Method of comparison equations for Schwarzschild black holes

We employ the method of comparison equations to study the propagation of a massless minimally coupled scalar field on the Schwarzschild background. In particular, we show that this method allows us to obtain explicit approximate expressions for the radial modes with energy below the peak of the effective potential which are fairly accurate over the whole region outside the horizon. This case can be of particular interest, for example, for the problem of black hole evaporation.

Figure 1

Comparison between the exact potential $-{\omega }^2(x)$ (solid line) and the approximate Morse potential $-{\Theta }^2[\sigma (x)]$ obtained after solving Eqs. (19a, 19b) (dotted line) for $\tilde{\omega }=1/2$ and $\ell =2$. The two turning points are $x_1\approx -3.13$ and $x_2\approx 1.23$.

Figure 2

Upper left graph: The function $\sigma =\sigma (x)$ obtained by integrating Eq. (16) numerically (solid line) and its approximate expression (24) with ${\kappa }_1=1$ and ${\kappa }_2=3$ (dashed line). Upper right graph: Comparison between the exact solution (solid line) and the approximate MCE solution (17) with $\sigma (x)$ evaluated numerically (dashed line) for $C_{+}\approx 0.517-0.228i$ and $C_{-}\approx 0.517+0.228i$. Panel (a) shows the percentage error for the same approximate function $\sigma$ given in the upper left graph. Panel (b) shows the percentage error for the approximate $\chi$ in the upper right graph for $x<x_1$ and panel (c) for $x_2<x$. All plots are for $\tilde{\omega }=1/2$, $\ell =2$ as in Fig. 1.

Figure 3

Comparison between the exact solution (solid line) and the approximate MCE solution (17) with $\sigma (x)$ given in Eq. (24) with ${\kappa }_1=1$ and ${\kappa }_2=3$ (dashed line) for $\tilde{\omega }=1/2$ and $\ell =2$ (same as in Figs. 1, 2). Panel (a) shows the result for $x<{\kappa }_1{x}_1$, panel (b) for ${\kappa }_1{x}_1<x<{\kappa }_2{x}_2$ and panel (c) for $x>{\kappa }_2{x}_2$.