The stress-energy tensor is decomposed as
or, equivalently,
,
, and
where
denotes the operation of projection onto the spatial
hypersurface and it acts on all free indices. Note that the
matter terms in Eqs. (12
), (14
), and (15
) are defined with respect to a normal observer. This is in
contrast to the usual definition of
with respect to the rest frame of the fluid in
hydrodynamics.