Limitations of the Standard Gravitational Perfect Fluid Paradigm

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. A set of wave mode rays that would all be geodesic in flat space (where there are no intrinsic length scales) and form a flat space perfect fluid would not all remain geodesic or of the perfect fluid form when the system is covariantized to curved space. 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. 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 Tμν = (ρ + p)UμUν + pgμν + qπμν where πμν is a symmetric, traceless rank two tensor which obeys Uπμν = 0. Such a qπμν type term is absent for an incoherently averaged steady state fluid in a spacetime where there are no intrinsic length scales, and would thus be missed in a covariantizing of a flat spacetime Tμν . However, in a general relativistic treatment of systems such as stars, such qπμν type terms cannot automatically be ignored, with their contribution in the strong gravity limit potentially being significant.