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I'm learning about the Fermat's principle of least time (FPLT) for how light refracts on entering water. Is gravitational lensing (around a star or galaxy) related to FPLT? From what I understand gravitational lensing is not refraction, because then we would see a rainbow in the bent light, but we don't. What is the relation between them?

Qmechanic
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foolishmuse
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1 Answers1

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Yes, there is a version of Fermat's principle adapted for general relativity! Like many situations in physics, it's an example of a principle of stationary action: Fermat is the name for the special case for light. You can derive the equations for the deflection of light around a mass, like a galaxy or star, using it.

This paper and this paper from the 90s were the ones which gave a derivation for an arbitrary spacetime. The tricky part to the idea of a Fermat principle for gravitational lensing is that the idea of "time" in general relativity is a bit more slippery than in Newtonian physics. In general we don't have a global concept of time; and light doesn't have a proper time. The remarkable thing that Kovner and Perlick showed was that we can still apply a version of Fermat, if we consider the idea of an "arrival time functional".

The general relativistic version of the Fermat principle states that

Let there be a null curve P from a fixed photon emission event to a possible observation event at its intersection with a timelike observer worldline. Then this P is a null geodesic (i.e. a light ray) if and only if the functional of its arrival time, measured along the observer worldline, is stationary with respect to variations of P restricted to null curves.

Basically, we are are considering possible arrival times measured by the observer, instead of a Newtonian time.

qwerty
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