Outside a spherically symmetric mass

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Newtonian gravitation as a field theory

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

Outside a spherically symmetric mass, the Newtonian potential Φ\Phi obeys ∇2Φ=4πGρm\nabla^2\Phi = 4\pi G\rho_{\rm m}, and in vacuum ρm=0\rho_{\rm m}=0, so δij∂i∂jΦ=0\delta^{ij}\partial_i\partial_j\Phi = 0: the tidal tensor ∂i∂jΦ\partial_i\partial_j\Phi is traceless in vacuum. Someone argues: 'Both sides of ∂i∂jΦ=13δij(δkl∂k∂lΦ)\partial_i\partial_j\Phi = \tfrac13\delta_{ij}(\delta^{kl}\partial_k\partial_l\Phi) carry two lower spatial indices i,ji,j and transform the same way under rotations, so the indices balance. Since the trace δkl∂k∂lΦ\delta^{kl}\partial_k\partial_l\Phi is zero in vacuum, index balance guarantees that ∂i∂jΦ=0\partial_i\partial_j\Phi = 0 for every pair i,ji,j separately, everywhere outside the mass.' ((a)) Take AiA^i and BjB^j to be two arbitrary constant vectors under rotations. Use the quotient rule to justify that ∂i∂jΦ\partial_i\partial_j\Phi is a genuine (0,2)(0,2) tensor under spatial rotations. ((b)) Find the error in the concluding claim, and refute it numerically: at the Earth's surface, r=R⊕=6.371×106r = R_\oplus = 6.371\times10^6 m, with GM⊕=3.990×1014 m3 s−2GM_\oplus = 3.990\times10^{14}\ \mathrm{m^3\,s^{-2}}, the point-mass tidal tensor along the radial direction n^\hat n is ∂i∂jΦ=(GM⊕/r3)(δij−3n^in^j)\partial_i\partial_j\Phi = (GM_\oplus/r^3)(\delta_{ij} - 3\hat n_i\hat n_j). Evaluate the radial component (i=j=i=j= radial) and one transverse component, and show neither is zero even though the trace is.

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