Fermat's and energy variation principles in field theory

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]

Fermat's principle

In case of conformally stationary spacetime [2] with coordinates a Fermat metric takes the form

,

where the conformal factor depends on time and space coordinates 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 and corresponds to stationary 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,[3] 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 [4] for closed system

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 δS/δxλ and leads to Euler–Lagrange equations

which is rewritten in form

After substitution of canonical momentum and forces they give [5] motion equations of lightlike particle in a free space

and

where are the Christoffel symbols of the first kind and indexes take values .

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 with the geodesic principle and the energy variational method as well as Fermat's principle give the same solution for the light propagation.

Generalized Fermat's principle

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 identity of the generalized Fermat principles and the stationary energy integral of a light-like particle for velocities is proved.[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 given by the first 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.

See also

References

  1. Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, p. 273, ISBN 9780750627689
  2. Perlik, Volker (2004), "Gravitational Lensing from a Spacetime Perspective", Living Rev. Relativ., 7 (9), Chapter 4.2
  3. D. Yu., Tsipenyuk; W. B., Belayev (2019), "Extended Space Model is Consistent with the Photon Dynamics in the Gravitational Field", J. Phys.: Conf. Ser., 1251 (012048)
  4. Landau, Lev D.; Lifshitz, Evgeny F. (1976), Mechanics Vol. 1 (3rd ed.), London: Butterworth-Heinemann, p. 14, ISBN 9780750628969
  5. D. Yu., Tsipenyuk; W. B., Belayev (2019), "Photon Dynamics in the Gravitational Field in 4D and its 5D Extension" (PDF), Rom. Rep. in Phys., 71 (4)
  6. V. P., Frolov (2013), "Generalized Fermat's Principle and Action for Light Rays in a Curved Spacetime", Phys. Rev. D, 88 (6), arXiv:1307.3291, doi:10.1103/PhysRevD.88.064039

Further reading

  • 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 requires |journal= (help)CS1 maint: ref=harv (link)
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