General Relativity: An Introduction for Physicists · Problem 9.24

The Schwarzschild geometry — Hobson, Efstathiou & Lasenby

Physics · photon-geodesics-equatorial-plane

Problem

Show that the geodesic equations for photon motion in the equatorial plane θ=π/2\theta = \pi/2 of the Schwarzschild geometry can be written in the form t˙=1bc(1−2μr)−1,ϕ˙=1r2,r˙2=1b2−U(r),\dot{t} = \frac{1}{bc}\left(1-\frac{2\mu}{r}\right)^{-1}, \quad \dot{\phi} = \frac{1}{r^2}, \quad \dot{r}^2 = \frac{1}{b^2} - U(r), where bb is a constant, the dots correspond to differentiation with respect to some affine parameter and U(r)=1r2(1−2μr).U(r) = \frac{1}{r^2}\left(1-\frac{2\mu}{r}\right). Suppose that a photon moving in the equatorial plane passes an observer at rest at a coordinate radius in the range 2μ<r<3μ2\mu < r < 3\mu. Show that the observer measures the radial and azimuthal components of the photon's velocity to be vr=±c[1−b2U(r)]1/2andvϕ=cb[U(r)]1/2.v_r = \pm c\left[1 - b^2 U(r)\right]^{1/2} \quad \text{and} \quad v_\phi = cb[U(r)]^{1/2}. If the observer emits a photon that makes an angle α\alpha with the outward radial direction, show that the photon will escape to infinity provided that sin⁡α<33μ[U(r)]1/2.\sin\alpha < 3\sqrt{3}\mu[U(r)]^{1/2}. Find the values of α\alpha at r=2μr = 2\mu and r=3μr = 3\mu.

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