A tower of height stands on Mars

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 Accelerated observers, Rindler coordinates and the equivalence principle

Problem

A tower of height ℓ=50 m\ell = 50\ \mathrm{m} stands on Mars, where the surface free-fall acceleration is g⋆=3.71 m s−2g_\star = 3.71\ \mathrm{m\,s^{-2}} and the radius is R⋆=3.390×106 mR_\star = 3.390\times10^{6}\ \mathrm{m}. ((a)) Using the equivalence principle (treating the tower's base as momentarily equivalent to a rocket with proper acceleration a=g⋆a=g_\star), predict the fractional clock-rate difference between the top and the bottom of the tower. ((b)) The Newtonian tidal field for a point mass is Err=∂2Φ/∂r2=−2GM/r3\mathcal{E}_{rr} = \partial^2\Phi/\partial r^2 = -2GM/r^3. Using g⋆=GM/R⋆2g_\star = GM/R_\star^2, write Err\mathcal{E}_{rr} in terms of g⋆g_\star and R⋆R_\star, and use it to estimate the fractional variation of the gravitational field over the tower's height, ∣δg/g⋆∣≈∣Err∣ ℓ/g⋆|\delta g/g_\star| \approx |\mathcal{E}_{rr}|\,\ell/g_\star. ((c)) Compare the two numbers from parts (a) and (b) and state what the second one tells about the reliability of the first.

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