Scalar field excited around a rapidly rotating black hole in Chern-Simons modified gravity

We discuss a Chern-Simons (CS) scalar field around a rapidly rotating black hole in dynamical CS modified gravity. The CS correction can be obtained perturbatively by considering the Kerr spacetime to be the background. We obtain the CS scalar field solution around the black hole analytically and numerically, assuming a stationary and axisymmetric configuration. The scalar field diverges on the inner horizon when we impose the boundary conditions that the scalar field be regular on the outer horizon and vanish at infinity. Therefore, the CS scalar field becomes problematic on the inner horizon.

Figure 1

(a) Scalar function ${\mathrm{\Theta }}_1$ and (b) its derivative $d{\mathrm{\Theta }}_1/d\tilde{r}$ with respect to $\tilde{r}$ when $\tilde{a}=0.1$. The red curve shows our result, while the dashed blue curve shows the result from the slow rotation approximation (SRA). The inner and outer horizons correspond to ${\tilde{r}}_{-}\simeq 0.005$ and ${\tilde{r}}_{+}\simeq 1.995$, respectively.

Figure 2

(a) Scalar function ${\mathrm{\Theta }}_1$ and (b) its derivative $d{\mathrm{\Theta }}_1/d\tilde{r}$ with respect to $\tilde{r}$ when $\tilde{a}=0.5$. The red curve shows our result, while the dashed blue curve shows the result from the slow rotation approximation (SRA). The inner and outer horizons correspond to ${\tilde{r}}_{-}\simeq 0.134$ and ${\tilde{r}}_{+}\simeq 1.866$, respectively.

Figure 3

(a) Scalar function ${\mathrm{\Theta }}_1$ and (b) its derivative $d{\mathrm{\Theta }}_1/d\tilde{r}$ with respect to $\tilde{r}$ when $\tilde{a}=0.8$. The red curve shows our result, while the dashed blue curve shows the result from the slow rotation approximation (SRA). The inner and outer horizons correspond to ${\tilde{r}}_{-}\simeq 0.4$ and ${\tilde{r}}_{+}\simeq 1.6$, respectively.

Figure 4

${\mathrm{\Theta }}_m(\tilde{r})$ $(m=1,3,5,\cdots ,15)$ obtained by numerical integration when $\tilde{a}=0.8$. The result of ${\mathrm{\Theta }}_1$ (solid curve) coincides with the analytic result (cross). The inner and outer horizons correspond to ${\tilde{r}}_{-}\simeq 0.4$ and ${\tilde{r}}_{+}\simeq 1.6$, respectively.

Figure 5

(a) Color map of ${\vartheta }^{(1)}$ for $\tilde{a}=0.8$. The vertical axis denotes $\tilde{r}\mu$, and the horizontal axis denotes $\tilde{r}\sqrt{1-{\mu }^2}$. (b) three-dimensional plot of ${\vartheta }^{(1)}(\tilde{r},\mu )$ for $\tilde{a}=0.8$. The inner horizon corresponds to ${\tilde{r}}_{-}\simeq 0.4$.