1. Fermat's principle

2. Principle of stationary integral of energy

3. Static spacetime

4. See also

5. References

6. Further reading

In general relativity the light is assumed to propagate in the vacuum along null geodesic in a pseudo-Riemannian manifold. Besides the geodesics principle in a classical field theory there exists the Fermat's principle for stationary gravity fields.^[1]

Fermat's principle

In more general case for conformally stationary spacetime ^[2] with coordinates a Fermat metric takes form

,

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 .

With and even if for one k the energy takes form

In both cases for the free moving particle the Lagrangian is

Its partial derivatives give the canonical momenta

and the forces

Momenta satisfy energy condition ^[3]

for closed system

and thus is Hamiltonian.

Standard variational procedure according to Hamilton's principle is applied to action

which is integral of energy. Stationary action is conditional upon zero variational derivatives {{math|δS/δx^λ }}

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

where are the second kind Christoffel symbols with respect to the metric tensor .

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.

See also

• Fermat's principle

References

Further reading

• {{Cite journal

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