In vacuum, consider the plane-wave perturbation h_ =(k_ _ +k_ _ ) ,e^ ik_ x^ , with…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Gravitational waves and gravitational-wave astronomy · Gravitational waves and gravitational-wave astronomy

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

Problem

In vacuum, consider the plane-wave perturbation hμν=(kμϵν+kνϵμ) eikαxα,h_{\mu\nu}=(k_\mu\epsilon_\nu+k_\nu\epsilon_\mu)\,e^{ik_\alpha x^\alpha}, with kαkα=0k^\alpha k_\alpha=0 and ϵμ\epsilon_\mu four arbitrary constants (real parts are taken at the end, as for any plane wave). ((a)) Find the trace hh and the trace-reversed hˉμν\bar h_{\mu\nu}, and show that hˉμν\bar h_{\mu\nu} satisfies both □hˉμν=0\Box\bar h_{\mu\nu}=0 and the Lorenz condition ∂μhˉμν=0\partial^\mu\bar h_{\mu\nu}=0 for every choice of ϵμ\epsilon_\mu. ((b)) Show that the whole perturbation is produced from hμν=0h_{\mu\nu}=0 by the coordinate change xμ→xμ+ζμx^\mu\to x^\mu+\zeta^\mu with ζν=iϵν eikαxα\zeta_\nu=i\epsilon_\nu\,e^{ik_\alpha x^\alpha}, and that this ζν\zeta_\nu satisfies □ζν=0\Box\zeta_\nu=0. ((c)) What does part (b) say about the physical content of such a wave? How many independent amplitudes of this form are there, and how does that number fit the count of plane-wave amplitudes in the chapter?

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