Light motion in curved spacetime
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.
Fermat's principle.
In case of conformally stationary spacetime with coordinates formula_1 a Fermat metric takes the form
formula_2
where the conformal factor formula_3 depends on time formula_4 and space coordinates formula_5 and does not affect the lightlike geodesics apart from their parametrization.
Fermat's principle for a pseudo-Riemannian manifold states that the light ray path between points formula_6 and formula_7 corresponds to stationary action.
formula_8
where formula_9 is any parameter ranging over an interval formula_10 and varying along curve with fixed endpoints formula_11 and formula_12.
Principle of stationary integral of energy.
In principle of stationary integral of energy for a light-like particle's motion, the pseudo-Riemannian metric with coefficients formula_13 is defined by a transformation
formula_14
With time coordinate formula_15 and space coordinates with indexes "k","q"=1,2,3 the line element is written in form
formula_16
where formula_17 is some quantity, which is assumed equal 1. Solving light-like interval equation formula_18 for formula_17 under condition formula_20 gives two solutions
formula_21
where formula_22 are elements of the four-velocity. Even if one solution, in accordance with making definitions, is formula_23.
With formula_24 and formula_25 even if for one "k" the energy takes form
formula_26
In both cases for the free moving particle the Lagrangian is
formula_27
Its partial derivatives give the canonical momenta
formula_28
and the forces
formula_29
Momenta satisfy energy condition for closed system
formula_30
which means that formula_17 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
formula_32
which is integral of energy. Stationary action is conditional upon zero variational derivatives "δS"/"δx"λ and leads to Euler–Lagrange equations
formula_33
which is rewritten in form
formula_34
After substitution of canonical momentum and forces they yields motion equations of lightlike particle in a free space
formula_35
and
formula_36
where formula_37 are the Christoffel symbols of the first kind and indexes formula_38 take values formula_39.
Energy integral variation and Fermat principles give identical curves for the light in stationary space-times.
Generalized Fermat's principle.
In the generalized Fermat’s principle 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. 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. The stationary energy integral principle gives a system of equations that has one equation more. It makes possible to uniquely determine canonical momenta of the particle and forces acting on it in a given reference frame.
Euler–Lagrange equations in contravariant form.
The equations
formula_40
can be transformed into a contravariant form
formula_41
where the second term in the left part is the change in the energy and momentum transmitted to the gravitational field
formula_42
when the particle moves in it. The force vector ifor principle of stationary integral of energy is written in form
formula_43
In general relativity, the energy and momentum of a particle is ordinarily associated with a contravariant energy-momentum vector formula_44. The quantities formula_45do not form a tensor. However, for the photon in Newtonian limit of Schwarzschild field described by metric in isotropic coordinates they correspond 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 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.
After replacing the affine parameter
formula_46
the expression for the momenta turned out to be
formula_47
where 4-velocity is defined as formula_48. Equations with contravariant momenta
formula_49 
are rewritten as follows
formula_50
These equations are identical in form to the ones obtained from the Euler-Lagrange equations with Lagrangian formula_51 by raising the indices. In turn, these equations are identical to the geodesic equations, which confirms that the solutions given by the principle of stationary integral of energy are geodesic. The quantities
formula_52
and
formula_53
appear as tensors for linearized metrics.
References.
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