Astron. Astrophys. 363, 425-439 (2000)

2. Boyer's condition for equilibrium configurations of test perfect fluid

We briefly summarize the well known results of a general theory of the equipotential surfaces inside any relativistic, differentially rotating, perfect fluid body (Boyer 1965; Abramowicz 1974), applied to test configurations of perfect fluid rotating in the stationary and axially symmetric spacetimes (Kozlowski et al. 1978; Abramowicz et al. 1978; Jaroszynski et al. 1980). In the standard coordinate system the spacetimes are described by the line element

where the metric coefficients depend neither on the time coordinate, t, nor the azimuthal coordinate, , i.e., the spacetimes contain timelike and azimuthal Killing vector fields and .

We shall consider test perfect fluid rotating in the direction. Its four velocity vector field has, therefore, only two non-zero components

which can be functions of the coordinates r, . The stress-energy tensor of the perfect fluid is

where and p denote the total energy density and the pressure of the fluid. The rotating fluid can be characterized by the vector fields of the angular velocity , and the angular momentum per unit mass (angular momentum density) , defined by

These vector fields are related by

In static spacetimes (), the relation (5) reduces to a simple formula

The surfaces of constant and are called von Zeipel's cylinders. The family of von Zeipel's cylinders does not depend on the assumed rotation law of the fluid, , in the static spacetimes, but it will depend on the rotation law in the stationary spacetimes (with ) (Kozlowski et al. 1978).

Projecting the energy conservation law onto the hypersurface orthogonal to the four velocity by the projection tensor , we obtain the relativistic Euler equation in the form

where

The solution of the relativistic Euler equation can be given by Boyer's condition determining the surfaces of constant pressure through the "equipotential surfaces" of the potential by the relations (Abramowicz et al. 1978)

the subscript "in" refers to the inner edge of the disk. For an alternative definition of Boyer's condition see (Abramowicz et al. 1978; Fishbone & Moncrief 1976; Fishbone 1977). The equipotential surfaces are determined by the condition

and in a given spacetime can be found from Eq. (10), if a rotation law is given. The surfaces of constant pressure are given by Eq. (9). The structure of thick accretion disks can be obtained also in the framework of a very practical and accurate Newtonian model for the gravitational field of a non-rotating black hole, known as the Paczynski-Wiita potential (Paczynski & Wiita 1980; Abramowicz et al. 1980).

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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