New Perspectives on the Irregular Singular Point of the Wave Equation for a Massive Scalar Field in Schwarzschild Space-Time

Abstract

For a massive scalar field in a fixed Schwarzschild background, the radial wave equation obeyed by Fourier modes is first studied. After reducing such a radial wave equation to its normal form, we first study approximate solutions in the neighborhood of the origin, horizon and point at infinity, and then we relate the radial with the Heun equation, obtaining local solutions at the regular singular points. Moreover, we obtain the full asymptotic expansion of the local solution in the neighborhood of the irregular singular point at infinity. We also obtain and study the associated integral representation of the massive scalar field. Eventually, the technique developed for the irregular singular point is applied to the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations in a Schwarzschild background.

1. Introduction

Almost a hundred and ten years after the creation of general relativity [1,2], even the study of just the scalar wave equation in a fixed curved background, which solves the Einstein equations, is still in the early stages. In the seventies, for example, very detailed papers were published on the radial wave equation in Schwarzschild space-time, but they claimed that it is not related to any known differential equation of mathematical physics [3,4,5]. Forty years after this work, the dissertation in Ref. [6] was still claiming that no exact solution of the radial wave equation is known in Schwarzschild space-time.

Every partial differential equation studied in physics and mathematics is a world of its own, and the choice of the appropriate techniques depends on the assumptions made: the nature of the physical phenomenon, its mathematical formulation, the ambient space-time manifold, the regularity required for the solutions and the existence or lack of symmetries. It has been, therefore, our aim to exploit the well-established tools of ordinary and partial differential equations to obtain both the exact solution of the radial wave equation for Fourier modes of a massive scalar field and the integral representation of the full scalar field in Schwarzschild geometry. The scalar field case is already of considerable physical interest in light of modern investigations of null infinity [7,8] and of its role in studying Hawking radiation [9] for quantum fields in curved space-time [10,11]. The spherical symmetry of the fixed space-time geometry makes it possible to exploit Fourier modes, which do not exist in a generic space-time without any symmetry.

The paper is organized as follows. Section 2 is devoted to an original synthesis of the radial wave equation and its reduction to normal form. Section 3 obtains approximate solutions of such an equation in the neighborhood of two regular singular points and of an irregular singular point. Section 4 gives the radial wave equation in the Heun equation form. Local solutions at the regular singular points are obtained in Section 5. A full local solution of the radial wave equation at infinity is obtained in Section 6. The resulting integral representation of the massive scalar field that solves the wave equation is studied analytically and numerically in Section 7. Section 8 is devoted to the application of Section 6 to the investigation of the homogeneous equation associated with the inhomogeneous Zerilli equation for gravitational perturbations of a Schwarzschild background. The results obtained and open problems are presented in Section 9, and relevant details are provided in the appendices.

2. Radial Wave Equation and Its Reduction to Normal Form

In agreement with our aims, our starting point is the scalar wave equation

$$(\square -{\mu }^2)\psi =0$$

(1)

for a scalar field $\psi$ of mass $\mu$ in a fixed Schwarzschild background (thus, we are not studying a coupled system of Einsten and scalar-field equations, nor are we studying a scalar self-force problem). This background space-time is the unique spherically symmetric (This means that space-time admits the group $SO\left(3\right)$ as a group of isometries, with the group orbits given by spacelike two-surfaces.) solution of the vacuum Einstein equations in a way made precise by the Birkhoff theorem [12]. In standard spherical coordinates $(t,r,\theta ,\varphi )$ where $t\in \mathbf{R}\cup \{-\infty \}\cup \{+\infty \}, r\ge 0, \theta \in [0,\pi ], \varphi \in [0,2\pi ]$, with $G=c=1$ units, on defining $f\left(r\right)=1-{\displaystyle \frac{2M}{r}}$, the space-time metric is diagonal:

$$g=-f\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{f\left(r\right)}}+r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2).$$

(2)

Since such components are t-independent, one can exploit a Fourier representation according to the expansion [6] ($Y_{lm}$ being the spherical harmonics on the 2-sphere, and $C_{lm}$ being coefficients of linear combination (The $C_{lm}$ coefficients can be derived by inverse Fourier transform following standard techniques, but we do not need their values in our analysis)).

$$\psi (t,r,\theta ,\varphi ;\mu )=\sum _{l=0}^{\infty }{\psi }_{l\mu }(t,r)\sum _{m=-l}^l{C}_{lm}{Y}_{lm}(\theta ,\varphi ),$$

(3)

where ${\psi }_{l\mu }$ are the integrated Fourier modes

$${\psi }_{l\mu }(t,r)={\int }_{-\infty }^{\infty }{e}^{-i\omega t}{R}_{l\omega \mu }\left(r\right){\displaystyle \frac{d\omega }{\sqrt{2\pi }}},$$

(4)

and $R_{l\omega \mu }$ solves the radial wave equation

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+{\displaystyle \frac{2(r-M)}{r^{2}f\left(r\right)}}{\displaystyle \frac{d}{dr}}+{\displaystyle \frac{1}{f\left(r\right)}}\left({\displaystyle \frac{{\omega }^2}{f\left(r\right)}}-{\displaystyle \frac{l(l+1)}{r^2}}-{\mu }^2\right)\right]R_{l\omega \mu }\left(r\right)=0.$$

(5)

This equation can be written in normal form by assuming the product

$$R_{l\omega \mu }\left(r\right)=\alpha \left(r\right){\beta }_{l\omega \mu }\left(r\right),$$

(6)

and looking for an $\alpha \left(r\right)$ such that the equation obeyed by ${\beta }_{l\omega \mu }\left(r\right)$ has vanishing coefficient of the first derivative. Indeed, upon setting in Equation (5)

$$p\left(r\right)={\displaystyle \frac{2(r-M)}{r^{2}f\left(r\right)}},q_{l\omega \mu }\left(r\right)={\displaystyle \frac{1}{f\left(r\right)}}\left({\displaystyle \frac{{\omega }^2}{f\left(r\right)}}-{\displaystyle \frac{l(l+1)}{r^2}}-{\mu }^2\right),$$

(7)

the general method for achieving normal form [13] yields

$$\alpha \left(r\right)=\exp \left(-{\displaystyle \frac{1}{2}}\int p\left(r\right)dr\right)={\displaystyle \frac{1}{\sqrt{r(r-2M)}}},$$

(8)

jointly with the second-order differential equation

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+J_{l\omega \mu }\left(r\right)\right]{\beta }_{l\omega \mu }\left(r\right)=0,$$

(9)

where the potential term is given by

$$\begin{array}{ccc}\hfill & & J_{l\omega \mu }\left(r\right)=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{dp}{dr}}-{\displaystyle \frac{1}{4}}{p}^2\left(r\right)+q_{l\omega \mu }\left(r\right)\hfill \\ & =& {\displaystyle \frac{1}{4r^2}}+{\displaystyle \frac{1}{4{(r-2M)}^2}}-{\displaystyle \frac{\left(\frac{1}{2}+l(l+1)\right)}{r(r-2M)}}\hfill \\ & +& {\displaystyle \frac{{\omega }^2{r}^2}{{(r-2M)}^2}}-{\displaystyle \frac{{\mu }^{2}r}{(r-2M)}}.\hfill \end{array}$$

(10)

We note that $J_{l\omega \mu }\left(r\right)$ has a second-order pole at $r=0$ and at $r=2M$, and hence these are regular singular points, whereas the point $r=\infty$ is an irregular singular point [14] as we are going to prove. In order to study the nature of the $r=\infty$ point, one defines the new independent variable $\rho ={\displaystyle \frac{1}{r}}$, which implies for ${\beta }_{l\omega \mu }\left(\rho \right)$ the differential equation

$$\left[{\displaystyle \frac{d^2}{d{\rho }^2}}+{\displaystyle \frac{2}{\rho }}+{\displaystyle \frac{J_{l\omega \mu }\left(\rho \right)}{{\rho }^4}}\right]{\beta }_{l\omega \mu }\left(\rho \right)=0,$$

(11)

where

$${\displaystyle \frac{J_{l\omega \mu }\left(\rho \right)}{{\rho }^4}}={\gamma }_{1l\omega }\left(\rho \right)+{\gamma }_{2\omega \mu }\left(\rho \right),$$

(12)

having defined

$${\gamma }_{1l\omega }\left(\rho \right)={\displaystyle \frac{1}{4{\rho }^2}}\left(1+{\displaystyle \frac{1}{{(1-2M\omega )}^2}}\right)-{\displaystyle \frac{\left(\frac{1}{2}+l(l+1)\right)}{{\rho }^2(1-2M\rho )}},$$

(13)

$${\gamma }_{2\omega \mu }\left(\rho \right)={\displaystyle \frac{1}{{\rho }^4}}\left({\displaystyle \frac{{\omega }^2}{{(1-2M\rho )}^2}}-{\displaystyle \frac{{\mu }^2}{(1-2M\rho )}}\right).$$

(14)

Since ${\gamma }_{2\omega \mu }\left(\rho \right)$ has a fourth-order pole at $\rho =0$, the point $r=\infty$ is indeed an irregular singular point for Equation (9).

The material in this section is standard but nevertheless useful for achieving a self-contained presentation of our subsequent analysis.

3. Approximate Solutions in the Neighborhood of r = 0, 2M and ∞

3.1. Fourier Modes as r → 0

As $r\to 0$, the form of $J_{l\omega \mu }\left(r\right)$ in Equation (10) implies that Equation (9) reduces to a form independent of l and $\omega$, i.e.,

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+{\displaystyle \frac{1}{4r^2}}\right]{\beta }_0\left(r\right)=0.$$

(15)

On looking for solutions in the form ${\beta }_0\left(r\right)=r^{\alpha }$, one finds that the square of $\left(\alpha -{\displaystyle \frac{1}{2}}\right)$ should vanish. Hence there exist two linearly independent integrals giving rise to

$${\beta }_0\left(r\right)=C_1\sqrt{r}+C_2\sqrt{r}\log \left(r\right).$$

(16)

On requiring regularity at the origin one finds $C_2=0$, and hence

$$R_0\left(r\right)={\displaystyle \frac{{\beta }_0\left(r\right)}{\sqrt{r(r-2M)}}}={\displaystyle \frac{C_1}{\sqrt{r-2M}}},$$

(17)

i.e., the Fourier modes are finite and constant at $r=0$. This agrees with the property in the massless case found in Ref. [3]: there exists one solution finite at the origin, which is physically preferable for this reason.

3.2. Fourier Modes as $r\to 2M$

As $r\to 2M$, by virtue of Equation (10) we find that

$$\underset{r\to 2M}{lim}{J}_{l\omega \mu }\left(r\right)=\underset{r\to 2M}{lim}\left[{\displaystyle \frac{{\chi }_2\left(\omega \right)}{{(r-2M)}^2}}+{\displaystyle \frac{{\chi }_1(l,\mu )}{(r-2M)}}+{\chi }_0\right],$$

(18)

where

$${\chi }_2\left(\omega \right)=4M^2{\omega }^2+{\displaystyle \frac{1}{4}},$$

(19)

$${\chi }_1(l,\mu )=-{\displaystyle \frac{1}{2M}}\left({\displaystyle \frac{1}{2}}+l(l+1)\right)-2M{\mu }^2,$$

(20)

$${\chi }_0={\displaystyle \frac{1}{16M^2}}.$$

(21)

This implies that Equation (9) reduces to

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+{\displaystyle \frac{{\chi }_2\left(\omega \right)}{{(r-2M)}^2}}\right]{\beta }_{\omega ,2M}\left(r\right)=0,$$

(22)

which (up to a multiplicative constant) is solved by

$${\beta }_{\omega ,2M}\left(r\right)={(r-2M)}^A,$$

(23)

where A is a root of the algebraic equation

$$A^2-A-{\chi }_2\left(\omega \right)=0.$$

(24)

Such roots are

$$A_{+}={\displaystyle \frac{1}{2}}+2iM\omega ,$$

(25)

$$A_{-}={\displaystyle \frac{1}{2}}-2iM\omega ,$$

(26)

and hence the Fourier modes have the limiting form

$$\begin{array}{ccc}\hfill & & R_{\omega ,2M}\left(r\right)={\displaystyle \frac{{\beta }_{\omega ,2M}\left(r\right)}{\sqrt{r(r-2M)}}}\hfill \\ & =& {\displaystyle \frac{1}{\sqrt{r}}}[D_1{(r-2M)}^{2iM\omega }+D_2{(r-2M)}^{-2iM\omega }],\hfill \end{array}$$

(27)

where $D_1$ and $D_2$ are suitable constants. Interestingly, this limiting behavior agrees with the massless case [3]: the two solutions, with coefficients $D_1$ and $D_2$, remain bounded on the horizon $r=2M$, and no solution approaches zero as $r\to 2M$. These properties are closely related to the possibility of destroying the black hole and to the radiation of higher multipole moments when a small scalar particle falls into the black hole [15].

3.3. Fourier Modes at Infinity

Last but not least, the potential term $J_{l\omega }$ in Equation (9) has the following Poincaré asymptotic expansion as $r\to \infty$:

$$J_{l\omega \mu }\sim {\omega }^2-{\mu }^2+{\displaystyle \frac{A_1(\omega ,\mu )}{r}}+{\displaystyle \frac{A_2(l,\omega ,\mu )}{r^2}}+\sum _{k=3}^{\infty }{\displaystyle \frac{A_k(l,\omega ,\mu )}{r^k}}$$

(28)

where we have defined

$$A_1(\omega ,\mu )=2M(2{\omega }^2-{\mu }^2),$$

(29)

$$A_2(l,\omega ,\mu )=4M^2(3{\omega }^2-{\mu }^2)-l(l+1).$$

(30)

Thus, as $r\to \infty$, ${\beta }_{l\omega \mu }\left(r\right)$ is (up to a multiplicative constant) the product of the leading order ${\beta }_{\omega ,\infty }\left(r\right)=e^{\pm r\sqrt{{\mu }^2-{\omega }^2}}$ times a function that we do not strictly need here, but is part of an analysis that we will perform in Section 6.

Note also that, by virtue of Equation (4), one has to integrate overall values of $\omega$ in order to evaluate the large-r scalar field. Thus, the Fourier modes at large r have to be considered in the following intervals for the $\omega$ variable:

$$[-\infty ,-\mu ],]-\mu ,\mu [\mathrm{and}[\mu ,\infty ].$$

This remark will be exploited in Section 7.

4. Connecting the Radial with the Heun Equation

The reduction to canonical form of Equation (5) helped us in searching for the singular points $r_0=0$, $r_{2M}=2M$ and for the one at infinity. Now we are going to see that Equation (5) belongs to the class of confluent Heun equations defined in Appendix A [16]. In order to prove it, we rely upon the change of independent variable frequently used in the literature, i.e.,

$$z={\displaystyle \frac{(r-r_{2M})}{(r_0-r_{2M})}}=1-{\displaystyle \frac{r}{2M}}.$$

(31)

In this way, the two finite singularities $r=\{0,2M\}$ correspond to $z=\{1,0\}$, respectively, and the physical requirement $r\ge 0$ leads to the restriction $z\le 1$. The Equation (5) henceforth takes the form

$$\left[{\displaystyle \frac{d^2}{d{z}^2}}-{\displaystyle \frac{(1-2z)}{z(1-z)}}{\displaystyle \frac{d}{dz}}+{\displaystyle \frac{4M^2{\omega }^2{(1-z)}^2}{z^2}}+{\displaystyle \frac{l(l+1)}{z(1-z)}}+{\displaystyle \frac{4M^2{\mu }^2(1-z)}{z}}\right]R_{l\omega \mu }\left(z\right)=0.$$

(32)

Thanks to this change of variable, such equation is nothing but a confluent Heun equation in the so-called non-symmetric canonical form. Such a form is obtained by exploiting

$$H_{\mathrm{NSCF}}\left(z\right)=e^{{\alpha }_{1}z}{z}^{{\alpha }_2}{(z-1)}^{{\alpha }_3}H\left(z\right),$$

(33)

where $H\left(z\right)$ solves the confluent Heun equation.

5. Local Solutions at the Regular Singular Points

In our specific case, upon defining the five Heun parameters (no confusion should arise with $\alpha$ and $\beta$ of Section 2)

$$\begin{array}{ccc}\hfill & & \alpha =4M\sqrt{{\mu }^2-{\omega }^2},\hfill \\ & & \beta =4iM\omega ,\hfill \\ & & \gamma =0,\hfill \\ & & \delta =4M^2({\mu }^2-2{\omega }^2),\hfill \\ & & \eta =-4M^2({\mu }^2-2{\omega }^2)-l(l+1),\hfill \end{array}$$

(34)

Equation (33) takes the form

$$H_{\mathrm{NSCF}}\left(z\right)=e^{{\displaystyle \frac{\alpha }{2}}z}{z}^{{\displaystyle \frac{\beta }{2}}}H\left(z\right),$$

(35)

since ${\alpha }_3=0$ for Equation (32). Starting from Equation (35), one can construct the complete Frobenius solution at $z=0$ (corresponding to $r=2M$), that reads (cf. Refs. [17,18,19,20,21])

$$R_0\left(z\right)=e^{{\displaystyle \frac{\alpha }{2}}z}{z}^{{\displaystyle \frac{\beta }{2}}}[C_1\mathrm{HeunC}(\alpha ,\beta ,\gamma ,\delta ,\eta ,z)+C_2{z}^{-\beta }\mathrm{HeunC}(\alpha ,-\beta ,\gamma ,\delta ,\eta ,z)],$$

(36)

whose leading order is

$$R_0\left(z\right)\sim e^{{\displaystyle \frac{\alpha }{2}}z}{z}^{{\displaystyle \frac{\beta }{2}}}(C_1+C_2{z}^{-\beta }).$$

(37)

The Frobenius solution at $z=1$ (i.e., at $r=0$) is given by

$$\begin{array}{ccc}\hfill R_1\left(z\right)& =& e^{{\displaystyle \frac{\alpha }{2}}z}{z}^{{\displaystyle \frac{\beta }{2}}}[C_1\mathrm{HeunC}(-\alpha ,\gamma ,\beta ,-\delta ,\eta +\delta ,-z)\hfill \\ & +& C_2{(z-1)}^{-\gamma }\mathrm{HeunC}(-\alpha ,-\gamma ,\beta ,-\delta ,\eta +\delta ,z)],\hfill \end{array}$$

(38)

whose leading order is

$$R_1\left(z\right)\sim e^{{\displaystyle \frac{\alpha }{2}}z}{z}^{{\displaystyle \frac{\beta }{2}}}(C_1+C_2{(z-1)}^{-\gamma }).$$

(39)

In order to visualize the results obtained so far, we plot in Figure 1 and Figure 2 the real and imaginary part of Equation (36), respectively. Similarly, we plot in Figure 3 and Figure 4 the real and imaginary part of Equation (38), respectively. In all these Figures, we make the following choice for the free parameters:

$$M=10,\mu =0.1,\omega =1,l=0,C_1=C_2=1.$$

(40)

In order to summarize the physical content of our results regarding the s-wave, the real part of the local solution slightly inside the black hole is very close to zero, while outside the event horizon, it exhibits strong oscillations, both in amplitude and in frequency. The imaginary part of the same solution follows the same behavior both qualitatively and quantitatively. Moreover, both the real and imaginary parts of the local solution at the origin possess a plateau very close to the z = 1 (i.e., $r=0$) point, and then decrease as z decreases.

6. Full Local Solution for Fourier Modes in the Neighborhood of r = ∞

When $r\to \infty$ one can evaluate ${\beta }_{l\omega \mu }\left(r\right)$, and hence the Fourier modes $R_{l\omega \mu }\left(r\right)$, by completing the analysis initiated in Equations (28)–(30). For this purpose, we recognize from such equations that the differential Equation (9) for ${\beta }_{l\omega \mu }\left(r\right)$ can be written in the form

$$\left({\displaystyle \frac{d^2}{d{r}^2}}+\sum _{k=0}^{\infty }{\displaystyle \frac{A_k(l,\omega ,\mu )}{r^k}}\right){\beta }_{l\omega \mu }\left(r\right)=0,$$

(41)

where $A_0={\omega }^2-{\mu }^2$, while $A_1$ and $A_2$ have been already defined in Equations (29) and (30), respectively. All subsequent coefficients can be computed from the asymptotic expansion of Equation (10) at large r. At this stage, Equation (41) suggests looking for ${\beta }_{l\omega \mu }\left(r\right)$ at large r in the form (setting $\epsilon =\pm 1$ and considering $\zeta$ as a parameter to be determined as shown below)

$${\beta }_{l\omega \mu }\left(r\right)\sim e^{\epsilon r\sqrt{-A_0}}{r}^{\zeta }\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right).$$

(42)

The insertion of Equation (42) into Equation (41) yields

$${\gamma }_0+{\displaystyle \frac{{\gamma }_1}{r}}+{\displaystyle \frac{{\gamma }_2}{r^2}}+\sum _{k=3}^{\infty }{\displaystyle \frac{{\gamma }_k}{r^k}}=0.$$

(43)

For this equation to be identically satisfied as $r\to \infty$, all ${\gamma }_k$ coefficients should vanish. Indeed, we find

$${\gamma }_0=A_0-A_0=0,$$

(44)

while

$${\gamma }_1=2\zeta \epsilon \sqrt{-A_0}+A_1,$$

(45)

$${\gamma }_2=(\zeta -1)(2\epsilon B_1\sqrt{-A_0}+\zeta )+A_1{B}_1+A_2,$$

(46)

jointly with infinitely many other equations for all subsequent ${\gamma }_k$ coefficients. Since they should all vanish, we obtain linear algebraic equations for $B_1,B_2,\dots ,B_{\infty }$. In addition, since we look for a bounded solution at large r, we restrict ourselves to the $\epsilon =-1$ case. (We are here studying a local solution which only holds at large r [3,5]. Thus, the $\epsilon =1$ mode cannot be avoided by studying conditions near the horizon $r=2M$.) For instance, from Equation (45) we evaluate the explicit form of $\zeta$, while from Equation (46) we obtain $B_1$, i.e.,

$$\zeta ={\displaystyle \frac{A_1}{2\sqrt{-A_0}}}={\displaystyle \frac{M(2{\omega }^2-{\mu }^2)}{\sqrt{{\mu }^2-{\omega }^2}}},$$

(47)

$$B_1={\displaystyle \frac{\zeta (\zeta -1)+A_2}{2(\zeta -1)\sqrt{-A_0}-A_1}}.$$

(48)

By virtue of Equation (6), we find eventually the large-r Fourier modes in the form (cf. Refs. [17,18,21])

$$R_{l\omega \mu ,\infty }\left(r\right)\sim {\displaystyle \frac{e^{-r\sqrt{-A_0}}}{\sqrt{r(r-2M)}}}{r}^{{\displaystyle \frac{A_1}{2\sqrt{-A_0}}}}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right).$$

(49)

Note that, by considering only the $k=0,1,2$ terms in Equation (41), one would obtain the Whittaker equation (a modified form of the confluent hypergeometric equation), solved by the two Whittaker functions. However, all negative powers of r occur in Equation (41), and hence one needs our factorization in Equation (49), where the exponential cancels the effect of $A_0$, and the $B_s$ functions of $l,\omega ,\mu$ compensate the negative powers of r in the operator provided that a non-integer power of r is included in our factorized ansatz. This procedure agrees with the general method of Poincaré for linear differential equations [22,23].

In Appendix C we offer a theoretical motivation for the ansatz given by Equation (42) and the explicit computation of $\zeta$ and the first $A_k$, $B_s$ coefficients. In order to visualize such a solution, we consider Equation (49) with the same choice for parameters given by Equation (40) (this time we do not have integration constants like $C_1$ and $C_2$), ending up with two plots: one for the real part (Figure 5) of the field and the other for the imaginary one (Figure 6). The relation with the local results by Persides in the massless case [3] is discussed in Appendix D.

The plots obtained in our large-r analysis clearly describe the vanishing of the field (both real and imaginary parts) at spacelike infinity due to damped oscillations. This is in agreement with our mathematical ansatz and physical requirements.

7. Large-r Solution of the Full Wave Equation

So far, we have studied the behavior of solutions for fixed frequencies. Since one of our goals is to study the spectrum of a massive field far away from the black hole, we now focus the attention on the study of the solution at spacelike infinity by including the contribution of all frequencies (neglecting the angular part). In the previous sections we have discovered that when $r\to \infty$, the Fourier modes take the approximate form (49). The large-r form of the scalar field $\psi$ which solves Equation (1) and satisfies the asymptotic condition

$$\underset{r\to \infty }{lim}\left|{\psi }_l(t,r)\right|<\infty$$

(50)

is therefore expressed by Equation (4), where, upon defining

$$\phi (t,r,\omega )=\omega t+r\sqrt{{\omega }^2-{\mu }^2}+M{\displaystyle \frac{(2{\omega }^2-{\mu }^2)}{\sqrt{{\omega }^2-{\mu }^2}}}\log \left(r\right),$$

(51)

we find, for ${\omega }^2>{\mu }^2$,

$$\begin{array}{ccc}\hfill & & {\psi }_{l\mu }^I(t,r)\sim {\int }_{-\infty }^{\infty }{e}^{-i\omega t}{R}_{l\omega \mu ,\infty }\left(r\right){\displaystyle \frac{d\omega }{\sqrt{2\pi }}}\hfill \\ & \sim & {\displaystyle \frac{1}{\sqrt{r(r-2M)}}}\left({\int }_{-\infty }^{-\mu }+{\int }_{\mu }^{\infty }\right){\displaystyle \frac{d\omega }{\sqrt{2\pi }}}{e}^{-i\phi (t,r,\omega )}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right).\hfill \end{array}$$

(52)

The integral is qualitatively different in the ${\omega }^2<{\mu }^2$ case, since the last two terms in Equation (51) become imaginary, leading to

$${\psi }_{l\mu }^{II}(t,r)\sim {\displaystyle \frac{1}{\sqrt{r(r-2M)}}}{\int }_{-\mu }^{\mu }{\displaystyle \frac{d\omega }{2\pi }}{e}^{-i\omega t}{e}^{r\sqrt{{\mu }^2-{\omega }^2}\left(1+M{\displaystyle \frac{{\mu }^2-2{\omega }^2}{\sqrt{{\mu }^2-{\omega }^2}}}{\displaystyle \frac{\log \left(r\right)}{r}}\right)}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right).$$

(53)

The investigation of these two integrals is a hard task. However, we can rely on analytical approximations and numerical analysis in order to extract the essential features from them. The starting point is given by studying the behavior in frequencies of the two integrand functions, which we label as

$${\mathrm{\Omega }}_{l\mu }^I(t,r,\omega )={\displaystyle \frac{1}{\sqrt{2\pi r(r-2M)}}}{e}^{-i\phi (t,r,\omega )}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right),$$

(54)

$${\mathrm{\Omega }}_{l\mu }^{II}(t,r,\omega )={\displaystyle \frac{1}{\sqrt{2\pi r(r-2M)}}}{e}^{-i\omega t}{e}^{r\sqrt{{\mu }^2-{\omega }^2}\left(1+M{\displaystyle \frac{{\mu }^2-2{\omega }^2}{\sqrt{{\mu }^2-{\omega }^2}}}{\displaystyle \frac{\log \left(r\right)}{r}}\right)}\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega ,\mu )}{r^s}}\right).$$

(55)

Notice that the l-dependence is encoded in the coefficients of the series expansion, requiring to add also the angular part (given by spherical harmonics) to study the superposition of several waves. We therefore limit ourselves to the study of the s-wave, given by the constraint $B_s(l,\omega ,\mu )=B_s(0,\omega ,\mu )$. Moreover, for numerical reasons, we consider only the first five negative powers of r, arriving at a simplified version of Equations (54) and (55). Upon setting

$$\mu =0.1,t=1,M=10,$$

(56)

we first consider the plots of the real (Figure 7) and imaginary (Figure 8) part of Equation (55).

It is possible to evaluate numerically the integral given by Equation (53) by expanding ${\mathrm{\Omega }}_{l\mu }^{II}(t,r,\omega )$ around $\omega =0$, since this region gives the largest contribution to it (as shown in Figure 7). We have performed such evaluation up to the sixth power of $\omega$ for the chosen parameters, arriving at the conclusion that such a piece diverges. In order to obtain also an analytical estimate of such a divergence, we consider the change of variable given by

$$\omega =\mu \sin \left(\tau \right)$$

(57)

applied to the leading term of Equation (55) and therefore obtain

$${\mathrm{\Omega }}_{l\mu }^{II}(t,r,\tau )\approx {\displaystyle \frac{1}{\sqrt{2\pi r(r-2M)}}}{r}^{\mu M(2\left|cos\left(\tau \right)\right|-\mathrm{sign}(cos\left(\tau \right))sec\left(\tau \right))}{e}^{\mu (r\left|cos\left(\tau \right)\right|-i \sin \left(\tau \right))}.$$

(58)

The requirement $\omega \ll 1$ yields $\tau \ll 1$, giving as leading term

$${\mathrm{\Omega }}_{l\mu }^{II}(t,r,\tau )\approx {\displaystyle \frac{1}{\sqrt{2\pi r(r-2M)}}}{r}^{\mu M}{e}^{\mu r}(i\mu t\tau -1)+\mathrm{O}\left({\tau }^2\right).$$

(59)

After a straightforward integration we arrive at the desired result, i.e.,

$${\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}d\tau \mu cos\left(\tau \right){\mathrm{\Omega }}_{l\mu }^{II}(t,r,\tau )\approx {\displaystyle \frac{1}{\sqrt{2\pi r(r-2M)}}}\mu e^{\mu r}{r}^{\mu M}+\mathrm{O}\left({\tau }^2\right).$$

(60)

Even though it is possible to perform more accurate computations (including more terms and/or the negative powers of r of the series expansion) it is clear that the dominant contribution is the one of the exponential term, which causes the divergence of the whole integral in the $r\to \infty$ limit, as shown by the purely numerical analysis given before. Notice that the field mass term, $\mu$, plays a key role: in the massless case the equivalent of such a piece vanishes (as it happens here in the $\mu \to 0$ limit). We have to, therefore, face a case where a field not only does not vanish at spacelike infinity, but furthermore it diverges exponentially. As far as we can see, this property is compatible with the findings in Ref. [24], where part (II) of Proposition 2.1 therein shows that the norm of a massive scalar field obeying the wave equation in Schwarzschild space-time is majorized by the product of two functions, one of which is indeed divergent in a suitable limit. This agreement merits explicit mention because the work in Ref. [24] has exploited advanced methods from the modern theory of hyperbolic equations, whereas we have used the standard techniques of classical mathematical physics.

The case of ${\mathrm{\Omega }}_{l\mu }^I(t,r,\omega )$ is strongly different. Upon relying on the same conventions, we show the plots of its real part (the ones for the phase are very similar) for $\omega \in [-{10}^3,-\mu ]$ in Figure 9 and for $\omega \in [\mu ,{10}^3]$ in Figure 10.

On both numerical and analytical grounds, we can say that, while the function ${\mathrm{\Omega }}_l^I(t,r,\omega )$ diverges, the associated integral (given by Equation (52)) remains finite and eventually vanishes at spacelike infinity. In fact, we have an oscillating term $e^{-i\phi }$ alleviated by a series of negative powers of r.

Interestingly, we find that a massive scalar field contains a term that diverges at spacelike infinity, while this feature is absent in the massless case. In conclusion, the mass plays a crucial role in the asymptotic description of a scalar field in spherical coordinates.

8. Application of Section 6 to the Zerilli Equation

The systematic algorithm of Section 6 can be applied to equations of even greater interest. For this purpose, we here consider the recent analysis in Ref. [25] of the Zerilli equation [26], which is obtained by perturbing a background Schwarzschild metric with a small parity-even perturbation expressed by a symmetric tensor field of type $(0,2)$. Eventually, the work in Ref. [25] obtains an inhomogeneous radial wave equation, that we express in the form

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+p\left(r\right){\displaystyle \frac{d}{dr}}+q(r,l,\omega )\right]u(r,l,\omega )=\tilde{I}(r,l,\omega ),$$

(61)

where

$$p\left(r\right)={\displaystyle \frac{2M}{r(r-2M)}},$$

(62)

$$q(r,l,\omega )={\displaystyle \frac{r^2}{{(r-2M)}^2}}({\omega }^2-V(r,l,\omega )),$$

(63)

having defined

$$V(r,l,\omega )=\left(1-{\displaystyle \frac{2M}{r}}\right){\displaystyle \frac{[2{\eta }^2(\eta +1)r^3+6{\eta }^{2}M{r}^2+18\eta M^{2}r+18M^3]}{r^3{(\eta r+3M)}^2}},$$

(64)

with $\eta ={\displaystyle \frac{(l-1)(l+2)}{2}}$. The work in Ref. [25] uses a Green-function method in order to solve the inhomogeneous differential equation (61) by using the well-established technique for obtaining the Green function of the operator on the left-hand side of Equation (61) from solutions of the associated homogeneus equation

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+p\left(r\right){\displaystyle \frac{d}{dr}}+q(r,l,\omega )\right]U(r,l,\omega )=0.$$

(65)

It has been our choice to focus on this particular aspect of the general problem since we have already written several sections on the massive scalar case, which is the main topic of our paper.

By repeating the reduction to normal form as we did in Section 2, we obtain $U(r,l,\omega )$ in the factorized form

$$U(r,l,\omega )=\gamma \left(r\right)\beta (r,l,\omega ),$$

(66)

where

$$\gamma \left(r\right)={\displaystyle \frac{1}{\sqrt{1-\frac{2M}{r}}}},$$

(67)

while $\beta (r,l,\omega )$ solves the equation

$$\left[{\displaystyle \frac{d^2}{d{r}^2}}+J(r,l,\omega )\right]\beta (r,l,\omega )=0,$$

(68)

having defined the potential term

$$\begin{array}{ccc}\hfill & & J(r,l,\omega )={\displaystyle \frac{1}{{(r-2M)}^2{r}^2{[(l(l+1)-2)r+6M]}^2}}[{\omega }^2{(l-1)}^2{(l+2)}^2{r}^6\hfill \\ & +& 12M{\omega }^2(l-1)(l+2)r^5+(-l^6-3l^5+l^4+7l^3-4l+36M^2{\omega }^2)r^4\hfill \\ & +& 2M{(l-1)}^3{(l+2)}^3{r}^3+9M^2\left(l(l+1)-{\displaystyle \frac{10}{3}}\right)(l+2)(l-1)r^2\hfill \\ & +& 36M^3(l-1)(l+2)r+36M^4].\hfill \end{array}$$

(69)

At this stage, in close analogy with Equation (41), we may expand Equation (68) as $r\to \infty$ in the form

$$\left({\displaystyle \frac{d^2}{d{r}^2}}+\sum _{k=0}^{\infty }{\displaystyle \frac{A_k(l,\omega )}{r^k}}\right)\beta (r,l,\omega )=0,$$

(70)

where the function $\beta$ has the asymptotic expansion

$$\beta (r,l,\omega )\sim e^{-r\sqrt{-A_0}}{r}^{\zeta }\left(1+\sum _{s=1}^{\infty }{\displaystyle \frac{B_s(l,\omega )}{r^s}}\right).$$

(71)

The formulae for expressing $\zeta$ and $B_s$ in terms of the $A_k(l,\omega )$ coincide with the ones in Section 6, whereas the $A_k$ are different. Indeed, we find for example that

$$A_0={\omega }^2,$$

(72)

$$A_1=4{\omega }^{2}M,$$

(73)

$$A_2=12M^2{\omega }^2-l(l+1),$$

(74)

$$A_3={\displaystyle \frac{2M[16M^2{\omega }^2(l(l+1)-2)-l^4-2l^3+5l^2+6l+4]}{l(l+1)-2}},$$

(75)

and hence, from Equations (47) and (48),

$$\zeta =-2iM\left|\omega \right|,$$

(76)

$$B_1=-M+{\displaystyle \frac{i[8M^2{\omega }^2-l(l+1)]}{2\left|\omega \right|}},$$

(77)

supplemented by a countable infinity of higher-order $A_k$ and $B_s$ coefficients.

Hereafter, we plot the real (Figure 11) and imaginary (Figure 12) part of Equation (66) upon considering the parameters

$$M=10,\omega =1,l=0,$$

(78)

up to the $B_5$ coefficient in Equation (71).

Figure 11 and Figure 12 show in a simple way that our findings are in full agreement with the plane-wave behavior as $r\to \infty$ obtained in Ref. [25] since, after the shrinking of the field at $r\simeq 140$, both real and imaginary part of the solution display constant oscillations through spacelike infinity.

9. Results Obtained and Open Problems

The scalar wave equation in Schwarzschild space-time has been studied intensively over more than half a century [3,4,5,7,8,17,18], until the very recent and highly valuable work in Ref. [27], but the material in our Section 6 and Section 7 remains original. More precisely, as far as we know, our original contributions deal with ordinary and partial differential equations for black hole space-times and are as follows:

(i)

For a massive scalar field in Schwarzschild geometry, our Equations (47)–(49) and (A30)–(A31) yield the full Poincaré asymptotic expansion of Fourier modes that solve the radial wave equation.

(ii)

Our Equations (52)–(53) yield the integral representation of the massive scalar field, with the resulting numerical analysis discussed therein.

Our formulae provide useful insights into the physical behavior of a massive scalar field in Schwarzschild space-time as the singular points of the radial wave equation are approached. Recognizing the radial equation as nothing but a particular case of the confluent Heun equation makes it pssible to rely upon established mathematical techniques without approximations. The algorithm developed in Section 6 and Section 7 allows, in principle, to compute every term of the series at infinity by reaching the desired level of accuracy. Moreover, even if we have shown plots only for specific values of parameters, our formulae can be easily implemented in every symbolic programming language.

For massive scalar fields in Schwarzschild space-time, the literature has mainly focused on late-time behavior [28] of Fourier modes, whereas rigorous results on solutions of the Cauchy problem are along the lines of Ref. [24]. We think that the next step will consist in applying analogous techniques to other spherically symmetric space-times, both in black-hole theory [21] and in cosmological backgrounds [29]. For this purpose, we have eventually studied the homogeneous equation associated with the inhomogeneous Zerilli equation in a Schwarzschild background. Our original Formulas (72)–(77) can be applied to build the Green function, which in turn leads to the solution of the inhomogeneous Zerilli equation as shown in Ref. [25].

More precisely, the following remarks are still in order:

(i)

The exponential divergence of the massive field at spacelike infinity arises from the integral in Equation (53). As far as we can see, no boundary condition can get rid of it. As we stressed in Section 7, it is encouraging that our result agrees with a more advanced mathematical analysis as the one performed in Ref. [24], which had been ignored in the physics-oriented literature. Maybe this property means that one has instead to investigate the coupled system consisting of Einstein and Klein–Gordon equations, but this would require a separate paper.

(ii)

The proof of convergence or divergence of the series in Equation (49) is an open and difficult technical problem because the $B_s$ coefficients therein contain rational functions of the parameters and products of such rational functions with square roots of polynomials in the parameters. However, as we say in Appendix E, an even more fundamental open problem is whether one should keep using the Poincaré definition of asymptotic expansion. This topic deserves a separate paper as well. Anyway, for example, the Poincaré asymptotic expansion of the logarithm of the $\mathrm{\Gamma }$-function is reliable at large values of the independent variable but is expressed by a divergent series [23]. Thus, good asymptotic estimates at large r in our paper would not be affected by divergence of the expansion.

(iii)

The cases ${\omega }^2<{\mu }^2$ and ${\omega }^2>{\mu }^2$ studied in Section 7 are both physically relevant, and we cannot foresee a viable regularization for the time being. Maybe one has to revert to what we have suggested at the end of item (i) in the present list.

(iv)

The familiarity with evaluation of $B_s$ coefficients gained in Section 6 will actually make it easier to complete the analysis of Section 8. More precisely, the $B_s$ coefficients in Equation (71) do not depend on the mass of any field and are simpler in this respect, but on the other hand, they have both real and imaginary parts, as is clear from Equation (77). Their evaluation by computer programs will lead to the evaluation of Fourier modes with the desired accuracy, and in turn, the Green function of the inhomogeneous radial wave equation (61) will be obtained with the same accuracy.

(v)

We cannot foresee any obstruction to studying wave equations in a Kerr background with our technique. The modern packages will help a lot in evaluating $B_s$ coefficients which depend also on angular momentum. Once more, we conclude that a separate paper is necessary to accomplish this task as well.