Addendum to "Absorption of a massive scalar field by a charged black hole"

TL;DR

This paper extends previous work on scalar field absorption by charged black holes, providing analytical results for low-frequency regimes where the field velocity is below a critical threshold, and discusses its physical implications.

Contribution

It offers the complementary analytical results for the low-velocity regime of scalar field absorption by black holes, filling a gap in prior high- and low-frequency analyses.

Findings

Analytical results for $v \,\lesssim\, v_c$ regime.

Discussion on physical relevance of low-velocity absorption.

Extension of previous numerical and analytical studies.

Abstract

In Phys.Rev.D89, 104053 (2014) we studied the absorption cross section of a scalar field of mass $m$ impinging on a static black hole of mass $M$ and charge $Q$. We presented numerical results using the partial-wave method, and analytical results in the high- and low-frequency limit. Our low-frequency approximation was only valid if the (dimensionless) field velocity $v$ exceeds $v_c = 2 \pi M m$. In this Addendum we give the complementary result for $v \lesssim v_c$, and we consider the possible physical relevance of this regime.

Equations

$${\sigma }_{lf}=\{\begin{array}{ll}{\sigma }_{lf}^{(1)}=\mathcal{A}/v,& v\gtrsim v_c,\\ {\sigma }_{lf}^{(2)}=\frac{4(\pi r_{+}{)}^2(2Mm)}{v^2},& v\lesssim v_c,\end{array}$$

$$\sigma =\frac{4\pi r_{+}^2{\rho }^2}{v},$$

$${\rho }^2=\frac{2\pi \eta }{e^{2\pi \eta }-1}$$

$${\rho }^2=-\frac{2\pi Mm(1+v^2)}{v\sqrt{1-v^2}}\frac{1}{\exp \left(-\frac{2\pi Mm(1+v^2)}{v\sqrt{1-v^2}}\right)-1}.$$

$${\sigma }_{lf}^{(2)}=\frac{4(\pi r_{+}{)}^2(2Mm)}{v^2}.$$

Full text

In Ref. bodc we analyzed the scenario of a neutral scalar field of mass $m$ and frequency $\omega$ absorbed by a Reissner-Nordström black hole of mass $M$ and charge $Q$. Our stated aim was to provide a quantitative full-spectrum description of absorption, by bringing together numerical methods and analytical approximations. However, due to a tacit assumption, we gave an incomplete description of the low-frequency regime; with this addendum we now fulfill our original objective.

The total absorption cross section $\sigma$ may be written as a sum of partial absorption cross sections $\sigma_{l}$. In the low-frequency limit $M\omega\ll 1$, the dominant contribution arises from the monopole sector, $\sigma_{lf}\approx\sigma_{l=0}$. In Ref. bodc we obtained $\sigma_{lf}=\mathcal{A}/v$ [see Eq. (63)], where $v=\sqrt{1-m^{2}/\omega^{2}}$ is the (dimensionless) velocity of the field, $\mathcal{A}=4\pi r_{+}^{2}$ is the area of the black hole, and $r_{+}$ is the areal coordinate of the event horizon location.

However, our original result did not encompass the limit of small velocities. After taking this into account, the completed low-frequency approximation is

[TABLE]

where $v_{c}=2\pi Mm$ is the velocity of the transition.

One may obtain the approximation valid for $v\lesssim v_{c}$ by starting from Eq. (62) of Ref. bodc , namely

[TABLE]

where we are considering only the first term in the denominator of Eq. (62). For this purpose, we write 111There are $2\pi$ factors (in front of $\eta$) missing in Eq. (57) of Ref. bodc .

[TABLE]

as

[TABLE]

Substituting Eq. (4) in Eq. (2) and considering the limit for $v\rightarrow 0$ we obtain

[TABLE]

In the uncharged case $Q=0$ we recover Unruh’s result for the case of a Schwarzschild black hole (cf. Eq. (97) of Ref. Unruh_1976 ).

In Fig. 1 we compare Eq. (1) with our numerical results. We can see that the numerical results present a transition between $\sigma_{lf}^{(1)}$ and $\sigma_{lf}^{(2)}$, which happens near $v=v_{c}$, with $v_{c}\approx 0.138$ in this case.

Recently a new dark matter candidate has been proposed by Hui et al. Hui:2016ltb , in the form of a scalar field with mass $m\approx 10^{-22}\,\text{eV}/c^{2}$ and de Broglie wavelength $\lambda_{B}\approx 1\,\text{kpc}$. Its corresponding velocity is $v\approx 4\times 10^{-4}$, found from $v=\left(1+\lambda_{B}^{2}/\lambda_{C}^{2}\right)^{-1/2}$, where $\lambda_{C}=h/mc\approx 0.4\,\text{pc}$ is the Compton wavelength. For a black hole mass $M_{1}=3.6\times 10^{6}M_{\odot}$ (e.g. Sgr. A* Schoedel:2009mv ), we find $v_{c}\approx 1.7\times 10^{-5}$ and thus $v>v_{c}$; whereas for a supermassive black hole of mass $M_{2}=2\times 10^{8}M_{\odot}$ (e.g. Andromeda’s supermassive BH has mass $(1.1-2.3)\times 10^{8}M_{\odot}$ Bender:2005rq ) we find $v_{c}=9.4\times 10^{-4}$ and thus $v<v_{c}$. This suggests that both regimes of Eq. (1) are potentially relevant in the scenario of Hui et al., and that disparate cross sections are possible. For example, $\sigma_{lf}^{(1)}\approx 3.6\times 10^{24}\text{m}^{2}$ for $M_{1}$ (Sgr. A*) and $\sigma_{lf}^{(2)}\approx 2.6\times 10^{28}\text{m}^{2}$ for $M_{2}$ (Andromeda).