Quasinormal modes of gravitational perturbation for uniformly accelerated black holes

We first show that the master equations for massless perturbations of accelerating rotating black holes can be transformed into the Heun’s equation. The quasinormal modes of the black holes can be easily calculated in the framework of the Heun’s equation. We identify three modes for the tensor perturbations: the photon sphere modes, which reduce to the quasinormal modes of Kerr black holes when the acceleration parameter vanishes; the near-extremal modes, which branch from the first set and become dominant when the spin is near extremal; and the acceleration modes, which are closely related to the acceleration horizon. We calculate the frequency spectrum of the quasinormal modes in various spin and acceleration parameters. We choose an angular boundary condition that keeps the angular function regular at $\theta =0$ and $\pi$, which is consistent with the boundary condition of the Kerr black hole. The conical singularity caused by the acceleration influences this boundary condition. We find that the $m_0=1$ modes have an anomalous behavior at particular accelerations.

Figure 1

Real (left) and imaginary (right) parts of the fundamental modes for all three sets with $s=-2$, $l=2$, $m_0=0$. The upper panels show the frequencies as a function of $a$ with fixed $A=0.05$. The real parts of the NE mode and A mode vanish, and we do not show them in the upper left panel. We only show part of the NE mode because its imaginary part decreases drastically when $a$ decreases. The bottom panels show the frequencies as a function of $A$ with fixed $a=0.99$. We can only reach $A=0.83$ during our numerical computation. The extremal value of $A$ is 0.876. Similarly, the imaginary part of the A mode decreases drastically with $A$ increasing, and we only show part of it.

Figure 2

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $a=0$. The black point represents the fundamental mode of the Schwarzschild black hole. The extremal value of $A$ is 0.5. The vertical line represents the value of $A$ that satisfies $\frac{P(\pi )}{P(0)}=2$.

Figure 3

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $a=0.5$. The extremal value of $A$ is 0.535. The vertical line represents the value of $A$ that satisfies $\frac{P(\pi )}{P(0)}=2$.

Figure 4

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $a=0.9$. The extremal value of $A$ is 0.696. The vertical line represents the value of $A$ that satisfies $\frac{P(\pi )}{P(0)}=2$.

Figure 5

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $A={10}^{-4}$.

Figure 6

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $A=0.2$.

Figure 7

Real (left) and imaginary (right) parts of $n=0$ PS modes for $s=-2$, $l=2$, $m_0=2,1,0,-1,-2$ with $A=0.4$.

Figure 8

Imaginary parts of $n=0$ and 1 NE modes for $s=-2$, $l=2$, $m_0=0$. The left panel shows $\mathrm{Im}({\omega }_{\mathrm{NE}})$ as a function of $a$ with $A=0.05$. The right panel shows $\mathrm{Im}({\omega }_{\mathrm{NE}})$ as a function of $A$ with $a=0.999$. We can only reach $A=0.93$ during our numerical computation. The extremal value of $A$ is 0.957.

Figure 9

Real (left) and imaginary (right) parts of $n=0$ A modes for $s=-2$, $l=2$, $m_0=2$, 1, 0 with $A=0.05$.

Figure 10

Real (left) and imaginary (right) parts of $n=0$ A modes for $s=-2$, $l=2$, $m_0=2$, 1, 0 with $a=0.2$.