holes[^1].

Thick discs
===========

Basic properties of thick discs are determined by equilibrium configurations of perfect fluid. Stress-energy tensor of perfect fluid is given by $$T^\mu_{\hphantom{\mu}\nu} = (p+\epsilon) U^\mu U_\nu + p\,\delta^\mu_\nu
\label{eq1}$$ where $\epsilon$ and $p$ denote total energy density and pressure of the fluid, $U^{\mu}$ is its four velocity. We shall consider test perfect fluid rotating in the $\phi$ direction, i.e., $U^{\mu} = \left( U^{t}, U^{\phi}, 0, 0\right)$. The rotating fluid can be characterized by the vector fields of the angular velocity $\Omega \left( r, \theta \right)$ and the angular momentum density $\ell \left( r, \theta \right)$, defined by $$\Omega = \frac{U^{\phi}}{U^{t}}, \qquad \ell = - \frac{U_{\phi}}{U_{t}}.
\label{eq2}$$ The vector fields are related by the metric coefficients of the KdS spacetime $$\Omega = - \frac{g_{t\phi} + \ell g_{tt}}{g_{\phi\phi} + \ell g_{t\phi}}.
\label{eq3}$$

Projecting the energy-momentum conservation law $T^{\mu\nu}_{\hphantom{\mu\nu};\nu} = 0$ onto the hypersurface orthogonal to the four velocity $U^\mu$ by the projection tensor $h_{\mu\nu} = g_{\mu\nu} +
U_\mu U_\nu$, we obtain the relativistic Euler equation in the form $$\frac{\partial_{\mu} p}{p+\epsilon} =
    -\partial_{\mu} (\ln U_t) +
    \frac{\Omega\,\partial_{\mu} \ell}{1- \Omega \ell},          \label{eqv6}$$ where $$(U_t)^2 =
    \frac{g^2_{t\phi} - g_{tt}\,g_{\phi\phi}}%
         {g_{\phi\phi} + 2 \ell g_{t\phi} + \ell^2 g_{tt}}.      \label{eqv7}$$

For barytropic perfect 