Fermat’s and energy variation principles in field theory

In case of conformally stationary spacetime [2] with coordinates ${\displaystyle (t,x^{1},x^{2},x^{3})}$ a Fermat metric takes the form

${\displaystyle g=e^{2f(t,x)}[(dt+\phi _{\alpha }(x)dx^{\alpha })^{2}-{\hat {g}}_{\alpha \beta }dx^{\alpha }dx^{\beta }]}$,

where the conformal factor ${\displaystyle f(t,x)}$ depends on time ${\displaystyle t}$ and space coordinates ${\displaystyle x^{\alpha }}$ 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 ${\displaystyle x_{a}=(x_{a}^{1},x_{a}^{2},x_{a}^{3})}$ and ${\displaystyle x_{b}=(x_{b}^{1},x_{b}^{2},x_{b}^{3})}$ corresponds to stationary action.

${\displaystyle S=\int _{\mu _{b}}^{\mu _{a}}\left({\sqrt {{\hat {g}}_{\alpha \beta }{\frac {dx^{\alpha }}{d\mu }}{\frac {dx^{\beta }}{d\mu }}}}+\phi _{\alpha }(x){\frac {dx^{\alpha }}{d\mu }}\right)d\mu }$,

where ${\displaystyle \mu }$ is any parameter ranging over an interval ${\displaystyle [\mu _{a},\mu _{b}]}$ and varying along curve with fixed endpoints ${\displaystyle x_{a}=x(\mu _{a})}$ and ${\displaystyle x_{b}=x(\mu _{b})}$.

In principle of stationary integral of energy for a light-like particle's motion [3], the pseudo-Riemannian metric with coefficients ${\displaystyle {\tilde {g}}_{ij}}$ is defined by a transformation

${\displaystyle {\tilde {g}}_{00}=\rho ^{2}{g}_{00},\,\,\,\,{\tilde {g}}_{0k}=\rho {g}_{0k},\,\,\,\,{\tilde {g}}_{kq}={g}_{kq}.}$

With time coordinate ${\displaystyle x^{0}}$ and space coordinates with indexes k,q=1,2,3 the line element is written in form

${\displaystyle ds^{2}=\rho ^{2}g_{00}(dx^{0})^{2}+2\rho g_{0k}dx^{0}dx^{k}+g_{kq}dx^{k}dx^{q},}$

where ${\displaystyle \rho }$ is some quantity, which is assumed equal 1 and regarded as the energy of the light-like particle with ${\displaystyle ds=0}$. Solving this equation for ${\displaystyle \rho }$ under condition ${\displaystyle g_{00}\neq 0}$ gives two solutions

${\displaystyle \rho ={\frac {-g_{0k}v^{k}\pm {\sqrt {(g_{0k}g_{0q}-g_{00}g_{kq})v^{k}v^{q}}}}{g_{00}v^{0}}},}$

where ${\displaystyle v^{i}=dx^{i}/d\mu }$ are elements of the four-velocity. Even if one solution, in accordance with making definitions, is ${\displaystyle \rho =1}$.

With ${\displaystyle g_{00}=0}$ and ${\displaystyle g_{0k}\neq 0}$ even if for one k the energy takes form

${\displaystyle \rho =-{\frac {g_{kq}v^{k}v^{q}}{2v_{0}v^{0}}}.}$

In both cases for the free moving particle the Lagrangian is

${\displaystyle L=-\rho .}$

Its partial derivatives give the canonical momenta

${\displaystyle p_{\lambda }={\frac {\partial L}{\partial v^{\lambda }}}={\frac {v_{\lambda }}{v^{0}v_{0}}}}$

and the forces

${\displaystyle F_{\lambda }={\frac {\partial L}{\partial x^{\lambda }}}={\frac {1}{2v^{0}v_{0}}}{\frac {\partial g_{ij}}{\partial x^{\lambda }}}v^{i}v^{j}.}$

Momenta satisfy energy condition [4] for closed system

${\displaystyle \rho =v^{\lambda }p_{\lambda }-L.}$

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

${\displaystyle S=\int _{\mu _{b}}^{\mu _{a}}Ld\mu =-\int _{\mu _{b}}^{\mu _{a}}\rho d\mu ,}$

which is integral of energy. Stationary action is conditional upon zero variational derivatives δS/δx^λ and leads to Euler–Lagrange equations

${\displaystyle {\frac {d}{d\mu }}{\frac {\partial \rho }{\partial v^{\lambda }}}-{\frac {\partial \rho }{\partial x^{\lambda }}}=0,}$

which is rewritten in form

${\displaystyle {\frac {d}{d\mu }}p_{\lambda }-F_{\lambda }=0.}$

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

${\displaystyle {\frac {dv^{0}}{d\mu }}+{\frac {v^{0}}{2v_{0}}}{\frac {\partial g_{ij}}{\partial x^{0}}}v^{i}v^{j}=0}$

and

${\displaystyle (g_{k\lambda }v_{0}-g_{0k}v_{\lambda }){\frac {dv^{k}}{d\mu }}+\left[v_{0}\Gamma _{0ij}-v_{\lambda }\Gamma _{\lambda ij}\right]v^{i}v^{j}=0,}$

where ${\displaystyle \Gamma _{kij}}$ are the Christoffel symbols of the first kind and indexes ${\displaystyle \lambda }$ take values ${\displaystyle 1,2,3}$.

For the isotropic paths a transformation to metric ${\displaystyle {\overline {g}}_{ij}=g_{ij}/{g_{00}}}$ is equivalent to replacement of parameter ${\displaystyle \mu }$ on ${\displaystyle d{\overline {\mu }}=d\mu /{\sqrt {g_{00}}}}$ to which the four-velocities ${\displaystyle {\overline {v}}^{i}=dx^{i}/d{\overline {\mu }}}$ correspond. The curve of motion of lightlike particle in four-dimensional spacetime and value of energy ${\displaystyle \rho }$ are invariant under this reparametrization. For the static spacetime the first equation of motion with appropriate parameter ${\displaystyle {\overline {\mu }}}$ gives ${\displaystyle {\overline {v}}^{0}=1}$ . Canonical momentum and forces take form

${\displaystyle {\overline {p}}_{\lambda }={\overline {v}}_{\lambda };\qquad {\overline {F}}_{\lambda }={\frac {1}{2}}{\frac {\partial {\overline {g}}_{ij}}{\partial x^{\lambda }}}{\overline {v}}^{i}{\overline {v}}^{j}.}$

Substitution of them in Euler–Lagrange equations gives

${\displaystyle {\frac {d}{d\mu }}\left({\overline {g}}_{\lambda k}{\overline {v}}^{k}\right)={\frac {1}{2}}{\frac {\partial {\overline {g}}_{ij}}{\partial x^{\lambda }}}{\overline {v}}^{i}{\overline {v}}^{j}}$.

After differentiation on the left side and multiplying by ${\displaystyle {\overline {g}}^{l\lambda }}$ this expression, after the summation over the repeated index ${\displaystyle \lambda }$, becomes null geodesic equations

${\displaystyle {\frac {d^{2}x^{l}}{d{\overline {\mu }}^{2}}}+\Gamma _{ij}^{l}{\frac {dx^{i}}{d{\overline {\mu }}}}{\frac {dx^{j}}{d{\overline {\mu }}}}=0,}$

where ${\displaystyle \Gamma _{ij}^{l}}$ are the second kind Christoffel symbols with respect to the metric tensor ${\displaystyle {\overline {g}}_{ij}}$.

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.

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.

• Fermat's principle

• 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)