  We show that the standard perfect fluid paradigm is not necessarily a valid
description of a curved space steady state gravitational source. Simply by
virtue of not being flat, curved space geometries have to possess intrinsic
length scales, and such length scales can affect the fluid structure. For modes
of wavelength of order or greater than such scales eikonalized geometrical
optics cannot apply and rays are not geodesic. Covariantizing thus entails not
only the replacing of flat space functions by covariant ones, but also the
introduction of intrinsic scales that were absent in flat space. In principle
it is thus unreliable to construct the curved space energy-momentum tensor as
the covariant generalization of a geodesic-based flat spacetime energy-momentum
tensor. By constructing the partition function as an incoherent average over a
complete set of modes of a scalar field propagating in a curved space
background, we show that for the specific case of a static, spherically
symmetric geometry, the steady state energy-momentum tensor that ensues will in
general be of the form
$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}+\pi_{\mu\nu}$ where the
anisotropic $\pi_{\mu\nu}$ is a symmetric, traceless rank two tensor which
obeys $U^{\mu}\pi_{\mu\nu}=0$. Such a $\pi_{\mu\nu}$ type term is absent for an
incoherently averaged steady state fluid in a spacetime where there are no
intrinsic length scales, and in principle would thus be missed in a
covariantizing of a flat spacetime $T_{\mu\nu}$. While the significance of such
$\pi_{\mu\nu}$ type terms would need to be evaluated on a case by case basis,
through the use of kinetic theory we reassuringly find that the effect of such
$\pi_{\mu\nu}$ type terms is small for weak gravity stars where perfect fluid
sources are commonly used.
