If we consider 2 shells in Schwarzschild spacetime their radial proper distance ##d s^2 = \frac{1}{1-\frac{r_s}{r}}d r^2## follows from the metric with ##dt=0## and ##d\Phi=0##. Integrating yields ##\Delta s## as a function of the r-coordinates of the shells. A reference with the formula for this is available, if of interest.(adsbygoogle = window.adsbygoogle || []).push({});

In contrast to the separation of the shells the radial proper distance between objects in free fall is increasing (accelerated, if I see it correctly). But how would we calculate ##\Delta s## in this case? Naively I was tempted to say the same way, but am not sure at all. So any help is appreciated.

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# I Radial proper distance between objects in free fall

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