Null integrability and photon phenomenology in the accelerated Schwarzschild black hole

Abstract

Uniformly accelerated black holes provide a controlled setting to isolate how translatory acceleration and conical defects modify photon dynamics and observable optics. We analyse null geodesics in the subextremal static C-metric, describing an accelerated Schwarzschild black hole pulled by a cosmic string/strut, and we keep the conicity parameter explicit throughout. By using dimensionless variables and a Mino-type parametrisation, the conformal Hamilton-Jacobi equation separates and both the radial and polar sectors reduce to Biermann-Weierstrass form, yielding compact closed solutions in terms of Weierstrass elliptic functions. This framework enables a systematic classification of photon trajectories, exhibits the loss of equatorial symmetry for nonzero acceleration via a fixed photon cone of constant-latitude null motion, and identifies a unique spherical photon surface shared by all latitudes. On the observational side, using the tetrad screen map, the circular shadow boundary follows transparently from the photon surface condition. We highlight the explicit cancellation of the conicity parameter and discuss that the angular radius depends on the acceleration and the observer's position; however, it is independent of the observer's inclination and of the conicity (because of the string tension). For nonstatic observers, a local Lorentz boost preserves circularity while changing the apparent angular size through aberration. We describe an algebraic inversion that infers the acceleration from a single shadow radius measurement once the mass-distance scale is fixed. Moreover, we derive closed expressions for the photon orbital frequency and Lyapunov exponent, providing Eikonal quasinormal mode estimates.

20 pages

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