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Looking for a metric of a black hole where the global time coordinate of the metric defines the proper time of co-moving observers. Are the Lemaître coordinates the only way to do this?

What's odd about these coordinates is that for a Schwazschield solution, the 3D curvature changes with time. i.e. the $g_{\rho,\rho}$ component depends on $\rho$ and $\tau$.

Thus instead of gravity being seen as 4D curvature. It seems it could be viewed as simply 3D curvature which changes with time? Would that be a fair analysis? (i.e. a 3x3 metric $g^{ij}(x,y,z,t)$ where the black hole solution must depend on time.)

And there would be no-way to view a black hole as a static 3D curvature of space only. (As is commonly demonstrated to high school students with a curved plastic sheet).

Unless maybe there is some other coordiate system in which $g_{\tau \tau}$ is constant and the other elmeents don't depend on $\tau$?

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  • $\begingroup$ The Gullstrand-Painlevé coordinates are not synchronous but are closely related to the Lemaître coordinates. They too use the proper time of a freely falling observer as the time coordinate. $\endgroup$ – John Rennie Nov 12 '19 at 10:49

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