$) vs the boost parameter $\alpha M$ of an accelerating Reissner-Nordström black hole with varying electric charge $Q/M$.

Figure 8

Top left: time evolution of a scalar field with $\ell =1$ on a Schwarzschild (blue curve) and Reissner-Nordtröm (red curve) black hole with $Q/M=0.5$. Top right: time evolution of a scalar field with $m_0=10$ on an accelerating black hole with $Q/M=0.3$, $\alpha M=0.1$ (blue curve) and a scalar field with $m_0=0$ on an accelerating black hole with $Q/M=0.999$, $\alpha M=0.5$ (red curve). The dominant QNMs in both cases belong to the photon surface family. Bottom left: time evolution of a scalar field with $m_0=0$ on an accelerating black hole with $Q/M=0.3$, $\alpha M=0.05$ (blue curve), and a scalar field with $m_0=1$ on an accelerating black hole with $Q=0$ and $\alpha M=0.03$ (red curve). The dominant QNMs in both cases belong to the acceleration family. Bottom right: time evolution of a scalar field with $m_0=0$ on an accelerating black hole with $Q/M=0.9995$, $\alpha M=0.6$ (blue curve), and a scalar field with $m_0=0$ on an accelerating black hole with $Q/M=0.999$, $\alpha M=0.3$ (red curve). The dominant QNMs in both cases belong to the near-extremal family. The QNMs for all cases are given in Table 3.