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I'm stuck on the following.

We know that the path taken by light to travel between 2 points A and B corresponds to the path which minimises time elapsed.

However, from relativity we also know light travels along geodesics in space time, i.e inertial world lines and these world lines correspond to world lines which maximise proper time between 2 events.

What am I getting wrong and how do I reconcile these?

Níckolas Alves
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Vishal Jain
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  • If I’m remembering right, I believe the proper time of an object moving at the speed of light is 0, i.e. in the reference frame of a photon going from point A to B, the photon would experience the trip instantaneously, but this obviously isn’t the case for an observer, so I think you’re talking about two different kinds of time here – Justin T May 19 '22 at 08:11
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    A geodesic is a path of minimal length in spacetime – SrJaimito May 19 '22 at 08:36
  • minimal length in space-time between 2 events == maximum proper time between 2 events. I get light takes the shortest route through space-time, but since our everyday observations tell us it also takes the path of the shortest time, I am not sure how to reconcile these 2 observations. – Vishal Jain May 19 '22 at 08:38
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    @VishalJain The thing is that it does not make sense to use the definition of proper time when describing a geodesic, since for light the variation of proper time is null. Therefore, proper time does not behave as a good parameter for mapping different points of the light's geodesic, and you have to use any other affine parameter. In order to reconcile both things, you can think that in non-relativistic theory light travels minimizing optical path instead of time (though it's correlated) – SrJaimito May 19 '22 at 08:46
  • Geodesics are critical - not necessarily minimal - with respect to length in a (pseudo-) Riemannian manifold. Light travels along what is called a null geodesic. https://physics.stackexchange.com/questions/188859/what-is-a-null-geodesic – kricheli May 19 '22 at 09:46
  • To visualize that geodesics need not be minimal consider circles on a sphere (the earth). You can travel between two not too far away points on the equator the short way (minimal distance) or the long way round along the equator. Either way you're on a geodesic. – kricheli May 19 '22 at 09:49
  • I think the geodesic also depend on the observer speed, and this also in classical physics. – QuantumPotatoïd May 24 '22 at 04:52

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