General Relativity: An Introduction for Physicists · Problem 11.9

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · painleve-gullstrand-coordinates

Problem

A massive particle is released from rest at infinity in the Schwarzschild geometry. Show that the covariant components of its subsequent 4-velocity at coordinate radius rr can be written as uμ=c2∂μTu_\mu = c^2\partial_\mu T, where T=t+1c∫∞r(2μr′)1/2(1−2μr′)−1dr′.T = t + \frac{1}{c}\int_\infty^r \left(\frac{2\mu}{r'}\right)^{1/2}\left(1-\frac{2\mu}{r'}\right)^{-1}dr'. Hence show that the line element of the Schwarzschild geometry in (T,r,θ,ϕ)(T, r, \theta, \phi) coordinates is given by ds2=c2dT2−(dr+2μc2/r dT)2−r2(dθ2+sin⁡2θ dθ2).ds^2 = c^2 dT^2 - \left(dr + \sqrt{2\mu c^2/r}\,dT\right)^2 - r^2\left(d\theta^2 + \sin^2\theta\,d\theta^2\right). Is this new form singular at r=2μr = 2\mu? What can you say about the hypersurface T=constantT = \text{constant}? Show that observers infalling radially from rest at infinity have T˙=1\dot{T} = 1 and hence give a physical interpretation of the TT coordinate.

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