By Barus C.
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Extra info for Acoustic Pressures in Case of Soap Bubbles
For such ωα the variational equation therefore reduces to ⎛ 1 (2π)4 ds Dk ⎞ (−i)n α⎝ n≥3 n! kγ · · · kβ t γ···βα ⎠ = 0 (108) with α = ExpA ωα . The t’s in the summation are those on which the orthogonality condition (100) was imposed. By (85) they are also symmetric on all the indices contracted with k’s. The method of Sect. 8 may therefore be applied to show that the k-space integral in (108) depends on ωα only through its value on (s), where as in that section, (s) is the hypersurface formed by all geodesics through z(z) orthogonal to n α .
The other contribution to the variational equation (81) is M αβ [ αβ ] given by (86), so with (68) that equation now takes the form F α ωα + (m βα + 21 L βα )∇β ωα (z) + m γβα ∇γβ ωα (z) ds = 0 (115) The New Mechanics of Myron Mathisson and Its Subsequent Development 31 for all smooth vector fields ωα on M of compact support. Interestingly, this is Mathisson’s original version in the case when only monopole and dipole moments are retained, but with the addition of external force and couple terms F α and L βα .
Choose a Minkowskian coordinate system on Tz(s) (M) such that n α = 0 only for α = 4. Then (56) and (61) show that m αβ (s, k) is independent of k4 . The inversion formula for the Fourier transform f (k) of a function f (x) of a single variable shows that f (k) dk = 2π f (0). (62) With this, the k4 integration in the s-integrand on the right of equation (55) can be performed to show that its value for fixed s depends on αβ (z(s), X ) only through its value on the hyperplane X 4 = 0, and hence on ϕαβ only through its value on (s), the image of this hyperplane under Expz .