The Schwarzschild radial coordinate as a measure of proper distance

It is shown that when time is measured in a Schwarzschild field by radially falling or rising geodesic clocks, the usual Schwarzschild radial coordinate $R$, defined by $d{s}^2=\frac{d{R}^2}{(1-\frac{2M}{R})}-(1-\frac{2M}{R})d{T}^2+R^{2}d{\Omega }^2$, has the physical significance that it is a measure of proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line.