Geodesic motions versus hydrodynamic flows in a gravitating perfect fluid:dynamical equivalence and consequences

Stimulated by the methods applied for the observational determination of ma sses in the central regions of the active galactic nuclei (AGN), we examine , in the context of the theory of general relativity, the exact conditions, under which, in the interior of a gravitating perfect-fluid source, the ge odesic motions and the adiabatic hydrodynamic Rows are dynamically equivale nt to each other. Dynamical equivalence rests on the functional similarity between the corresponding, covariantly expressed differential equations of motion and is obtained with the aid of a conformal transformation between t he metric tensors of the original fluid, on the one hand, and the so-called virtual fluid on the other. In the latter, the hydrodynamic flow motions a re formally the same as the geodesic motions. The conformal factor so obtai ned is written in terms of the specific enthalpy of the original fluid, and hence it is attributed a clear physical interpretation. The components of the virtual fluid's energy-momentum tensor are determined, through the inva riant held equations, in terms of the original fluid's corresponding quanti ties, the conformal factor and its spacetime derivatives. In the Newtonian limit, the extra contribution to the original energy density results in an extra inertial-energy density and hence in an extra mass, both of which are always non-vanishing. The associated results indicate that, in the determi nation of the masses in the central regions of the AGNs, the observationall y determined nuclear mass is being underestimated with respect to the real physical one, Accordingly, we evaluate the corresponding mass deficit, whic h, in typical cases of AGNs, is not always negligible compared with the mas s of the central dark object, and it can be comparable to the total rest ma ss of the circumnuclear gas involved. Finally, the implications of the resu lts are discussed, on the assumed form of the mass-density distribution law for the circumnuclear gas and the corresponding form of the extra inertial -energy density. We find that, under certain conditions, the density index is directly related to the polytropic index in the fluid's adiabatic equati on of state.