Our notation is the following: denotes a multi-index, made of (spatial) indices. Similarly, we write for
instance (in practice, we generally do not need to write explicitly the “carrier” letter or ), or
. Always understood in expressions such as Eq. (34) are summations over the indices
ranging from 1 to 3. The derivative operator is a short-hand for . The function (for any space-time
indices ) is symmetric and trace-free (STF) with respect to the indices composing . This means that for any pair
of indices , we have and that (see Ref. [403*] and
Appendices A and B in Ref. [57*] for reviews about the STF formalism). The STF projection is denoted with a hat,
so , or sometimes with carets around the indices, . In particular, is the
STF projection of the product of unit vectors , for instance and
; an expansion into STF tensors is equivalent to the usual
expansion in spherical harmonics , see Eqs. (75) below. Similarly, we denote where
, and . The Levi-Civita antisymmetric symbol is denoted (with ).
Parenthesis refer to symmetrization, . Superscripts indicate successive time
derivations.