where conformal factor depending on time and space coordinates 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 and corresponds to zero variation of action
,
where is any parameter ranging over an interval and varying along curve with fixed endpoints and .
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 is defined by a transformation
With time coordinate and space coordinates with indexes k,q=1,2,3 the line element is written in form
where is some quantity, which is assumed equal 1 and regarded as the energy of the light-like particle with . Solving this equation for under condition gives two solutions
where are elements of the four-velocity. Even if one solution, in accordance with making definitions, is .
which is integral of energy. Stationary action is conditional upon zero variational derivatives δS/δxλ
and leads to Euler–Lagrange equations
which is rewritten in form
After substitution of canonical momentum and forces they give motion equations of lightlike particle in a free space
and
Static spacetime
For the isotropic paths a transformation to metric is equivalent to replacement of parameter on to which the four-velocities correspond. The curve of motion of lightlike particle in four-dimensional spacetime and value of energy are invariant under this reparametrization.
For the static spacetime the first equation of motion with appropriate parameter gives . Canonical momentum and forces take form
Substitution of them in Euler–Lagrange equations gives
.
After differentiation on the left side and multiplying by this expression, after the summation over the repeated index , becomes null geodesic equations
So in case of the static spacetime the geodesic principle and the energy variational method as well as Fermat's principle give the same solution for the light propagation.
Belayev, W. B. (2011). "Application of Lagrange mechanics for analysis of the light-like particle motion in pseudo-Riemann space". arXiv:0911.0614. Bibcode:2009arXiv0911.0614B. {{cite journal}}: Cite journal requires |journal= (help); Invalid |ref=harv (help)CS1 maint: postscript (link)