A radio transmitter sits at the base of a m tower and a receiver at the top

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · The equivalence principle and gravity as geometry · The equivalence principle and gravity as geometry

The Geometry of Gravity, Ch Redshift, local inertial frames and the failure of flat spacetime

Problem

A radio transmitter sits at the base of a 45.045.0 m tower and a receiver at the top. Both are at rest in the Earth's weak, static field, so the metric component g00=−(1+2Φ/c2)g_{00} = -(1 + 2\Phi/c^2) applies with Φ=g⊕ℓ\Phi = g_\oplus \ell for height ℓ\ell measured from the base, g⊕=9.81 m s−2g_\oplus = 9.81\ \mathrm{m\,s^{-2}}. ((a)) Find the fractional frequency shift Δν/ν≡(νreceived−νemitted)/νemitted\Delta\nu/\nu \equiv (\nu_{\rm received}-\nu_{\rm emitted})/\nu_{\rm emitted} between the two ends of the tower. ((b)) State whether the received frequency is higher or lower than the emitted one, and say why.

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