In some weak-field coordinates the trace-reversed perturbation has a single non-zero…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation

Problem

In some weak-field coordinates the trace-reversed perturbation has a single non-zero component, hˉ11=κx1\bar h_{11}=\kappa x^1, with κ=1.0×10−6 m−1\kappa=1.0\times10^{-6}\ \mathrm{m^{-1}}, over the region ∣x1∣≤500 m|x^1|\le500\ \mathrm{m}, and nothing depends on x0x^0. Under a small coordinate change xμ→xμ+ζμx^\mu\to x^\mu+\zeta^\mu the trace-reversed perturbation shifts as hˉμν→hˉμν−∂μζν−∂νζμ+ημν∂λζλ.\bar h_{\mu\nu}\to\bar h_{\mu\nu}-\partial_\mu\zeta_\nu-\partial_\nu\zeta_\mu+\eta_{\mu\nu}\partial^\lambda\zeta_\lambda. ((a)) Show that under this shift ∂μhˉμν\partial^\mu\bar h_{\mu\nu} changes by −□ζν-\Box\zeta_\nu. ((b)) Evaluate ∂μhˉμν\partial^\mu\bar h_{\mu\nu} for the given perturbation, and say whether it satisfies the Lorenz condition. ((c)) Find a ζν\zeta_\nu that depends only on x1x^1, with only ζ1\zeta_1 non-zero, which imposes the Lorenz condition. List the non-zero components of hˉμν\bar h_{\mu\nu} afterwards, check that they satisfy the condition, and evaluate the largest of them in magnitude at x1=500 mx^1=500\ \mathrm{m}.

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