Interior structure of rotating black holes. II. Uncharged black holes

A solution is obtained for the interior structure of an uncharged rotating black hole that accretes a collisionless fluid. The solution is conformally stationary, axisymmetric, and conformally separable, possessing a conformal Killing tensor. The solution holds approximately if the accretion rate is small but finite, becoming more accurate as the accretion rate tends to zero. Hyper-relativistic counter-streaming between collisionless ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. As ingoing and outgoing streams approach the inner horizon, they focus into twin narrow beams directed along the ingoing and outgoing principal null directions, regardless of the initial angular motions of the streams. The radial energy-momentum of the counter-streaming beams gravitationally accelerates the streams even faster along the principal directions, leading to exponential growth in the streaming density and pressure, and in the Weyl curvature and mass function. At exponentially large density and curvature, inflation stalls, and the spacetime collapses. As the spacetime collapses, the angular motions of the freely-falling streams grow. When the angular motion has become comparable to the radial motion, which happens when the conformal factor has shrunk to an exponentially tiny scale, conformal separability breaks down, and the solution fails. The condition of conformal separability prescribes the form of the ingoing and outgoing accretion flows incident on the inner horizon. The dominant radial part of the solution holds provided that the densities of ingoing and outgoing streams incident on the inner horizon are uniform, independent of latitude; that is, the accretion flow is “monopole.” The subdominant angular part of the solution requires a special nonradial pattern of angular motion of streams incident on the inner horizon. The prescribed angular pattern cannot be achieved if the collisionless streams fall freely from outside the horizon, so the streams must be considered as delivered ad hoc to just above the inner horizon.

Figure 1

Comparison of the condition on the average ratio of $P_t$ to $P_{\varphi }$ at the inner horizon required by conformal separability, Eq. (160), to the constraint on the same ratio imposed by the condition that particles free-fall from outside the outer horizon, Eq. (161), for the case of an uncharged black hole with angular momentum parameter $a=0.96M_{•}$. In the equatorial region $\cos \theta \sim 0$, outgoing particles (positive $P_t$ and $P_{\varphi }$) satisfy the constraint but ingoing particles (negative $P_t$ and $P_{\varphi }$) do not. Conversely in the polar regions $|\cos \theta |\sim 1$, ingoing particles satisfy the constraint but outgoing particles do not.