Fermat's principle in general relativity

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Dated: 2 january 2015

I wrote this upon my discovery that Fermat's principle works perfectly well in general relativity. Usually, light paths in a Schwarzschild metric are determined by writing down their Lagrangean equations of motion (aka geodesic equations). Fermat's principle turns out to be equivalent. I would formulate some things differently today (2018) from what I wrote then. For example, I would not talk about length contraction in the Schwarzschild metric, which is somewhat of a misnomer. There is a factor different from one between the local radial proper length increment and the local radial coordinate increment. But this should really not be called length contraction (nor spatial expansion). We also do not consider the fact that the azimuthal coordinate gets more densely spaced near the poles on a sphere as indicative of length dilation, why should we take the reduction of density of the radial coordinate towards the center of the Schwarzschild metric as indicative of length contraction? Also, I might less emphasize the anisotropy of space in that metric nowadays. But by and large, I still find this essay well-made.

Note added 07.12.2023: My remark about fallacious arguments sometimes leading to correct results proved prophetic in view of the fact that my own calculation in the article Light deflection by the sun contained errors, even though the final result was correct.

Note added: Having Fermat's principle for wave phenomena in anisotropic "media" at hand, it is also possible to discuss in some detail how light bending at small distance from the sun (with light moving almost perpendicular to the radial coordinate direction) is dominated by the equivalence principle, wheras at large distances from the sun (with light moving almost parallel to the radial coordinate), the main effect of the equivalence principle is not bending but a frequency shift, with bending mostly caused by spatial curvature.

Fermat's principle in general relativity and light deflection by the sun, 02.01.2015

Next: Derivation of the Schwarzschild geometry   Up: Introduction science education project    Previous: Light deflection by the sun

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