General Relativity: An Introduction for Physicists · Problem 9.7

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · lemaitre-coordinates-schwarzschild

Problem

Show that the 'Lemaitre' line element ds2=c2 dw2−49[9μ2(z−cw)]2/3dz2−[9μ2(z−cw)2]2/3dΩ2,ds^2 = c^2\,dw^2 - \frac{4}{9}\left[\frac{9\mu}{2(z-cw)}\right]^{2/3}dz^2 - \left[\frac{9\mu}{2}(z-cw)^2\right]^{2/3}d\Omega^2, where dΩ2=dθ2+sin⁡2θ dϕ2d\Omega^2 = d\theta^2 + \sin^2\theta\,d\phi^2, describes the Schwarzschild geometry. Show that observers with fixed spatial coordinates (z,θ,ϕ)(z,\theta,\phi) are in free fall and had zero velocity at infinity, and that the proper time of such observers is ww.
Intuitive explanation

The Lemaître coordinates rewrite Schwarzschild geometry in a form where the radial coordinate zz is comoving with freely falling observers—those who drop from rest at infinity follow surfaces of constant zz. The key insight is that by transforming away the coordinate singularity at the Schwarzschild radius, we expose the physical equivalence principle: observers at fixed spatial coordinates experience only gravitational acceleration (geodesic motion), not coordinate artifacts. Verifying this requires checking that the worldlines are geodesics and that ww itself measures proper time along these trajectories.

Intuitive explanation

The Lemaître coordinates rewrite Schwarzschild geometry in a form where the radial coordinate zz is comoving with freely falling observers—those who drop from rest at infinity follow surfaces of constant zz. The key insight is that by transforming away the coordinate singularity at the Schwarzschild radius, we expose the physical equivalence principle: observers at fixed spatial coordinates experience only gravitational acceleration (geodesic motion), not coordinate artifacts. Verifying this requires checking that the worldlines are geodesics and that ww itself measures proper time along these trajectories.

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