B Proving Proper Time of Photon in Friedman Metric

exmarine
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If I understand it correctly, the proper time differential for a photon in flat space is zero. That is evident if the velocity of light is equal to c, so the right hand side of the Minkowski metric is equal to zero. Therefore the left side must also be zero.

My question: Is the same true for the Schwarzschild and Freidman metrics? I think yes, but I don’t know how to prove it. Thanks for any information about that.
 
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Yes, but technically it is the spacetime interval that is 0, not proper time. Proper time is only defined for timelike world lines
 
Well, it is a matter of definition. A null path is defined as one with zero interval along it. If you are asking why light follows such a path, this follows from a photon having no mass, which is subject to experiment, plus that if it is true in SR, it must true locally - as differential statement - everywhere in GR by the principle of equivalence.
 
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