Interior structure of rotating black holes. I. Concise derivation

This paper presents a concise derivation of a new set of solutions for the interior structure of accreting, rotating black holes. The solutions are conformally stationary, axisymmetric, and conformally separable. Hyper-relativistic counter-streaming between freely-falling collisionless ingoing and outgoing streams leads to mass inflation at the inner horizon, followed by collapse. The solutions fail at an exponentially tiny radius, where the rotational motion of the streams becomes comparable to their radial motion. The papers provide a fully nonlinear, dynamical solution for the interior structure of a rotating black hole from just above the inner horizon inward, down to a tiny scale.

Figure 1

Partial Penrose diagram illustrating why the Kerr geometry is subject to the inflationary instability. Ingoing and outgoing streams just outside the inner horizon must pass through separate ingoing and outgoing inner horizons into causally separated pieces of spacetime where the timelike Kerr time coordinate $t$ goes in opposite directions. To accomplish this, the ingoing and outgoing streams must exceed the speed of light through each other, which physically they cannot do. In reality, hyper-relativistic counter-streaming between the ingoing and outgoing streams ignites and then drives the exponentially growing inflationary instability. The inset shows the direction of coordinate time $t$ in the various regions. Proper time of course always increases upward in a Penrose diagram.

Figure 2

Contours of constant radius $x$ and latitude $y$ in an uncharged black hole with spin parameter $a=0.96M_{•}$. The thicker contours mark the outer and inner horizons. The left panel depicts a Kerr black hole. The right panel depicts a black hole of the kind considered in the present series of papers, which undergoes inflation just above the inner horizon, then collapses. In the Kerr geometry, surfaces of constant radius are confocal ellipsoids in Boyer-Lindquist coordinates, while surfaces of constant latitude are confocal hyperboloids, with a ring singularity at their focus. In the inflationary geometry, the streaming energy density and pressure, and Weyl curvature, inflate to exponentially huge values at (just above) the inner horizon, which is destroyed. In the conformally separable solutions presented here, the geometry then collapses radially to exponentially tiny size without changing shape.

Figure 3

Evolution of the geometry and energy-momenta from electrovac through inflation and collapse. (Left) The parameter $U_x\equiv -{\Delta }_{x}d\xi /dx$ and the horizon function ${\Delta }_x$, Eqs. (28), as a function of the inflationary exponent $\xi$, for parameters $v=0.001$, $u=0.002$, and ${\Delta }_x^{\prime }=1$ (the solutions in this paper apply in the limit of tiny $v$ and $u$; small finite values are adopted in this plot to avoid numerical overflow). Inflation ignites as the horizon function $|{\Delta }_x|$ decreases below $U_x$. Inflation ends as the absolute value of the horizon function goes through a minimum, and the geometry proceeds to collapse. Once $|{\Delta }_x|\gtrsim 1$, the angular components of the collisionless streams exceed their radial components, and the solution breaks down, but this happens only after the geometry has collapsed to exponentially tiny scale. (Right) The tetrad-frame radial, radial-angular, and angular collisionless energy-momenta $T_{xx}\propto {\rho }^{-2}{U}_x/|{\Delta }_x|$, $T_{xy}\propto {\rho }^{-2}{U}_x/\sqrt{|{\Delta }_x|}$, and $T_{yy}\propto {\rho }^{-2}{U}_x$. The energy-momenta grow exponentially huge despite their small initial values. Indeed, the smaller the initial energy-momenta, the faster and larger they grow. The dashed line is minus the polar (real) spin-0 component of the Weyl curvature, $-C\propto {\rho }^{-2}{U}_x^2/|{\Delta }_x|$. The axial (imaginary) spin-0 Weyl component is comparable to $T_{yy}$.

Figure 4

Angular flow pattern of freely-falling particles that produces the conditions (22) at the inner horizon required if conformal separability is imposed to subdominant radial-angular order, for an uncharged black hole with spin parameter $a=0.96M_{•}$. The thicker contours mark the outer and inner horizons, the latter being destroyed by inflation. Ingoing and outgoing particles fall along the same trajectories in the $x\mathrm{\text{−}}y$ plane, but have opposite motions in the azimuthal $\varphi$ coordinate. Trajectories near the equatorial plane change from ingoing to outgoing, or vice versa, inside the outer horizon; the transition is marked by the lines changing from dashed to solid. The angular flow pattern cannot be achieved with collisionless streams that fall from outside the outer horizon. In the equatorial region, outgoing but not ingoing particles can fall from outside the outer horizon, while in the polar region, ingoing but not outgoing particles can fall from outside the outer horizon.