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

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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

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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. Solving light-like interval 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   which means that   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   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 yields [5] motion equations of lightlike particle in a free space   and   where   are the Christoffel symbols of the first kind and indexes   take values  . Energy integral variation and Fermat principles give identical curves for the light in stationary space-times.[5]

Generalized Fermat's principle

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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. 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

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The equations   can be transformed [3][5] into a contravariant form   where the second term in the left part is the change in the energy and momentum transmitted to the gravitational field   when the particle moves in it. 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  . The quantities  do not form a tensor. 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.

After replacing the affine parameter   the expression for the momenta turned out to be   where 4-velocity is defined as  . Equations with contravariant momenta   are rewritten as follows   These equations are identical in form to the ones obtained from the Euler-Lagrange equations with Lagrangian   by raising the indices.[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. The quantities   and   appear as tensors for linearized metrics.

See also

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References

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  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. ^ a b c 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 (12048): 012048, Bibcode:2019JPhCS1251a2048T, doi:10.1088/1742-6596/1251/1/012048
  4. ^ Landau, Lev D.; Lifshitz, Evgeny F. (1976), Mechanics Vol. 1 (3rd ed.), London: Butterworth-Heinemann, p. 14, ISBN 9780750628969
  5. ^ a b c d e 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): 064039, arXiv:1307.3291, Bibcode:2013PhRvD..88f4039F, doi:10.1103/PhysRevD.88.064039, S2CID 118688144
  7. ^ V. I., Ritus (2015), "Lagrange equations of motion of particles and photons in the Schwarzschild field", Phys. Usp., 58: 1118, doi:10.3367/UFNe.0185.201511h.1229
  8. ^ R. C., Tolman; P., Ehrenfest; B., Podolsky (1931), "On the Gravitational Field Produced by Light", Phys. Rev., 37 (5): 602, Bibcode:1931PhRv...37..602T, doi:10.1103/PhysRev.37.602
  9. ^ Tolman, R. C. (1987), Relativity, Thermodynamics and Cosmology, New York: Dover, pp. 274–285, ISBN 9780486653839
  10. ^ Belayev, V. B. (2017), The Dynamics in General Relativity Theory: Variational Methods, Moscow: URSS, pp. 89–91, ISBN 9785971043775
  11. ^ Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, pp. 315–323, ISBN 9780716703440