Generalized Planck–Einstein Relation in Curved Spacetimes: Implications for Light Propagation Near Black Holes

By applying Maxwell’s equations to curved spacetimes, the Planck–Einstein energy–frequency relation for photons, originally formulated in Minkowski space, is generalized for application in Riemann space. According to this relation, photon energy depends not only on the photon frequency but also on the physical speed of photons, which may vary when locally measured in non-inertial static frames. In Minkowski space, the energy of free photons is conserved as neither frequency shifts nor changes in photon speed are observed. In Riemann space, energy of free photons also remains conserved as gravitational redshift is compensated by a corresponding variation in photon speed. The generalized Planck–Einstein relation may have significant astrophysical implications, particularly for gravitational lensing, observations of neutron star mergers, supernovae and quasars, the propagation of light near black holes, and expanding cosmologies.