General Relativity: An Introduction for Physicists · Problem 13.21

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-sagnac-effect-axisymmetric-metric

Problem

The general axisymmetric stationary metric can be written in the form ds2=A dt2−B(dϕ−ω dt)2−C dr2−D dθ2,ds^2 = A\, dt^2 - B(d\phi - \omega\, dt)^2 - C\, dr^2 - D\, d\theta^2, where AA, BB, CC, DD and ω\omega are functions only of the coordinates rr and θ\theta. Alice is an astronaut in a powered spaceship that maintains fixed (r,ϕ)(r, \phi) coordinates in the equatorial plane θ=π/2\theta = \pi/2 (at a position for which gtt>0g_{tt} > 0). She simultaneously emits two photons in opposite tangential directions in the equatorial plane and uses a prearranged system of mirrors to cause each photon to move along a circular (non-geodesic) path of constant rr. Show that the coordinate angular velocities of the two photons are given by dϕdt=ω±(AB)1/2.\frac{d\phi}{dt} = \omega \pm \left(\frac{A}{B}\right)^{1/2}. Hence show that the two photons do not arrive back with Alice simultaneously but are separated by a time interval Δτ=4πωBc(A−Bω2)1/2,\Delta\tau = \frac{4\pi\omega B}{c(A - B\omega^2)^{1/2}}, as measured by Alice's on-board clock. Comment on the physical significance of this result.

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