General Relativity: An Introduction for Physicists · Problem 11.2

Schwarzschild black holes — Hobson, Efstathiou & Lasenby

Physics · schwarzschild-isotropic-coordinates

Problem

By introducing the new coordinate ρ\rho defined by r=ρ(1+μ2ρ)2,r = \rho\left(1+\frac{\mu}{2\rho}\right)^2, show that the line element for the Schwarzschild geometry can be written in the isotropic form ds2=c2(1−μ2ρ)2(1+μ2ρ)−2dt2−(1+μ2ρ)4(dρ2+ρ2 dθ2+ρ2sin⁡2θ dϕ2).ds^2 = c^2\left(1-\frac{\mu}{2\rho}\right)^2\left(1+\frac{\mu}{2\rho}\right)^{-2}dt^2 - \left(1+\frac{\mu}{2\rho}\right)^4\left(d\rho^2+\rho^2\,d\theta^2+\rho^2\sin^2\theta\,d\phi^2\right). Show that g00≈1−2μ/ρg_{00}\approx 1-2\mu/\rho in the weak-field limit μ≪ρ\mu\ll\rho.

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