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 edit
In case of conformally stationary spacetime[2] with coordinates a Fermat metric takes the form
Fermat's principle for a pseudo-Riemannian manifold states that the light ray path between points and corresponds to stationary action.
Principle of stationary integral of energy edit
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
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
Momenta satisfy energy condition [4] for closed system
Standard variational procedure according to Hamilton's principle is applied to action
After substitution of canonical momentum and forces they yields [5] motion equations of lightlike particle in a free space
Generalized Fermat's principle edit
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 edit
The equations
After replacing the affine parameter
See also edit
References edit
- ^ Landau, Lev D.; Lifshitz, Evgeny F. (1980), The Classical Theory of Fields (4th ed.), London: Butterworth-Heinemann, p. 273, ISBN 9780750627689
- ^ Perlik, Volker (2004), "Gravitational Lensing from a Spacetime Perspective", Living Rev. Relativ., 7 (9), Chapter 4.2
- ^ 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
- ^ Landau, Lev D.; Lifshitz, Evgeny F. (1976), Mechanics Vol. 1 (3rd ed.), London: Butterworth-Heinemann, p. 14, ISBN 9780750628969
- ^ 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)
- ^ 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
- ^ 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
- ^ 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
- ^ Tolman, R. C. (1987), Relativity, Thermodynamics and Cosmology, New York: Dover, pp. 274–285, ISBN 9780486653839
- ^ Belayev, V. B. (2017), The Dynamics in General Relativity Theory: Variational Methods, Moscow: URSS, pp. 89–91, ISBN 9785971043775
- ^ Misner, Charles W.; Thorne, Kip. S.; Wheeler, John A. (1973), Gravitation, W. H. Freeman, pp. 315–323, ISBN 9780716703440