Mass inflation followed by Belinskii-Khalatnikov-Lifshitz collapse inside accreting, rotating black holes

Numerical evidence is presented that the Poisson-Israel mass inflation instability at the inner horizon of an accreting, rotating black hole is generically followed by Belinskii-Khalatnikov-Lifshitz oscillatory collapse to a spacelike singularity. The computation involves following all 6 degrees of freedom of the gravitational field. To simplify the problem, the computation takes as initial conditions the conformally separable solutions of Andrew J. S. Hamilton and Gavin Polhemus [Interior structure of rotating black holes. I. Concise derivation, Phys. Rev. D 84, 124055 (2011)] and Andrew J. S. Hamilton [Interior structure of rotating black holes. II. Uncharged black holes, Phys. Rev. D 84, 124056 (2011)] just above the inner horizon of a slowly accreting, rotating black hole and integrates the equations inward along single latitudes.

Figure 1

Evolution of (top panels) the lengths $a_a$ and (bottom panels) spatial rotation $U_{ab}$ [Eq. (50)] of the three axes of the spatial vierbein during BKL collapse inside an accreting, rotating black hole as a function of the mean scale factor $\overline{a}\equiv (a_1{a}_2{a}_3{)}^{1/3}$ (the cube root of the spatial volume element), along a trajectory into the black hole at 42° latitude. (Left and right panels) The two gauges considered in this paper, principal and BSSN (Sec. 3h). The accretion rates and spin of the black hole are given by Eqs. (51) and (53). The dashed lines are the approximate conformally separable solution from [17, 18], while the solid lines are from the numerical computation. The behavior is characteristic of BKL collapse: there are always two collapsing axes and one expanding axis. Power-law Kasner epochs are punctuated by BKL bounces in which one of the collapsing axes turns into expansion and the previously expanding axis turns into collapse.

Figure 2

Evolution of the Kasner exponents $q_a$ in the same model as Fig. 1, for the BSSN and principal gauges.

Figure 3

Hamiltonian constraint $(\mathrm{\Pi }-\kappa \tilde{T}{)}_{123\alpha \beta \gamma }$ [Eq. (56)] in the same model as Fig. 1. Shown for comparison are the (modified) energy-momentum tensor ${\tilde{T}}_{123\alpha \beta \gamma }$ and the potential energy term ${(\frac{1}{4}\mathit{e}\wedge [\mathbf{\Gamma },\mathbf{\Gamma }])}_{123\alpha \beta \gamma }$ in Eq. (8b). Lines are short dashed where values are negative.

Figure 4

(Left panel) Conformal radius $r$ in units of the conformal black hole mass $M$, and (right panel) horizon function ${\mathrm{\Delta }}_r$ (which is negative), in the model of Fig. 1, in BSSN gauge. The results in principal gauge are practically indistinguishable from those in BSSN gauge.

Figure 5

Spatial components $p_a^{\pm }$ of tetrad-frame momenta and densities $N^{\pm }$ of outgoing and ingoing collisionless null streams, in the same model used in Fig. 1. Lines are short dashed where values are negative. Momenta and densities of principal outgoing and ingoing null streams are also shown, designated (pn). By construction, in principal gauge, the principal outgoing and ingoing directions lie along the tetrad radial direction (the 1-direction).

Figure 6

Evolution of the scale factors $a_a$ at various latitudes, in BSSN gauge. The accretion rates and spin of the black hole are given by Eqs. (51) and (53). The dashed lines are the approximate conformally separable solution from [17, 18], while the solid lines are from the numerical computation.

Figure 7

Evolution of the Kasner exponents $q_a$, for the same models as in Fig. 6, again in BSSN gauge.