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

The hollow sphere 169and introduced the normalization constant C nl , which is fixed by the orthogonalityproperties ∫(u S n ′ l ′ m ′ ) ∗ · (u S nlm )ϱ 0 d 3 x = Mδ nn ′δ ll ′δ mm ′ (11.90)Vwhere M is the mass of the hollow sphere:M = 4π 3 ϱ 0 R 3 (1 − ς 3 ), ς ≡ a R≤ 1. (11.91)Equation (11.90) fixes the value of C nl through the radial integral∫ Rς R[A 2 nl (r) + l(l + 1)B2 nl (r)]r 2 dr = 4π 3 ϱ 0(1 − ς 3 )R 3 (11.92)as can be easily verified by using well known properties of angular momentumoperators and spherical harmonics. We shall later specify the values of thedifferent parameters appearing in the above expressions as required in eachparticular case which will in due course be considered. As seen in [9], ascalar–tensor theory of GWs such as JBD predicts the excitation of the sphere’smonopole modes as well as the m = 0 quadrupole modes. In order to calculatethe energy absorbed by the detector according to that theory it is necessary tocalculate the energy deposited by the wave in those modes, and this in turnrequires that we solve the elasticity equation with the GW driving term included inits right-hand side. The result of such a calculation was presented in full generalityin [9], and is directly applicable here because the structure of the oscillationeigenmodes of a hollow sphere is equal to that of the massive sphere—only theexplicit form of the wavefunctions needs to be changed. We thus haveE osc (ω nl ) = 1 2 Mb2 nll∑m=−l|G (lm) (ω nl )| 2 (11.93)where G (lm) (ω nl ) is the Fourier amplitude of the corresponding incoming GWmode, andb n0 = − ϱ ∫ R0A n0 (r)r 3 dr (11.94)M ab n2 = − ϱ ∫ R0[A n2 (r) + 3B n2 (r)]r 3 dr (11.95)M afor monopole and quadrupole modes, respectively, and A nl (r) and B nl (r) aregiven by (11.87). Explicit calculation yieldsb n0R = 3 C n04π 1 − ς 3 [(R) − ς 3 (a)] (11.96)b n2R = 3 C n24π 1 − ς 3 [(R) − ς 3 (a)] (11.97)