The connection between entropy and the absorption spectra of Schwarzschild black holes for light and massless scalar fields

TL;DR

This paper explores the relationship between entropy and the absorption spectra of Schwarzschild black holes, showing that large wavelength waves are reflected, consistent with thermodynamic principles and numerical simulations.

Contribution

The study combines heuristic thermodynamic arguments with numerical simulations to analyze wave reflection and absorption by black holes at wavelengths comparable to the Schwarzschild radius.

Findings

Large wavelength waves are substantially reflected by black holes.

The critical wavelength for reflection matches entropy-based predictions.

Propagation speed differs from c near the Schwarzschild radius for large wavelengths.

Abstract

We present heuristic arguments suggesting that if EM waves with wavelengths somewhat larger than the Schwarzschild radius of a black hole were fully absorbed by it, the second law of thermodynamics would be violated, under the Bekenstein interpretation of the area of a black hole as a measure of its entropy. Thus, entropy considerations make the well known fact that large wavelengths are only marginally absorbed by black holes, a natural consequence of thermodynamics. We also study numerically the ingoing radial propagation of a scalar field wave in a Schwarzschild metric, relaxing the standard assumption which leads to the eikonal equation, that the wave has zero spatial extent. We find that if these waves have wavelengths larger that the Schwarzschild radius, they are very substantially reflected, fully to numerical accuracy. Interestingly, this critical wavelength approximately coincides with the one derived from entropy considerations of the EM field, and is consistent with well known limit results of scattering in the Schwarzschild metric. The propagation speed is also calculated and seen to differ from the value $c$, for wavelengths larger than $R_{s}$, in the vicinity of $R_{s}$. As in all classical wave phenomena, whenever the wavelength is larger or comparable to the physical size of elements in the system, in this case changes in the metric, the zero extent 'particle' description fails, and the wave nature becomes apparent.