Tensor Spherical and Pseudo-Spherical Harmonics in Four-Dimensional Spaces

Abstract

Recently the large-scale anisotropy in the 2_7 K radiation has been detected in serveral astronomical observations. I) It seems to be due to primordial perturbations in Friedmann universe models_ As was shown by Lifshitz;) they can be classified into three types, that is, acoustic, rotational and gravitationalwave perturbations, and expanded in terms of scalar, vector and second-order tensor spherical harmonics, respectively, which were defined on the 3-sphere in the four-dimensional Euclidean space E •. Lifshitz and Khalatnikov ) derived the equations satisfied by each kind of spherical harmonics and gave the explicit expressions of the solutions only in the simplest cases. In order to derive them in all cases, it is necessary to treat them systematically from the group-theoretical standpoint. The tensor harmonics on the 2sphere in the three-dimensional Euclidean space E3 have already been derived by Regge and Wheeler,.) and others:) In this note the vector and second-order tensor harmonics in the four-dimensional space are derived by the use of Regge and Wheeler's technique. N ow we consider a three-dimensional manifold M3 which is a subspace of E. and at the same time also a spacelike section (the cosmic time t=const) in a Friedmann closed universe. The line-element in M3 is expressed as ds 2= Yapdxadx p = dx2 +sinx( d82+ sin28d<p2), (1) where x1=x, x 2=8 and x 3 =<p. The lineelement in E. is given by dr2 + r2 ds2 (the fourth coordinate xo= r). The conditions for the harmonics are shown as follows :3) For scalar harmonics Q