# Light takes the path of most time elapsed?

1. Nov 22, 2013

### AntiElephant

Hi, I'm sorry if this is the wrong section. My Dad keeps telling me that light takes the path of most time elapsed between two points, and he remembers hearing this because it's counter intuitive. I'm a Physics undergrad myself but I can't see if and how there's any truth behind this. I've searched and found nothing even equivalent to this fact. So if it were to be true I can only imagine it's from post-graduate/research level work or from a module I haven't taken. The only thing I can guess is that light is a quantum mechanical particle and hence takes every path from A to B, but the probability of taking anything but path of shortest time elapsed is so low that it practically never takes it? Again, the statement wouldn't make sense to everything I already know, but that doesn't exclude the fact there may be some technicality where we can observe light taking the path of most time elapsed?

Any links to sources would be appreciated.

2. Nov 22, 2013

### jbriggs444

The straight-line (aka geodesic) path between any two events is the path that maximizes the perceived time (aka proper time) of someone travelling from the one event to the other. [If you deviate from the straight line path, time dilation reduces the total amount of time experienced].

Light follows straight line paths.

In the context of curved space, a geodesic is only locally straight. Accordingly, a geodesic path may only locally maximize proper time. For example, consider the analogy that there are two "straight line" paths from New York to Boston on the surface of the earth. Both are "locally straight" in that they do not deviate to the right or left. But one goes the long way around.

Last edited: Nov 22, 2013
3. Nov 22, 2013

### Andy Resnick

It's the *least* time, not the most time. Google "Fermat's principle".

4. Nov 22, 2013

Staff Emeritus
Andy is right - it's the least time. If you think about it, you can always make a longer path, so we wouldn't see anything at all if it were the most time.

5. Nov 22, 2013

### WannabeNewton

This is for time-like geodesics. It has no relevance whatsoever to null geodesics.

Andy had it right anyways: light takes the path of least time between two points in space.

6. Nov 22, 2013

### BruceW

I think light will take any path that has an extremal total time. So this could be minimum or maximum time. But in most situations, we will have minimum-time paths. In fact, it is difficult to think of a possible situation where light could take a maximum-time path... maybe if there was a path along which there is a higher refractive index than the surrounding space. remember that the path only needs to be locally extremal.

edit: this is a half-guess, since the wikipedia page on Fermat's principle seems to imply that light can also take the path of maximum time between two points in space. Now I'm actually trying to think if it is true... maybe if a path of maximum time between two points in space could be shown to have zero change of proper time, then it would show that light can also take paths with maximum time between two points in space...

edit again: ah, but light does not have zero change in proper time when it is moving through a material with a refractive index not equal to 1.

Last edited: Nov 22, 2013
7. Nov 22, 2013

### Staff: Mentor

I would bet long odds that he's misremembering a different counter-intuitive statement, namely that an object at rest is following a space-time path of maximum elapsed time.

This is one of the ways of approaching the famous twin paradox: No matter how you tweak and tease the traveling twin's journey, the elapsed time ends up being less than that experienced by the stay-at-home twin.

8. Nov 22, 2013

### BruceW

To add more to what I was saying about light also taking path of maximum time, here: http://scienceworld.wolfram.com/physics/FermatsPrinciple.html Eric Weisstein says the same, but he doesn't give any examples. But anyway, I'd agree with Nugatory, I'm guessing this is not what AntiElephant's Dad had in mind anyway.

9. Nov 22, 2013