Scalar field perturbations of the Schwarzschild black hole in the Gödel universe

We investigate the scalar field perturbations of the $4+1$-dimensional Schwarzschild black hole immersed in a Gödel universe, described by the Gimon-Hashimoto solution. This may model the influence of the possible rotation of the universe upon the radiative processes near a black hole. In the regime when the scale parameter $j$ of the Gödel background is small, the oscillation frequency is linearly decreasing with $j$, while the damping time is increasing. The quasinormal modes are damping, implying stability of the Schwarzschild-Gödel space-time against scalar field perturbations. The approximate analytical formula for large multipole numbers is found.

Figure 1

Real part of $\omega$ as a function of the Gödel background scale parameter $j$ (diamond $k=2$, $m=1$, $l=1$), (box $k=1$, $m=1$, $l=1$), (star $k=0$, $m=1$, $l=1$).

Figure 2

Imaginary part of $\omega$ as a function of the Gödel background scale parameter $j$ (star $k=2$, $m=1$, $l=1$), (box $k=1$, $m=1$, $l=1$), (diamond $k=0$, $m=1$, $l=1$).

Figure 3

Imaginary part of $\omega$ as a function of the mass $\mu$ for $l=1$, $k=1$, $m=1$ (box), $l=1$, $k=0$, $m=1$ (star), $l=1$, $k=0$, $m=0$ (diamond); $j=1/8$.

Figure 4

Real part of $\omega$ as a function of the mass $\mu$ for $l=1$, $k=1$, $m=1$ (box), $l=1$, $k=0$, $m=1$ (star), $l=1$, $k=0$, $m=0$ (diamond); $j=1/8$.