Relativistic Hydrodynamics 2025, 16 - Geometrically Thick Tori

Assume a non-geodesic, perfect fluid and relativistic tori orbiting a black hole on a metric corresponding to a Kerr black hole in Boyer-Lindquist coords written in general form. The compatibility condition is automatically fulfulled from the Zeipel theorem. The effective potential defines the constant sepcific angular momentum, which is correlated to a strictly positive constant. U. W - W_i = -∫dp' / (ρh)

l = ±U = const, l_ms < U < l_mb

W(r, θ) = ln|u_t|

The classification of tori discerns between the energy gap at the innder edge ΔW_in. If ΔW_in ≥ 0, then a small amount of mass can be lost through the cusp, leading to an instability, as the tori's mass decreases, shifting its equilibrium position. This is referred to as the 'runaway instability'. This is generally the case for relativistic tori around Schwarzschild and Kerr black holes, if they're not self-gravitating. Assuming the sound speed is invariant under changes of the polytropic constant K, over the Schwarzschild metric in cylindrical coords, and retaining the 0th order terms. The accretion then only depends on kinematics and geometry of the accretion disc, only characterized by the angular coordinate. The resulting polytropic equation of state is p = KΠ^N_p+1, ψ = K(N_p + 1)Π. The sound speed becomes, invariant under transformations Π → Π/α. In Schwarzschild-de Sitter spacetime

For fixed cosmological parameters and angular momentum, the polytropic equation of state can be integrated analytically for all r ≤ r_s.