Notice that the condition of
being divergence-free depends on the metric
itself, which is not known before
actually solving the nonlinear wave equation (4.5), and the latter, in turn, depends on
. Therefore, one
cannot specify
by hand, except in the vacuum case,
, or in the case
with
the cosmological constant. In the more general case, the stress-energy tensor has to be computed from a
diffeomorphism-independent action for the matter fields and one has to consistently solve the coupled Einstein-matter
system.