Greybody factors at large imaginary frequencies

TL;DR

The paper studies greybody factors for massless fields scattering off Schwarzschild and Reissner-Nordström black holes in the limit of large imaginary frequency, extending previous work on quasinormal modes. Using a monodromy analysis of the 1D wave equation with a black-hole potential, Neitzke derives explicit asymptotic expressions for the transmission and reflection coefficients: for Schwarzschild in $d\ge4$, $T(\omega)\approx \frac{e^{\beta \omega}-1}{e^{\beta \omega}+3}$ and $R(\omega)\approx \frac{2i}{e^{\beta \omega}+3}$, while for RN in $d=4$, $T(\omega)\approx \frac{e^{\beta \omega}-1}{e^{\beta \omega}+2+3 e^{-\beta_I \omega}}$ and $R(\omega)\approx i\sqrt{3}\,\frac{1+e^{-eta_I \omega}}{e^{\beta \omega}+2+3 e^{-\beta_I \omega}}$, with further relations for $T(-\omega)$ and $R(-\omega)$. Interpreting these results in the Hawking radiation framework yields greybody-modified spectra that continue to bear the imprint of horizon structure, and suggests a conjectured boundary description with exotic statistics, potentially involving both inner and outer horizons for RN. The work is complemented by numerical comparisons that corroborate the asymptotic formulas and illuminate the behavior of quasinormal frequencies in the RN case, including rational vs irrational horizon-parameter regimes. Overall, the results provide a precise link between high-imaginary-frequency scattering and black-hole microphysics, and point to fruitful directions for extending the analysis to rotating or higher-dimensional spacetimes.

Abstract

Extending a computation which appeared recently in hep-th/0301173, we compute the transmission and reflection coefficients for massless uncharged scalars and gravitational waves scattered by d>=4 Schwarzschild or d=4 Reissner-Nordstrom black holes, in the limit of large imaginary frequencies. The transmission coefficient has an interpretation as the "greybody factor" which determines the spectrum of Hawking radiation. The result has an interesting structure and we speculate that it may admit a simple dual description; curiously, for Reissner-Nordstrom the result suggests that this dual description should involve both the inner and outer horizons. We also discuss some numerical evidence in favor of the formulas of hep-th/0301173.

Figures (3)

• Figure 1: The complex $r$-plane. The $\textrm{Re\ }(x) < 0$ region is colored gray, and the points $r=0$, $r = r_H$ and a contour $\gamma$ for calculation of the transmission and reflection coefficients are marked. $\gamma$ runs along the line $\textrm{Re\ }(x) = 0$ to $\lvert\omega x\rvert \gg 1$ in each direction. The symbol $\bigstar$ represents the direction in which the boundary conditions "at infinity" are imposed.
• Figure 2: Inspiraling behavior of the asymptotic formula \ref{['asymptotic-qnf-reissner-nordstrom-4d-rewritten']} for the $n=5$ quasinormal frequency as the charge is increased toward extremality. The axes are $\textrm{Re\ } GM \omega$ and $\textrm{Im\ } GM \omega$. The center of the spiral is at $GM\omega = (11/8)i + \log 3 / 8\pi$.
• Figure 3: Real parts (left) and imaginary parts (right) of quasinormal frequencies for the Reissner-Nordstrøm black hole, at mode number (top to bottom) $n=30,5000,10000$. The red line is the numerical data of Berti:2003zu and the black line is the asymptotic formula \ref{['asymptotic-qnf-reissner-nordstrom-4d-rewritten']}. The vertical axis is (Re or Im) $GM \omega$; the horizontal is $Q$, in units where $Q=1$ is extremal.