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

156 Detection of scalar gravitational waveswhere N collectively denotes the set of quantum numbers identifying the mode.The basic equation governing the response of the detector is [12]Ä N (t) + τN −1 ȦN (t) + ω 2 N A N (t) = f N (t). (11.14)We assume that the gravitational interaction obeys the principle of equivalencewhich has been experimentally supported to high accuracy. In terms of the socalledelectric components of the Riemann tensor E ij ≡ R 0i0 j , the driving forcef N (t) is then given by [13]∫f N (t) =−M −1 E ij (t)ψ i∗N (⃗x)x j ρ d 3 x (11.15)where M is the sphere mass and we consider the density ρ as a constant. In anymetric theory of gravity E ij is a (3×3) symmetric tensor, which depends on time,but not on spatial coordinates.Let us now investigate which sphere eigenmodes can be excited by a metricGW, i.e. which sets of quantum numbers N give a non-zero driving force.(a) Toroidal modesThe eigenmode vector, ψnlm T can be expressed as in equation (11.6). Up to anadimensional normalization constant C, the driving force is∫f (T)N (t) =−e−iω N t 3C R∫ π4π R 3 drr 3 j l (k (T)nlr) dθ sin θ00∫ 2π{ (Eyy − E xx× dφsin θ sin 2φ ∂Y lm∗02∂θ+ cos θ cos 2φ ∂Y lm∗ )∂φ(+ E xy sin θ cos 2φ ∂Y lm∗− cos θ sin 2φ ∂Y ∗ )lm∂θ∂φ[+ E xz − sin φ cos θ ∂Y lm∗+ (sin θ cos φ − cos2 θ∗ ]∂θsin θ cos φ)∂Y lm∂φ[+ E yz cos φ cos θ ∂Y lm∗+ (sin θ sin φ − cos2 ∗ ]θlmsin φ)∂Y∂θsin θ ∂φ()+cos θ ∂Y ∗ }. (11.16)Using the equationsE zz − E xx + E yy2lm∂φand∂Y ∗lm∂θ∂Y ∗lm= (−) m [ 2l + 14π] 1(l − m)! 2 ∂ Pm(l + m)![ 2l + 1 (l − m)!∂φ =−im(−)m 4π (l + m)!l(cos θ)e −imφ (11.17)∂θ] 12Pml (cos θ)e −imφ (11.18)