General Relativity: An Introduction for Physicists · Problem 9.13

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · radial-infall-with-initial-velocity

Problem

A particle at infinity in the Schwarzschild geometry is moving radially inwards with coordinate speed u0u_0. Show that at any coordinate radius rr the coordinate velocity is given by (drdt)2=(1−2GMc2r)2c2[1−1γ02(1−2GMc2r)],\left(\frac{dr}{dt}\right)^2 = \left(1-\frac{2GM}{c^2 r}\right)^2 c^2\left[1 - \frac{1}{\gamma_0^2}\left(1-\frac{2GM}{c^2 r}\right)\right], where γ0=(1−u02/c2)−1/2\gamma_0 = \left(1-u_0^2/c^2\right)^{-1/2}. Determine the velocity relative to a stationary observer at rr, and show that this velocity tends to cc as rr tends to 2GM/c22GM/c^2, irrespective of the value of u0u_0.
Intuitive explanation

The key is to use energy conservation in curved spacetime: the particle's four-velocity norm is invariant, so the energy per unit rest mass measured at infinity determines the motion everywhere else. Start by writing the normalization condition gμνuμuν=−c2g_{\mu\nu}u^\mu u^\nu = -c^2 in Schwarzschild coordinates, using the fact that at infinity the particle has coordinate speed u0u_0 and the metric is flat. This gives you a conserved quantity that relates the coordinate velocity at radius rr to the initial conditions—no integration needed, just algebra. For the local velocity, transform to the rest frame of a stationary observer at rr using the metric's time-dilation factor.

Intuitive explanation

The key is to use energy conservation in curved spacetime: the particle's four-velocity norm is invariant, so the energy per unit rest mass measured at infinity determines the motion everywhere else. Start by writing the normalization condition gμνuμuν=−c2g_{\mu\nu}u^\mu u^\nu = -c^2 in Schwarzschild coordinates, using the fact that at infinity the particle has coordinate speed u0u_0 and the metric is flat. This gives you a conserved quantity that relates the coordinate velocity at radius rr to the initial conditions—no integration needed, just algebra. For the local velocity, transform to the rest frame of a stationary observer at rr using the metric's time-dilation factor.

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