- #36

Dale

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OK, so let’s walk though the process of actually doing something like this so that you understand what needs to be done and why nobody wants to do it.My query isn't about how certain parts of the equations can be usefully applied, it is to do with the arbitrariness of which objects were at rest.

1) you need to start with a mathematical model of your spacetime. The standard model here is the Schwarzschild spacetime.

2) in that spacetime you need to write down an equation representing the clock’s worldline.

3) Here is where you make a choice, either you do things the easy way and simply skip to step 6) or you do things the hard way and continue on to step 4) which is a lot of work and completely unnecessary

4) Then you need to come up with a coordinate transform. This coordinate transform must be smooth and invertible everywhere and the clock’s worldline should have constant spatial coordinates in these new coordinates. It is especially easy to get non-invertible coordinates.

5) take that transformation and apply it to the Schwarzschild metric to get the metric in the new coordinate system

6) calculate the length of the worldline by integrating the metric along the worldline.

The math is set up so that you are guaranteed to get the same result whether you do steps 4) and 5) or not.