Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment.

I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must move along a timelike geodesic. And a timelike geodesic can be defined as a geodesic that lies within the light cone.

I want to know why exactly null geodesics define the light cone. Or, why null geodesics define the path of light.

Also, if there's a better explanation why matter follows timelike geodesics, that would also be welcome.

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Even in curved spacetime, you can perform a coordinate transformation at any location ("move to a freely falling frame") such that your metric is locally flat , and takes the form $$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

If you consider a null trajectory where $ds^2$ is set to 0, then the above equation is the statement that "the differential physical distance traveled along the trajectory, as measured by an observer in a freely falling frame at the location in consideration, is equal to the speed of light times the differential time interval measured by that observer." From Einstein's equivalence principle, this is precisely the way that light must behave.

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+1 for coord transformation. – P O'Conbhui Mar 31 '12 at 1:21

Suppose there is a flash event that we can represent as a light cone as the flash expands.

There are three shutters with detectors around this flash event. The shutters open and close only once and this is almost instantaneous. One shutter opens in spacetime outside the light cone. One Shutter opens in spacetime inside the lightcone and the third opens exactly on the edge of the light cone.

The first shutter misses the flash because it opens before the flash reaches the shutter. According to the spacetime diagram the separation between the flash and the shutter is spacelike because there is a distance of space between the shutter opening and the flash reaching the shutter. The second shutter misses the flash because the flash has already passed over it in the past. Thus the separation between the shutter and the flash is timelike. The shutter that catches the flash has no separation. There is no difference in space or time between the shutter opening and the flash entering the shutter. You could say the difference is Null. Thus you can think of it as a null geodesic.

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