Thread: Coordinate-free relativity View Single Post
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 Quote by JDoolin Very well done! Because the lengths matched, in fact, you did not have to decide which point you were measuring "from" and which point you were measuring "to". I admit this is one scenario that hadn't occurred to me. But how would you modify this process if you needed to measure lengths of things that were not exactly 8.5" long?
Get another standard rod that is as long as needed, or (more commonly) get a large number of very small standard rods and count how many are used.

 Quote by JDoolin Yes.
That is all we are saying. Those caveats are acceptable. The geometry we are interested in spacetime is the spacetime interval.

 Quote by JDoolin But if you get into general relativity; for instance, the Schwarzschild metric, even your choice of origin will affect measurement of distance, time, and space-time intervals.
Not really. The origin can be moved in time as desired without even changing the components of the metric tensor. And you can do a diffeomorphism to a coordinate system with any arbitrary origin. Such a transformation will cause the components of the metric to change, but all measurements of spacetime intervals will be unchanged. Since such quantities do not depend on the choice of coordinate system you can express them without reference to any coordinate system if you wish. That is the point of coordinate-free relativity.