(ed.). Gravitational waves (IOP, 2001)(422s).

Using the TT gauge to understand gravitational waves 17To understand wave propagation we look for the easiest solution of the vacuumgravitational field equations:( )£ ¯h αβ ≡ − ∂2∂t 2 +∇2 ¯h αβ = 0. (2.6)Plane waves have the form:¯h αβ = e αβ exp(ik γ x γ ) (2.7)where the amplitude , polarization tensor e αβ and wavevector k γ are allconstants. (As usual one has to take the real part of this expression.)The Einstein equations imply that the wavevector is ‘light-like’, k γ k γ = 0,and the gauge condition implies that the amplitude and the wavevector areorthogonal: e αβ k β = 0.Linearized theory describes a classical gravitational field whose quantumdescription would be a massless spin 2 field that propagates at the speed oflight. We expect from this that such a field will have only two independentdegrees of freedom (helicities in quantum language, polarizations in classicalterms). To show this classically we remember that h αβ is symmetric, so it hasten independent components, and that the Lorentz gauge applies four independentconditions to these, reducing the freedom to six. However, the Lorentz gauge doesnot fully fix the coordinates. In fact if we perform another infinitesimal coordinatetransformation (x µ → x µ + ξ µ with ξ µ ,ν = O(h)) and impose £ξ µ = 0, weremain in Lorentz gauge. We can use this freedom to demand:e 0α = 0 ⇒ e ij k j = 0 (transverse wave), (2.8)e i i = 0 (traceless wave). (2.9)These conditions can only be applied outside a sphere surrounding the source.Together they put the metric into the transverse-traceless (TT) gauge. We willexplicitly construct this gauge in chapter 5.2.2 Using the TT gauge to understand gravitational wavesThe TT gauge leaves only two independent polarizations out of the original ten,and it ensures that h αβ = h αβ . In order to understand the polarization degrees offreedom, let us take the wave to move in the z-direction, so that k z = ω, k 0 = ω,k x = 0, k y = 0; the TT gauge conditions in equations (2.8) and (2.9) lead toe 0α = e zα = 0 and e xx =−e yy . This leaves only two independent componentsof the polarization tensor, say e xx and e xy (which we denote by the symbols⊕, ⊗).A wave for which e xy = 0 (pure ⊕ polarization) produces a metric of theform:ds 2 =−dt 2 + (1 + h + ) dx 2 + (1 − h + ) dy 2 + dz 2 , (2.10)