# How does Fermat's principle of least time for light apply to curved spacetime?

In a region of space which has no massive object light rays travel parallel to each other or ,simply, in a straight line. However, in a positively curved region of space (like near a planet or a star), like in this image light rays "bend" if that body is in there path.

Supposedly, rays of light are traveling from some body but there's a star between there path and they get curved so that the light reaches us, the observers. Am I correct if I say that the reason light rays bend is because that path in the curved region takes the least time for the light rays?

The way I see it, if I trace the path of light in absence of the star and then superimpose that path when the star is present, the new path gets curved but the distance is least in this case which is why light will take that path.

Fermat's principle does indeed still hold, in the following form$$^1$$:

Let $$S$$ be an event (the source) and $$\ell$$ a timelike world line (the observer) in a spacetime $$(M, g_{\alpha\beta})$$. Then a smooth null curve $$\gamma$$ from $$S$$ to $$\ell$$ is a light ray (null geodesic) if, and only if, its arrival time $$\tau$$ on $$\ell$$ is stationary under first-order variations of $$\gamma$$ within the set of smooth null curves from $$S$$ to $$\ell$$.

In other words, this says that given a source event and an observer, out of all the possible trajectories that move at light speed, the actual trajectory will be the one for which the time of arrival is stationary (which includes minimum). What this shows is that all the effects from the gravitational field are simply encapsulated into the time of arrival, like you say.

$$^1$$ Schneider, Ehlers and Falco, Gravitational Lenses, section 3.3, page 100

• I'm not sure that I've understood this properly because I'm still a high school student. Let me know if I'm correct in my understanding - Out of all the possible paths that light can take, it takes the one with which it arrives first? Is that right? Jun 21, 2020 at 18:26
• @DarkLumiere There are some technicalities, but essentially yes. The theorem concerns a ray of light being emitted from a certain point at a certain time, and eventually arriving at someone waiting for it. There are many possible paths it could take while still moving at lightspeed; the actual path is the one that arrives first. Jun 21, 2020 at 19:26
• I see. Thanks a lot! Jun 22, 2020 at 10:33
1. Well, it is not possible to write a stationary action principle (SAP) for null geodesics/massless particles without the use of auxiliary variables, cf. e.g. this related Phys.SE post. This makes any interpretation of the action as (proportional to) proper time challenging.

2. Nevertheless, it is possible to use such SAP to derive a Fermat's principle, at least for some curved spacetimes, cf. my Phys.SE answer here.

3. See also Ref. 1 and this & this related Phys.SE posts.

References:

1. V.P. Frolov, Generalized Fermat's principle and action for light rays in a curved spacetime, arXiv:1307.3291. (Hat tip: Physics_Et_Al.)

Light travels along paths called geodesics. Geodesics have the property that they are the paths of minimal distance between two points. Therefore, it takes less time to travel along a geodesic than any other path between two points. The reason light travels along a geodesic is that these paths also have the property that they are locally "straight." Basically, the light thinks that it is travelling in a straight line the whole time.