Dynamical evolution of a scalar field coupling to Einstein’s tensor in the Reissner-Nordström black hole spacetime

We study the dynamical evolution of a scalar field coupling to Einstein’s tensor in the background of a Reissner-Nordström black hole. Our results show that the coupling constant $\eta$ imprints in the wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase of the coupling constant $\eta$ the real parts of the fundamental quasinormal frequencies decrease and the absolute values of imaginary parts increase for fixed charge $q$ and multipole number $l$. In the strong coupling, we find that for $l\ne 0$ the instability occurs when $\eta$ is larger than a certain threshold value ${\eta }_c$ which deceases with the multipole number $l$ and charge $q$. However, for the lowest $l=0$, we find that there does not exist such a threshold value and the scalar field always decays for arbitrary coupling constant.

Figure 1

Variety of the effective potential $V(r)$ with the polar coordinate $r$ for fixed $l=0$ (left), $l=1$ (middle), and $l=2$ (right). The solid, dashed, dashed-dotted, and dotted lines are corresponding to the cases with $\eta =0$, 10, 100, and 200, respectively. We set $2M=1$ and $q=0.2$.

Figure 2

Variety of the real parts of the fundamental quasinormal modes with $q$ for scalar field coupling to Einstein’s tensor in the Reissner-Nordström black hole spacetime. The figures from left to right are corresponding to $l=0$, 1 and 2. The solid, dashed, dashed-dotted, and dotted lines are corresponding to the cases with $\eta =0$, 2, 4, 6, respectively. We set $2M=1$.

Figure 3

Variety of the absolute value of imaginary parts of the fundamental quasinormal modes with $q$ for scalar field coupling to Einstein’s tensor in the Reissner-Nordström black hole spacetime. The figures from left to right are corresponding to $l=0$, 1, and 2. The solid, dashed, dashed-dotted, and dotted lines are corresponding to the cases with $\eta =0$, 2, 4, 6, respectively. We set $2M=1$.

Figure 4

The dynamical evolution of a scalar field coupling to Einstein’s tensor in the background of a Reissner-Nordström black hole spacetime. The figures from left to right are corresponding to $l=0$, 1, and 2. The solid, dashed, dashed-dotted, and dotted lines are corresponding to the cases with $\eta =1$, 60, 80, and 100, respectively. We set $2M=1$. The constants in the Gauss pulse (13) are $v_c=10$ and $\sigma =3$.

Figure 5

The change of the threshold value ${\eta }_c$ with the inverse multipole number $l^{-1}$ for fixed $q$. The points $l=1$, 2, 3, 4, 32, 100 were fitted by the function ${\eta }_c=a{l}^{-1.12}+b$. The values of $(a,b)$ for $q=0.1$, 0.2, and 0.3 are (338.71, 96.00), (73.72, 21.10), and (24.52, 7.31), respectively.

Figure 6

The changes of the numerical constants $a$, $b$ with the charge $q$, which were fitted by the functions $a=q^{-2.6}$ and $b=r_{+}^4/q^2$, respectively.