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In the Schwarzchild metric, $t$ is the time on the clock of an observer at infinity, and $r$ is the related to the area of a sphere by $A=4\pi r^2$. Are there more physical coordinates one could use, where $t$ is the time on the clock of an observer at finite $r$, and $r$ refers to distances this observer could actually measure with a meter stick? I know the metric will depend on the motion of the observer, but perhaps there are some "preferred frames" that are more relevant than others?

Eric David Kramer
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"More physical" is a little vague, and depending on the precise meaning there are indeed other alternatives.

For instance, spatially isotropic coordinates exist: search for "isotropic coordinates" or check Buchdahl's paper, and take a look at this question.

A simple way for the time coordinate to reflect proper time for a static observer at a radius $r_0$ is simply to rescale the Scharzschild's metric $t$-coordinate by an appropriate amount $L$: $t' = Lt$.

This is what's actually done in the definition of Terrestrial Time TT coordinate, of which International Atomic Time TAI (and therefore Universal Coordinated Time UTC, besides leap seconds) is a realization:

For many Earth-related astronomy applications, the geocentric celestial reference system GCRS of coordinates $(T,X,Y,Z)$ is used. It is a truncated series expansion of isotropic coordinates for the Schwarzschild solution (outside the event horizon). Its time coordinate $T$ is called Geocentric Coordinate Time TCG, and represents the proper time of a non-rotating observer at infinity. Terrestrial time TT is related to TCG by a linear transformation, so that it equals the proper time measured (in an averaged sense) by a stationary observer on Earth's surface. The scaling in this linear transformation takes into account that the observer is not at infinity, and is moreover moving. For these matters see e.g.

A similar rescaling can be done for the $r$ coordinate.

pglpm
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