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

0Testing theories of gravity 157the integration over φ can be performed. Equation (11.16) then contains integralsover θ of the form:∫ π[(sin 2 θ − cos 2 θ)Pl ±1 (cos θ)− sin θ cos θ ∂ ]P±1 l(cos θ)dθ (11.19)∂θand∫ π0[2 sin θ cos θ Pl ±2 (cos θ)+ sin 2 θ ∂ P±2∂θl(cos θ)]dθ. (11.20)After integration by parts, the derivative terms in equations (11.19) and (11.20)exactly cancel the non-derivative ones. The remaining boundary terms vanish too,thanks to the periodicity of the trigonometric functions and to the regularity of theassociated Legendre polynomials. The vanishing of these integrals has a profoundphysical consequence. It means that in any metric theory of gravity the toroidalmodes of the sphere cannot be excited by GW and can thus be used as a veto inthe detection.(b) Spheroidal modesThe forcing term is given by:∫f (S)N(t) =−M−1 E ij (t)− B N (r)e ink x nr L kY lm (θ, ϕ)( xx j ir A N (r)Y lm (θ, ϕ))ρ d 3 x. (11.21)One is thus led to compute integrals of the following types∫x j x i Y lm (θ, ϕ) d 3 x (11.22)and∫x j x i L k Y lm (θ, ϕ) d 3 x (11.23)Since the product x i x j can be expressed in terms of the spherical harmonics withl = 0, 2 and the angular momentum operator does not change the value of l,one immediately concludes that in any metric theory of gravity only the l = 0, 2spheroidal modes of the sphere can be excited. At the lowest level there are a totalof five plus one independent spheroidal modes that can be used for GW detectionand study.11.2.3 Measurements of the sphere vibrations and wave polarization statesFrom the analysis of the spheroidal modes active for metric GW, we now want toinfer the field content of the theory. For this purpose it is convenient to expressthe Riemann tensor in a null (Newman–Penrose) tetrad basis [7].