A physicist at a mountain observatory of height m above sea level sends a light…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Symmetry: Lie derivatives, Killing vectors and conserved quantities

Problem

A physicist at a mountain observatory of height ℓ=1000\ell=1000 m above sea level sends a light signal straight down to a detector at sea level. Both the observatory and the detector are static (at rest at fixed rr) in the weak-field metric ds2=−(1+2Φc2)c2dt2+dr2+r2dΩ2,Φ(r)=−GM⊕r,ds^2=-\Big(1+\frac{2\Phi}{c^2}\Big)c^2dt^2+dr^2+r^2d\Omega^2,\qquad \Phi(r)=-\frac{GM_\oplus}{r}, with dΩ2d\Omega^2 the line element of the unit two-sphere. Using the photon's four-momentum pμp^\mu, the fact that ξ=∂t\xi=\partial_t is a Killing vector of this metric, and the local relation between a static observer's proper time and the coordinate time tt, find the fractional frequency shift Δν/ν=(νsea−νmountain)/νmountain\Delta\nu/\nu=(\nu_{\rm sea}-\nu_{\rm mountain})/\nu_{\rm mountain} recorded at sea level, to first order in g⊕ℓ/c2g_\oplus\ell/c^2, taking g⊕=9.8 m s−2g_\oplus=9.8\ \mathrm{m\,s^{-2}} as the local gravitational acceleration.

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