Higher-dimensional BKL dynamics in AdS black holes

Abstract

Chaotic BKL dynamics provides a canonical description of the approach to spacelike singularities as a sequence of Kasner epochs grouped into eras. While this paradigm is well established for cosmological singularities, explicit realizations inside black holes have been scarce, despite renewed interest from holography. Here, we construct a broad class of asymptotically AdS black holes in $D\ge 4$ whose interiors exhibit bona fide BKL dynamics as the singularity is approached. In the near-singularity regime, the evolution reduces to billiard-like motion in a compact domain that forms a regular $(D-2)$-simplex. We derive closed-form bouncing rules for the Kasner exponents in arbitrary dimension and prove the ensuing chaotic dynamics. A key novelty for $D\ge 5$ is a richer internal organization of eras: inequivalent transitions between epochs lead to distinct Kasner seasons, yielding new patterns of epoch/era structure for both electric and gravitational walls. Finally, we investigate a holographic diagnostic, the thermal $a$-function, whose monotonic flow captures individual epochs and eras and can display near-walking behavior in suitable Kasner regimes.

Figures (13)

• Figure 1: Appearance of BKL dynamics inside a four-dimensional AdS black hole. We show the interior-time evolution of the three independent diagonal components of the metric, $ds^{2}=g_{zz}\,dz^{2}+\sum_{i=0}^{2} g_{ii}\,(dx^{i})^{2}$.
• Figure 2: Left: Effective Kasner exponents in the black hole interior \ref{['eq:intbh4']}. The different plateaus of the effective Kasner exponents correspond to Kasner epochs, which undergo very rapid transitions into new Kasner epochs. Observe the presence of a first (long) Kasner era, which is the result of some appropriate boundary conditions that produce this illustrative result --- other choices of boundary conditions are possible, giving rise to shorter initial eras. We notice that, around $\rho \sim 2900$ and $\rho \sim 3200$, the system undergoes changes of Kasner eras. Right: Approach to the singularity in the configuration space defined by $\left ( \frac{f_0}{\rho}, \frac{f_1}{\rho} \right)$. It is explicitly checked that it consists of straight lines (Kasner epochs) interrupted by abrupt bounces to the edges of an equilateral triangle. Kasner eras correspond to those collections of bounces between the same two edges.
• Figure 3: Mixmaster dynamics inside a five-dimensional AdS black hole. Left: The plot shows how the four independent metric components evolve with interior time. At late times, the evolution breaks up into the familiar BKL sequence of Kasner epochs, organized into eras. This is the direct generalization of the plots that may be found in the four-dimensional case DeClerck:2023fax. Right: We explicitly observe that the dynamical evolution towards the singularity may be conceived as a set of free trajectories in the configuration space $\left\lbrace \frac{f_0}{\rho},\frac{f_1}{\rho},\frac{f_2}{\rho} \right\rbrace$ confined to the interior of a tetrahedron.
• Figure 4: Effective Kasner exponents for two different five-dimensional black holes fitting into the ansatz \ref{['eq:metricbh']}. Left: We observe three complete Kasner eras. The last two are quite short and only contain one epoch, while the first one is quite long and features 20 epochs in Kasner season I --- see Definition \ref{['def:ks']}. Right: We see one complete era and a transition into a new one. We explicitly identify a Kasner epoch in Kasner season II.
• Figure 5: Points $(u_1,u_2) \in \mathcal{D}_2$ may be mapped to the interior of the ellipse $u_1^2+u_1 u_2+u_2^2=1$ by defining $(v_1,v_2)=\frac{1}{u_1^2+u_2^2+u_1 u_2}(u_1,u_2)$. In this figure, we present the initial epochs of a number of consecutive eras in the five-dimensional BKL dynamics triggered by electric walls, in terms of the parametrization $(v_1,v_2)$. As it turns out, all such points $(v_1,v_2)$ lie within the region $\mathcal{I}$ delimited by $v_1^2+v_1 v_2+v_2^2 \leq 1$, $v_2\geq v_1$ and $v_1^2 + v_2^2 + v_1 v_2 + v_2 + 2 v_1 \leq 0$. We have plotted around 11k eras in the left figure and around 45k eras in the right plot. We observe how the interior of the ellipse gets uniformly filled, which conforms a hallmark of chaotic behavior. Observe that the density of initial points of eras change drastically around the dotted blue line $v_2=v_1+1$. Interestingly enough, we have checked that this is due to the fact that the initial epochs of eras whose previous era featured Kasner season II (see Definition \ref{['def:ks']}) are only allowed to be below the line $v_2=v_1+1$. Amusingly, numerically one may compute that average number of epochs in a given Kasner era is around $\sim$ 4.4.

Theorems & Definitions (1)

• Definition