Regular scalar charged clouds around a Reissner-Nordstrom black hole and no-hair theorems

In this work, we reanalyze the possibility of finding bound states (scalar clouds) of a test, charged and complex-valued scalar field with mass $\mu$ and charge $q$ in the background of a Reissner-Nordstrom black hole (RNBH). In order to determine the existence of such scalar clouds, we impose suitable regularity conditions for the scalar field at the event horizon. We find numerical evidence for the absence of such clouds in the subextremal and extremal RNBH when the field is massive but not self-interacting. More importantly, we put forward a theorem that proves that such clouds cannot exist. On the other hand, when a suitable self-interacting potential is included, the theorem no longer applies, providing a heuristic justification behind the existence of charged clouds (dubbed Q-clouds) that were reported recently.

Figure 1

The scalar-field potential $\tilde{U}$ (48) as a function of $\psi$, assuming the values $\mu =1=\lambda$ and $\nu =9/32$ as in Ref. [11].

Figure 2

The function $\mathrm{\Lambda }(\psi )$ (56) for different values of $q$, taking $\mu =1=\lambda$ and $\nu =9/32$ as in Fig. 1. Here, we assume $Q=0.09$ and $r_H=0.15$. The minima are located at ${\psi }_{\pm }^{\mathrm{\Lambda }}$ where $\mathrm{\Lambda }(\psi )$ is negative.

Figure 3

The function $\mathrm{\Sigma }(\psi )$ (59), taking $\mu =1=\lambda$ and $\nu =9/32$ as in Fig. 1. The minima are located at ${\psi }_{\pm }^{\mathrm{\Sigma }}\approx \pm 1.0887$ where $\mathrm{\Sigma }(\psi )$ is negative.

Figure 4

The function ${\mathrm{\Lambda }}_{\mathrm{ext}}(\psi )$, taking $\mu =1=\lambda$ and $\nu =9/32$ for different values of $q$ in the extremal Reissner-Nordstrom scenario.

Figure 5

The effective potential $U_{\mathrm{eff}}$ (68), taking $\mu =1=\lambda$, $\nu =9/32$, $Q=0.09$, and $r_H=0.15$ for different values of $q$.

Figure 6

Q-cloud solutions $\psi (r)$ for a RNBH with $Q=0.09$, $r_H=0.15$, and $\mu =1=\lambda$, $\nu =9/32$, taking different values for $q$.

Figure 7

The quantity $\mathrm{\Lambda }(\psi )$ associated with the solutions of Fig. 6. Notice that this quantity can be negative and contributes nontrivially to the integral (57).

Figure 8

Solutions $\psi (r)$ (solid lines) with $q=1$ for three different values of ${\psi }_H$ within a RNBH with $Q=0.09$, $r_H=0.15$, $\mu =1=\lambda$, and $\nu =9/32$. The three solutions correspond to ${\psi }_H$: 1.3820 (purple line), 1.381958 (green line), and 1.381959590 (red line). The horizontal dotted lines represent the two values ${\psi }_{\max }\approx \pm 0.6175$ associated with the local maxima of the effective potential $U_{\mathrm{eff}}$ depicted in Fig. 5 for $q=1$. The solution corresponding to an asymptotically vanishing field, which is the relevant for the current analysis, is marked in red color.

Figure 9

Q-cloud solutions $\psi (r)$ for near extremal RNBH $Q\approx M$, taking $M=2$, $\mu =1=\lambda$, $\nu =9/32$, and $q=0.8$. The charge of the black hole is taken to be $Q=(1-ϵ)M$ with $ϵ={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$.

Figure 10

The effective potential $U_{\mathrm{eff}}$ (68) associated with the radial solutions for the near extremal RNBH (Fig. 9), taking $\lambda =1$, $\nu =9/32$, $\mu =1$, and $q=0.8$. The charge of the black hole is taken to be $Q=(1-ϵ)M$ with $ϵ={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, and ${10}^{-5}$. For reference, the exact extremal case $r_H=M=Q$ is depicted by the black dashed line.

Figure 11

The effective potential $U_{\mathrm{eff}}^{\mathrm{ext}}$ (68), taking $\mu =1=\lambda$, $\nu =9/32$, and $Q=r_H=0.15$ for different values of $q$.

Figure 12

Q-cloud solutions $\psi (r)$ for near extremal RNBH $Q\approx M$, taking $M=2$, $\mu =1=\lambda$, and $\nu =9/32$. The charge of the black hole is taken to be $Q=(1-ϵ)M$ with $ϵ={10}^{-4}$ and for values $q$ approaching $1/3$.

Figure 13

The effective potential $U_{\mathrm{eff}}$ (68) associated with the radial solutions for the near extremal RNBH (Fig. 12), taking $\lambda =1$, $\nu =9/32$, and ${\mu }^2=1$ and for different values of $q$ that approach the positive value $q_{\min }^{\mathrm{ext}}\approx 1/3$ with $Q=(1-ϵ)M$ and $ϵ={10}^{-4}$.

Figure 14

The effective potential $U_{\mathrm{eff}}^{\mathrm{ext}}$ (68) for an extremal RNBH ($r_H=|Q|=M$), taking $\mu =1=\lambda$ and $\nu =9/32$ and for ${\mu }_{\text{eff},\text{ext},\infty }^2={\lambda }^2/3\nu =32/27$.