Fermat's and energy variation principles in field theory explained

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.

Fermat's principle

(t,x ^{1,x} ^{2,x} ^{3)} a Fermat metric takes the form$$g\; =\; e^[(dt+\backslash phi\_\{\backslash alpha\}(x)dx^\{\backslash alpha\})^\{2\}-\backslash hat\{g\}\_\{\backslash alpha\backslash beta\}\; dx^\{\backslash alpha\}\; dx^\{\backslash beta\}],$$where the conformal factor f(t,x) depends on time t and space coordinates x ^{\alpha} 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

x ^{3} _{a)} and x _{b} = (x ^{3} _{b)} corresponds to stationary action.$$S=\backslash int^\_\backslash left(\backslash sqrt+\backslash phi\_(x)\backslash frac\; \backslash right)\; d\backslash mu,$$where \mu is any parameter ranging over an interval [\mu _{a,} \mu _{b]} and varying along curve with fixed endpoints x _{a=x(\mu} _{a)} and x _{b=x(\mu} _{b)} .

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

\tilde{g} _{ij} is defined by a transformation$$\backslash tilde\_\; =\backslash rho\; ^\_,\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash tilde\_=\backslash rho\_,\backslash ,\backslash ,\backslash ,\backslash ,\; \backslash tilde\_\; =\_\; .$$

With time coordinate

x ^{0} and space coordinates with indexes k,q=1,2,3 the line element is written in form$$ds^2=\backslash rho^2\; g\_(dx^)^+\; 2\backslash rho\; g\_dx^dx^+g\_dx^dx^,$$where \rho is some quantity, which is assumed equal 1. Solving light-like interval equation ds =0 for \rho under condition g _{00} \ne 0 gives two solutions$$\backslash rho\; =\backslash frac,$$where v ^{i} = dx ^{i/d\mu} are elements of the four-velocity. Even if one solution, in accordance with making definitions, is \rho =1 .

With

g _{00} =0 and g _{0k} \ne 0 even if for one k the energy takes form$$\backslash rho\; =-\backslash frac.$$

In both cases for the free moving particle the Lagrangian is$$L=\; -\backslash rho.$$

Its partial derivatives give the canonical momenta$$p\_=\backslash frac=\backslash frac$$and the forces$$F\_=\backslash frac=\backslash frac\backslash fracv^v^.$$

Momenta satisfy energy condition for closed system$$\backslash rho=v^p\_-L,$$which means that

\rho 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$$S=\backslash int^\_L\; d\backslash mu=-\backslash int^\_\backslash rho\; d\backslash mu,$$which is integral of energy. Stationary action is conditional upon zero variational derivatives and leads to Euler–Lagrange equations$$\backslash frac\backslash frac-\backslash frac=0,$$which is rewritten in form$$\backslash frac\; p\_-F\_=0.$$

After substitution of canonical momentum and forces they yields motion equations of lightlike particle in a free space$$\backslash frac+\backslash frac\; \backslash frac\; v^i\; v^j\; =\; 0$$and$$(g\_\; v\_0\; -\; g\_\; v\_)\; \backslash frac+\backslash left[v\_\{0\}\backslash Gamma\_\{0ij\}-v\_\{\backslash lambda\}\; \backslash Gamma\_\{\backslash lambda\; ij\}\backslash right]\; v^i\; v^j=0,$$where

\Gamma _{kij} are the Christoffel symbols of the first kind and indexes λ take values 1 , 2 , 3 .Energy integral variation and Fermat principles give identical curves for the light in stationary space-times.

Generalized Fermat's principle

In the generalized Fermat’s principle 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. 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

The equations$$\backslash frac\; p\_-F\_=0$$can be transformed into a contravariant form$$\backslash frac+g^\; \backslash frac\; v^\; p^\; =F^,$$where the second term in the left part is the change in the energy and momentum transmitted to the gravitational field$$\backslash frac=g^\; \backslash frac\; v^\; p^$$when the particle moves in it. The force vector ifor principle of stationary integral of energy is written in form$$F^k\; =\; g^\backslash frac\backslash fracv^v^.$$In general relativity, the energy and momentum of a particle is ordinarily associated with a contravariant energy-momentum vector

p ^{k} . The quantities F ^{k} do not form a tensor. However, for the photon in Newtonian limit of Schwarzschild field described by metric in isotropic coordinates they correspond 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 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$$d\backslash acute=v\_0v^0d\backslash mu$$the expression for the momenta turned out to be$$p^=\backslash acute^,$$where 4-velocity is defined as

\acute{v} ^{λ} = dx ^{λ} /d\acute{\mu} . Equations with contravariant momenta$$\backslash frac+g^\; \backslash frac\; v^\; p^\; =\; g^\backslash frac\backslash fracv^v^$$ are rewritten as follows$$\backslash frac+g^\; \backslash frac\; \backslash acute^\; p^\; =\; g^\backslash frac\backslash frac\backslash acute^\backslash acute^.$$These equations are identical in form to the ones obtained from the Euler-Lagrange equations with Lagrangian L = (1)/(2) g _{ij} (dx _{i})/(ds) (dx _{j})/(ds) by raising the indices. In turn, these equations are identical to the geodesic equations, which confirms that the solutions given by the principle of stationary integral of energy are geodesic. The quantities$$\backslash frac=g^\; \backslash frac\; \backslash acute^\; p^$$and$$\backslash acute^k\; =\; g^\backslash frac\backslash frac\backslash acute^\backslash acute^$$appear as tensors for linearized metrics.

See also

• Fermat's principle
• Variational methods in general relativity