Beyond Kasner Epochs: Ordered Oscillations and Spike Dynamics Inside Black Holes with Higher-Derivative Corrections

TL;DR

The paper addresses how near-singularity dynamics inside black holes, traditionally described by chaotic BKL Kasner evolution, are modified by higher-derivative quantum corrections. By constructing a scalar-tensor framework with a tower of curvature terms up to order $N_{\text{max}}$ and a scalar potential $V(\psi)$, the authors analyze interior dynamics under a plane-symmetric ansatz and derive regimes determined by the relative growth of $V(\psi)$ to $\psi^{\frac{2N_{\text{max}}}{N_{\text{max}}-1}}$. They identify three distinct phases—modified Kasner eons, persistent periodic oscillations, and oscillatory spike dynamics—plus a finite-volume singularity mechanism when $V(\psi)$ diverges too quickly, and demonstrate the universality of these phases across Einstein-Gauss-Bonnet and Lovelock theories, with phase boundaries governed by couplings like $\alpha_2$, $\alpha_3$, and quartic coefficients $c_4$. The work provides a comprehensive framework for gravitational nonlinearity in extreme regimes and suggests that quantum corrections can enforce structured interior dynamics rather than driving chaotic BKL behavior. These insights have potential implications for black-hole interiors and early-universe cosmology under higher-derivative corrections.

Abstract

Building upon the long-standing paradigm that dynamics near a spacelike singularity are governed by a sequence of Kasner epochs, we demonstrate that this picture is fundamentally altered when higher-curvature or quantum gravitational corrections are included. By incorporating such terms alongside a minimally coupled scalar field, we discover three distinct dynamical phases near the singularity: modified Kasner eons, persistent periodic oscillations, and oscillatory spike dynamics with growing amplitude. In particular, the Kasner-like geometry persisting only in highly constrained situations. The latter two regimes represent a clean departure from classical Kasner phenomenology, revealing a richer and more ordered landscape of behaviors in the deep interior of black holes beyond Einstein gravity. This work establishes a comprehensive approach for understanding the gravitational nonlinearity in the most extreme gravitational environment.

Figures (10)

• Figure 1: Effective Kasner exponents for hairy Einstein-GB black hole interiors in $d=7$ and $\alpha_2=10^{-2}$ with $\psi(z_H)=0.10$. Upper panel: $V(\psi)=-42-3\psi^2-2\psi ^4$, for which the plateau corresponds to a hairy GB eon of Case I with $p_t=p_x=0.224$. Bottom panel: $V(\psi)=-42-3\psi^2+2\psi ^4$, where the Kasner eon is replaced by periodic oscillations.
• Figure 2: Phase diagram in the plane of GB coupling $\alpha_2$ versus $c_4$ for quartic potential $V\sim c_4 \psi^4$ and $d=4$. The interior dynamics separate into modified Kasner eons (Case I) and periodic oscillations (Case II), with the boundary given by \ref{['pbc4']}. The Case I region shrinks for larger $d$. The inset displays the evolution of $\widetilde{p}_x$ for $V=-12-\frac{3}{2}\psi^2+c_4\psi^4$ with $\psi(z_H)=0.30$ and $\alpha_2=10^{-2}$, where curves from bottom to top corresponds to $c_4=2.47$, 2.485 and 2.49. The purple dashed line marks the critical boundary at $p_x=3/4$, near which a transient Kasner-like regime emerges from Case II side.
• Figure 3: Far‑interior evolution of $\widetilde{p}_t$ and $\widetilde{p}_x$ for $d=4$ and $\alpha_2=10^{-2}$. In contrast to a Kasner plateau, they display oscillatory growth. Upper panel: Potential $V(\psi)=-12-\frac{3}{2}\psi^2+2\psi ^6$ with $\psi(z_H)=0.3$. The peak positions share a common period in the scaled coordinate $({z/z_H})^{0.50}$, while their amplitudes follow a well‑defined envelope. Bottom panel: Potential $V(\psi)=-12-\frac{3}{2}\psi^2+\frac{1}{10}\cosh(\psi)$ with $\psi(z_H)=0.001$. Here the peaks are periodic in $({z/z_H})^{1.68}$.
• Figure S1: Far-interior evolution of $\widetilde{p}_x$ in the range $z/z_H\in[10^4,10^{20}]$ obtained from the full EoMs \ref{['EQ:FulEoMrho']} (solid red curve) and from the approximate equations (11) in the main text (dashed blue curve). The results are in quantitative agreement. We consider $V(\psi)=-42-3\psi^2+2\psi^4$ with $d=7$, $\alpha_2=10^{-2}$, and $\psi(z_H)=0.10$.
• Figure S2: Positions of the $n$-th interior peak in a hairy Einstein–Gauss‑Bonnet black hole with $V(\psi)=-42-3\psi^2+2\psi ^4$, $d=7$, $\alpha_2=10^{-2}$, and $\psi(z_H)=0.10$. Blue points mark the positions extracted from the bottom panel of Fig. \ref{['fig:case12']}; the red line shows a linear fit in the index $n$. The fit $\ln{(z/z_H)}=3.67+5.02n$ confirms a logarithmic periodicity.