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
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,
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 ):
The unit vector satisfies the following integral identities:
Living Rev. Relativity 14, (2011), 7
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