Physics:Fermat’s and energy variation principles in field theory

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

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

where the conformal factor [math]\displaystyle{ f(t,x) }[/math] depends on time [math]\displaystyle{ t }[/math] and space coordinates [math]\displaystyle{ x^{\alpha} }[/math] 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 [math]\displaystyle{ x_a=(x^1_a,x^2_a,x^3_a) }[/math] and [math]\displaystyle{ x_b=(x^1_b,x^2_b,x^3_b) }[/math] corresponds to stationary action.

[math]\displaystyle{ S=\int^{\mu_a}_{\mu_b}\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 }[/math],

where [math]\displaystyle{ \mu }[/math] is any parameter ranging over an interval [math]\displaystyle{ [\mu_a, \mu_b] }[/math] and varying along curve with fixed endpoints [math]\displaystyle{ x_a=x(\mu_a) }[/math] and [math]\displaystyle{ x_b=x(\mu_b) }[/math].

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 [math]\displaystyle{ \tilde{g}_{ij} }[/math] is defined by a transformation

[math]\displaystyle{ \tilde{g}_{00} =\rho ^{2}{g}_{00} ,\,\,\,\, \tilde{g}_{0k}=\rho{g}_{0k} ,\,\,\,\, \tilde{g}_{kq} ={g}_{kq} . }[/math]

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

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

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

[math]\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} }, }[/math]

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

With [math]\displaystyle{ g_{00}=0 }[/math] and [math]\displaystyle{ g_{0k} \ne 0 }[/math] even if for one k the energy takes form

[math]\displaystyle{ \rho =-\frac{g_{kq} v^{k} v^{q} }{2v_{0} v^{0}}. }[/math]

In both cases for the free moving particle the Lagrangian is

[math]\displaystyle{ L= -\rho. }[/math]

Its partial derivatives give the canonical momenta

[math]\displaystyle{ p_{\lambda}=\frac{\partial L}{\partial v^{\lambda}}=\frac{v_{\lambda}}{v^{0}v_{0}} }[/math]

and the forces

[math]\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}. }[/math]

Momenta satisfy energy condition for closed system

[math]\displaystyle{ \rho=v^{\lambda}p_{\lambda}-L, }[/math]

and thus [math]\displaystyle{ \rho }[/math] is Hamiltonian.

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

[math]\displaystyle{ S=\int^{\mu_a}_{\mu_b}L d\mu=-\int^{\mu_a}_{\mu_b}\rho d\mu, }[/math]

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

[math]\displaystyle{ \frac{d}{d\mu}\frac{\partial \rho }{\partial v^{\lambda}}-\frac{\partial \rho }{\partial x^{\lambda}}=0, }[/math]

which is rewritten in form

[math]\displaystyle{ \frac{d}{d\mu} p_{\lambda}-F_{\lambda}=0. }[/math]

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

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

and

[math]\displaystyle{ (g_{k\lambda} v_{0}-g_{0k}v_{\lambda})\frac{dv^{k}}{d\mu}+\left[\frac{1}{2v_{0}}\frac{\partial g_{ij}}{\partial x^{0}}(g_{00}v^{0}v_{\lambda}+ g_{k\lambda}v^{k}v_{0})-\frac{1}{2}\frac{\partial g_{ij}}{\partial x^{\lambda}}v_{0} +\frac{\partial g_{i\lambda}}{\partial x^{j}}v_0- \frac{\partial g_{0i}}{\partial x^{j}}v_{\lambda}\right]v^i v^j=0. }[/math]

Static spacetime

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

[math]\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}. }[/math]

Substitution of them in Euler–Lagrange equations gives

[math]\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} }[/math].

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

[math]\displaystyle{ \frac{d^2 x^l}{d\overline{\mu}^2}+\Gamma^l_{ij} \frac{dx^i}{d\overline{\mu}}\frac{dx^j}{d\overline{\mu}}=0, }[/math]

where [math]\displaystyle{ \Gamma^l_{ij} }[/math] are the second kind Christoffel symbols with respect to the metric tensor [math]\displaystyle{ \overline{g}_{ij} }[/math].

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.

See also

• Fermat's principle

Further reading

• Belayev, W. B. (2011). Application of Lagrange mechanics for analysis of the light-like particle motion in pseudo-Riemann space. Bibcode: 2009arXiv0911.0614B.