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

Gravitational wave radiation in the JBD theory 161In equation (11.35), T µν is the matter stress–energy tensor and t µν is thegravitational stress–energy pseudotensor, that is a function of quadratic order inthe weak gravitational fields θ µν and ξ. The reason why we have written the fieldequations at the quadratic order in θ µν and ξ is that in this way, as we will seelater, the expressions for θ µν and ξ include all the terms of order (v/c) 2 , where vis the typical velocity of the source (Newtonian approximation).Let us now compute τ 00 and S at the order (v/c) 2 . Introducing theNewtonian potential U produced by the rest-mass density ρ∫ ρ(⃗x ′ , t)U(⃗x, t) =|⃗x −⃗x ′ | d3 x ′ (11.37)the total pressure p and the specific energy density (that is the ratio of energydensity to rest-mass density) we get (for a more detailed derivation, see [7]):τ 00 = 1 ρ, (11.38)ϕ 0TS ≃−(1 − 1 2(2ω BD + 3) 2 θ − 2 ξ )ϕ 0(ρ=1 + − 3 p 2(2ω BD + 3ρ + 2ω )BD + 1ω BD + 2 U . (11.39)Far from the source, equations (11.33) and (11.34) admit wave-like solutions,which are superpositions of terms of the formθ µν (x) = A µν (⃗x,ω)exp(ik α x α ) + c.c. (11.40)ξ(x) = B(⃗x,ω)exp(ik α x α ) + c.c. (11.41)Without affecting the gauge condition (11.32), one can impose h =−2ξ/ϕ 0 (sothat θ µν = h µν ). Gauging away the superflous components, one can write theamplitude A µν in terms of the three degrees of freedom corresponding to stateswith helicities ±2 and 0 [19]. For a wave travelling in the z-direction, one thusobtains⎛⎞0 0 0 0⎜ 0 eA µν = 11 − b e 12 0 ⎟⎝⎠ , (11.42)0 e 12 −e 11 − b 00 0 0 0where b = B/ϕ 0 .11.3.2 Power emitted in GWsThe power emitted by a source in GWs depends on the stress–energy pseudotensort µν according to the following expression∫∫P GW = r 2 d = r 2 〈t 0k 〉ˆx k d (11.43)