Strong-Field Amplitude Modulation of Massless Klein--Gordon Waves near a Schwarzschild Horizon
Abstract
Abstract We study massless Klein--Gordon waves outside a Schwarzschild black hole. The aim is to understand how a wave mode can depart from the null-geodesic, or ray-optics, description when its wavelength is not negligible compared with the gravitational length scale. The work is motivated by gravitational length stretching (GLS), in which the amplitude factor of a curved-spacetime mode is interpreted as a geometry-induced modulation of the spatial probability profile. We use Cartesian Schwarzschild coordinates and select a beam-like solution sector. The stationary mode is written with a complex wave number, $k(z)=a(z)+ib(z)$. The real part $a(z)$ controls the phase, while $b(z)$ controls the amplitude factor $B(z)=\exp[-\int b(z')dz']$. We solve the resulting amplitude--phase equations numerically in the exterior region. Far from the horizon, and in the short-wavelength regime, the solution agrees with the standard null-geodesic result. For low-frequency modes, the phase gradient differs strongly from the eikonal wave number, and the amplitude develops clear spatial modulation. The group velocity is computed from a small family of nearby frequency solutions, $a(z;\omega)$, through $v_g=(\partial a/\partial\omega)^{-1}$. The near-horizon growth of this Schwarzschild-coordinate group velocity is interpreted as a sign that the eikonal approximation has broken down, not as a violation of local causality. These results indicate that the GLS amplitude mechanism has a…
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