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 consider the carrier letter
or
), or
. Always understood in expressions such as Equation (25) are
summations over the
indices
ranging from 1 to 3. The derivative operator
is a short-hand for
. The function
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. [210
] and Appendices A and B in Ref. [26
] 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
; an
expansion into STF tensors
is equivalent to the usual expansion in spherical harmonics
.
Similarly, we denote
and
. Superscripts like
indicate
successive
time-derivations.