The internal structure of black holes represents a frontier in general relativity and strong‑field gravitational dynamics. While the exterior properties of spontaneously scalarized black holes—such as their shadows, photon rings, and thermodynamics—have been extensively studied in recent years, their interiors remain largely unknown.

A team of researchers, including Professor Li Li from the Institute of Theoretical Physics of the Chinese Academy of Sciences (ITP-CAS), Ze Sun (a graduate student jointly affiliated with the ITP-CAS and the Hangzhou Institute for Advanced Study, UCAS, supervised by Li Li), and Fu-Guo Yang (a postdoctoral fellow at ICTP‑AP, who completed his PhD at the ITP-CAS from 2017 to 2022), has systematically investigated the interior behaviour of spontaneously scalarized black holes in asymptotically flat spacetimes within the framework of Einstein–Maxwell–scalar theory.

A charged Reissner–Nordström black hole possesses a smooth inner Cauchy horizon and a timelike singularity. However, the study finds that, under very general conditions, spontaneously scalarized black holes no longer harbour an inner Cauchy horizon. The scalar field triggers an instability of the inner horizon, leading to a rapid collapse of the Einstein–Rosen bridge near that horizon, and eventually the spacetime ends at a spacelike Kasner singularity. The Kasner singularity is characterised by a set of exponents that reflect the anisotropic contraction and expansion of spacetime in its vicinity.

Figure 1: Critical behaviour of the Kasner parameter β versus the black hole’s charge‑to‑mass ratio q . Left panel: results for different scalar–electromagnetic coupling functions. Right panel: dependence on the coupling strength. The different symbols represent numerical results, and the solid lines show the universal scaling relation.

Further analysis reveals that near the critical point where the spontaneously scalarized black hole branches off from the Reissner–Nordström solution, the Kasner parameter β—which characterises the interior singularity structure—exhibits a clear scaling relation with the charge‑to‑mass ratio q : β~(q/q_c -1)^-γ ,where q_c is the critical charge‑to‑mass ratio for spontaneous scalarization, and the critical exponent γ=0.5 is independent of the specific form of the coupling function, demonstrating the universality of the critical phenomenon. As shown in Figure 1, this scaling behaviour remains robust whether one varies the functional form of the coupling or tunes its strength. This finding suggests that even in the highly nonlinear dynamics of black hole interiors, simple and universal critical laws can emerge.

This work supports the strong cosmic censorship conjecture, deepens our understanding of the interior spacetime structure of black holes, and provides a new theoretical example of universal behaviour under nonlinear effects in general relativity. The results have been published in Physical Review Letters. Following the common practice in high‑energy physics, the authors are listed in alphabetical order by surname. Professor Li Li and his collaborators have recently conducted systematic studies on black hole interiors, including proving no‑inner‑horizon theorems [1,2], classifying the transitions between different Kasner geometries inside black holes [3,4], and revealing the significant influence of higher‑derivative corrections on interior dynamics [5], among others.

This research was supported by the National Key Research and Development Program of China and the National Natural Science Foundation of China.

References

[1] Rong-Gen Cai, Li Li and Run-Qiu Yang, No Inner-Horizon Theorem for Black Holes with Charged Scalar Hairs, JHEP 03, 263 (2021) [arXiv:2009.05520 [gr-qc]].

[2] Yu-Sen An , Li Li and Fu-Guo Yang Yang, No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs, Phys. Rev. D 104, no.2, 024040 (2021) [arXiv:2106.01069 [gr-qc]].

[3] Rong-Gen Cai, Mei-Ning Duan, Li Li and Fu-Guo Yang, Towards classifying the interior dynamics of charged black holes with scalar hair, JHEP 02, 169 (2024) [arXiv:2312.11131 [gr-qc]].

[4] Rong-Gen Cai, Mei-Ning Duan, Li Li and Fu-Guo Yang, Clarifying Kasner dynamics inside anisotropic black hole with vector hair, JHEP 04, 179 (2025) [arXiv:2408.06122 [gr-qc]].

[5] Mei-Ning Duan, Li Li, Yu-Xuan Li and Fu-Guo Yang, Beyond Kasner Epochs: Ordered Oscillations and Spike Dynamics Inside Black Holes with Higher-Derivative Corrections, [arXiv:2601.21658 [gr-qc]].

Original link:

https://journals.aps.org/prl/abstract/10.1103/l4tf‑kqyl

Contributor:Li Li