I was looking at the Static Weak Field Metric, which Hartle gives as:(adsbygoogle = window.adsbygoogle || []).push({});

##ds^2 = (1- \frac{2\Phi(x^i)}{c^2})(dx^2 + dy^2 + dz^2)##

For a fixed time.

Where, for example, ##\Phi(r) = \frac{-GM}{r}##

I was trying to figure out how the coordinates (x, y, z) could be defined. Clearly, they can't be defined by measurements of length. Hartle says nothing about this.

I suspect that the ##r## in the second equation is probably the measurable distance, and not ##(x^2 + y^2 + x^2)^{1/2}##

The best explanation I could come up with myself is that if you measured ##r## and ##\Phi(r)## at every point and knew ##G## and ##M## then you could define ##x, y, z## precisely so that the equation for ##ds^2## holds!

Does that sound right and/or can anyone shed any light on this?

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# I Defining Spacetime Coordinates

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