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

TL;DR

This paper explores the conditions under which geodesic motions and hydrodynamic flows in a gravitating perfect fluid are mathematically equivalent, revealing implications for mass estimates in active galactic nuclei.

Contribution

It demonstrates the dynamical equivalence between geodesic and hydrodynamic motions via conformal transformations and discusses its impact on mass measurements in AGNs.

Findings

Geodesic and hydrodynamic equations are functionally similar under conformal transformations.

Solution spaces for both motions are isomorphic, allowing one to be derived from the other.

Mass overestimation in AGNs can be significant, sometimes exceeding the mass of the circumnuclear gas.

Abstract

Stimulated by the methods applied for the observational determination of masses in the central regions of the AGNs, we examine the conditions under which, in the interior of a gravitating perfect fluid source, the geodesic motions and the general relativistic hydrodynamic flows are dynamically equivalent to each other. Dynamical equivalence rests on the functional similarity between the corresponding (covariantly expressed) differential equations of motion and is obtained by conformal transformations. In this case, the spaces of the solutions of these two kinds of motion are isomorphic. In other words, given a solution to the problem "hydrodynamic flow in a perfect fluid", one can always construct a solution formally equivalent to the problem "geodesic motion of a fluid element" and vice versa. Accordingly, we show that, the observationally determined nuclear mass of the AGNs is being overestimated with respect to the real, physical one. We evaluate the corresponding mass-excess and show that it is not always negligible with respect to the mass ofthe central dark object, while, under circumstances, can be even larger than the rest-mass of the circumnuclear gas involved.