Interior structure of black holes with nonlinear terms

TL;DR

The paper addresses how the internal structure of black holes in a holographic s-wave superconductor is influenced by higher-order nonlinear terms, focusing on the oscillations of the Kasner exponent $p_t$ near the critical temperature. It introduces nonlinear self-interactions with coefficients $λ$ and $τ$, derives the Kasner interior regime, and analyzes $p_t$ numerically. The key finding is that the nonlinear coefficient $λ$ linearly controls the oscillation period, with positive $λ$ expanding and negative $λ$ compressing the oscillatory region, while $τ$ has a milder effect away from the critical point; transforming the temperature variable reveals a well-defined periodic pattern. This work demonstrates active control of interior black-hole dynamics via model parameters, offering insights into Kasner-type interiors and potential connections to black-hole information and quantum chaos in holographic settings.

Abstract

We investigate the oscillation of the Kasner exponent $p_t$ near critical point of the hairy black holes dual to holographic superfluid and reveal a clear inverse periodicity $f(T_c/(T_c-T))$ in a large region below the critical temperature. We first introduce the fourth-power term with a coefficient $λ$ to adjust the oscillatory behavior of the Kasner exponent $p_t$ near the critical point. Importantly, we show that the nonlinear coefficient $λ$ provides accurate control of this periodicity: a positive $λ$ stretches the region, while a negative $λ$ compresses it. By contrast, the influence of another coefficient $τ$ is more concentrated in regions away from the critical point. This work provides a new perspective for understanding the complex dynamical structure inside black holes and extends the actively control from the fourth- and sixth-power term into the black hole interior region.

Figures (6)

• Figure 1: The dependence of the condensates on the nonlinear term $\lambda$ with parameter $\tau=0$. Solid lines represent the condensed solutions, with different colors indicating different values of $\lambda$. The black dashed line represents the normal solution.
• Figure 2: The Josephson oscillations behavior for $T=0.981T/T_c$ with $\lambda=$0.4, 0 and $-0.4$.
• Figure 3: The Kasner exponents $p_t$ as function of temperature $T$. Here, red, blue, and green correspond to $\lambda$ equal to 0.4, 0, and $-0.4$, respectively.
• Figure 4: The relationship between the Kasner exponent $p_t$ and $T_c/(T_c-T)$. The solid lines represents the original $p_t$ data, and the black dashed line denote the data derived from $p_t(\alpha_0)$. The parameter values are $\lambda=0.4~(A=1.73923,B=0.884,C=2.7)$, $\lambda=0(A=1.73923,B=0.478,C=2)$, and $\lambda=-0.4~(A=1.85,B=0.0725,C=10.8)$
• Figure 5: The relationship between periodical length $L_p$ and nonlinear parameters $\lambda$ and $\tau$. The solid lines are the original data, and the dashed lines are the fitting data.