Resonant excitation of black holes by massive bosonic fields and giant ringings

We consider the massive scalar field, the Proca field, and the Fierz-Pauli field in the Schwarzschild spacetime and we focus more particularly on their long-lived quasinormal modes. We show numerically that the associated excitation factors have a strong resonant behavior and we confirm this result analytically from semiclassical considerations based on the properties of the unstable circular geodesics on which a massive particle can orbit the black hole. The conspiracy of (i) the long-lived behavior of the quasinormal modes and (ii) the resonant behavior of their excitation factors induces intrinsic giant ringings, i.e., ringings of a huge amplitude. Such ringings, which are moreover slowly decaying, are directly constructed from the retarded Green function. If we describe the source of the black hole perturbation by an initial value problem with Gaussian initial data, i.e., if we consider the excitation of the black hole from an extrinsic point of view, we can show that these extraordinary ringings are still present. This suggests that physically realistic sources of perturbations should generate giant and slowly decaying ringings and that their existence could be used to constrain ultralight bosonic field theory interacting with black holes.

Figure 1

The $(\ell =2,n=0)$ QNM of the massive scalar field. (a) The complex quasinormal frequency $2M{\omega }_{20}$ for $\tilde{\alpha }=0,0.05,\dots ,1.55,1.60$. (b) The resonant behavior of the excitation factor ${\mathcal{B}}_{20}$. (c) The excitation factor ${\mathcal{B}}_{20}$ for $\tilde{\alpha }=0,0.01,\dots ,1.60,1.61$. (d) and (e) Some intrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{20}$. We compare them with the ringing corresponding to the massless scalar field. The results are obtained from (11) with $r=50M$ and $r^{\prime }=10M$.

Figure 2

The $(\ell =3,n=0)$ QNM of the massive scalar field. (a) The complex quasinormal frequency $2M{\omega }_{30}$ for $\tilde{\alpha }=0,0.05,\dots ,2.15,2.20$. (b) The resonant behavior of the excitation factor ${\mathcal{B}}_{30}$. (c) The excitation factor ${\mathcal{B}}_{30}$ for $\tilde{\alpha }=0,0.01,\dots ,2.18,2.19$. (d) and (e) Some intrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{30}$. We compare them with the ringing corresponding to the massless scalar field. The results are obtained from (11) with $r=50M$ and $r^{\prime }=10M$.

Figure 3

The $(\ell =3,n=1)$ QNM of the massive scalar field. (a) The complex quasinormal frequency $2M{\omega }_{31}$ for $\tilde{\alpha }=0,0.05,\dots ,2.50,2.55$. (b) The resonant behavior of the excitation factor ${\mathcal{B}}_{31}$. (c) The excitation factor ${\mathcal{B}}_{31}$ for $\tilde{\alpha }=0,0.01,\dots ,2.44,2.45$. (d) and (e) Some intrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{31}$. We compare them with the ringing corresponding to the massless scalar field. The results are obtained from (11) with $r=50M$ and $r^{\prime }=10M$.

Figure 4

The odd-parity $(\ell =2,n=0)$ QNM of the Proca field. (a) The complex quasinormal frequency $2M{\omega }_{20}$ for $\tilde{\alpha }=0,0.05,\dots ,1.40,1.45$. (b) The resonant behavior of the excitation factor ${\mathcal{B}}_{20}$. (c) The excitation factor ${\mathcal{B}}_{20}$ for $\tilde{\alpha }=0,0.01,\dots ,1.44,1.45$. (d) and (e) Some intrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{20}$. We compare them with the ringing corresponding to the massless vector field. The results are obtained from (11) with $r=50M$ and $r^{\prime }=10M$.

Figure 5

The odd-parity $(\ell =1,n=0)$ QNM of the Fierz-Pauli field. (a) The complex quasinormal frequency $2M{\omega }_{10}$ for $\tilde{\alpha }=0,0.05,\dots ,1.00,1.05$. (b) The resonant behavior of the excitation factor ${\mathcal{B}}_{10}$. (c) The excitation factor ${\mathcal{B}}_{10}$ for $\tilde{\alpha }=0,0.01,\dots ,1.05,1.06$. (d) and (e) Some intrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{10}$. We compare them with the ringing corresponding to the odd-parity $(\ell =2,n=0)$ QNM of the massless spin-2 field. The results are obtained from (11) with $r=50M$ and $r^{\prime }=10M$.

Figure 6

The quasinormal excitation factor of the $(\ell =3,n=0)$ QNM of the scalar field. Exact and asymptotic behaviors.

Figure 7

The quasinormal excitation factor of the $(\ell =3,n=1)$ QNM of the scalar field. Exact and asymptotic behaviors.

Figure 8

The quasinormal excitation factor of the $(\ell =4,n=0)$ QNM of the scalar field. Exact and asymptotic behaviors.

Figure 9

The quasinormal excitation factor of the odd-parity $(\ell =2,n=0)$ QNM of the Proca field. Exact and asymptotic behaviors.

Figure 10

Resonant behavior of the excitation coefficient ${\mathcal{C}}_{20}$ of the $(\ell =2,n=0)$ QNM of the massive scalar field. It is obtained from (24) by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=10M$.

Figure 11

Resonant behavior of the excitation coefficient ${\mathcal{C}}_{30}$ of the $(\ell =3,n=0)$ QNM of the massive scalar field. It is obtained from (24) by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=10M$.

Figure 12

The $(\ell =2,n=0)$ QNM of the massive scalar field. (a) and (b) Some extrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{20}$ and comparison with the ringing associated with the massless scalar field. Here the results are obtained from (23) with $r=50M$ and by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=10M$.

Figure 13

The $(\ell =3,n=0)$ QNM of the massive scalar field. (a) and (b) Some extrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{30}$ and a comparison with the ringing associated with the massless scalar field. Here the results are obtained from (23) with $r=50M$ and by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=10M$.

Figure 14

The odd-parity $(\ell =1,n=0)$ QNM of the Fierz-Pauli field. (a) and (b) Some extrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{10}$ and a comparison with the ringing associated with the odd-parity $(\ell =2,n=0)$ QNM of the massless spin-2 field. Here the results are obtained from (23) with $r=50M$ and by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=10M$.

Figure 15

The odd-parity $(\ell =1,n=0)$ QNM of the Fierz-Pauli field. (a) and (b) Some extrinsic ringings corresponding to values of the mass near and above the critical value ${\tilde{\alpha }}_{10}$ and a comparison with the ringing associated with the odd-parity $(\ell =2,n=0)$ QNM of the massless spin-2 field. Here the results are obtained from (23) with $r=50M$ and by using (20) with ${\varphi }_0=1$, $a=1$, and $r_0=20M$.