Upper bound about cross-sections inside black holes and complexity growth rate

This paper proposes a new universal geometrical inequality between the spacelike cross-section inside event horizon and the total mass for static asymptotically AdS black holes. This inequality implies that complexity growth rate in"complexity-volume conjecture"satisfies the upper bound argued by quantum information theory and the planar Schwarzschild-AdS black hole has fastest complexity growth at late-time limit. This offers a new universal constraint for the inner geometry of black holes as well as makes a first step toward the proof of conjecture that black hole is the fastest quantum"computer"in nature.


Background and motivation
Black holes, as ultra dense objects in universe, exhibit many fascinating physical and mathematical properties. Particularly, many such properties can be presented by some universal inequalities, such as the positive mass theorem [1,2], the second law of black holes [3,4], the Penrose inequality [5][6][7][8][9] and so on. These universal inequalities do not only show some beautiful mathematical aspects of general relativity but also deepen the physical understanding about foundations of gravity.
From classical viewpoint of general relativity, the inner regions of event horizon are nonobservable. Therefor, most of universal inequalities focus on the quantities which involve only horizon and outside black holes. The universal inequalities about the inner structures of black holes are still lack of exploring. However, the recent developments of holographic duality show that the inner structures of black holes may also have boundary correspondences and can play important roles in considering the boundary physics. For example, in the studies of holographic computational complexity [10][11][12][13][14][15], it has been showed that the complexity growth rate in a Schwarzschild-AdS black hole is determined completely by the inner geometry [16]. This paper makes a first step to explore a universal inequality about inner geometry of static black holes and exhibits an interesting application in holographic duality. The inequality arises from following simple question. Consider the (d + 1)-dimensional Schwarzschild-AdS black hole with the metric Here f (r) = 1/ 2 AdS − f 0 z d and AdS is the AdS radius. Inside horizon, z is time but t is spatial coordinate. For a class of special slices which are fixed "time" z, the volume reads Geometrically, Σ can be interpreted as the "size" of a crosssection as it integration with respective to t gives us the volume of the slice. Different z will give us different size of cross-section. The directly computation shows that max Σ = 8πE AdS /(d−1). Here E is the total energy/mass of the spacetime. This shows an inequality for these special cross-sections inside a Schwarzschild-AdS black hole After a few of straightforward computations, one can verify that above inequality is still true for non-extreme Reissner-Nordström (RN)-AdS black holes. This paper conjectures that inequality (1.3) is still true for general spatial slices of general static black holes. More detailed, it conjectures that: if (i) outermost horizon is a connected Killing horizon with positive surface gravity and nonnegative sectional curvature, (ii) the spacetime is asymptotically Schwarzschild-AdS 1 and (iii) weak energy condition and Einstein equation are satisfied, then inequality (1.3) is always true and the saturation may happen only in planar Schwarzschild black holes. To support this conjecture and as a very primary study, this paper gives the proof in planar/spherically symmetric cases. It also consideres non-planar/non-spherical perturbations in planar/spherically symmetric backgrounds and shows that inequality (1.3) is till true.
Though it is not the original motivation of inequality (1.3), this paper finds that the inequality (1.3) has important application in holographic duality. It has been argued from quantum information theory that the complexity growth rate satisfies Lloyd's bound [18] This bound describes the ultimate speed of quantum computations [18]. We will show that, if the inequality (1.3) is true, then the complexity growth rate in "complexity-volume" (CV) conjecture [10,11] always satisfies the Lloyd's bound (1.4) and the upper bound of Eq. (1.4) is saturated only in planar Schwarzschild-AdS black holes when τ → ∞. Any regular matter (satisfies weak energy condition) in the bulk always slows complexity growth. This supports the conjecture that black holes may be fastest "computers" in nature [13,14]. The similar growth rate bound was once conjectured in "complexity=action" (CA) conjecture [13,14] but has been found to be false even in Schwarzschild black holes [16,[19][20][21]. As inequality (1.4) is an important property of complexity in quantum information theory, inequality (1.3) gives us a new viewpoint to compare CV and CA conjectures.

Cross-section inside the black hole
Let us begin our discussion from giving the precise definition about "size of cross-section". Consider a (d + 1)-dimensional static black hole with an outermost non-degenerated connected Killing horizon. Assume ξ I = (∂/∂t) I is Killing vector field and timelike outside outermost Killing horizon. A cross-section S d−1 is an arbitrary spacelike (d−1)-dimensional submanifold inside the black hole (If there are inner horizons, then "inside black hole" means the region between the outermost horizon and next-outermost horizon). The size of this cross-section is defined as Here dΣ IJ is the directed surface element 2-form of S d−1 and n I is a unit normal covector of S d−1 which satisfies n I ξ I = 0. Geometrically, Σ stands for the projected area of S d−1 along the direction ξ I , see Fig. 1 for an intuitive explanation. The cross-section S d−1 is trivial if ξ I tangent to S d−1 as the size is zero. For nontrivial cross-section, normal covector n I is timelike and unique upon the origination. This paper conjectures that the integration (2.1) always satisfies inequality (1.3) under the conditions mentioned above In a general static (d+1)-dimensional spacetime, we can always choose local coordinates so that the metric reads Here f = f (z), h ij and χ are functions of {z, x i }. We assume that the outermost horizon is connected and locates at z = z h . A nontrivial cross-section S d−1 inside black hole can be parameterized by z = z S (x i ) and t = t S (x i ). Then size of S d−1 is See appendixes A and B for mathematical details.

Relationship to the complexity growth rate
Though it is completely based on geometrical considerations, the "size of cross-section" has directly relationship to the complexity growth rate in CV conjecture. The CV conjecture (see Refs. [10][11][12] for more details) states that the complexity of a boundary CFT state is proportional to the maximal volume of space-like codimension-one surfaces W d connecting boundary time slices t L and t R , i.e., Here G N is the Newton constant and we set G N = 1 for convenience, is a length scale associated with the bulk geometry such as the horizon radius or AdS radius and so on. We take = 4π 2 AdS /(d − 1). Globally, the time coordinate t in metric (2.2) is ambiguous when we compute the volume of W d as a same t can stand for the region inside horizon or outside horizon. This ambiguous can be cured by introduce a new parameter s such that W d is parameterized by Here the " " stands for the partial derivative with respective to s. As the spacetime is static, the maximal volume depends on only the value of t L + t R . Note the time direction of t L is opposite to the direction of bulk time t at the left boundary. For two symmetric time slices, i.e., t L = t R , the extremal surface contains two parts: t > 0 and t < 0. One of them is just the mirror symmetry of the other. Their intersection is denoted by A, which is given by t(s A ) = 0 and z = z(s A , x i ) = z A (x i ). See Fig. 1 for a schematic explanation. Due to mirror symmetry between left (t < 0) and right parts The extremal surface is obtained by variation on volume functional. The maximal volume V , i.e., the on-shell value of V , is only the function t L + t R = τ and so we have V on-shell = V on-shell (τ ). The growth rate of complexity then reads Recall Hamilton-Jacobi equation in classical mechanics that the partial derivative of onshell action with respective to canonical coordinate gives us canonical momentum. Thus, Here ∂ t z = z /t . As the volume functional does not depend on canonical coordinate t explicitly, its "momentum" will be independent of s. The integration in Eq. (3.4) will be same if we compute it in the surface of s = s A (i.e., surface A), so we have Here we have used the fact ∂ t z| A = 0. The complexity growth rate is given by size of cross-section A. In fact Eq. (3.5) will be still true even if the coordinates (2.2) can only cover a neighborhood of A. In Schwarzschild-AdS and RN-AdS black holes, it recovers the result reported by Ref. [16] after we setting z(s, x i ) = z(s) and χ = 0. We see that inequality (1.3) implies inequality (1.4) and only the planner AdS black hole may saturated Lloyd's bound (1.4). All regular matters (satisfy weak energy condition) will slow the complexity growth of black holes.

Proof in planar/spherically symmetric cases
We assume that χ is only the function of z and h ij dx . We also require that χ = O(z d ) and f (z)e −χ/2 = kz 2 + 1/ 2 AdS − f 0 z d + O(z d+1 ) as z → 0 so that the spacetime asymptotically Schwarzschild-AdS. The total energy/mass and entropy S reads Here V k,d−1 := √ hd d−1 x. The proof contains 3 steps. In the first step, we note that in the inner region 0 <h ≤ h. This shows Note that z in above integration is the function of x i . Let us assume that z =z maximizes the function √ −f e −χ/2 z −d . Then we find and R(z) := −f (z)e −χ(z) z −2d . We now need to find upper bound of R.
In the second step, we use Einstein's equations. The equation for χ reads d dz Here T IJ is the energy-momentum tensor and w I = f −1 e χ/2 ∂ ∂t I + ∂ ∂z I is a futuredirected null vector. The weak energy condition contains null energy condition and so insures χ ≥ 0. Because boundary conditions show χ(0) = 0, we find χ ≥ 0 and so e χ ≥ 1.
On the other hand, we use Einstein's equations for f and χ and obtain When z > z h , (∂/∂z) I is timelike and weak energy condition insures T zz ≥ 0 and so T zz f ≤ 0. Consider the partner of Eq. (4.5) in vacuum case with same horizon radius Here ξ(z h ) is the solution of

The weak energy condition and
The last inequality is saturated only if k = 0.
At the third step, we refer to Penrose inequality, which exhibits how the horizon area is bounded by the total energy of the black hole. It was first proposed for asymptotically flat spacetime [5][6][7][8][9] and has been generalized into (d + 1)-dimensional asymptotically AdS spacetime and has been proven in planar/spherical cases (see, e.g., Refs. [22][23][24][25][26]). Here a(S) := 4S/V k,d−1 = 1/z d−1 h is the entropy density. It is saturated only in AdS black holes. Eq. (4.8) (4.10) The inequality (1.3) is obtained if we combine inequalities (4.9) and (4.10). The saturation may appear only if inequalities (4.10) and (4.9) are both saturated, which will happen only if the spacetime is Schwarzschild black hole with k = 0.

Non-planar/non-spherical perturbations
We assume is infinitesimal and all the functions are independent of t. h (1) ij , f (1) and χ in general will be functions of {z, x i }. We also impose a gauge tr(h In the planar Schwarzschild-AdS black hole, we have known that maximal size of crosssection is given by z(x i ) =z. With the perturbation, the maximal size of cross-section is given by the cross-section z(x i ) =z + ζ (1) (x i ). Taking them into Eq. (2.3) and only considering linear order of , one can find Assume the energy-momentum tensor of matters fields to be T IJ . The Einstein equation for χ in the linear order reads Here ∂ 2 := ∂ i ∂ i . Integrating it and using the fact T IJ w I w J ≥ 0, then we obtain as the second integration can be converted into a boundary integration and V 0,d−1 = ∞. This tells us that ∀z > 0 and so max Σ ≤ V 0,d−1 f 0 AdS /2. With the perturbation, the total energy/mass now reads Thus, we conclude that the inequality (1.3) is still satisfied under non-planar/non-spherical static perturbations. This result combining with the proof in planar/spherical symmetric cases offer us strong evidence for the inequality (1.3).

Discussion
To conclude, this paper proposed and discussed a new universal inequality for the inner geometry of black holes and found an interesting application in holographic duality. Except for seeking the proof about Eq. (1.3) in more general cases, there are many aspects which are worthy of exploring in detailed in the future.
It is interesting to check if we can use stationary spacetime to replace the static spacetime. An example is BTZ black hole with nonzero angular momentum [27,28].
One can verify that max Σ ≤ 8πE AdS and the saturation appears only if J = 0. This implies that we may only need the spacetime to be stationary. On the other hand, the stationary condition seems to be necessary as we need Killing vector ξ I to define Σ, and it has been reported that the complexity growth rate in CV conjecture can violate Lloyd's bound in dynamic spacetimes, e.g. see Ref. [29].
Consider that there is a next-outermost Killing horizon at z =z h > z h with nonzero surface gravity. In the limit z h →z h , i.e., the Hawking temperature T H → 0, we can find max Σ → 0 but the total energy and entropy can be arbitrarily large. This implies that, in low temperature limit, there may be an upper bound which is controlled by temperature and much tighter than inequality (1.3). For example, in BTZ black hole (6.1) one can verify Σ ≤ max Σ = 4πT H S AdS , which is much smaller then E in low temperature limit. It is interesting to check if a similar bound is also true in general cases.
Our proof is invalid for cases with hyperbolic AdS boundary, i.e., k = −1. Our discussion is still correct up to Eq. (4.6). However, as (d − 2) 2 AdS kz 2 + d may be negative for large z, we cannot insure thatR(z) ≥ R(z) for all z ≥ z h . This is the reason that why the conjecture in this paper requires that horizon has nonnegative sectional curvature. In fact hyperbolically AdS black hole can have negative energy so Eq.
with a positive number c d . The inequality may be saturated only in Schwarzschild black holes. It is worth exploring this result in detail in the future.

A About Eq. (2.3)
Let us first explain how to obtain Eq. (2.3). For the nontrivial cross-section S d−1 , we can always embed it into the a d-dimensional spacelike manifold M d of which unite normal covectorñ I satisfiesñ I | S d−1 = n I . We parameteriz the surface M d to be z = z(t, x i ). The normal covector readsñ The conditionñ I | S d−1 = n I and ξ I n I = (∂/∂t) I n I = 0 implieṡ Here dot means the partial derivative with respective to t. The induced metric on this surface then reads, . (A. 3) It can be rewritten in following form Hereh ij := h ij + f −1 ∂ i z∂ j z is the induced metric of co-dimensional 2 surface and it defines As M d is spacelike, the metrich ij is positive-definite and N 2 > 0. The volume of M d then reads Using Eq. (B.1) and setting w i = 0, we find that Assume the cross-section S d−1 is given by t = t(x i ), then we have following (d − 1)dimensional induced metric The normalized factor α is  On the other hand, we haveñ I | S d−1 = n I and so we haveñ I ξ I | S d−1 = 0. This shows Then we obtain Finally, we find that the size of cross-section then reads

B A mathematical identity
Let us present a mathematical proposition: for a n-dimensional non-singular metric B ij , two n-dimensional vectors v i and w i , a number c, we define w i = B ij w j and Then we have following relationship Here W and B are the determinants of W ij and B ij , respectively. The proof is as follow. Let us define H ij (J) := J 2 (c + w i w i )v i v j + J(w i v j + v i w j ) and B ij (J) = B ij + H ij (J) .
Then we see that W =B(1). HereB(J) is the determinant ofB ij (J). One can verify following three equations HereB ij is the inverse ofB ij . By combining these three equations, one can find