General Relativity: An Introduction for Physicists · Problem 9.27

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · schwarzschild-with-cosmological-constant

Problem

If the cosmological constant Λ\Lambda is non-zero, show that the line element outside a static spherically symmetric matter distribution is given by ds2=c2(1−2μr−Λr23)dt2−(1−2μr−Λr23)−1dr2−r2(dθ2+sin⁡2θ dϕ2).ds^2 = c^2\left(1-\frac{2\mu}{r}-\frac{\Lambda r^2}{3}\right)dt^2 - \left(1-\frac{2\mu}{r}-\frac{\Lambda r^2}{3}\right)^{-1}dr^2 - r^2(d\theta^2 + \sin^2\theta\,d\phi^2). Hence show that, in the weak-field Newtonian limit, a spherically symmetric mass MM produces a gravitational field strength g⃗\vec{g} given by g⃗=(−GMr2+c2Λr3)r^.\vec{g} = \left(-\frac{GM}{r^2} + \frac{c^2\Lambda r}{3}\right)\hat{r}. Show that the shapes of massive particle orbits in the above geometry differ from those in the Schwarzschild geometry, but that the shapes of photon orbits do not.

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