Complexity equals anything for multi-horizon black holes

Complexity equals anything for multi-horizon black holes investigates the complexity equals anything proposal using codimension-one and codimension-zero gravitational observables to probe more complete black hole interiors than previous volume complexity proposals

Introduction to Holographic Complexity Proposals

Holographic complexity is widely regarded as an excellent probe of the interior of black holes It is most commonly explored in the context of the thermofield double (TFD) state, which is dual to an eternal two-sided black hole There are three important proposals regarding holographic complexity: complexity equals volume (CV) [11], complexity equals action (CA) [16], and complexity equals spacetime volume (CV 2.0) [17] Each of these proposals satisfies the basic characteristics of quantum circuit complexity First, complexity grows linearly at late times [10]. Second, the response of complexity to perturbations exhibits a universal time delay, commonly referred to as the switchback effect [18] In addition, all of the holographic complexity proposals have ambiguous definitions, such as the ill-defined length scale l in CV and CV 2.0 It was pointed out in Refs. [19–22] that the ambiguities in quantum circuit complexity can be naturally matched by those in its holographic dual, i.e., the “complexity equals anything” proposal (Cgen)

Codimension-One Observables and Generalized Volume Complexity

The expression of complexity with codimension-one observables is given by CgenF1, ΣF2 = 1/GNl Z ΣF2 d dσ √h F1(gµν; Xµ) This proposal is commonly referred to as the “generalized volume complexity” When F1 = F2, Eq. (3.3) can be rewritten as Cgen = max ∂Σ=Στ 1/GNl Z Σ d dσ √h a(r) Finding the extremal surface is analogous to solving a classical mechanics problem where the action and Lagrangian are identified as S ∼ Cgen and L ∼ √h a(r) Due to the symmetries of the spherical black hole, we can parametrize Σ in terms of (v(σ), r(σ), omegad−1), then, Cgen can be written as Cgen = Vd−1/GNl Z Σ dσrd−1p −f(r) ˙v squared + 2 ˙vr a˙ (r) For a given conserved momentum Pv, we can determine the minimum radius (˙rmin = 0) of the extremal surfaces and inversely solve for the boundary time at which it is anchored, i.e., τ = -2/Z ∞ rmin dr Pv / f(r) p P squared v − U0(r) An important conclusion is that at late times, the extremal surfaces approach constant−r surfaces, at which point the effective potential reaches a local maximum, i.e., U0(rf) = P 2∞, U'0(rf) = 0, U''0(rf) < 0

Codimension-Zero Observables and Multi-Horizon Probing

The expression of complexity with codimension-zero observables is given by Cgen h G1, F1±,MG2,F2± i (ΣCFT) = 1/GNl Z Σ+[G2,F2+] d dσ √h F1+ (gµν; Xµ+) + 1/GNl Z Σ−[G2,F2-] d dσ √h F1- (gµν; Xµ-) + 1/GNl 2 Z M[G2,F2±] d d+1x √−g G1(gµν) Thanks to the Stokes’ theorem, Eq. (3.18) can always be recast as two independent extremizations, i.e δX± Σ± d dσ √h F2± (gµν; Xµ±) ± Ge2(gµν; Xµ±) = 0 When G2 and F2 are constants, the two boundaries of the extremal region M becomes two constant mean curvature (CMC) slices [89, 90] KΣ+ = −αB/α+l, KΣ− = αB/α-l We can verify this conclusion at late times (r = rf): nµ± r=rf=(0, 1/p−f(rf), 0, 0) KΣ± r=rf=4f(rf) + rf f'(rf) 2/rp−f(rf), where rf can be obtained by Eq. (3.37)

Probing Different Interior Regions of Multi-Horizon Black Holes

The generalized volume complexity enables the probing of a more complete black hole interior, that is, all spacetime regions where the blackening factor f(r) < 0 In terms of the codimension-zero gravitational observables, we show that the corresponding extremal region can also extend into the spacetime region inside the third horizon, provided the outer curvature of its upper boundary is sufficiently large Moreover, by exploiting the distinct physical characteristics of different gravitational observables, we are able to successfully distinguish the various black hole interiors This extension highlights the robustness of the “complexity equals anything” proposal and its potential as a general diagnostic tool for probing the internal structure of a black hole with complex horizon structure

Conclusion on Multi-Horizon Black Holes

In conclusion, we have demonstrated the unique advantages of the generalized volume complexity in exploring the internal structure of multi-horizon black holes, using a three-horizon black hole as an example In Sec. 6, we will extend this result to a more general case Due to the unique flexibility, the “complexity equals anything” proposal can always probe the spacetime regions where f(r) < 0

Appendix on Matter-Built Constructions

Nonlinear electrodynamics is the key factor for the emergence of multi-horizon black holes, which has been discussed in detail in Ref. [95] Then, a natural question is whether the gravitational observables constructed from the matter sectors can influence the geometric properties of holographic complexity discussed in the main text To address this, we consider the following matter-built weights a(r) to be inserted in the generalized-volume functional: a1(r) = 1 + λ1 α L(F, α) − λ2 α squared L(F, α) squared a2(r) = 1 + λ1 α L(F, α) − λ2 α cubed L(F, α) cubed The corresponding effective potentials U1,2(r) = −f(r)a1,2(r) 2/r 4 exhibit additional local maxima inside the Cauchy horizon for suitable parameters, allowing the extremal surface to extend into deeper interior regions However, for the Bardeen-AdS class black hole, L(F, α) → Const. as r → 0, so the matter term cannot drive the leftmost peak of U(r) arbitrarily close to the singularity In other words, these constructions allow the extremal surfaces to extend into the Reigon III, but they cannot be tuned to approach r = 0 Therefore, while matter-built constructions permit penetration beyond the inner horizon, curvature-built choices remain more effective as singularity probes

References

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[3] S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep-th/0603001 [hep-th]].

[17] J. Couch, W. Fischler and P. H. Nguyen, “Noether charge, black hole volume, and complexity,” JHEP 03, 119 (2017) [arXiv:1610.02038 [hep-th]].

[24] M. T. Wang, H. Y. Jiang and Y. X. Liu, “Generalized Volume-Complexity for Reissner-Nordstr¨om-AdS Black Hole,” JHEP 07, 178 (2023) [arXiv:2304.05751 [hep-th]].

[95] C. Gao, “Black holes with many horizons in the theories of nonlinear electrodynamics,” Phys. Rev. D 104, no.6, 064038 (2021) [arXiv:2106.13486 [gr-qc]].

1. Enhanced Black Hole Interior Characterization: AI systems will be able to distinguish between different regions of a multi-horizon black hole interior by analyzing characteristic complexity growth rates derived from different scalar function constructions a(r).

2. Multi-Horizon Probing Capability: The improved system can analyze the three-horizon case, where it can determine whether extremal surfaces probe Region I, Region III, or both by examining the behavior of effective potentials.

3. Singularity and Horizon Localization: AI systems will be able to predict whether an extremal surface approaches the singularity or the Cauchy horizon based on parameters like alpha B and determine if a region is impossible to be probed by holographic complexity when f(r) is positive.

4. Robustness Across Singularities: The system can use different constructions of the scalar function a(r), such as a1(r) and a2(r), to ensure that the local maximum of the effective potential appears in any region where f(r) is negative, allowing for probing across different spacetime regions.

5. Late Time Growth Rate Prediction: The improved system can compute late time complexity growth rates using formulas like lim tau to infinity dCgen/dtau = Vd-1 GNl Pv, allowing prediction based on conserved momenta P+v and P-v.