Hello! I am a bit confused about the definition of the radius in Schwarzschild metric. In the Schutz book on GR (pg. 264, General rules for integrating the equations) he says: "A tiny sphere of radius ##r = \epsilon## has circumference ##2\pi\epsilon##, and proper radius ##|g_{rr}|^{1/2}\epsilon## (from the line element). Thus a small circle about ##r = 0## has ratio of circumference to radius of ##2\pi|g_{rr}|^{−1/2}##". I am a bit confused. Is the proper radius the one measured by a local observed (walking from the center to that radius) and the actual radius, the same distance measured by an observer at infinity? And the same for the circle circumference? Thank you!