Computational Cosmic Censorship

Abstract: We propose a computational formulation of weak cosmic censorship in AdS/CFT. Using the complexity=action proposal, we evaluate the Wheeler-DeWitt action for overcharged Reissner- Nordström-AdS spacetimes containing naked timelike singularities. We show that the bulk, null, and joint contributions remain finite, while the Gibbons-Hawking-York term at the singularity diverges. More generally, for any static and spherically symmetric geometry with near-origin scaling $f(r)\sim a r^{-p}$, the singularity term diverges whenever $p>D-3$. This implies divergent holographic complexity and, even relative to the logarithmically divergent extremal charged sector, leaves an infinite complexity gap. This suggests an operational form of censorship: naked singularities are excluded not by geometry alone, but by an infinite computational cost arising from their local near-singularity structure.

• The paper shows that naked singular spacetimes correspond to an infinite computational complexity barrier using the complexity=action proposal.
• It details how the GHY term near the singularity, with its dominant divergence, distinguishes overcharged RN–AdS geometries from regular black hole sectors.
• The analysis highlights that local geometric features near singularities, rather than global causal structures, determine the accessibility of holographic states.

Computational Complexity as a Mechanism for Cosmic Censorship in AdS/CFT

Introduction and Context

The paper "Computational Cosmic Censorship" (2604.20170) introduces a novel operational perspective on the weak cosmic censorship conjecture using the AdS/CFT correspondence and the complexity=action (CA) proposal. Traditional formulations of cosmic censorship are geometric, hinging on causal structure and the existence of event horizons to cloak singularities. However, the paper shifts focus to whether such spacetime configurations are accessible via finite physical or computational processes, leveraging holographic duality to recast access to gravitational singularities in computational terms.

Within AdS/CFT, bulk geometries correspond to boundary quantum states, with the complexity of state preparation conjecturally mapped to the on-shell action of Wheeler–DeWitt (WdW) patches in the bulk. Under the CA proposal, this bridges geometric features of the bulk and computational resources required on the boundary. The core conjecture is that naked singularities—spacetimes without event horizons—are segregated from regular black hole sectors by an infinite computational complexity barrier, rendering them operationally inaccessible.

Analysis of Wheeler–DeWitt Action Divergences

The technical approach involves explicit evaluation of the WdW action for overcharged Reissner–Nordström–AdS (RN–AdS) spacetimes, which lack an event horizon and exhibit a naked timelike singularity. The CA proposal equates holographic complexity $\mathcal{C}$ with the gravitational action:

$\mathcal{C} = \frac{I_{\text{WdW}}}{\pi\hbar}$

For the overcharged RN–AdS metric, the components of the WdW action are dissected to identify the source of divergence:

• Bulk terms (Einstein–Hilbert and Maxwell): Despite locally divergent fields near $r=0$, the spherical volume element $r^{D-2}$ sufficiently damps the integrand, ensuring finiteness of the bulk integrals.
• Null boundary and joint contributions: Finiteness is maintained by construction (e.g., via affine parametrization for null surfaces) and through the dominance of suppressing factors in the joint terms.
• Gibbons–Hawking–York (GHY) term at the singularity: The critical result is that the extrinsic curvature evaluated at a timelike regulator surface $r = \epsilon \to 0$ introduces a divergence scaling as $-\frac{q^2}{\epsilon^{D-3}}$, which dominates all other contributions.

All finite contributions can be robustly subtracted using suitable reference spacetimes, such as the extremal charged black hole sector, but the divergence from the GHY term persists, leading to an infinite relative complexity even compared to the known logarithmic divergence in the extremal sector [Carmi et al., (Carmi et al., 2017)].

Universality and Geometric Criteria for Divergence

The divergence structure is shown to be universal for a broad class of static, spherically symmetric metrics with near-singularity scaling $f(r) \sim a r^{-p}$:

• Finiteness of the WdW time-width: The time-width $\Delta t(r)$ remains finite for $p > -1$.
• Divergent GHY term: The boundary term diverges as $\epsilon^{D-3-p}$ if $\mathcal{C} = \frac{I_{\text{WdW}}}{\pi\hbar}$0.

This criterion encompasses a wide array of charged black hole geometries in diverse dimensions, as well as their multi-charge and dilatonic generalizations. The local behavior of the metric near the singularity, rather than global or infrared properties, solely determines the complexity divergence.

Implications and Theoretical Significance

The findings substantiate a form of computational censorship: geometries with naked singularities are excluded not by geometric or causal prohibitions, but due to an insurmountable barrier in the holographic complexity landscape. In the AdS/CFT context, this translates to physical inaccessibility of boundary states dual to such singular spacetimes within any finite sequence of quantum operations, assuming finite complexity growth.

The results delineate a hierarchy:

• Regular finite-temperature black holes: No complexity divergence.
• Charged extremal black holes: Logarithmic divergence in complexity of formation.
• Overcharged (naked singular) geometries: Power-law divergence, resulting in an infinite relative complexity with respect to the extremal sector.

The divergence is manifestly local—stemming from the extrinsic curvature at the singular boundary and indifferent to large-scale features or modifications to the null-boundary structure. This sharply distinguishes it from divergences arising in, for example, negative-mass Schwarzschild–AdS spacetimes, and indicates robustness under modifications of UV-insensitive bulk prescriptions [Katoch et al., (Katoch et al., 2023)].

Potential extension beyond the AdS setting is suggested, contingent on the persistence of the local geometric origin of the divergence when the framework is appropriately generalized.

Future Directions

Several follow-up questions present themselves:

• Beyond the CA proposal: It remains to be seen if similar divergent complexity barriers arise in the complexity=volume (CV) proposal, or other candidate holographic complexity measures.
• Quantum corrections: The persistence of the divergence in the presence of quantum or stringy effects is an open issue, with implications for the ultimate robustness of complexity-based censorship in fully quantum gravity.
• Direct boundary characterization: While the bulk calculation is classical and geometric, a direct boundary theory realization of this infinite complexity separation, possibly through quantum circuit analysis or boundary CFT diagnostics, would strengthen the conjecture.

Conclusion

The paper establishes a precise and robust connection between naked singularities in asymptotically AdS spacetimes and infinite computational cost in the holographic dual, as computed via the complexity=action proposal. This operational reformulation of the weak cosmic censorship conjecture introduces a new perspective: certain regions of the gravitational phase space may be fundamentally excluded not solely by geometric or dynamical laws, but by computational infeasibility. This distinguishes overcharged, nakedly singular spacetimes from regular black holes and even their extremal limits, and suggests a deeper interrelation between spacetime geometry, quantum information, and the landscape of physically accessible states. The results motivate further investigations into the computational structure of quantum gravity and its implications for the fundamental limitations of state preparation in holographic settings.