Analytic derivation of photon ring geometry, redshift, and travel time for a Schwarzschild black hole in an inhomogeneous pressureless plasma, with comparisons to vacuum and homogeneous plasma cases.
As first step we take (40) and write down the conditional equation for the roots
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Gravitational Lensing in the Schwarzschild Spacetime: Photon Rings in Vacuum and in the Presence of a Plasma
Analytic derivation of photon ring geometry, redshift, and travel time for a Schwarzschild black hole in an inhomogeneous pressureless plasma, with comparisons to vacuum and homogeneous plasma cases.