Scalar field in Reissner-Nordstr\"om spacetime: Bound state and scattering state

TL;DR

This paper investigates the behavior of a massive scalar field in Reissner-Nordström spacetime, deriving wave functions, eigenvalues, and phase shifts for bound and scattering states, with a focus on the tortoise coordinate.

Contribution

It provides explicit solutions for scalar field bound and scattering states in Reissner-Nordström spacetime, including phase shifts using the integral equation method.

Findings

Derived bound-state wave functions and eigenvalues.

Obtained explicit scattering phase shift expressions.

Introduced the tortoise coordinate for analysis.

Abstract

In this paper, we solve the massive scalar field in the Reissner-Nordstr\"om spacetime. The scalar field in the Reissner-Nordstr\"om spacetime has both bound states and scattering states. For bound states, we solve the bound-state wave function and the eigenvalue spectrum. For scattering states, we solve the scattering wave function and give an explicit expression for scattering phase shift by the integral equation method. Especially, we introduce the tortoise coordinate for the Reissner-Nordstr\"om spacetime.

Equations

$$\left(\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^{\mu }}\sqrt{-g}{g}^{\mu \nu }\frac{\partial }{\partial x^{\nu }}-{\mu }^2\right)\Phi =0$$

$$d{s}^2=-\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)d{t}^2+\frac{1}{1-\frac{2M}{r}+\frac{Q^2}{r^2}}d{r}^2+r^{2}d{\Omega }^2.$$

$$\{\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\frac{1}{r^2}\frac{d}{dr}{r}^2\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\frac{d}{dr}+{\omega }^2-\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)\left[\frac{l\left(l+1\right)}{r^2}+{\mu }^2\right]\}{R}_l\left(r\right)=0,$$

$$r_{\pm }=M\pm \sqrt{M^2-Q^2}.$$

$$\{\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)\frac{1}{r^2}\frac{d}{dr}{r}^2\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)\frac{d}{dr}+\left[{\eta }^2-\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)\frac{l\left(l+1\right)}{r^2}-\left(-\frac{r_{+}}{r}-\frac{r_{-}}{r}+\frac{r_{+}{r}_{-}}{r^2}\right){\mu }^2\right]\}{R}_l\left(r\right)=0,$$

$$z=\frac{2r-\left(r_{+}+r_{-}\right)}{r_{+}-r_{-}}$$

$$p$$

$$\beta$$

$$m$$

$$s$$

$$\lambda$$

$$\frac{d}{dz}\left(z^2-1\right)\frac{d}{dz}y\left(z\right)+\left[-p^2\left(z^2-1\right)+2p\beta z-\lambda -\frac{m^2+s^2+2msz}{z^2-1}\right]y\left(z\right)=0.$$

$$R_l\left(r\right)={y\left(z\right)|}_{z=\left[2r-\left(r_{+}+r_{-}\right)\right]/\left(r_{+}-r_{-}\right)}.$$

$$r_{\ast }$$

$$=r+\frac{r_{+}^2}{r_{+}-r_{-}}\ln \left(\frac{r}{r_{+}}-1\right)-\frac{r_{-}^2}{r_{+}-r_{-}}\ln \left(\frac{r}{r_{-}}-1\right).$$

$$\frac{d^2{u}_l\left(r\right)}{d{r}_{\ast }^2}+\{\left({\omega }^2-{\mu }^2\right)-\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)\left[\frac{l\left(l+1\right)}{r^2}+\left(\frac{r_{+}+r_{-}}{r^3}-\frac{2r_{+}{r}_{-}}{r^4}\right)\right]-{\mu }^2\left(\frac{r_{+}{r}_{-}}{r^2}-\frac{r_{+}+r_{-}}{r}\right)\}{u}_l\left(r\right)=0.$$

$$\left|R_l\left(r=r_{\pm }\right)\right|<\infty .$$

$$\left|R_l\left(r\to \infty \right)\right|=0.$$

$$\left|R_l\left(r\to \infty \right)\right|<\infty .$$

$$\frac{d^2{u}_l\left(r\right)}{d{r}_{\ast }^2}+{\omega }^2{u}_l\left(r\right)\stackrel{r\to r_{+}}{\sim }0.$$

$$u_l\left(r\right)\stackrel{r\to r_{+}}{\sim }{e}^{\pm i\omega r_{\ast }}.$$

$$R_l\left(r\right)\stackrel{r\to r_{+}}{\sim }\frac{1}{r_{+}}{e}^{\pm i\omega r_{\ast }}\sim e^{\pm i\omega r_{\ast }}.$$

$$\frac{d^2{u}_l\left(r\right)}{d{r}_{\ast }^2}+{\omega }^2{u}_l\left(r\right)\stackrel{r\to r_{-}}{\sim }0.$$

$$u_l\left(r\right)\stackrel{r\to r_{-}}{\sim }{e}^{\pm i\omega r_{\ast }}.$$

$$R_l\left(r\right)\stackrel{r\to r_{-}}{\sim }\frac{1}{r_{-}}{e}^{\pm i\omega r_{\ast }}\sim e^{\pm i\omega r_{\ast }}.$$

$$R_l\left(r\right)\stackrel{r\to \infty }{\sim }0;$$

$$\frac{d^2{u}_l\left(r\right)}{d{r}_{\ast }^2}+{\eta }^2{u}_l\left(r\right)\stackrel{r\to \infty }{\sim }0.$$

$$u_l\left(r\right)\stackrel{r\to \infty }{\sim }{e}^{\pm i\eta r_{\ast }}.$$

$$R_l\left(r\right)\stackrel{r\to \infty }{\sim }\frac{1}{r}{e}^{\pm i\eta r_{\ast }}.$$

$$\left|y\left(z=\pm 1\right)\right|<\infty .$$

$$\left|y\left(z\to \infty \right)\right|$$

$$\eta =ik,k>0$$

Full text

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https://doi.org/ \TitleScalar field in Reissner-Nordström spacetime: Bound state and scattering state \AuthorShi-Lin Li 1, Yuan-Yuan Liu 1,3 , Wen-Du Li 2,3,‡ , and Wu-Sheng Dai 1,‡

\firstnoteliwendu@tjnu.edu.cn. \secondnotedaiwusheng@tju.edu.cn.

\titlecontentssection[0.6em] \thecontentslabel    \contentspage \titlecontentssubsection[2.1em] \thecontentslabel    \contentspage

1. 1 Introduction

2. 2 Scalar field in Reissner-Nordström spacetime: bound state and scattering state

3. 2.1 Radial equation

4. 2.2 Confluent Heun equation

5. 2.3 Tortoise coordinate for Reissner-Nordström spacetime

6. 2.4 Boundary condition

7. 2.5 Bound-state solution

8. 2.6 Scattering solution

9. 3 Scattering phase shift: Integral equation

10. 3.1 Effective potential

11. 3.2 Green function and integral equation

12. 3.3 Scattering boundary condition

13. 3.4 Scattering phase shift and scattering cross section

14. 4 Conclusion

1 Introduction

Scattering on black hole backgrounds provides insights in the evolution of perturbations “at infinity” angelopoulos2020non . Scattering by black holes can help us to understand the nature of black holes anderson2002scattering . Scattering in the Reissner-Nordström spacetime, including scalar field scattering benone2014absorption ; crispino2009scattering , spinor field scattering cotuaescu2016partial ; sporea2017scattering ; thierry2010time , and electromagnetic field scattering crispino2009electromagnetic is an important issue. Some methods are developed for the calculation of scattering on black holes, such as the Born approximation batic2012born , the integral equation method li2018scalar , and the partial wave method crispino2009scattering . In scattering problems, exact solutions play important roles, e.g., the scalar scattering in the Schwarzschild spacetime vieira2016confluent and in the Kerr-Newman spacetime vieira2014exact , the solution of the Regge-Wheeler equation, and the solution of the Teukolsky radial equation fiziev2011application . Moreover, various kinds of fields, e.g., scalar fields macedo2014absorption ; batic2012orbiting ; brito2013massive , spinor fields ahn2008black , and vector fields rosa2012massive ; batic2012orbiting scattered on various kinds of spacetime, e.g., the Kerr spacetime brito2013massive , the dyonic black hole vieira2016confluent , the deformed non-rotating spacetime pei2015scattering , and the arbitrary dimensional black holes okawa2011super are studied. The scattering method is important in the calculation of the Hawking radiation, such as the Hawking radiation of the Reissner-Nordström-de Sitter black hole zhao2010hawking and of the Kerr-Newman black hole zhou2008hawking , the spinor particle Hawking radiation li2008hawking ; li2008dirac , the Hawking radiation of the acoustic black hole zhang2011hawking , and the species problem in the Hawking radiation chen2018entropy . In this paper, we determine the scattering boundary condition by the asymptotic property of the confluent Heun function. The Heun function is used in many physical problems, especially in exactly solving dynamical equations hortaccsu2013heun ; ishkhanyan2016schrodinger ; batic2015potentials ; li2016exact ; batic2013potentials ; batic2018semicommuting ; li2019scattering .

A scalar field in the Reissner-Nordström spacetime is described by the scalar equation

[TABLE]

under the Reissner-Nordström metric:

[TABLE]

The Reissner-Nordström spacetime is a charged spacetime with the charge $Q$ and the mass $M$.

A scalar field in the Reissner-Nordström spacetime has both bound states and scattering states. We first solve the bound-state solution and the scattering-state solution by solving the scalar field equation in the Reissner-Nordström spacetime directly. By constructing the tortoise coordinate of the Reissner-Nordström spacetime, we convert the scalar radial equation into a one-dimensional stationary Schrödinger equation. We solve the bound-state wave function and the eigenvalue and the scattering wave function.

Especially, we introduce the tortoise coordinate for the Reissner-Nordström spacetime for analyzing the asymptotic behavior of the field.

Moreover, in order to seek an explicit expression of the scattering phase shift, we construct the Green function of the radial equation. Using the Green function, we construct an integral equation for the scattering wave function. Solving the integral equation with the scattering boundary condition, we calculate the scattering phase shift explicitly.

Scattering in the Reissner-Nordström spacetime is long-range scattering. We need to first determine the scattering boundary condition. Determining the scattering boundary condition for long-range scattering is often difficult, since different long-range scatterings have different boundary conditions hod2013scattering ; liu2014scattering ; li2016scattering ; li2016exact . As a contrast, the scattering boundary condition for short-range scatterings is the same. In this paper, we first rewrite the radial equation as a confluent Heun equation, and then we determine the scattering boundary condition based on the asymptotic behavior of the confluent Heun function.

Furthermore, when calculating scattering cross sections, we encounter a difficulty in the partial sum approximation. An incorrect oscillation appears when truncating the partial wave expansion. We show how to eliminate such an oscillation in li2021eliminating .

In section 2, we solve bound-state and scattering-state solutions directly from the scalar field equation in the Reissner-Nordström spacetime. In section 3, we solve the scattering phase shift explicitly by the integral equation method. The conclusions are summarized in section 4.

2 Scalar field in Reissner-Nordström spacetime: bound state and

scattering state

2.1 Radial equation

The scalar equation (1) with the Reissner-Nordström metric (2) leads the radial equation:

[TABLE]

{paracol}

2 \switchcolumnwhere $R_{l}\left(r\right)$ is the radial wave function, $\omega^{2}$ is energy of the incident particle, and $\mu$ is the mass of the particle.

The Reissner-Nordström spacetime has two horizons at

[TABLE]

The radial equation (3) can be reexpressed by $r_{+}$ and $r_{-}$ as

[TABLE]

{paracol}

2 \switchcolumnwhere $\eta^{2}=\omega^{2}-\mu^{2}$.

2.2 Confluent Heun equation

The radial equation (5), by the replacement

[TABLE]

and

[TABLE]

can be rewritten as

[TABLE]

This is just the confluent Heun equation ronveaux1995heun .

The radial wave function, by Eq. (6), can be obtained by

[TABLE]

2.3 Tortoise coordinate for Reissner-Nordström spacetime

We introduce the tortoise coordinate for the Reissner-Nordström spacetime as

[TABLE]

Introducing $R_{l}\left(r\right)=u_{l}\left(r\right)/r$ and rewriting the radial equation (3) under the tortoise coordinate give

[TABLE]

{paracol}

2 \switchcolumn

2.4 Boundary condition

For the Reissner-Nordström spacetime, we need to impose three boundary conditions at the outer and inner horizons $r=r_{\pm}$ and at $r\rightarrow\infty$, respectively.

At the outer and inner horizons, $r=r_{\pm}$, we require that $\left|R_{l}\left(r_{\pm}\right)\right|$ is finite:

[TABLE]

The boundary condition at $r\rightarrow\infty$ needs to be considered in two cases: scattering states and bound states. For bound states, the boundary condition is

[TABLE]

For scattering states, the boundary condition is determined by the asymptotics behavior and satisfies

[TABLE]

We determine the boundary conditions at $r\rightarrow\infty$ and $r\rightarrow r_{\pm}$ by analyzing the asymptotic behavior of the radial equation, respectively.

The boundary condition at the outer horizon. At the outer horizon, $r=r_{+}$, the asymptotics of the radial equation for $r\rightarrow r_{+}$ by Eq. (11) reads

[TABLE]

The solution of the asymptotic equation (15) is

[TABLE]

That is, the scattering boundary at $r\rightarrow r_{+}$ is

[TABLE]

*The boundary condition at the inner horizon. *At the inner horizon, $r=r_{-}$, the asymptotics of the radial equation for $r\rightarrow r_{-}$ by Eq. (11) reads

[TABLE]

The solution of the asymptotic equation (18) is

[TABLE]

That is, the scattering boundary at $r\rightarrow r_{-}$ is

[TABLE]

*The bound-state boundary condition. *The bound-state boundary condition is

[TABLE]

that is, $u_{l}\left(r\right)$ is finite at $r\rightarrow\infty$.

*The scattering boundary condition. *The asymptotics of the radial equation for $r\rightarrow\infty$ by Eq. (11) reads

[TABLE]

The solution of the asymptotic equation (22) is

[TABLE]

That is, the scattering boundary at $r\rightarrow\infty$ is

[TABLE]

In section 2.2, we convert the radial equation (3) into the confluent Heun equation (8), so the boundary conditions of $R_{l}\left(r\right)$ should be converted into the boundary conditions of $y(z)$.

The boundary conditions at the outer and inner horizons, Eq. (12), becomes

[TABLE]

The boundary conditions at $r\rightarrow\infty$, Eqs. (14) and (13) become

[TABLE]

2.5 Bound-state solution

In section 2.2, we convert the radial equation (5) into the confluent Heun equation (8).

For bound states

[TABLE]

and

[TABLE]

Here $\eta=-k^{2}$ in Eq. (27) is the eigenvalue of the radial equation (5). The bound-state boundary condition for the radial equation $R\left(r\right)\overset{r\rightarrow\infty}{\sim}0$ corresponds to the boundary condition for the confluent Heun equation (8) $y\left(z\right)\overset{z\rightarrow\infty}{\sim}0$. The boundary conditions at the horizons $r_{\pm}$ correspond to the boundary conditions of the confluent Heun equation (8) at $z=\pm 1$. That is, we need a solution of the confluent Heun equation (8) satisfying $\left|\Pi\left(p,\beta,\pm 1\right)\right|<\infty$ and $\Pi\left(p,\beta,z\right)\overset{z\rightarrow\infty}{\sim}0$. Such a solution of Eq. (8) is ronveaux1995heun

[TABLE]

{paracol}

2 \switchcolumnwhere $\sigma=\lambda+2p\left(-2\beta+m+s+1\right)-m\left(m+1\right)$ and $\operatorname*{Hc}{}^{\left(a\right)}\left(a,b,c,d,e;z\right)$ is the confluent Heun function. Then the solution of the radial equation reads

[TABLE]

{paracol}

2 \switchcolumnwhere

[TABLE]

{paracol}

2 \switchcolumnThe eigenvalue of the radial equation (5), $k^{2}$, relates the eigenvalue of the confluent Heun equation (8), $\lambda_{n}$, through the relation (32). By the relation (32), we arrive at

[TABLE]

For a large $p=\frac{1}{2}k\left(r_{+}-r_{-}\right)$, the eigenvalue of the confluent Heun equation (8), $\lambda_{n}$, has the following asymptotics ronveaux1995heun ,

[TABLE]

where

[TABLE]

with $n$ an integer.

Then the eigenvalue of the radial equation $k^{2}$ can be solved from Eqs. (36) and (37):

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

The eigenvalue of the radial equation then reads

[TABLE]

For large $k$, the expansion of Eqs. (32) and (37) gives

[TABLE]

and the eigenvalue

[TABLE]

{paracol}

2 \switchcolumnFor a large $n$, i.e., the high-energy case, the eigenvalue becomes

[TABLE]

That is, the contribution of the charge $Q$ is proportional to $n^{-4}$, while the contribution of the mass $M$ is proportional to $n^{-2}$. For a large $n$ or a small $Q$, the eigenvalue (46) becomes

[TABLE]

This agrees with the result of a Klein-Gordon particle in the Coulomb potential in the high-energy case. The eigenvalue of the Klein-Gordon equation with the Coulomb potential is $\eta_{\text{Coulomb}}^{2}=\mu^{2}\left[1+\alpha^{2}/\left(n+\beta\right)^{2}\right]^{-1/2}-\mu^{2}$, where $\alpha$ and $\beta$ are some constants schwabl2013advanced . For a large $n$, $\eta_{\text{Coulomb}}^{2}\sim-1/n^{2}$ li2019scattering .

For small $n$, i.e., the low-energy case, for a large $Q$, consequently a large $M$ for $Q$ must be less than $M$, the eigenvalue becomes

[TABLE]

This agrees with the result of a Klein-Gordon particle in the Coulomb potential in the low-energy case. For small $n$, the eigenvalue of the Coulomb potential is $\eta_{\text{Coulomb}}^{2}=A-Bn$ li2019scattering .

When the charge $Q\rightarrow 0$, the eigenvalue (45) becomes

[TABLE]

which agrees with the corresponding result in the Schwarzschild spacetime given by Ref. li2019scattering .

For illustration, we rearrange the expression of the eigenvalue, Eq. (45), as

[TABLE]

In Figs. (1), (2), and (3) we plot the eigenvalue as a function of the mass and the charge.

\switchcolumn

Figure1. The bound-state eigenvalue $\omega/\mu$ as a function of $M\mu$ and $Q/M$. \switchcolumn

\switchcolumn

**Figure 2. The bound-state eigenvalue $\omega/\mu$ as a function of $M\mu$ with $Q/M=10^{-3}$. ** \switchcolumn

\switchcolumn

Figure 3. The bound-state eigenvalue $\omega/\mu$ as a function of $Q/M$ with $M\mu=1$. \switchcolumn

2.6 Scattering solution

The scattering solution can be obtained by performing the replacement

[TABLE]

in Eq. (34):

[TABLE]

{paracol}

2 \switchcolumnwhere

[TABLE]

{paracol}

2 \switchcolumnThis is the exact scattering wave function. Note that in scattering the eigenvalue spectrum is a continuous spectrum.

3 Scattering phase shift: Integral equation

The above scattering solution is an exact solution. In this section we present an explicit approximate expression of the scattering phase. We construct an integral equation by the Green function for the scattering wave function and solve the scattering phase shift from the integral equation.

3.1 Effective potential

The scalar field equation (3) is more complicated. In this section, with the help of the tortoise coordinate, by introducing an effective potential, we convert the radial equation (3) into a one-dimensional stationary Schrödinger equation.

By defining an effective potential

[TABLE]

we rewrite Eq. (11) as

[TABLE]

It can be seen that under the tortoise coordinate $r_{\ast}$ the radial equation (11) is a one-dimensional stationary Schrödinger equation with the potential $V_{l}^{\text{eff}}$.

Note that for scattering, we only consider the solution at $r>r_{+}$.

3.2 Green function and integral equation

In the following we first construct the Green function from the solution of the radial equation and then construct a scattering integral equation.

First, we construct the Green function from two linearly independent solutions of the radial equation. Before solving the radial equation (55), we first solve a solution with $V_{l}^{\text{eff}}=0$:

[TABLE]

It can be checked directly that Eq. (56) has two linearly independent solutions:

[TABLE]

where the solution is outside the horizon, i.e., $r_{+}<r<\infty$.

By these two linearly independent ”free” solutions, we can construct the Green function arfken2013mathematical :

[TABLE]

Now we determine the continuity condition of the Green function.

The Green function for Eq. (55) is defined by

[TABLE]

Integrating both sides of Eq. (61),

[TABLE]

we have

[TABLE]

Performing the integral gives

[TABLE]

In order to satisfy Eq. (64), we require that $G\left(r,r^{\prime}\right)$ is continuous at $r^{\prime}$ so that the second term vanishes when $\varepsilon\rightarrow 0$, i.e.,

[TABLE]

and the first term have a jump satisfying

[TABLE]

i.e.,

[TABLE]

Then we arrive at a continuity condition of the Green function:

[TABLE]

This gives

[TABLE]

Solving Eqs. (70) and (71) gives the coefficients

[TABLE]

Then we obtain the Green function

[TABLE]

By the Green function, we can construct an integral equation for $u_{l}\left(r\right)$:

[TABLE]

{paracol}

2 \switchcolumnor,

[TABLE]

This is an integral equation of $u_{l}\left(r\right)$.

3.3 Scattering boundary condition

In this section, we determine the scattering boundary condition by virtue of the asymptotic behavior of the confluent Heun function.

For scattering, we only concern the large-distance asymptotic behavior of the radial equation. In the following, we consider the asymptotic equation of the radial equation.

The confluent Heun equation (8), with $p=ic$ and $\beta=i\gamma$, has an asymptotic solution ronveaux1995heun

[TABLE]

This gives an asymptotic solution of the radial equation (5):

[TABLE]

{paracol}

2 \switchcolumnFor high energy scattering, $\mu/\eta\ll 1$, the asymptotics of the radial function can be written as

[TABLE]

{paracol}

2 \switchcolumnIntroducing $R_{l}\left(r\right)=u_{l}\left(r\right)/r$, we arrive at

[TABLE]

where $\delta_{l}$ is the scattering phase shift and

[TABLE]

Now we obtain the asymptotic solution. This requires that the asymptotics of the solution of the radial equation (3) should take the form of the asymptotics (81); that is, Eq. (81) is just the scattering boundary condition.

3.4 Scattering phase shift and scattering cross section

Now we calculate the scattering phase shift.

In order to compare to the asymptotic solution (81), we seek for an asymptotic equation of the integral equation (77). Rewrite the integral equation (77) as

[TABLE]

{paracol}

2 \switchcolumnThen taking $r\rightarrow\infty$ gives the asymptotics

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

By the tortoise coordinate (10), Eq. (84) can be written as

[TABLE]

The scattering phase shift can be obtained by comparing two asymptotic solutions, Eqs. (81) and (89). Direct comparison gives

[TABLE]

The scattering phase shift then reads

[TABLE]

{paracol}

2 \switchcolumnwhere $\displaystyle\phi=\arctan\frac{\beta\left(\eta\right)}{\alpha\left(\eta\right)}$ is used.

Now we determine the constants $A$ and $B$.

For $V_{l}^{\text{eff}}=0$ with $M=0$ and$\ Q=0$, Eq. (77) becomes

[TABLE]

The boundary condition requires that at the horizon $r=r_{+}$ with $r_{+}=M+\sqrt{M^{2}-Q^{2}}=0$, we must have $u_{l}\left(0\right)=0$ so that the radial wave function is finite. Consequently, $B=0$ and $A$ is an arbitrary constant. Taking $A=1$ gives the scattering phase shift

[TABLE]

{paracol}

2 \switchcolumnThe zeroth-order phase shift is then

[TABLE]

The first-order phase shift, by substituting the zeroth-order wave function

[TABLE]

into Eq. (93), reads

[TABLE]

The scattering amplitude then reads pike2008scatteringPage

[TABLE]

and the differential scattering cross section is

[TABLE]

In Fig. (4), we compare three differential scattering cross sections up to the second-order scattering phase shift: the Schwarzschild case $Q/M\simeq 0$, the typical Reissner-Nordström case $Q/M=1/2$, and the extremal Reissner-Nordström case $Q/M\simeq 1$. Here the phase shift $\delta_{l}\simeq\delta_{l}^{\left(0\right)}+\delta_{l}^{\left(1\right)}$ and sums to $l=6$.

The scattering amplitude $f\left(\theta\right)$ is given by the series (96). The scattering amplitude $f\left(\theta\right)$ in the interval from [math] to $\pi$ has no oscillations. However, often, the sum in Eq. (96) cannot be performed exactly, and one has to approximately replace the exact sum by a partial sum consisting of the first several terms.  The partial sum has an incorrect oscillation which does not appear in the exact sum. The incorrect oscillation cannot be eliminated by simply keeping more terms into account. We suggest an approach to eliminate such an oscillation in the partial sum approximation in li2021eliminating .

\switchcolumn

Figure 4. Differential scattering cross sections of the typical Reissner-Nordström case $Q/M=1/2$, the extremal Reissner-Nordström case $Q/M\simeq 1$, and the Schwarzschild case $Q/M\simeq 0$. \switchcolumn

4 Conclusion

In this paper, we solve the massive scalar field in the Reissner-Nordström spacetime. The solutions of bound states and scattering states are presented. The bound-state wave function, the bound-state eigenvalue and the scattering wave function are calculated by directly solving the radial equation. In order to obtain an explicit expression of the scattering phase shift, we use the integral approach.

The scattering boundary condition of long range potentials is difficult to determine, since different long range potentials have different scattering boundary conditions. Scattering on the Reissner-Nordström spacetime is essentially a long range potential. In this paper, we determine the scattering boundary condition based on the asymptotic behavior of the confluent Heun function.

Moreover, it is worthy to note here that the result given in the present paper recovers the result of the Schwarzschild spacetime when the charge $Q=0$.

In the calculation of scattering cross sections, we encounter an incorrect oscillation in the partial sum approximation. We suggest an approach for eliminating such oscillations in the appendix.

The scattering phase shift also plays an important role in quantum field theory through the scattering spectral method graham2009spectral . The heat kernel method is another important method in quantum field theory barvinsky1987beyond ; barvinsky1990covariant ; barvinsky1990covariant3 ; dai2009number ; dai2010approach ; mukhanov2007introduction . In virtue of the relation between the scattering spectral method and the heat kernel method pang2012relation ; li2015heat , the result of the scattering phase shift can also be applied to the heat kernel theory in quantum field theory.

Acknowledgements.

We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supported in part by Special Funds for theoretical physics Research Program of the NSFC under Grant No. 11947124, and NSFC under Grant Nos. 11575125 and 11675119. \reftitleReferences