Details

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.