# Does light take the path of least time because it travels in straight lines or vice versa?

My question is which of these two feats is a consequence of the other? Light travels in straight lines, mostly. Does it do that as a result of Fermat's principle of least time? and if so, is there a reason as to why it follows the path of least time? or is this another "that's the way the universe works" question? And by reason I mean a physical explanation not mathematical deduction. Or is it the other way around? meaning light taking the path of least time is just an obvious manifestation of the fact that it goes in straight lines?

• Light only goes in straight lines when there is no gravity. In other words, light NEVER goes in straight lines, it only approximately does. The path that light follows falls from the lightlike geodesic equations. It has to do with the curvature of spacetime. I'd post an answer but this is the extent of my knowledge and this isn't clear enough. However, if you don't find an answer, look into the lightlike geodesic equations. – Vivid Kraig May 20 '19 at 0:10
• I recommend that you get a copy of "QED" by Feynman. It's a thin little paperback, and contains a nice intuitive explanation of why light takes the shortest (least time) path. Actually it doesn't necessarily take the shortest path; it's just most likely to take a path that's very nearly the shortest possible path. – S. McGrew May 20 '19 at 4:53
• Physics is mostly about how not why. – Qmechanic May 20 '19 at 8:55

The fact that light travels at straight lines has the same reason as any particle moving on straight line if there is no external agent to change its path - the fact that spacetime is homogeneous and isotropic in any inertial frame of reference (actually that is the definition of inerial frame of reference).

In general relativity, there are (in general) only local inertial frames, and then the straight lines are called geodesics.

Light would move on straight lines due to symmetry even without fermats principle. The fermats principle is usefull when there is no symmetry and the actual paths must be computed

Light travels in straight lines because it takes the path of least time. The reason it takes the path of least time is because this is the path where the phase, which is proportional to the time, is a minimum. Since it is minimum, its derivative is zero with respect to small changes in the path. Thus, decomposing the light into many coherent wavelets, each with amplitude $$A \sim e^{i \vec{k}\cdot{x} - ik t}$$ the path of minimum phase, i.e. the path of least time, is the path where the wavelets all add up in phase because the derivative there (of the phase) is zero. Thus, for this path, all the wavelets add up constructively, and we have a nice light ray. Away from this path, the wavelets add up destructively because all the phases are different, and we get zero.

From Feynman QED, pp. 38-45

Feynman used to say that a good physicist knows many different ways to arrive at the same result. If you assume Fermat's Principle, you will get that light travels at straight line. If you assume light travels in straight lines and snell's law, you will get Fermat's Principle. This equivalence makes it hard to decide which one is the "fundamental" description.

I personally think that the Differential Equation is always the correct form to consider as the fundamental one. A differential equation describe the evolution of a state through time, which is what one observes. This view also lets you avoid some relativity problems.

So to your question, the answer will be that both results are direct byproducts of Maxwell's Equations, making neither one of them more fundamental than the other.