A Comprehensive Examination of Shadow Derivations in Spherically Symmetric Black Holes: From Synge's Historical Insights to Modern Methodologies

الملخص

The present work constitutes an erudite discussion of a precise derivation of shadows for spherically symmetric black holes. A concise overview of the historical Synge solution is also furnished, accompanied by a thorough exposition of the contemporary methodology that can be employed with more intricate metrics. The methodology will subsequently be applied to a selection of the most significant black hole solutions. The present work provides a thoroughgoing analysis of black hole shadows for spherically symmetric solutions, with a particular focus on the Schwarzschild and Reissner-Nordström (anti-)de Sitter metrics. The study commences with a historical overview of Synge's pioneering work on the escape of photons from gravitationally intense stars, which laid the foundation for understanding black hole shadows. This employs the technique of Lagrange formalism in order to derive the constants of motion and the trajectory equation for photons within spherically symmetric geometries. This approach facilitates the efficient computation of shadows by determining the turning points of photon orbits and the angular radius of the shadow. The methodology is applied to the Schwarzschild, Reissner-Nordström, and Reissner-Nordström-Kottler solutions, revealing the dependence of the shadow on the black hole's charge and the cosmological constant. The subsequent discourse herein entails an exhaustive examination of its relationship to the frequently employed concepts of escape cone and critical impact parameter. Moreover, the impact of gravitational collapse as well as plasma on the shadow of a black hole is a concomitant consideration in this theoretical framework.