In general relativity, light is assumed to propagate in a vacuum along a null geodesic in a pseudo-Riemannian manifold.
Besides the geodesics principle in a classical field theory there exists Fermat's principle for stationary gravity fields.
[1] In case of conformally stationary spacetime[2] with coordinates
a Fermat metric takes the form
Fermat's principle for a pseudo-Riemannian manifold states that the light ray path between points
and varying along curve with fixed endpoints
In principle of stationary integral of energy for a light-like particle's motion,[3] the pseudo-Riemannian metric with coefficients
and space coordinates with indexes k,q=1,2,3 the line element is written in form
Solving light-like interval equation
In both cases for the free moving particle the Lagrangian is
Its partial derivatives give the canonical momenta
Momenta satisfy energy condition [4] for closed system
is the energy of the system that combines the light-like particle and the gravitational field.
Standard variational procedure according to Hamilton's principle is applied to action
Stationary action is conditional upon zero variational derivatives δS/δxλ and leads to Euler–Lagrange equations
After substitution of canonical momentum and forces they yields [5] motion equations of lightlike particle in a free space
Energy integral variation and Fermat principles give identical curves for the light in stationary space-times.
[5] In the generalized Fermat’s principle [6] the time is used as a functional and together as a variable.
It is applied Pontryagin’s minimum principle of the optimal control theory and obtained an effective Hamiltonian for the light-like particle motion in a curved spacetime.
It is shown that obtained curves are null geodesics.
The stationary energy integral for a light-like particle in gravity field and the generalized Fermat principles give identity velocities.
[5] The virtual displacements of coordinates retain path of the light-like particle to be null in the pseudo-Riemann space-time, i.e. not lead to the Lorentz-invariance violation in locality and corresponds to the variational principles of mechanics.
The equivalence of the solutions produced by the generalized Fermat principle to the geodesics, means that the using the second also turns out geodesics.
It makes possible to uniquely determine canonical momenta of the particle and forces acting on it in a given reference frame.
where the second term in the left part is the change in the energy and momentum transmitted to the gravitational field
The force vector ifor principle of stationary integral of energy is written in form
In general relativity, the energy and momentum of a particle is ordinarily associated [7] with a contravariant energy-momentum vector
However, for the photon in Newtonian limit of Schwarzschild field described by metric in isotropic coordinates they correspond[3][5] to its passive gravitational mass equal to twice rest mass of the massive particle of equivalent energy.
This is consistent with Tolman, Ehrenfest and Podolsky result [8][9] for the active gravitational mass of the photon in case of interaction between directed flow of radiation and a massive particle that was obtained by solving the Einstein-Maxwell equations.
[10] In turn, these equations are identical to the geodesic equations,[11] which confirms that the solutions given by the principle of stationary integral of energy are geodesic.