Any Cartesian tensor field depending on two angles spanning a sphere can be expanded in a unique decomposition in symmetric trace-free tensors. Such a decomposition is equivalent to a decomposition in tensorial harmonics, but it is sometimes more convenient. It begins with the fact that the angular dependence of a Cartesian tensor can be expanded in a series of the form

where and denote multi-indices and , angular brackets denote an STF combination of indices, is a Cartesian unit vector, , and . This is entirely equivalent to an expansion in spherical harmonics. Each coefficient can be found from the formula where the double factorial is defined by . These coefficients can then be decomposed into irreducible STF tensors. For example, for , we have where the ’s are STF tensors given by Similarly, for a symmetric tensor with , we have where These decompositions are equivalent to the formulas for addition of angular momenta, , which results in terms with angular momentum ; the superscript labels in these formulas indicate by how much each term’s angular momentum differs from .By substituting Eqs. (B.3) and (B.7) into Eq. (B.1), we find that a scalar, a Cartesian 3-vector, and the symmetric part of a rank-2 Cartesian 3-tensor can be decomposed as, respectively,

Each term in these decompositions is algebraically independent of all the other terms.We can also reverse a decomposition to “peel off” a fixed index from an STF expression:

In evaluating the action of the wave operator on a decomposed tensor, the following formulas are useful:

In evaluating the -component of the Lorenz gauge condition, the following formula is useful for finding the most divergent term (in an expansion in ):

And in evaluating the -component, the following formula is useful for the same purpose: where we have defined and .The unit vector satisfies the following integral identities:

where the curly braces indicate the smallest set of permutations of indices that make the result symmetric. For example, .

Living Rev. Relativity 14, (2011), 7
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