Which Is More Fundamental: Time or Distance?

[Albertgauss]

Is the following logic correct?

When spacetime is flat, we say that light travels in a straight line. A planet also would travel in a straight line because that would take the shortest travel time between two points.

Does light seek out the shortest time to travel between two points, or the shortest distance? Which one is more fundamental?

Now if spacetime is curved, would I say light travels on the curved path because that curved path represents the shortest time it takes light to travel between two points, but not necessarily the shortest distance?

And same thing with a planet orbiting a star. Is this logic correct? --> the orbit of the planet represents the shortest time a planet can go from one coordinate to another.

 

[PeterDonis]

When spacetime is flat, we say that light travels in a straight line.

Strictly speaking, it travels on a null geodesic. See below.

A planet also would travel in a straight line because that would take the shortest travel time between two points.

No; it takes the longest travel time between two events. You have to think of spacetime, not space. An object that is not subject to any forces travels on a geodesic of spacetime, and a geodesic is a curve of maximal proper time (maximum time elapsed) between two given events (points in spacetime).

Does light seek out the shortest time to travel between two points, or the shortest distance? Which one is more fundamental?

Neither. Rather, both are the same thing as far as light is concerned. Light travels on null geodesics, which do not have an invariant "time" or "distance" associated with them (whereas timelike geodesics, the curves that things like planets travel on, do have an invariant time--the proper time according to the planet's clock).

f spacetime is curved, would I say light travels on the curved path because that curved path represents the shortest time it takes light to travel between two points, but not necessarily the shortest distance?

Neither. See above.

And same thing with a planet orbiting a star. Is this logic correct? --> the orbit of the planet represents the shortest time a planet can go from one coordinate to another.

No. The planet's orbit represents the longest proper time between two given events--but in curved spacetime there is a caveat, that the orbit might not be the globally longest proper time, but only the longest within a particular class of curves. In the case of the planet, if we consider the two given events to be two successive perhelion passages of the planet, its orbit will be the longest proper time in the class of curves that all go around the star once between those two events. But there will be another geodesic with even longer proper time, which doesn't go around the star at all--it just goes radially outward and then falls back again in such a way that it passes through both successive perihelion events of the planet.

 

[Albertgauss]

It looks like I ran into a website that has bad information. I was only looking at the image and failed to actually read what was on the website. I'll be more careful than that next time. I think I got everything I need from this. I understand the above as much as I need to. In fact, all of this is on Wiki. I didn't go to WIki first because I didn't think I needed to.