Shadow response of a Schwarzschild black hole in a gravitational standing wave

Abstract

Motivated by Mandel (2005 Phys. Rev. D 72 084025) and Szybka et al (2019 Phys. Rev. D 100 064025), we develop a perturbative theory of the shadow response of a Schwarzschild black hole to an auxiliary weak standing-wave perturbation. The shadow is defined geometrically as the capture–escape separatrix of the null Hamiltonian flow on an observer’s screen. We use the standing field as a time-symmetric boundary-value approximation rather than as a generic equilibrium state of an isolated black hole. The physical radiative field is obtained by adding the radiation-reaction sector, which restores outgoing behavior at future null infinity and downgoing behavior at the future horizon. The standing-wave sector is characterized covariantly by an effective high-frequency stress tensor with equal counter-propagating null fluxes. We derive a first-order transfer functional from the metric perturbation to the displacement of the critical curve. In a fixed observer-screen and areal-radius representative, the leading response is set by the contraction of the perturbation with the tangent to the unstable photon-sphere congruence at r=3M. A Regge–Wheeler–Zerilli decomposition yields mode-resolved transfer coefficients and parity selection rules. Reflection-symmetric polar modes stretch or squeeze the shadow, whereas reflection-antisymmetric modes generate leading centroid shifts. The linear transfer functional also maps the standing and radiation-reaction sectors to the physical critical curve. Damped quasinormal forcing gives a coherent decaying modulation, while persistent standing forcing can split the photon-sphere separatrix and generate chaotic capture–escape structure.