Effective and Inertial Masses of a Photon Near a Black Hole for a Family of Envelope Orbits
Photons move along geodesic paths, which are bent in a gravitational field. This allows one to assume that the photon has a finite inertial mass. This Demonstration shows how the effective and inertial masses change as functions of the distance to the center of gravity and the motion constant for the photon's orbits near a black hole with minimum radius . The unit of distance is chosen as the Schwarzschild radius, such that .
The normalized effective mass of a photon can be defined by the proportionality between the canonical momentum and the four-velocity , such that . In a central gravitational field, the normalized inertial mass is found from , where is the radial component of the canonical force vector. The relation between and is obtained from the condition , leading to the constant of motion .
The decrease of the coordinate velocity in most of its trajectory in the gravitational field is known as the Shapiro time-delay effect. It can be pictured as a negative inertial mass for the photon. By contrast, a positive mass particle accelerates as it approaches the center of gravity.
For further technical details, please refer to W. Belayev, "Application of Lagrange Mechanics for Analysis of the Light-Like Particle Motion in Pseudo-Riemann Space," 2009, http://arxiv.org/abs/0911.0614.