# Gravitational Lensing Derivations

Gold Member
Hey,
I just had the chance to extract the gravitational lensing caused by a massive point using Fermat's principle.

I was wondering though, is there any other way to do that?

Also is the light's time delation induced by the "refraction index" $n$ (Saphiro delay) connected to "gravitational time delation"?

## Answers and Replies

Hey,
[..] gravitational lensing caused by a massive point using Fermat's principle [..]
Also is the light's time delation induced by the "refraction index" $n$ (Saphiro delay) connected to "gravitational time delation"?
Answering your second question: yes indeed, although a bit indirectly. Fermat's principle is related to Huygen's principle and gravitational time dilation follows from the reduced speed of light, which plays a role in calculating the bending of light by that means - see p.821 (and for context p.820) here:
https://en.wikisource.org/wiki/The_...Perihelion-motion_of_the_paths_of_the_Planets.

PS Jonathan Scott provided fitting commentary to that link about the "double bending" here: https://www.physicsforums.com/threads/bending-of-light.787335/#post-4944166

Last edited:
PAllen
Science Advisor
As to alternative derivations, it is worth noting the Hugen's and Fermat's principle are derived results in GR, even more so than the geodesic principle (which can be derived from the field equations, so it need not be assumed separately). Einstein used this principle in his derivations without proof (so far as I know) - he assumed it must be true based physical intuition. (Pauli, in his 1921 work, provides a justification for this based on work by Levi-Civita and Weyl; at that time, it was only proved for spherical symmetry, and unknown whether such a principle had any validity in a more general spacetime. MTW justifies Fermat's principle for any static solution).

Most modern books derive lensing directly treating null geodesic paths, without bothering to use (let alone justify) Huygen's principle or Fermat's principle and a varying speed of light (which is coordinate dependent). For example the following all use pure null geodesic analysis for light bending with no mention at all of Huygen's or Fermat's principle or varying light speed:

P. G. Bergmann's 1942 text (enthusiastically endorsed by Einstein)
J.L. Synge 1960 General Relativity book
James L. Anderson's 1967 Principles of Relativity Physics

Misner, Thorne, & Wheeler derive light bending directly from null geodesic analysis, but for Shapiro time delay they introduce and justify Fermat's principle for static fields, and imply it isn't true [or even definable] in more general cases.

ChrisVer