On-axis scattering of scalar fields by charged rotating black holes

TL;DR

This paper studies how massless scalar fields scatter off charged rotating black holes, using numerical methods and comparing results with classical and semiclassical theories to understand the scattering behavior.

Contribution

It provides new numerical analysis of on-axis scalar field scattering by Kerr-Newman black holes and validates results against classical and semiclassical approximations.

Findings

Numerical scattering cross sections match analytical predictions.

Excellent agreement between numerical and classical/semiclassical results.

Results vary with different black hole parameters.

Abstract

We investigate the scattering of a massless scalar field by a Kerr-Newman black hole, considering the case of on-axis incidence. We use the partial wave method to find numerical results for the scattering cross section, which we compare with classical and semiclassical analytical results, obtaining excellent agreement. We present a selection of plots for different values of the black hole parameters.

Equations

$$d{s}^2$$

$$(\sqrt{-g}{)}^{-1}{\partial }_{\mu }\left(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }\psi \right)=0,$$

$${\psi }_{\omega lm}(t,r,\theta ,\varphi )=\sum _{l=0}^{+\infty }\sum _{m=-l}^{+l}\frac{{\mathcal{U}}_{\omega lm}(r)}{\sqrt{r^2+a^2}}{S}_{\omega lm}(\theta )e^{im\varphi -i\omega t},$$

$$\left(\frac{d^2}{d{r}_{\star }^2}+{\mathcal{V}}_{\omega lm}\right){\mathcal{U}}_{\omega lm}(r_{\star })=0,$$

$${\mathcal{V}}_{\omega lm}(r)$$

$$\frac{d{r}_{\star }}{dr}=\left(\frac{r^2+a^2}{\Delta }\right).$$

$$(\sin \theta {)}^{-1}\frac{d}{d\theta }\left(\sin \theta \frac{d{S}_{\omega lm}}{d\theta }\right)$$

$$+\left({\lambda }_{\omega lm}+a^2{\omega }^2{\cos }^2\theta -\frac{m^2}{{\sin }^2\theta }\right)S_{\omega lm}=0,$$

$$\int d\theta \sin \theta {\left|S_{\omega lm}(\theta )\right|}^2=(2\pi {)}^{-1}.$$

$$\mathcal{U}_{\omega lm}(r_{\star})\sim\left\{\begin{array}[]{c l}\mathcal{U}_{I}+{{\cal R}_{\omega lm}}\mathcal{U}_{I}^{*}&(r_{\star}\rightarrow+\infty),\\ {\mathcal{T}_{\omega lm}}\mathcal{U}_{T}&(r_{\star}\rightarrow-\infty),\end{array}\right.$$

$${\mathcal{U}}_T=e^{-i\left(\omega -m\frac{a}{a^2+r_{+}^2}\right)r_{\star }}\sum _{j=0}^N{g}_j(r-r_{+}{)}^j,$$

$${\mathcal{U}}_I=e^{-i\omega r_{\star }}\sum _{j=0}^N\frac{h_j}{r^j},$$

$$\frac{d\sigma }{d\Omega }=|f(\theta ,\varphi ){|}^2,$$

$$f(\theta )=\frac{2\pi }{i\omega }\sum _{l=0}^{\infty }{S}_{\omega l0}(0)S_{\omega l0}(\theta )\left[(-1{)}^{l+1}{\mathcal{R}}_{\omega l0}-1\right],$$

$${\rho }^2\dot{t}$$

$${\rho }^4{\dot{r}}^2$$

$${\rho }^2\dot{\varphi }$$

$${\rho }^4{\dot{\theta }}^2$$

$$b=\sqrt{\mathcal{C}+a^2}.$$

$${\left(\frac{dr}{d\theta }\right)}^2=\frac{(a^2+r^2{)}^2-b^2\Delta }{b^2-a^2\sin \theta }.$$

$${\left(\frac{du}{d\theta }\right)}^2=u^4\left[\frac{(a^2+\frac{1}{u^2}{)}^2-b^2{\Delta }_u}{b^2-a^2\sin \theta }\right],$$

$$\Theta (b)=2\beta (b)-\pi ,$$

$$\beta (b)={\int }_0^{u_0}du{\left\{u^4\left[\frac{(a^2+\frac{1}{u^2}{)}^2-b^2{\Delta }_u}{b^2-a^2\sin \theta }\right]\right\}}^{-1/2},$$

$$\frac{d\sigma }{d\Omega }=\frac{1}{\sin \theta }\sum _j{b}_j(\theta )\left|\frac{d{b}_j}{d\theta }\right|,$$

$${\frac{d\sigma }{d\Omega }|}_{\theta \approx \pi }=\mathcal{A}{J}_0^2\left(\omega b_g\sin \theta \right),$$

$$\mathcal{A}\equiv 2\pi \omega b_g{\left|\frac{db}{d\theta }\right|}_{\theta =\pi },$$

$$f(\theta )=\sum _{j=0}^{+\infty }{F}_{\omega j}{Y}_{j0}(\theta ),$$

$$F_{\omega j}\equiv \frac{2\pi }{i\omega }\sum _{l=0}^{+\infty }{b}_{\omega jl}{S}_{\omega l0}(0)\left[(-1{)}^{l+1}{\mathcal{R}}_{\omega l0}-1\right],$$

$$f(\theta )=\frac{1}{(1-\cos \theta {)}^n}\sum _{j=0}^{+\infty }F{}_{\omega j}^{(n)}{Y}_{j0}(\theta ),$$

$$F{}_{\omega j}^{(n+1)}$$

Full text

††journal: Physics Letter B

1 Introduction

In the past few years, the detection of gravitational waves (GWs) has been reported by the LIGO–Virgo Collaborations [1, 2, 3, 4, 5, 6]. Most of the GWs signals cataloged, so far, have been produced during the merger of binary systems composed by black holes (BHs). These detections give us compelling evidence of the existence of black holes with tens of solar masses and open the way to the observation of the strong regime of gravity. The existence of supermassive BHs has been recently supported by the Event Horizon Telescope observations, which captured the first image of a BH shadow [7].

Scattering by objects plays a key role in physics. In general relativity, in particular, the measurement of light deflection led to the first experimental verification of a prevision by this theory [8, 9]. Moreover, by studying the scattering of null geodesics one can find the BH shadow, which can be used, for instance, to investigate deviations from the Kerr metric [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and to search for quantum fluctuations close to the event horizon [21].

The scattering of fields by compact objects gives rise to interesting effects such as glories [22], orbiting [23] and rainbow scattering [24]. All these phenomena can be related to characteristics of the classical deflection function [25, 26, 27, 28]. Through a semiclassical analysis one can find analytical approximations, what helps to understand such effects, besides acting as consistency checks for the full numerical analysis.

Another key feature of scattering by BHs is superradiance [29], which allows low-frequency waves to experience an amplification, while the BH suffers a decrease in its parameters. This effect implies in a reflection coefficient greater than the unity, within the superradiance threshold, what accounts for a negative absorption cross section [30, 31, 32].

Wave scattering by BHs has been the subject of many investigations. In particular, for scalar fields there are numerous works on the subject [33, 34, 35, 36, 37, 38, 39], what can be associated mainly to the simplicity of the scalar field equations compared to fermionic, electromagnetic and gravitational cases. One can see scalar fields as proxy models, used to highlight important features of scattering by compact objects. The detection of the Higgs boson, however, has brought extra experimental motivation for the study of scalar fields [40]. Moreover, there are models of dark matter which consider scalar fields with light masses [41].

The scattering of waves with spin 1/2 [42, 43, 44, 45], 1 [46, 47, 48, 49] and 2 [50, 51, 52, 53] by BHs has been considered in a number of works. For these cases we identify reminiscent behaviors of the scalar case, e.g., an oscillatory pattern coming from the orbiting and a glory pattern in the backward direction. However, when we consider non-zero spin, new effects arise, such as spin precession, polarization and helicity-reversal.

In the context of Einstein-Maxwell theory, asymptotically flat (rotating and electrically charged) BHs are described by the Kerr-Newman solution. The evolution and stability of scalar perturbations in Kerr-Newman background have already been studied [54, 55, 56, 57, 58]. Investigations concerning bound states formed of scalar fields (scalar clouds) around Kerr-Newman BHs have also been carried out [59, 60, 61]. As far as we are aware of, a study of the time-independent scattering of scalar fields by Kerr-Newman BHs is still lacking in the literature, and constitutes the aim of the present Letter. We investigate the scattering of a massless scalar field by a charged and rotating (Kerr-Newman) BH, considering the case of a massless scalar wave impinging along the BH’s rotation axis. In order to compute the (differential) scattering cross section, we consider a partial wave approach. Moreover, considering the properties of null geodesics, we compute the classical differential cross section and find a semiclassical approximation for the glory scattering, which we compare with our numerical results, finding excellent agreement.

The remaining of this Letter is organized as follows: in Sec. 2, we review the treatment given to the massless scalar field in the Kerr-Newman background. In Sec. 3, we present an expression for the massless scalar scattering cross section of Kerr-Newman BHs, obtained via the partial-wave approach. We review some classical (Subsec. 3.1) and semiclassical (Subsec. 3.2) results of the scattering cross section, focusing in the case of Kerr-Newman BHs. Section 4 is devoted to the discussion about the numerical methods used to obtain the results presented in Sec. 5. We finish this Letter presenting some final remarks in Sec. 6. We adopt metric signature $(+,-,-,-)$, and natural units, such that $c=G=\hslash=1$.

2 Scalar field in the Kerr-Newman spacetime

In electrovacuum, BH solutions are given by the Kerr-Newman metric, which is described by the following line element written in Boyer-Lindquist coordinates

[TABLE]

where $\Delta\equiv r^{2}-2Mr+a^{2}+Q^{2}$, $\rho^{2}\equiv r^{2}+a^{2}\cos^{2}\theta$, and $\xi\equiv(r^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta$. If the relation $a^{2}+Q^{2}<M^{2}$ is satisfied by the BH’s mass $M$, electric charge $Q$, and rotation parameter $a$, this solution possesses two horizons, namely the event horizon, $r_{+}$, and the Cauchy horizon, $r_{-}$, which are given explicitly by $r_{\pm}=M\pm\sqrt{M^{2}-(a^{2}+Q^{2})}$. Here, we shall be interested in such a BH spacetime.

We shall consider a massless and chargeless scalar field in this spacetime, described by the Klein-Gordon equation, which in its covariant form reads

[TABLE]

where $g$ and $g^{\mu\nu}$ are the determinant and the contravariant components of the metric, respectively. In order to find a solution for Eq. (2), we use the following ansatz

[TABLE]

such that the radial function $\mathcal{U}_{\omega lm}$ obeys the differential equation

[TABLE]

with

[TABLE]

The coordinate $r_{\star}$ is the tortoise coordinate, which can be found by integrating the equation

[TABLE]

The angular solution is given by the scalar spheroidal harmonics, $S_{\omega lm}$, which satisfy the equation

[TABLE]

and are normalized by

[TABLE]

Note that the angular eigenvalues $\lambda_{lm}$ are necessary to solve both the radial [Eq. (4)] and angular [Eq. (7)] equations.

Since we are dealing with a scattering problem, we consider an incoming wave from infinity, which is partially transmitted through the event horizon and partially scattered to infinity. The boundary conditions that are in agreement with these requirements are given by

[TABLE]

with

[TABLE]

and

[TABLE]

where $g_{j}$ and $h_{j}$ are constants which can be fixed by imposing that Eq. (9) obeys Eq. (4). In order to simplify our equations we assume $g_{j}=h_{j}=1$. The quantities $\mathcal{R}_{\omega lm}$ and $\mathcal{T}_{\omega lm}$ are related to the reflection and transmission coefficients, respectively.

3 Scattering cross section

Our main goal is to numerically compute the differential scattering cross section, which can be expressed as

[TABLE]

where $f(\theta,\phi)$ is the scattering amplitude, that for on-axis incidence 111Note that $m=0$ for on-axis incidences (see Ref. [32]). is given by the following partial wave series

[TABLE]

in which $\theta$ represents the scattering angle.

3.1 Geodesics scattering

Let us first analyze the null geodesic scattering prior to show our numerical results for the scalar scattering cross section. Massless particles follow null geodesics in general relativity, which for the Kerr-Newman spacetime are described by [62]

[TABLE]

where $\mathcal{L}_{z}=L_{z}/E$ is the azimuthal angular momentum per unit energy and $\mathcal{C}=C/E^{2}$ is the Carter’s constant divided by the squared energy.

For on-axis incidence, we have $\mathcal{L}_{z}=0$, and the impact parameter is simply

[TABLE]

Considering Eqs. (15) and (17), we can find an orbital equation, namely

[TABLE]

Defining $u\equiv 1/r$, we can rewrite Eq. (19) as follows:

[TABLE]

where $\Delta_{u}\equiv(1/u^{2}-2M/u+Q^{2}+a^{2})$.

The deflection function can be written as [63]

[TABLE]

in which

[TABLE]

where $u_{0}=1/r_{0}$, and $r_{0}$ is the radius of closest approach for a null geodesic.

For a stream of particles, the classical differential scattering cross section is [63]

[TABLE]

where the sum takes into account the fact that massless particles with $b$ close to the critical impact parameter $b_{c}$ can orbit the BH many times before being scattered. Note that the scattering angle $\theta$ is related to the deflection angle $\Theta$ via $\Theta=\pm\theta-2N\pi$, with $N=0,1,2,\ldots$.

3.2 Glory scattering

Scattering at large angles is associated to the glory effect, which is characterized by a bright spot or halo in the backward direction [64]. Using a semiclassical approach, an analytical formula for the glory scattering in the BH case can be obtained, and is given by [65, 66, 22, 23]

[TABLE]

where $J_{0}$ is the Bessel function of the first kind of order 0, and we have defined the intensity of the glory peak as

[TABLE]

and the glory impact parameter $b_{g}\equiv b(\pi)$.

4 Numerical methods

In this section we make a brief discussion about the numerical methods used to compute the scattering cross section for the massless scalar field.

From Eqs. (12) and (13) we can see that the basic ingredients to compute the scattering cross section are the coefficients $\mathcal{R}_{\omega l0}$, as well as the angular functions $S_{\omega l0}(\theta)$ and their eigenvalues $\lambda_{\omega l0}$.

In order to obtain the coefficients $\mathcal{R}_{\omega l0}$, we numerically integrate the radial equation, Eq. (4), using the numerical procedure described in Refs. [30, 32]. We point out that, for the results presented here, the series in Eqs. (10) and (11) were truncated at $N=3$. The scalar spheroidal harmonics $S_{\omega l0}(\theta)$ and their eigenvalues $\lambda_{\omega l0}$ are obtained via spectral decomposition as described in Refs. [67, 68, 69]. Due to a divergence at $\theta=0^{\circ}$, the series in Eq. (13) is poorly convergent. We can circumvent this problem using a series reduction technique, as described in A.

5 Numerical results

In Fig. 1, we show the massless scalar scattering cross section of Kerr-Newman BHs considering a fixed value for the electric charge $Q=0.3M$ (top panel) and for the rotation parameter $a=0.8M$ (bottom panel) as well. We note that the width of the oscillations become larger as we increase the value of the rotation parameter (top panel). A similar behavior is observed when we consider different values of the electric charge (bottom panel), so that as one increases the value of $Q$, the interference fringes become wider. We observe that the scattering cross section of Kerr-Newman BHs presents a high peak at $180^{\circ}$ (see Fig. 1), the so-called glory peak.

In Fig. 2, we exhibit the intensity of the glory peak of Kerr-Newman BHs with $Q=0.3M$ (top panel) and $a=0.8M$ (bottom panel) as a function of both the BH’s rotation parameter (top panel) and electric charge (bottom panel), respectively. We choose different values for the wave-frequency $\omega M=1.0$, $2.0$ and $3.0$, and we also compare our numerical results with the analytical ones obtained via the semiclassical glory scattering formula presented in Sec. 3.2. We see that the glory peak exhibits a non-monotonic behavior around the semiclassical result, as we vary both the rotation parameter (top panel) and the electric charge (bottom panel).

We finish this section by comparing our numerical results for the differential scattering cross section with the analytical ones. In Fig. 3, we compare the geodesic with the scalar scattering cross sections of Kerr-Newman BHs with $a/M=0.9$, $Q/M=0.3$ and $M\omega=3.0$, together with the semiclassical glory approximation. We note that both the geodesic and the scalar scattering cross sections present a divergent behavior for small scattering angles. Moreover, the scalar scattering cross section presents a great agreement with the glory approximation near the backward direction (${{\theta}}\approx 180^{\circ}$).

6 Final remarks

We have numerically computed the scalar scattering cross section of Kerr-Newman BHs, focusing in the case of on-axis incidence. Our numerical results have been obtained using a series reduction technique, exhibiting excellent agreement with the analytical (classical and semiclassical) ones. As in the case of Kerr BHs [36], the (on-axis) cross section of a Kerr-Newman BH presents a behavior similar to the corresponding static (Reissner-Nordstrom [37]) case. We have shown that the angular widths of the spiral oscillations become larger as one increases the BH’s rotation and/or the electric charge. We can interpret these effects through a semiclassical reasoning, as follows: from the glory formula given by Eq. (24), the width of the glory oscillations is proportional to $1/{\omega b_{g}}$, and one can show that the glory impact parameter diminishes as the values of $a$ and/or $Q$ increase. We have seen that the flux scattered in the backward direction is a non-monotonic function of both the rotation parameter and the electric charge of Kerr-Newman BHs. This behavior is not revealed by the semiclassical analysis.

Acknowledgments

The authors would like to acknowledge Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) – Finance Code 001, from Brazil, for partial financial support. This research has also received funding from the European Union’s Horizon 2020 research and innovation programme under the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740.

Appendix A Series reduction technique

In this Appendix, we outline the series reduction technique used to compute the massless scalar scattering cross section. This method is based mainly on Ref. [70] and has been successfully adapted to BH cases (for an example, see Ref. [52]).

We start rewriting the partial wave series $f(\theta)$, using the spectral decomposition of $S_{\omega l0}(\theta)$, as a sum over spherical harmonics, namely

[TABLE]

where $Y_{j0}$ are the standard scalar spherical harmonics. In Eq. (26) we have defined

[TABLE]

with $b_{\omega jl}$ being the spectral decomposition coefficients (for more details, see Refs. [67, 69]).

In order to improve the convergence properties of the partial wave series, one can define a new series as follows:

[TABLE]

in which $F{{}^{(0)}_{\omega j}}=F_{\omega j}$, with $F_{\omega j}$ given by Eq. (27). The reduced series $f^{(n)}(\theta)$ is less divergent at the forward direction ($\theta=0^{\circ}$). Using the properties of the spherical harmonics, one can show that the coefficients $F{{}^{(n+1)}_{\omega j}}$ can be obtained from the coefficients $F{{}^{(n)}_{\omega j}}$ via the following recurrence formula:

[TABLE]

For the results presented in Sec. 5, we have computed the scattering cross section with the help of Eq. (28), by applying three ($n=3$) iterations. Typically, we terminate the series at $l_{\rm{max}}\sim 40$ and $j_{\rm{max}}\sim 40$, for $\omega M=3.0$.