Fermat's and energy variation principles in field theory

In case of conformally stationary spacetime{{Citation| last= Perlik | first = Volker |title=Gravitational Lensing from a Spacetime Perspective|journal= Living Rev. Relativ.| volume=7| date=2004| issue=9| at=Chapter 4.2}} with coordinates $(t,x^1,x^2,x^3)$ a Fermat metric takes the form

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

where the conformal factor $f(t,x)$ depends on time $t$ and space coordinates $x^\{\backslash alpha\}$ 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 $x\_a=(x^1\_a,x^2\_a,x^3\_a)$ and $x\_b\; =\; (x^1\_b,x^2\_b,x^3\_b)$ corresponds to stationary action.

$$S=\backslash int^\{\backslash mu\_a\}\_\{\backslash mu\_b\}\backslash left(\backslash sqrt\{\backslash hat\{g\}\_\{\backslash alpha\backslash beta\}\; \backslash frac\{dx^\{\backslash alpha\}\}\{d\backslash mu\}\; \backslash frac\{dx^\{\backslash beta\}\}\{d\backslash mu\}\}+\backslash phi\_\{\backslash alpha\}(x)\backslash frac\{dx^\{\backslash alpha\}\}\{d\backslash mu\}\; \backslash right)\; d\backslash mu,$$

where $\backslash mu$ is any parameter ranging over an interval $[\backslash mu\_a,\; \backslash mu\_b]$ and varying along curve with fixed endpoints $x\_a=x(\backslash mu\_a)$ and $x\_b=x(\backslash mu\_b)$.

In principle of stationary integral of energy for a light-like particle's motion,{{Citation| last1=D. Yu. | first1= Tsipenyuk|last2=W. B. | first2=Belayev | title=Extended Space Model is Consistent with the Photon Dynamics in the Gravitational Field|journal= J. Phys.: Conf. Ser.| volume=1251| date=2019| issue=12048 | page= 012048| doi= 10.1088/1742-6596/1251/1/012048| bibcode= 2019JPhCS1251a2048T| doi-access=free}} the pseudo-Riemannian metric with coefficients $\backslash tilde\{g\}\_\{ij\}$ is defined by a transformation

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

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

$$ds^2=\backslash rho^2\; g\_\{00\}(dx^\{0\})^\{2\}+\; 2\backslash rho\; g\_\{0k\}dx^\{0\}dx^\{k\}+g\_\{kq\}dx^\{k\}dx^\{q\},$$

where $\backslash rho$ is some quantity, which is assumed equal 1. Solving light-like interval equation $ds=0$ for $\backslash rho$ under condition $g\_\{00\}\; \backslash ne\; 0$ gives two solutions

$$\backslash rho\; =\backslash frac\{-g\_\{0k\}\; v^\{k\}\; \backslash pm\; \backslash sqrt\{(g\_\{0k\}\; g\_\{0q\}\; -g\_\{00\}\; g\_\{kq\})v^\{k\}\; v^\{q\}\; \}\; \}\{g\_\{00\}\; v^\{0\}\; \},$$

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

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

$$\backslash rho\; =-\backslash frac\{g\_\{kq\}\; v^\{k\}\; v^\{q\}\; \}\{2v\_\{0\}\; v^\{0\}\}.$$

In both cases for the free moving particle the Lagrangian is

$$L=\; -\backslash rho.$$

Its partial derivatives give the canonical momenta

$$p\_\{\backslash lambda\}=\backslash frac\{\backslash partial\; L\}\{\backslash partial\; v^\{\backslash lambda\}\}=\backslash frac\{v\_\{\backslash lambda\}\}\{v^\{0\}v\_\{0\}\}$$

and the forces

$$F\_\{\backslash lambda\}=\backslash frac\{\backslash partial\; L\; \}\{\backslash partial\; x^\{\backslash lambda\}\}=\backslash frac\{1\}\{2v^\{0\}v\_\{0\}\}\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^\{\backslash lambda\}\}v^\{i\}v^\{j\}.$$

Momenta satisfy energy condition {{Citation| last1 = Landau | first1 = Lev D. | author-link = Lev Landau | last2 = Lifshitz | first2 = Evgeny F. | author2-link=Evgeny Lifshitz | title= Mechanics Vol. 1 | edition = 3rd | location = London | publisher=Butterworth-Heinemann| date=1976 | isbn=9780750628969| page = 14}} for closed system

$$\backslash rho=v^\{\backslash lambda\}p\_\{\backslash lambda\}-L,$$

which means that $\backslash rho$ 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

$$S=\backslash int^\{\backslash mu\_a\}\_\{\backslash mu\_b\}L\; d\backslash mu=-\backslash int^\{\backslash mu\_a\}\_\{\backslash mu\_b\}\backslash rho\; d\backslash mu,$$

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

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

which is rewritten in form

$$\backslash frac\{d\}\{d\backslash mu\}\; p\_\{\backslash lambda\}-F\_\{\backslash lambda\}=0.$$

After substitution of canonical momentum and forces they yields {{Citation| last1=D. Yu. | first1= Tsipenyuk| last2=W. B. | first2=Belayev | title=Photon Dynamics in the Gravitational Field in 4D and its 5D Extension |journal= Rom. Rep. In Phys.| volume=71| date=2019| issue=4 | url=https://rrp.nipne.ro/2019/AN71109.pdf }} motion equations of lightlike particle in a free space

$$\backslash frac\{dv^0\}\{d\backslash mu\}+\backslash frac\{v^0\}\{2v\_0\}\; \backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^0\}\; v^i\; v^j\; =\; 0$$

and

$$(g\_\{k\backslash lambda\}\; v\_0\; -\; g\_\{0k\}\; v\_\{\backslash lambda\})\; \backslash frac\{dv^k\}\{d\backslash mu\}+\backslash left[v\_\{0\}\backslash Gamma\_\{0ij\}-v\_\{\backslash lambda\}\; \backslash Gamma\_\{\backslash lambda\; ij\}\backslash right]\; v^i\; v^j=0,$$

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

Energy integral variation and Fermat principles give identical curves for the light in stationary space-times.{{Citation| last1=D. Yu. | first1= Tsipenyuk| last2=W. B. | first2=Belayev | title=Photon Dynamics in the Gravitational Field in 4D and its 5D Extension |journal= Rom. Rep. In Phys.| volume=71| date=2019| issue=4 | url=https://rrp.nipne.ro/2019/AN71109.pdf }}

In the generalized Fermat’s principle {{Citation| last1=V. P. | first1= Frolov | title=Generalized Fermat's Principle and Action for Light Rays in a Curved Spacetime |journal= Phys. Rev. D| volume=88| date=2013| issue=6 | page= 064039 | doi=10.1103/PhysRevD.88.064039 | arxiv=1307.3291| bibcode= 2013PhRvD..88f4039F | s2cid= 118688144 }} 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.

The equations

$$\backslash frac\{d\}\{d\backslash mu\}\; p\_\{\backslash lambda\}-F\_\{\backslash lambda\}=0$$

can be transformed {{Citation| last1=D. Yu. | first1= Tsipenyuk|last2=W. B. | first2=Belayev | title=Extended Space Model is Consistent with the Photon Dynamics in the Gravitational Field|journal= J. Phys.: Conf. Ser.| volume=1251| date=2019| issue=12048 | page= 012048| doi= 10.1088/1742-6596/1251/1/012048| bibcode= 2019JPhCS1251a2048T| doi-access=free}} into a contravariant form

$$\backslash frac\{dp^\{k\}\; \}\{d\backslash mu\; \}+g^\{k\backslash lambda\; \}\; \backslash frac\{\backslash partial\; g\_\{\backslash lambda\; i\}\; \}\{\backslash partial\; x^\{j\}\; \}\; v^\{j\}\; p^\{i\}\; =F^\{k\},$$

where the second term in the left part is the change in the energy and momentum transmitted to the gravitational field

$$\backslash frac\{d\backslash stackrel\{\backslash leftrightarrow\}\{p^\{k\}\}\; \}\{d\backslash mu\; \}=g^\{k\backslash lambda\; \}\; \backslash frac\{\backslash partial\; g\_\{\backslash lambda\; i\}\; \}\{\backslash partial\; x^\{j\}\; \}\; v^\{j\}\; p^\{i\}$$

when the particle moves in it. The force vector ifor principle of stationary integral of energy is written in form

$$F^k\; =\; g^\{k\backslash lambda\}\backslash frac\{1\}\{2v^\{0\}v\_\{0\}\}\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^\{\backslash lambda\}\}v^\{i\}v^\{j\}.$$

In general relativity, the energy and momentum of a particle is ordinarily associated {{Citation| last1=V. I.| first1= Ritus | title=Lagrange equations of motion of particles and photons in the Schwarzschild field|journal=Phys. Usp. | volume=58| date=2015| page=1118 | doi=10.3367/UFNe.0185.201511h.1229 | doi-access=free}} 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{{Citation| last1=D. Yu. | first1= Tsipenyuk|last2=W. B. | first2=Belayev | title=Extended Space Model is Consistent with the Photon Dynamics in the Gravitational Field|journal= J. Phys.: Conf. Ser.| volume=1251| date=2019| issue=12048 | page= 012048| doi= 10.1088/1742-6596/1251/1/012048| bibcode= 2019JPhCS1251a2048T| doi-access=free}} 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 {{Citation| last1=R. C. | first1= Tolman|author-link=Richard C. Tolman|last2=P. | first2=Ehrenfest |author2-link=Paul Ehrenfest|last3=B. | first3=Podolsky |author3-link=Boris Podolsky| title=On the Gravitational Field Produced by Light|journal= Phys. Rev.| volume=37| date=1931| issue= 5|page= 602| doi=10.1103/PhysRev.37.602 | bibcode= 1931PhRv...37..602T| doi-access=free}}{{Citation|author=Tolman, R. C. |author-link=Richard C. Tolman|title=Relativity, Thermodynamics and Cosmology|location=New York|publisher=Dover|date=1987|isbn=9780486653839| pages = 274–285}} 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

$$d\backslash acute\{\backslash mu\}=v\_0v^0d\backslash mu$$

the expression for the momenta turned out to be

$$p^\{\backslash lambda\}=\backslash acute\{v\}^\{\backslash lambda\},$$

where 4-velocity is defined as $\backslash acute\{v\}^\{\backslash lambda\}=dx^\{\backslash lambda\}/d\backslash acute\{\backslash mu\}$. Equations with contravariant momenta

$$\backslash frac\{dp^\{k\}\; \}\{d\backslash mu\; \}+g^\{k\backslash lambda\; \}\; \backslash frac\{\backslash partial\; g\_\{\backslash lambda\; i\}\; \}\{\backslash partial\; x^\{j\}\; \}\; v^\{j\}\; p^\{i\}\; =\; g^\{k\backslash lambda\}\backslash frac\{1\}\{2v^\{0\}v\_\{0\}\}\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^\{\backslash lambda\}\}v^\{i\}v^\{j\}$$

are rewritten as follows

$$\backslash frac\{dp^\{k\}\; \}\{d\backslash acute\{\backslash mu\}\; \}+g^\{k\backslash lambda\; \}\; \backslash frac\{\backslash partial\; g\_\{\backslash lambda\; i\}\; \}\{\backslash partial\; x^\{j\}\; \}\; \backslash acute\{v\}^\{j\}\; p^\{i\}\; =\; g^\{k\backslash lambda\}\backslash frac\{1\}\{2\}\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^\{\backslash lambda\}\}\backslash acute\{v\}^\{i\}\backslash acute\{v\}^\{j\}.$$

These equations are identical in form to the ones obtained from the Euler-Lagrange equations with Lagrangian $L=\backslash frac\{1\}\{2\}g\_\{ij\}\backslash frac\{dx\_i\}\{ds\}\backslash frac\{dx\_j\}\{ds\}$ by raising the indices.{{Citation|author= Belayev, V. B.|title=The Dynamics in General Relativity Theory: Variational Methods|location=Moscow|publisher=URSS|date=2017|isbn=9785971043775| pages = 89–91}} In turn, these equations are identical to the geodesic equations,{{Citation

|first1=Charles W.| last1=Misner | author-link=Charles W. Misner | first2=Kip. S.|last2=Thorne|author2-link=Kip Thorne|first3=John A.|last3=Wheeler

|author3-link=John A. Wheeler|title=Gravitation|publisher= W. H. Freeman|date=1973|isbn=9780716703440| pages = 315–323|title-link=Gravitation (book)}} which confirms that the solutions given by the principle of stationary integral of energy are geodesic. The quantities

$$\backslash frac\{d\backslash stackrel\{\backslash leftrightarrow\}\{p^\{k\}\}\; \}\{d\backslash acute\{\backslash mu\}\}=g^\{k\backslash lambda\; \}\; \backslash frac\{\backslash partial\; g\_\{\backslash lambda\; i\}\; \}\{\backslash partial\; x^\{j\}\; \}\; \backslash acute\{v\}^\{j\}\; p^\{i\}$$

and

$$\backslash acute\{F\}^k\; =\; g^\{k\backslash lambda\}\backslash frac\{1\}\{2\}\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; x^\{\backslash lambda\}\}\backslash acute\{v\}^\{i\}\backslash acute\{v\}^\{j\}$$

appear as tensors for linearized metrics.

• Fermat's principle
• Variational methods in general relativity

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{{DEFAULTSORT:Fermat's and energy variation principles in field theory}}

Category:General relativity

Category:Variational principles