Contents
Abstract

We discuss the way in which field theory quantities assemble the spatial geometry of three-dimensional anti-de Sitter space (AdS). The field theory ingredients are the entanglement entropies of boundary intervals. A point in AdS corresponds to a collection of boundary intervals, which is selected by a variational principle we discuss. Coordinates in AdS are integration constants of the resulting equation of motion. We propose a distance function for this collection of points, which obeys the triangle inequality as a consequence of the strong subadditivity of entropy. Our construction correctly reproduces the static slice of AdS and the Ryu-Takayanagi relation between geodesics and entanglement entropies. We discuss how these results extend to quotients of AdS – the conical defect and the BTZ geometries. In these cases, the set of entanglement entropies must be supplemented by other field theory quantities, which can carry the information about lengths of non-minimal geodesics.

Nuts and Bolts for Creating Space



Bartłomiej Czech and Lampros Lamprou

czech, llamprou -AT- stanford -DOT- edu




Stanford Institute for Theoretical Physics, Stanford University

382 Via Pueblo Mall, Stanford, CA 94305-4060, USA

1 Introduction

The gravitational force is the most familiar force of Nature. It was the first force to become an object of scientific inquiry [1, 2] and the earliest one to be understood quantitatively, at both medium [3] and large distance scales [4]. Yet at small scales, it remains the only known interaction still shrouded in mystery. There are, of course, good reasons for this: since gravity is the dynamical theory of space and time, it is not a priori clear what it would mean to understand gravity at arbitrarily small distances. Indeed, the very notion of a point in space – one that underlies our microscopic understanding of other physical interactions captured by the Standard Model – may well be a semiclassical construct, which on the level of the fundamental theory gives way to other, less familiar objects.

The present paper collects some lessons, which can be drawn about the microscopic nature of space from the holographic duality [5]. The holographic setting is convenient for several reasons. First, it translates gravitational problems into the language of gauge theory, which is in principle understood at all scales. In this way, holography grants access to and control over both the semiclassical spacetime and the microscopic degrees of freedom from which it is built. In particular, classical geometric objects on the gravity side – minimal surfaces – are conjecturally related to a set of fundamentally quantum quantities – entanglement entropies of spatial regions in the field theory [6, 7]. This relation allows one to go beyond gravitational perturbation theory and to study the fundamental theory of gravity in a background independent way. Finally, in the AdS/CFT correspondence the minimal surfaces that compute entanglement entropies are spacelike geodesics – simple objects, which in many cases are known analytically. Although the converse is not true – not all spatial geodesics compute boundary entanglement entropies [8, 9, 10] – much is known about their interpretation in the dual field theory [11, 12, 13, 14]. We will use these facts to explain how the spatial slice of AdS arises as a geometric description of the vacuum state of the dual two-dimensional conformal field theory (Secs. 3-4). We will further discuss how our reasoning extends to static quotients of AdS – the conical defect geometry (Sec. 5) and the non-rotating BTZ black holes (Sec. 6).

Our starting point is the Ryu-Takayanagi (RT) proposal [6, 7]. Its profound consequences for the emergence of spacetime were first pointed out in [15, 16] (see also [17, 18, 19, 20, 21, 22, 23, 24, 25]), which argued that reducing the entanglement between complementary half-spaces in the field theory is dual to lengthening a bridge in the spacetime until the latter pinches off into disconnected components. Extrapolating from this reasoning, it was conjectured that a holographic spacetime arises as a geometrization of the entanglement structure of a quantum state [19] (see also [23]). The present paper relies on a quantitative version of this statement, which appeared in [26] (see also [27, 28, 29, 30, 31]). It generalized the Ryu-Takayanagi proposal from spacelike geodesics to arbitrary differentiable curves on a spatial slice of AdS. Instead of entanglement entropy, the length of a non-geodesic curve computes a novel, UV-finite combination of field theory entanglement entropies, called differential entropy:

(1)

The differential entropy is a functional of a family of boundary intervals determined by one function , which implicitly picks out a differentiable curve on a spatial slice of AdS. We review the derivation of and useful facts about eq. (1) in Sec. 2. Eq. (1) will allow us to understand quantitatively how two-dimensional hyperbolic space – the spatial slice of AdS in static coordinates – emerges as a geometric encoding of entanglement entropies in the boundary field theory.

What does it mean for a spatial manifold to emerge? We answer this question and state our main results in Sec. 3. In essence, a geometry is a set of points and a distance function, which satisfies the triangle inequality. Traditionally, we relate spacetime points to boundary quantities using the famous HKLL construction [32, 33, 34], but that relation is limited to bulk low energy effective field theory. Here we wish to understand the emergence of space, so our goal is a non-perturbative definition of a bulk point, which assumes no prior knowledge of the metric. Sec. 3 states just that: a physically motivated, abstract definition of a bulk point in field theory. We then use eq. (1) to define a distance function on our abstractly defined bulk points and verify that its obeys the triangle inequality.111Our ab initio construction of points and distances should be distinguished from [35, 36, 37, 38], who use entanglement entropies / minimal surfaces to reconstruct numerically a metric assuming a certain ansatz.

In Sec. 4 we explain the physical and geometric reasons why our definition of points and distances correctly reproduces the static spatial slice of AdS. To do so, we detail several aspects of the differential entropy formula, which have not been previously discussed in the literature. The discussion is geometric and shows that bulk points are essentially limits of differentiable bulk curves shrunk to zero size. But the geometric statements have an intimate connection with information theory: for example, the strong subadditivity of entropy is an essential premise of the construction.

We discuss how our results extend to static quotients of AdS, the conical defect geometry and the non-rotating BTZ black holes, in Secs. 5 and Sec. 6. On the geometry side the construction is identical as in pure AdS, but on the field theory side it requires new ingredients. This is related to the fact that away from pure AdS not all geodesics compute entanglement entropies. The results raise several interesting issues, such as the emergence of locality on sub-AdS scales, which we briefly discuss. Comparing how our construction is applied to pure AdS and to its quotients highlights a salient fact: that the boundary definition of a spacetime point is state-dependent. For example, the boundary object, which in the field theory vacuum defines a bulk point in pure AdS, in the thermal state with defines a curve, which wraps around the black hole horizon with a berth of order . In the final section we discuss the significance of our findings and how they might be extended to holographic spacetimes beyond AdS and its quotients.

2 Review of hole-ography

We work in the context of the AdS / CFT duality, though the results of [26] extend to more general setups [27, 29, 30]. Here and in Secs. 3 and 4 we concentrate on pure AdS, which corresponds to the vacuum of the boundary CFT. We represent pure AdS in static coordinates:

(2)

We will distinguish the bulk coordinate from the angular coordinate on the boundary, denoted with .

We will make frequent use of spacelike geodesics. In coordinates (2), a spacelike geodesic centered at takes the form:

(3)

The regulated length of this geodesic in Planck units is

(4)

where is a gravitational infrared cutoff. According to the Ryu-Takayanagi proposal [6, 7], this quantity also computes the entanglement entropy of an interval of length in the vacuum of the dual CFT. On the field theory side, is an ultraviolet cutoff and [39], where is the central charge of the CFT.

Bulk curves

Consider a closed, smooth bulk curve on the slice. For every point on the curve, there is a unique geodesic tangent to the curve at that point. This geodesic, which also lives on the slice, has both endpoints on the asymptotic boundary of AdS. We denote the angular coordinates of the endpoints with , with marking the tangency point on the bulk curve ; see Fig. 1. In pure AdS the parameters defining the geodesics are:

(5)
(6)

Eqs. (5-6) can be inverted [30, 31]. The inverse mapping, which reconstructs the bulk curve from the boundary function , is:

(7)
(8)

Here and throughout this paper primes represent derivatives with respect to the boundary coordinate :

(9)

Note that a boundary function defines a bulk curve only when:

(10)
Figure 1: The notation introduced in Sec. 2. We consider the set of spacelike geodesics tangent to a given curve , where is the bulk angular coordinate. The geodesics are centered at and have width . The variable is reserved for the boundary coordinate.
Differential entropy

The main result of [26] is that the circumference of the bulk curve is computed on the field theory side by a novel quantity called differential entropy, which is a combination of (derivatives of) boundary entanglement entropies:

(11)

Importantly, the measure of integration is uniform not in the bulk coordinate , but in the boundary coordinate .

This result is easily verified for a central circle , but in other cases its proof is nontrivial. Eq. (11) represents an equality of two closed integrals for all choices of contours. Thus, the difference between the integrands

must be an exact form . The correct choice of is the length of the geodesic contained in the angular wedge between and , which in pure AdS takes the form:

(12)

We will interpret in a different way in Sec. 4.5. The total derivative term vanishes when , which is why for a circle the bulk formula (11) can be confirmed directly by inspection. An immediate corollary is a relation between the differential entropy and lengths of open curves in the bulk:

(13)

Eq. (13) computes the length of an open differentiable curve that stretches in the bulk angular wedge . The boundary integral extends from to – the midpoints of the intervals subtended by those geodesics, which are tangent to the curve at its endpoints.

A graphical explanation

Formula (11) can be motivated with the following heuristic argument. Rewrite the differential entropy by adding yet another total derivative term

(14)

and “discretize” the resulting integrand by Taylor expanding to first order in :

(15)

Let be the boundary interval subtended by the geodesic that is tangent to the bulk curve and centered at . Explicitly, we have:

(16)
(17)

Thus, eq. (15) computes the difference between the entanglement entropies of and of the overlap interval . In this way, the differential entropy can be expressed as

(18)

for a family of intervals with uniformly distributed centers, each defining a geodesic that is tangent to the target curve in the bulk.222In this form, the argument applies only when the bulk curve is convex. See [27] and Sec. 4.4 for a more general discussion. A graphical representation of eq. (18) is shown in Fig. 2.

Figure 2: A graphical representation of eq. (18). The black, continuous geodesics are tangent to the bulk curve; their lengths are . The purple, dotted geodesics subtend . We illustrate the effect of the limit by displaying finite combinations of geodesics at and for a circle. We also display the finite sum for the curve shown in Fig. 1.

A corollary of this graphical argument is that the bulk curve localizes on intersections of infinitesimally separated geodesics. This observation, which was emphasized in [40], gives the simplest method to obtain eqs. (7-8). It also explains the necessary condition (10) for a boundary function to describe a bulk curve: when (10) is not satisfied, infinitesimally separated geodesics do not intersect [27].

3 Points and distances

We now explain how in AdS the most basic geometric objects – points and geodesics connecting them – emerge from field theory data. The input from field theory is the complete set of entanglement entropies of spatial regions. We show how to use this data to construct a Euclidean geometry. The resulting space will by construction obey the relations reviewed in Sec. 2, including the Ryu-Takayanagi relation (4). In other words, the construction reproduces the spatial slice of AdS in static coordinates.

In this section we emphasize the conceptual basis of our construction. A more technical discussion, including proofs of some results, is relegated to Sec. 4.

3.1 Bulk Points

What is a bulk point? We present three answers to this question that are supplied by hole-ography, proceeding in order of increasing abstraction and robustness. The first two answers are motivational; we include them for pedagogical purposes. The final answer is a constructive definition of a bulk point – a definition stated entirely in the boundary theory.

3.1.1 Points as limits of curves

Consider a circle of radius centered at an arbitrary point on the slice of AdS. We can compute its corresponding boundary function using eqs. (5-6). From the bulk point of view it is then straightforward to take the limit and shrink the circle to a point – a curve of zero length located at the center of the circle. By taking the same limit for the boundary function we obtain a function, which depends parametrically only on the bulk coordinates of the limiting point. It is a simple exercise to verify that for an arbitrary bulk point with coordinates and we obtain:

(19)

Inserting into the differential entropy formula (1) we get identically zero. This is consistent with what happens to the length of a bulk circle as it is shrunk to zero size.

In light of this discussion one may be tempted to identify the set of bulk points with the set of zero differential entropy functions on the boundary. The rational would be to select curves of zero circumference, that is – points. We will explain in Sec. 4.6 that this would be incorrect, because eq. (1) computes signed lengths of oriented curves (see also [30]). Consequently, there are infinitely many curves of finite length but changing orientation, whose differential entropy evaluates to zero. This fact means that a hole-ographic definition of a bulk point must be more subtle.

3.1.2 Points from intersections of geodesics

The function in eq. (19) selects geodesics, which pass though the bulk point . We may attempt to turn this observation into a definition of a bulk point. To do so, we would need to formulate a condition that two distinct geodesics intersect at a given point. It is sufficient to state this condition for two infinitesimally separated geodesics (centered at and ) and then impose it over the full range of .

Two infinitesimally separated geodesics intersect at the bulk angle if they satisfy

(20)

where is given by eq. (3). We solve this equation with respect to and then demand that the intersection point remain constant as a function of . The latter condition is a second order differential equation, which selects , the boundary functions corresponding to bulk points:

(21)

After a redefinition eq. (21) becomes a harmonic oscillator differential equation for . The solutions are precisely the functions (19) derived in the previous section. It is not surprising that eq. (21) is second order: the integration constants are the coordinates and of the bulk point, which lives in a two-dimensional spatial slice of AdS.

Eq. (21) uniquely picks the boundary projections of bulk points. It was derived using the explicit form of AdS geodesics, but it is a useful guide toward a more general boundary definition of a bulk point. Consider the reciprocal of the left hand side of eq. (21): tracks the evolution of the boundary image of a point traveling along the curve, which is parameterized using the bulk angular coordinate . This means that its sign agrees with the sign of the extrinsic curvature of the curve. In this way, eq. (21) diagnoses the orientation of the curve: when it is positive the curve is convex and when it is negative it is concave. Eq. (21) tells us that a point is a “curve” which is neither convex nor concave: its orientation is undefined. In the next subsection we will see that this last statement can be cast directly in the language of the field theory.

3.1.3 Points as extrema of extrinsic curvature

In a negatively curved space the Gauss-Bonnet theorem states that:

(22)

On the left hand side is the length element along a curve and is its extrinsic curvature; the integral is taken over the length of the curve. In the middle expression, the Ricci scalar is integrated over the area enclosed by the curve. As a consequence, if a differentiable curve in a negatively curve space is shrunk to a point, the integral in (22) approaches . This means that if we extremize the left hand side of (22) over the set of all curves, we will have found points – “curves,” which circle around a point infinitesimally.

In Appendix A we have calculated the extrinsic curvature of a bulk curve . Using eqs. (5-6) the result can be re-expressed as:333We remind the reader that in our notation .

(23)

The numerator of eq. (23) can be rationalized in a simple way: it is the unique reparameterization invariant expression that vanishes when is proportional to – which, as we shall see in Sec. 4.1, is the case for curves, i.e. geodesics. We comment on the meaning of the denominator below.

If points extremize the integral in eq. (22), we can treat eq. (23) as a Lagrangian. The extrema of the resulting action will be bulk points. Indeed, the Euler-Lagrange equation of the action

(24)

is precisely eq. (21) and its complete set of solutions are the given in eq. (19).

Eq. (24) and its generalization (26) below are the main results of our paper. Action (24) defines bulk points purely in terms of the boundary theory: it can be stated and extremized without reference to the bulk, even though we arrived at it by studying curvatures in the bulk using eqs. (5-6).

3.1.4 Toward a background-independent definition

Action (24) correctly picks out points in pure AdS, but it does not work in other asymptotically AdS spacetimes such as the conical defect or the BTZ black hole. In particular, the denominator of eq. (23) was specifically chosen to reproduce the extrinsic curvature of a bulk curve in AdS and not in other asymptotically AdS spacetimes. We can fix this problem, at least for the conical defect and BTZ geometry, by observing that:

(25)

Substituting this in eq. (24) yields our final boundary definition of a bulk point: it is a “curve,” which extremizes the action:

(26)

Note that so long as the consistency condition (10) is satisfied, the expression under the square root is positive as a consequence of the strong subadditivity of entropy. The Euler-Lagrange equation reads:

(27)

It reduces to eq. (19) in the case of pure AdS. We shall see in Secs. 5 and 6 that eq. (27) correctly selects points in the bulk of the conical defect and BTZ spacetime.

Action (26) as a length in an auxiliary space

Eq. (26) has a DBI form: it computes the length of a curve in an auxiliary Lorentzian space coordinatized by and with metric

(28)

This auxiliary space is well known in integral geometry:444We would like to thank Michael Freedman for explaining to us the connection with integral geometry at the Aspen Center for Physics. its volume element is called the kinematic measure. Points in the kinematic space are geodesics on a spatial slice of the bulk. Eq. (27) tells us that the “inverse” statement also holds: geodesics in the kinematic space are points in the bulk. Metric (28) is special, because it is the unique metric (up to overall scale) on the space of geodesics, which is invariant under rotations and boosts in the bulk. In the case of AdS, the kinematic space is de Sitter space. We comment briefly on the relevance of integral geometry to holography in the Discussion.

3.2 Bulk Distances

3.2.1 Definition of geodesic distance

Given two points and with associated boundary functions and , we give the following boundary definition of the geodesic distance between them:

Definition 1

Let for all . The distance between and (measured in units of ) is:

(29)

To explain why this formula is correct, we will need several additional results, which we present in Sec. 4. In a nutshell, given two closed, convex bulk curves with boundary functions and , their pointwise minimum is the boundary image of their joint convex cover in the bulk. This result is developed in Sec. 4.3 and illustrated in Fig. 6 therein. When the two bulk “curves” are points and , their convex cover is the “closed curve,” which runs from to along a geodesic and then returns from to . The circumference of the cover, which is double the distance between and , is then given by .

The simplest example of eq. (29) is the distance of any point on the slice of AdS from the origin. The origin has and its vanishes everywhere. For a point located on , in eq. (19) is greater than outside the interval . Thus, formula (29) and eq. (19) give equal to:

(30)

We will present another example of formula (29) at work in Sec. 4.1.

3.2.2 Properties of distance

Let us confirm that Definition 1 obeys the axioms of a distance function. We remarked below eq. (19) that , so . Reflexivity follows directly from the definition. To confirm positivity , write

(31)

and note that the integrand in the first term is greater than or equal to the integrand in the second term, because . It remains to prove:

The triangle inequality.

We wish to verify that Definition 1 satisfies:

(32)

We have illustrated the computation for a generic triple of points in Fig. 3. The functions that define the bulk points are shown in the left panel. In the middle panel, we present the boundary functions that compute the pairwise distances between and according to eq. (29). The terms on the left hand side of (32) are marked in continuous brown while the right hand side is drawn in dashed red. In many places the dashed red overlaps with a continuous brown line and we have a direct cancellation. However, there is one interval over which we have two identical contributions from the left hand side: it is the interval where is smaller than and . Using the fact that we can trade one of these two contributions for the differential entropy of over the complementary interval, dressed with an extra minus sign. This leads to further cancellations, with the final result displayed in the right panel of Fig. 3.

Figure 3: The differential entropy proof of the triangle inequality (32). Terms on the left hand side are drawn in continuous brown while the right hand side is in dashed red. See text below for more explanation.

After the cancellations, both sides of the inequality are differential entropies computed over the same two intervals, but for different boundary functions:

(33)

Because the boundary function on the left hand side is strictly smaller than the boundary function the right hand side, the result again follows from the strong subadditivity of entropy: .

The proof is considerably simpler if we exploit conformal symmetry to place in the center of AdS. We chose not to do this and presented instead a proof, which relies only on two robust facts: the zero differential entropy of the point functions and the concavity of entropy . In this form, the proof carries over to other holographic spacetimes discussed in the present paper – the conical defect geometry (Sec. 5) and the static BTZ black hole (Sec. 6).

4 Hole-ography of curves, points and distances

In this section we discuss aspects of hole-ography, which lie at the root of the material in Sec. 3. The first three subsections explain our definition of geodesic distance given in eq. (29). The results also make it convenient to discuss nonconvex curves, which we do in Sec. 4.4. The last two subsections are relevant to understanding our definition of a point as an extremum of action (24).

4.1 Recovering the Ryu-Takayanagi formula

Another consistency check on the distance function (29) is that it correctly reproduces the lengths of geodesics anchored at the boundary. Consider two points and at some fixed angular locations and at equal radial coordinate . Sending it to infinity, the distance should approach the entanglement entropy (4) for the interval on the boundary. To corroborate this, examine the behavior of as approaches infinity:

(34)

This limit is illustrated in Fig. 4. Recall that our definition (29) of geodesic distance involves the pointwise minimum of and . In the limit (34) this pointwise minimum becomes the boundary causal diamond of the interval and of its complement. To make contact with formula (4), we must therefore examine the differential entropy (11) of a boundary causal diamond.

Figure 4: The functions of two points and , which approach the boundary at fixed angular coordinates . The pointwise minimum approaches the causal diamond (shaded) of the interval and of its complement.

This appears problematic, because eq. (34) does not satisfy the consistency condition (10). But it is useful to include boundary causal diamonds as a special case. Although formulas (7-8) do not recover the correct bulk curve (the geodesic connecting the boundary points at ), the differential entropy of a causal diamond can still be evaluated. Let and . For the causal diamond of the interval , the differential entropy is:

(35)

Thus, in AdS the differential entropy of a causal diamond equals the entanglement entropy of the boundary interval supporting it. Because we are working in a geometry dual to a pure state of the field theory, the causal diamond of gives a contribution equal to (35). Applying this conclusion to , we get , so that the distance between boundary points and correctly reproduces the entanglement entropy of interval .

4.2 Discontinuity in and finite pieces of bulk geodesics

As a step toward understanding eq. (29), consider a boundary function , which is only piecewise differentiable; see Fig. 5. Suppose jumps at some . Let us examine separately in the ranges , where it is differentiable. Each range defines a bulk curve with an endpoint. Both endpoints live on a common geodesic given by and . But looking at eqs. (7-8), we see that the two endpoints are a finite distance apart, because their locations depend on . In particular, eq. (8) gives us the bulk angular coordinates of the endpoints and , which are necessarily distinct.

Figure 5: Upper left: (green and blue), whose derivative is discontinuous at a point (red). Upper right: the geodesics defined by and their (color coded) joint outer envelope. The two bulk open curves have the same tangent geodesic at their endpoints (red), because is continuous. The thickened part of this geodesic stretches between the endpoints of the open curves; its length is , viz. eq. (36). The differential entropy of in the upper left panel is the length of this outer envelope. Lower panels: the individual bulk open curves and their tangent geodesics.

We would like to understand the differential entropy formula evaluated on our piecewise differentiable . To this end, we use eq. (13) for and individually:

(36)

In the final line we used the fact that the term computes the length of the geodesic with opening angle centered at , which is contained in the angular wedge . Likewise, is the length of the same geodesic contained in ; the minus sign arises to give a negative contribution if . Thus, the two -terms combine to form the length of the geodesic that connects the endpoints of the two open curves. Overall, computes the length of a continuously differentiable bulk curve, which is formed by joining the left and right open curves with the geodesic tangent at their endpoints; see the top right panel of Fig. 5.

The final conclusion is that a discontinuity in corresponds to a finite stretch of the bulk curve following one geodesic. The length over which the geodesic is followed depends on the jump in via eq. (8). We encountered one example of this finding in the previous subsection, where we saw that the differential entropy of a causal diamond (whose jumps at the top) computes the full length of a geodesic.

4.3 Pointwise minimum and the convex cover of curves

The discussion above has an interesting consequence. Consider two closed, convex, differentiable curves and in the bulk; see Fig. 6. Let their boundary functions be and and call their pointwise minimum . First, suppose that and intersect. We discussed this situation in Sec. 4.2 and illustrated it in Fig. 5. Generically, the derivative jumps at the intersection point and the bulk curve defined by first follows curve , then follows a geodesic which is tangent to both and (this is the geodesic selected by the intersection of and ), then follows curve . This is nothing but the convex cover of and . A second, special case is if everywhere, but then properly encloses in the bulk. The final conclusion is that the operation of taking a pointwise minimum on the boundary corresponds to taking the convex cover in the bulk.

This means, in particular, that if and describe two points and in the bulk, then defines the closed curve, which circumscribes the convex cover of the set . This closed curve follows the geodesic from to and then returns the same way from to , which agrees with our prescription (29) for geodesic distance. Fig. 6 illustrates this construction, though one must imagine shrinking the bulk circles to points, as described in Sec. 3.1.1.

Figure 6: Left: two closed bulk curves and (dashed blue and dotted orange) and their joint convex cover (thick green). We have drawn all geodesics tangent to the convex cover; the special geodesics that are tangent to both curves are drawn in black and red. Right: the functions and corresponding to the two bulk curves (dashed blue and dotted orange), their pointwise minimum (thick green) and intersection points (black and red.)

4.4 Inflection points and nonconvex curves

In the preceding subsection we assumed that the curves were convex. In the context of AdS, convexity means that the curve does not cross its tangent geodesics. We devote this subsection to consider a nonconvex curve – one that crosses its own geodesic in at least one point; see Fig. 7. In analogy to flat space geometry, such a point is an inflection point. We shall denote the bulk angular coordinate of the inflection point .

The top panels of Fig. 7 show that as we approach the inflection point, the centerpoints of the tangent geodesics approach from the same side. In other words, attains an extremum at . This means that, strictly speaking, we cannot speak of as a function of : as we trace the bulk curve, approaches the value and then reverses direction.555This was previously noted in [27, 28]. This is shown in the top right panel of Fig. 7.

In the bulk, as the inflection point is approached from either side, the coordinate approaches . Recall that eqs. (7-8) reconstruct the bulk curve from boundary data. In particular, eq. (8) implies that is well defined and equal for both branches of . We can say that the plot of develops an infinitely sharp cusp.

Nonconvex curves

A closed differentiable bulk curve defines an , which must return to itself after a rotation in the bulk (or its multiple). This means that inflection points can only occur in pairs. After adding a second inflection point (middle panel of Fig. 7), we obtain a nonconvex bulk curve and shown in the bottom of Fig. 7. The plot in the bottom right is typical for nonconvex curves.

Figure 7: Top: As we approach the concavity from the left, the associated boundary intervals move to the far right (green continuous plots) until the first inflection point (red), where they turn back (orange dashed plots). Middle row: As we leave the concavity, the intervals shift to the left (orange dashed plots) until the second inflection point (red), whereafter they return to moving forward (green continuous plots). Bottom: An aggregate plot of the bulk curve and of its , with three special geodesics and their tangency points singled out. The two red ones are tangent at the two inflection points; the green one occurs at the self-intersection of the plot of and is tangent to the curve at two distinct points. This geodesic traces the convex cover of the curve (black dashed; see also Sec. 4.3.)

Interestingly, the circumference of the curve is still computed using eq. (1), with the caveat that the segment of the boundary between the inflection points is first traversed forward, then backward, then forward again. Thus, in this case it is more correct to write:

(37)

We will comment more on the application of this formula to nonconvex curves in Sec. 4.6.

Convex cover of a nonconvex curve

The bottom panel of Fig. 7 shows that the plot of for a nonconvex curve self-intersects. Recall that every point on the plot defines a geodesic tangent to the curve. If the plot self-intersects, one geodesic must be tangent to the curve at two distinct points. We have marked that geodesic along with its two tangency points in the bottom panel of Fig. 7. This feature occurs whenever the bulk curve develops a concavity.

Using Sec. 4.3, we now know how to find the convex cover of a nonconvex curve. All we have to do is take the pointwise minimum of the multi-branched plot of . The resulting discontinuity in is responsible for the finite segment, along which the convex cover follows the special geodesic with two tangency points. See the bottom panels of Fig. 7 for an illustration.

4.5 Bulk curves with endpoints and corners

In this and the next subsection we gather some results, which clarify our hole-ographic construction of bulk points in Sec. 3.1.

We start with eq. (13), which gives a differential entropy formula for the length of a bulk curve with endpoints. The function featured in eq. (13) is dissatisfying in a hole-ographic context, because it depends on the bulk coordinate . We would like to relate to the function , which defines the endpoint of the bulk curve according to Sec. 3.1.

Figure 8: An open bulk curve with an endpoint at . We trade the inhomogeneous term in eq. (13) for an extension of given by in eq. (40). The extension describes all geodesics (shown in dashed blue) that pass through the endpoint of the bulk curve until , where the inhomogeneous term is set to zero.

Suppose that the endpoint of an open bulk curve falls at . The boundary function of the curve (not of the endpoint) terminates at some . Generically and is nonvanishing; see eq. (12). Now take over the range and concatenate it with of the curve. The bulk meaning of this operation is shown in Fig. 8: it adds to the set of geodesics tangent to curve a further set of geodesics, which pass through endpoint . Extending the boundary function in this way does not affect the bulk curve at all. Using Sec. 4.2, this directly implies that

(38)

for any discontinuity in would add to the bulk curve a finite piece of a geodesic. Eqs. (38) express the fact that point lives on the bulk curve ; the tangent to at is centered at .

We now compute the differential entropy of the concatenated range of . Referring to eq. (3), the geodesic with opening angle centered at can be recast in the form:

(39)

Because endpoint lives on that geodesic, we can express of eq. (19) as:

(40)

It is now straightforward to confirm that:

(41)

This gives a new perspective on the function : it is the differential entropy of the zero length extension of the open bulk curve, which resets to . A consequence of the resetting is that is set to zero. Note that when the integral in eq. (41) “runs backwards” and is negative.

A corner in the bulk

Putting together two curves with coincident endpoints we can construct a bulk curve with a “corner.” Two open curves and have a common endpoint if:

(42)
(43)

The values are centerpoints of the two geodesics, which are tangent to and to at . The location of , the corner of , is set by the choice of , which satisfies conditions (42-43). To obtain the circumference of such a curve, we substitute in the differential entropy formula (1) the boundary function, which is obtained by concatenating the boundary functions of curve , corner point , and curve .

In summary, a piecewise differentiable bulk curve can be represented on the boundary by a continuous and differentiable . A corner – that is, a discontinuity in – occurs when follows over a finite range of one of our point functions defined in eq. (19).

4.6 Orientation and signed lengths

In addition to the corners discussed in Sec. 4.5, bulk curves may develop another, qualitatively different type of singularity. As an example, consider:

(44)

Using eqs. (7-8) we find the bulk curve shown in Fig. 9. It has several surprising features. First, it contains three cusps, even though it does not agree with eq. (19) on any finite size interval. Next, as varies from 0 to , the curve is traversed twice. Finally, plugging eq. (44) into the differential entropy formula (1) gives , even though the curve has nonvanishing length.

Figure 9: The bulk curve defined by eq. (44). Left: The color coding is with reference to eq. (45). Right: We display geodesics tangent to the curve.
Figure 10: An example corner (discussed in Sec. 4.5, left panel) and cusp (Sec. 4.6, right). The normal vectors, which define an orientation on the curve, point toward – the centerpoints of the boundary intervals selected by in eq. (6).
Left: is discontinuous at a corner. In Sec. 4.5 we explained how to define a continuous boundary function around a corner. The discontinuity in is filled with , the point function of the corner (eq. 19). The “normal vectors” in this range are shown in blue; they preserve the orientation on the curve.
Right: is continuous at a cusp. The function is continuous and differentiable, because the tangent geodesics vary continuously along the curve. However, the orientation on the curve (relative to ) is reversed.

To understand this, we read off from eq. (44) and draw the set of geodesics tangent to the curve. In contrast to the cases discussed in Sec. 4.5, the geodesics tangent to the curve vary smoothly in the neighborhood of each cusp. This is equivalent to saying that the cusp is infinitely sharp: both segments incident on the cusp tend to the same limiting gradient. Yet another way to phrase this is that and simultaneously vanish, so is well defined and not equal to zero. A corner and a cusp are contrasted in Fig. 10.

Signed length of oriented curves

For each , draw at the corresponding tangency point an arrow aimed toward on the boundary. This procedure endows the bulk curve with an orientation. Fig. 10 shows that cusps reverse the orientation while corners, treated with our prescription of Sec. 4.5, preserve it. One easily confirms that after passing a cusp, the differential entropy formula computes the length of the curve with a relative minus sign. In other words, formula (1) computes the signed length of an oriented curve.666This was first observed in [30]. For example, for the bulk curve of eq. (44) we have:

(45)

The color-coding follows the left panel of Fig. 9. Each term appears twice (albeit with a different sign), because as ranges from 0 to , the bulk curve is traversed twice. Example (44) illustrates why does not work as a boundary definition of a bulk point. Instead, identifying points involves the extrinsic curvature of the curve.

Orientation

How can we diagnose a reversal of orientation? It occurs at common zeroes of and . Differentiating eqs. (7-8), we identify the common factor:

(46)

In this diagnostic, the upper sign selects the outward orientation for a circle . Comparing (46) with eq. (21), we see that points are precisely those “curves” whose orientation is everywhere undefined. Recall that we derived eq. (21) in Sec. 3.1.2 by demanding that the intersection point of two infinitesimally separated geodesics remain at constant in the bulk. Condition (46) distinguishes when the said point moves in the direction of increasing or decreasing . Because intersections of neighboring geodesics are what defines the bulk curve, this is equivalent to deciding in which direction we scan the bulk curve as on the boundary increases.

Discretized differential entropy

When the orientation of the bulk curve changes, the discrete approximation to differential entropy given in eq. (18) must be supplemented by a second case:

(47)

This equation was first observed in [27] and explained formally in [30]. To check that the second case correctly captures the extra minus sign occasioned by an orientation flip, define so that . We now note that for infinitesimally separated intervals the differences between and and between and can be ignored.

A reversal of orientation could be seen as replacing a boundary interval with its complement. This is because the normal to the curve, which used to point toward the center of , gets flipped toward the center of . In a pure state such as the vacuum dual to AdS, this allows us to rewrite eq. (47) in an arguably more symmetric fashion:

(48)
Nonconvex curves

Eq. (47) was first observed in [27] to apply to nonconvex curves. Specifically, the second case applies in the region between two inflection points. We first verify this statement using eq. (46). We then explain why the differential entropy formula correctly computes circumferences of nonconvex curves, despite a change of sign in (46).

First, look at condition (46). We observed in Sec. 4.4 that in a neighborhood of an inflection point both and are continuous. However,

(49)

blows up at inflection points, because they are characterized by a reversal of direction of so . The blow-up means that the sign of in eq. (46) alone sets the orientation of the curve. Because attains a maximum or minimum at an inflection point, necessarily changes sign, so the orientation of the curve as defined by (46) is reversed between inflection points. This confirms that a concave stretch of a bulk curve falls under the second case of eq. (47).

And yet, as we remarked in eq. (37), the differential entropy of a nonconvex curve reproduces exactly the circumference of the curve, without an extra minus sign. To understand why the length of the curve between inflection points does not contribute negatively, return to condition (46). As already remarked, it specifies in which direction the bulk curve is scanned as increases. However, in Sec. 4.4 we found that between inflection points is traversed from right to left. This means that after orientation reversal the bulk curve is again scanned in the correct direction – from left to right. In this way, eq. (37) is a consequence of two compensating sign changes, which are illustrated in Fig. 11.

= -
Figure 11: The differential entropy formula computes the total length of a nonconvex curve thanks to two sign changes, one from a flip in orientation and one from reversing the direction of integration.

5 Conical defect geometry

From the bulk point of view, Sec. 3.1.2 constructs a point in AdS by selecting the complete set of geodesics that pass through it. The input to our construction must therefore encompass (boundary avatars of) all spatial geodesics in the bulk. When the spacetime contains geodesics, which cannot be interpreted as entanglement entropies, it is necessary to supplement the set of all entanglement entropies with additional boundary data.

This section presents the simplest example, where data beyond entanglement entropy first becomes necessary. The example is the conical defect geometry and the new data was named entwinement in [14]. In the next section we will discuss the BTZ geometry, where the input to the hole-ographic construction will include yet another novel ingredient.

5.1 Review of the conical defect geometry and entwinement

The conical defect metric (see e.g. [41, 42, 43]) is:

(50)

with ranging from to . To see the conical defect singularity at , change coordinates according to

(51)

to obtain metric (2), but with in the range from 0 to . When is an integer, this is AdS. Applying the inverse of this coordinate change to the spatial geodesic (3) in AdS, we obtain the geodesics in the conical defect: