Complexity equals anything for multi-horizon black holes

Introduction to the show: ident: High Energy Physics: Theory. The day in high-energy theory: gravity, strings, fields and the questions left open.

Kai: We're looking at the paper: "Complexity equals anything for multi-horizon black holes".

Mira: Here's the gist of the paper. 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 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: .

Kai: 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 Cgen

F1, ΣF2: (ΣCFT) = 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.

Kai: (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.

Paper summary: Kai: So, this paper investigates whether "complexity equals anything" can probe all spacetime regions where the blackening factor f(r) is negative, which means deeper inside the black hole structure.

Mira: That's the key advantage they point out; it’s a significant improvement over previous volume complexity proposals because of that flexibility in its definition.

Lev: I wonder how feasible this is for real hardware; if we were trying to measure something like this, what would be the overhead involved just to get those observables?

Kai: Well, they use the Bardeen-AdS class black hole as their example system to test these ideas against complexity equals volume and other proposals.

Mira: They show that the generalized volume complexity allows for probing all spacetime regions where f(r) is negative, which means extending the probe into what's inside the third horizon.

Lev: That extension into Region III, that's a big deal if it means we can actually probe beyond just the outer horizon or the inner one.

Kai: And they use codimension-zero gravitational observables to show that this region can be reached provided you have a sufficiently large outer curvature for the upper boundary of your extremal surface.

Conclusion: Mira: So, looking at "Complexity equals anything for multi-horizon black holes," the main point is that this proposal offers a more flexible way to explore the interior of black holes by using these two types of observables.

Kai: It means we can probe all regions where the blackening factor f(r) is negative, which was a big improvement because it addresses some ambiguities in older complexity definitions.

Lev: For us building error correction hardware, this suggests that if we could construct observables based on these ideas, the theory allows for probing deeper structures than previous models predicted.

Mira: The authors are showing that by using the generalized volume complexity and those codimension-zero observables, you can successfully distinguish different black hole interiors based on their unique gravitational properties.

Kai: It’s not just a mathematical exercise; it suggests this "complexity equals anything" idea could be a more general diagnostic tool for understanding the internal structure of any black hole with complex horizon structure.