General Relativity: An Introduction for Physicists · Problem 10.6

Experimental tests of general relativity — Hobson, Efstathiou & Lasenby

Physics · radar-echo-coordinate-time

Problem

In the radar-echoes test, a photon travels from radial coordinate rr to r0r_0, which is the radial coordinate of the closest approach of the photon to the Sun. Verify that, to first order in μ/r\mu/r, the elapsed coordinate time is given by t(r,r0)=(r2−r02)1/2c+2μcln⁡[r+(r2−r02)1/2r0]+μc(r−r0r+r0)1/2.t(r, r_0) = \frac{(r^2 - r_0^2)^{1/2}}{c} + \frac{2\mu}{c}\ln\left[\frac{r + (r^2 - r_0^2)^{1/2}}{r_0}\right] + \frac{\mu}{c}\left(\frac{r - r_0}{r + r_0}\right)^{1/2}.

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