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

162 Detection of scalar gravitational waveswhere r is the radius of a sphere which contains the source, is the solid angle, is the energy flux and the symbol 〈···〉implies an average over a region of sizemuch larger than the wavelength of the GW. At the quadratic order in the weakfields we find〈t 0z 〉=−ẑ ϕ [ ]0c 4 4(ωBD + 1)〈(∂ 0 ξ)(∂ 0 ξ)〉+〈(∂ 0 h αβ )(∂ 0 h αβ )〉 . (11.44)32πϕ 2 0Substituting (11.40) and (11.41) into (11.44), one gets〈t 0z 〉=−ẑ ϕ [0c 4 ω 2 2(2ωBD + 3)|B| 2 + A αβ∗ A αβ − 1 ]16π2 |Aα α| 2 , (11.45)and using (11.42)〈t 0z 〉=−ẑ ϕ 0c 4 ω 28πϕ 2 0[|e 11 | 2 +|e 12 | 2 + (2ω BD + 3)|b| 2] . (11.46)From (11.46) we see that the purely scalar contribution, associated with b and thetraceless tensorial contribution, associated with e µν , are completely decoupledand can thus be treated independently.11.3.3 Power emitted in scalar GWsWe now rewrite the scalar wave solution (11.41) in the following wayξ(⃗x, t) = ξ(⃗x,ω)e −iωt + c.c. (11.47)In vacuo, the spatial part of the previous solution (11.47) satisfies the Helmholtzequation(∇ 2 + ω 2 )ξ(⃗x,ω)= 0. (11.48)The solution of (11.48) can be written asξ(⃗x,ω)= ∑ jmX jm h (1)j(ωr)Y jm (θ, ϕ) (11.49)where h (1)j(x) are the spherical Hankel functions of the first kind, r is the distanceof the source from the observer, Y jm (θ, ϕ) are the scalar spherical harmonicsand the coefficients X jm give the amplitudes of the various multipoles whichare present in the scalar radiation field. Solving the inhomogeneous waveequation (11.34), we find∫X jm = 16πiω j l (ωr ′ )Ylm ∗ (θ, ϕ)S(⃗x,ω)dV (11.50)Vwhere j l (x) are the spherical Bessel functions and r ′ is a radial coordinate whichassumes its values in the volume V occupied by the source.