In the original version of general relativity of 1914 Einstein proposed that light follows null geodesic. Considering Fermat's principle and ccomparing it with geodesics principle, Pauli, General relativity, 1921, notes that Palatini and De-Zuani showed that it is necessary to distinguish static case from more general stationary case. This is probably why Einstein  has ceased to include geodesics principle for light propagation in subsequent works. In the English edition of the Pauli book, 1958, there is no reference to these articles.

There is no generally accepted variational principle for obtaining a null geodesic equations. The commonly used path variation admits violation of the condition s=0, which means that with certain coordinates variations the interval becomes time-like or space-like. This approach exceeds the limits of variational principle in mechanics that virtual motions of the system are compared with cinematically possible motions. Fock in The theory of space, time and gravitation wrote that variation of null path does not make sense. Alas, the definition of a geodesic in four-dimensional space-time using the variation of the time component of the 4-velocity vector is also erroneous, since the varying functional must differ from the variables that define it.

 Fermat's principle for stationary gravity fields gives same results as principle of stationary integral of energy of the light particle https://www.researchgate.net/publication/317101911_Application_of_Lagrange_mechanics_for_analysis_of_the_light-like_particle_motion_in_pseudo-Riemann_space_additional_version

that applies also to non-stationary gravity fields, in which motion of the light particle is free. Its application for Gödel metrics shows that Fermat's principle provides solution that is different from the null geodesic in this space-time, see also

https://www.researchgate.net/publication/317102173_The_dynamics_in_general_relativity_theory_variational_methods .

Article Application of Lagrange mechanics for analysis of the light-...

Book The dynamics in general relativity theory: variational methods

The Fermat principle relies on least time.

Geodesics rely on least distance

Not the same in optical refraction!

For light rays in space  more properly take

d(s)=0 as a condition. Ie the speed is always c locally in the distorted space-time

That is over very short distances there is always a local Lorenz frame.

In general mechanics relies on least action.

No, Gödel's spacetime isn't a counterexample-it's an example. Gödel's spacetime admits closed time-like curves, that's all. But these are geodesics in that spacetime and this is equivalent to Fermat's principle, that was formulated for light rays. 

Stam,

It would be nice if you could give a link to an article that confirms your claim, or its proof. I mean identity of geodesics and curves following from Fermat's principle, as it is formulated in Landay and Lifshitz, The Classical Theory of Fields, p. 273 or Perlik, Gravitational Lensing from a Spacetime Perspective, Living Rev. Relativ., 7 (9), Chapter 4.2.

If you make a study of my article, then you will be convinced of the opposite. At the same velocities along the coordinates y and z the velocities along the coordinate r will always be different, see Eqs. (10.19)-(10.24) .

It's a straightforward exercise. The claim about the coordinates is irrelevant to this point. Fermat's principle describes the definition of the geodesic, because any non-trivial dependence on the velocity, beyond dimensional analysis, that controls the constant part of the velocity, which is the conversion coefficient between distance and time, simply defines the refraction index in the context of Fermat's principle.   And this is in all textbooks-it was discovered, in fact, by Hamilton (and, independently, by Jacobi) in the 19th century. And both stressed the correspondence with the least action principle in mechanics, which is the same that leads to the geodesic equation. Once more, the words may change, but their meaning only matters; and the mathematics is unambiguous. 

 The Fermat principle looks for an extremal on a set of isotropic paths. A null geodesic is an extremal on a set of isotropic and non-isotropic paths whose value approaches zero. These two sets do not coincide. Therefore, these curves can be different. According to the principle of least action in mechanics, variations should not lead to a qualitative change in the system, and in the case of a null geodesic this rule is violated.

The statement about the Fermat principle is wrong; the statement about the null geodesic is wrong, too , as are the statements that follow. Once more, the correct statements are known, so there's no point wasting time over wrong statements.

Light rays follow the null geodesics in  spacetime-that's exactly what Fermat's principle states, since the refraction index defines the spacetime in that case (it defines what's known, sometimes, as the ``optical metric'').

The Fermat principle: least time in N-dimensional space. Geodesics: least distance in N+1-dimensional space-time. This is not the same thing. This is proved unambiguously.

Of course they are-they are related by the product of the refractive index-that defines the non-trivial properties of spacetime, as probed by the light ray-and a numerical constant that has the dimensions of velocity. 

Fermat's principle does describe situations with a non-trivial refractive index.

The statement about N+1 vs N-dimensional spaces is wrong. That Fermat didn't use the notion of spacetime is irrelevant, the mathematics matter.

In Fermat's principle the varied function is time. It is variable with respect to N space coordinates of a stationary or quasi-stationary space. For nonstationary spaces, it is not applicable. The geodesic is the result of a variation of the path in N+1 dimensional space-time in N+1 coordinates.

The various viewpoints are very interesting; And the poser most of all.

Fermat Principle in mechanics and in Relativity (SR or GR) are different.

In mechanics one uses Jacobi principle; and for gravity you can obtain 'deflection of light' value (Soldner) that is half of GR value. To obtain red shift you have to use Fermat  principle;Or you imagine a vertical cylinder on the sun in which light goes to and fro to constitute a clock and apply  Soldner method to obtain Red Shift. See,for instance (my)  Matscience Report # 84/ webb site @imsc.res.in/library;section on GR(Vectors,Tensors and Relativity).  In my Physics Reports (1975)/DYN. SYMM. in MECH./I show that in mechanics the Jacobi and Fermat principles  define same geodesic paths.

In case of Relativity, the velocity space is a space of negative curvature,represented as interior of a circle of radius  "c".The Interior is complete by itself; "c" lies on the 2-sphere,much like the "poisson kernel" in the Dirichlet problem.In terms of complex variable z, linear fractional transformations of z, with complex coefficients is transitive as Lorentz group-i.e. as sl(2,C).

Maxwell Fields Correspond to (1,0)+(0,1) representations of Poincare group for zero first Casimir invariant (4-m0mentum square=mass square). For a bounded domain in space-time ( De Sitter/ anti De Sitter space) the automorphisms are subgroups of conformal conformal group (see e.g my article in Boulder lectures or my Physics Reports). For DE Sitter case these leave paths of Uniform Acceleration unchanged.And the group is:Spatial euclidean group+dilation+conformal as acceleration transformations. For velocity of light you most go the boundary where the full conformal group acts.

Geodesic equation in GR is invariant only under Projective changes;this includes adding a Vector(times Identity tensor) to the connection, or change in proper-time parameter ;i.e. one may add fibers without changing the system of geodesics. For null geodesics, which define characteristics of a Hyperbolic system one requires conformal symmetry.  But one has to be careful: because conformal group SO(4,2) has continuous mass(both positive/negative)representations that are ever present in GR -as  for instance in cosmological constants ;spherically symmetric mass distributions like Robertson-walker are all conformaly flat. It is already too long.

Gravitational Waves/disturbances are NOT GRAVITONS(i.e like photons) but much like PHONONS/energy-quasi particles and should be called GRAVINONS. These are fourth order disturbances and are energy-wise represented by double     divergence of the Conformal tensor; it has rank two and has zero divergence.

Dear Kishin,

Thanks for the detailed answer. I will read your article in Physics Reports as soon as I can.

I read your article but did not find a mention of Fermat's principle, although your report certainly deserves attention.

Dear Stam,

You was right. I checked the calculations and made sure Fermat's and geodesics principles gives the same solution for the light propagation for Gödel metrics https://www.researchgate.net/publication/317101911_Application_of_Lagrange_mechanics_for_analysis_of_the_light-like_particle_motion_in_pseudo-Riemann_space_additional_version. Furthermore, Frolov has proposed generalized Fermat's principle https://arxiv.org/abs/1307.3291 for an arbitrary metrics that yields null geodesics. The original equation is consistent with the equation obtained by principle of stationary integral of energy of the light-like particle. Therefore, the correspondence of the null geodetics to the variational principles in mechanics can be considered proven.

Kishin Mariwalla I love that you mentioned gravitational wave quantification as Gravinons! I can see where you are at. Could you point to some reference also ? Thanks a lot.

It was used in Gravitational waves Gravinons in GR in presentation to Indian Assoc. for GR (28 Jan 2009, Calcutta). See also Physics Reports 20c(1975)-sec 10. Einstein suggested grav waves in 1916, but by 1936 he and Rosen had found

nil anology to em case. See Pauli- Relativity- Notes 17,18. GR admits no open characteristic solutions. ;black holes are compact . Gravinons are analogues of

phonons, Their Energy tensor is double divergence of conformal tensor. SO grav waves are are excitations in GR with locally arbitrary dark energy. See my Gravity

Sans Singularities arXiv:gr-qc/0205113v1 28 May 2002.

There is a slightly different way to creating a theory of quantum gravity

Article Extended space model is consistent with the photon dynamics ...

although it is possible that they intersect.

Relativity mandates that all velocities lie in a sphere of radius c ; light alone reigns

over the pe!riphery, making incursions inside to enable perception. So all perception

is opto-mechanical, as in gravity wave detection. The projection on spacetime is a

space with Lorentz-Riemann curvature locally arbitrary with covariant constant

addition. Thus there is dark matter everywhere locally. These all have equivalent

paths. This via Feynman gives quantization. Equivalence of paths is Equivalence

Principle ! And we have an already unified theory with dark matter if we recognize

projectively equivalent paths as an equivalent class as in Feynman's method.

Comments and critcism invited.