^{*}

Edited by: Jakub Mielczarek, Jagiellonian University, Poland

Reviewed by: Tadashi Takayanagi, Kyoto University, Japan; Jan De Boer, University of Amsterdam, Netherlands

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In a recent note [

What is it that takes place in the holographic representation of a theory when an object in the bulk is gravitationally attracted to a massive body? Consider a holographic theory representing a region of empty space. By operating with a simple boundary operator

and becomes increasingly complex. If expanded in simple boundary operators the average number of such operators will increase and one says the size of the operator grows [

Let's say that the particle is moving away from the boundary so that the size is increasing. It seems plausible that velocity is related to the rate of change of size. This is oversimplified but it roughly captures the idea that size, and its rate of change, holographically encode the motion of the particle.

Now suppose there is a heavy mass at the center of the bulk region. The gravitational pull of the heavy mass will accelerate the particle away from the boundary. We may expect that the growth of

In [

Newton's second law is summarized by the familiar equation,

or its generalization,

Newton's third law—the law of action and reaction—says that the force exerted by

Newton's law of attraction,

My arguments are a heuristic mix of quantum information and gravitation and involve some guesswork, but a more formal basis has been found by Lin et al. [

The size of an operator is roughly the average number of elementary constituent degrees of freedom that appear in the expansion of the operator. While in general it is a somewhat imprecisely defined notion, in the SYK context in which we will be working it can be very precisely defined in terms of the average number of fermions in the operator [

If the considerations of this paper are to be useful, it will be necessary to generalize the concept of size to more general cases, in particular to higher dimensional gauge-gravity dualities. At the present time I do not know of any precise definition of size in strongly coupled CFTs. This is a serious hole in our knowledge that I hope will be filled.

Many of the equations in this paper are correct up to numerical factors relating SYK quantities to NERN quantities. These factors are in-principle computable using numerical SYK techniques, and depend on the locality parameter

The bulk dual of the SYK model is usually taken to be a version of the (1 + 1)-dimensional Jackiw-Teitelboim dilaton-gravity system. But that description (of a system with no local degrees of freedom) does not do justice to the spectrum of excitations of the SYK system. In many ways SYK is similar to the long throat of a near-extremal charged black hole whose geometry is approximately

To keep the paper self-contained, in this section I will review near-extremal black holes, and then in section 3, the dictionary relating SYK and near-extremal black holes will be explained. I will closely follow the discussion of NERN black holes in [

The metric of the (3 + 1)-dimensional Reissner-Nordstrom black hole is,

The inner (−) and outer (+) horizons are located at,

Define

The temperature is given by,

or

The extremal limit is defined by ^{2} = ^{2} at which point the horizon radii are equal, _{+} = _{−}. Our interest will be in near-extremal Reissner-Nordstrom (NERN) black holes, for which

In the NERN limit the temperature is small (β ≫ _{+}) and the near-horizon region develops a “throat” whose length is much longer than _{+}. The throat is an almost-homogeneous cylinder-like region in which the gravitational field is uniform over a long distance.

The exterior geometry consists of three regions shown in

The

The Rindler region has proper length ~ _{+} which means that it's about as long as it is wide.

The gravitational field (i.e., the proper acceleration

Proceeding outward, the next region is the

The throat is long and of almost constant width. The geometry in the throat region is approximately _{+}, which we will soon see is the location of a potential barrier which separates the throat from the far region. The throat is a feature of charged black holes and is absent from the Schwarzschild black hole.

For most purposes the geometry in the throat can be approximated by the extremal geometry with _{+} = _{−}.

The proper length of the throat is,

giving

We will assume that

Next is the

The far region lies beyond _{+}. The far region will not be of much interest to us. We will cut it off and replace it by a boundary condition at _{+}.

The three regions outside a near-extremal charged black hole. Unlike for uncharged black holes, there is now a “throat” separating the Rindler and far regions.

The black hole is effectively sealed off from the far region by a potential barrier. Low energy quanta in the throat are reflected back as they try to cross from the throat to the far region, or from the far region to the throat. The barrier height for a NERN black hole is much higher than the temperature and provides a natural boundary of the black hole region. It may be thought of as the holographic boundary in a quantum description. It is also the so-called Schwarzian boundary that appears in current literature on SYK theory [

The S-wave potential barrier has the form

and for a NERN black hole it is given by,

The width of the barrier in proper distance units is of order _{+} and for near extremal RN it is much narrower than the length of the throat. It therefore forms a fairly sharp boundary separating the black hole from the rest of space.

At the top of the barrier the potential is,

where

The top of the potential barrier serves as an effective boundary of the black hole. It occurs at,

We may eliminate reference to the entire region beyond the boundary and replace it by a suitable boundary condition^{1}_{+}. This is accomplished by the introduction of a boundary term in the gravitational action.

We define a radial proper-length coordinate ρ measured from the black hole boundary^{2}

In the throat

At the boundary ρ = 0, and at the beginning of the Rindler region ρ = _{+} log (β/_{+}). Note that ρ has a large variation over the throat region which makes it a more suitable radial coordinate than

The black hole boundary, defined as the place where _{+}, is not a rigid immovable object. Fluctuations or dynamical back reaction can change the metric so that the distance from the horizon to the boundary varies. This can be taken into account by allowing the boundary to move from its equilibrium position at ρ = 0.

In

Penrose diagram for a NERN black hole. The curved lines represent the trajectory of the black hole boundary at _{+}. On the left side the boundary is shown in its equilibrium location while on the right it is moving in reaction to some matter.

The equation of motion of the boundary is generated by the Hawking-Gibbons-York boundary term (Schwarzian action in SYK literature) needed to supplement the Einstein-Maxwell action in the presence of a boundary. For small slow perturbations the boundary motion is non-relativistic with a large mass of order _{+} (^{3}

Consider a particle dropped at _{+}, which corresponds to an energy

A particle is introduced at the top of the potential, and subsequently rolls down the potential.

Under the influence of a uniform gravitational field it accelerates^{4}

So far a small but important effect has been ignored. There is a back reaction that occurs when the particle falls off the potential. The potential exerts a force on the particle, which in turn exerts an equal and opposite force on the boundary. The result is that the boundary recoils with a small velocity. [With some effort this can be seen in the Schwarzian analysis [

The boundary recoils when the particle is accelerated. At all times the particle and the boundary have equal and opposite momentum.

Once the particle falls off the potential it quickly becomes relativistic. In the throat region its trajectory is given by

Thus, the particle trajectory satisfies,

or

The total time to fall from ρ = 0 to the beginning of the Rindler region is β. During that time the distance traveled is

Let's consider the relation between the Schwarzschild coordinate _{+} = _{−} and that

may be replaced by its extremal value

Recall that ρ is the proper distance measured from the boundary at _{+},

or,

The so-called surface gravity κ will play an important role in what follows. At the horizon the surface gravity is related to the temperature of the black hole by,

More generally it is defined at any radial position

which in the throat is approximated by,

The purpose of the tilde notation is to indicate a local quantity, i.e., one that may vary throughout the throat. Corresponding variables without the tilde indicate the value of the quantity at the horizon. We may also define

(Except at the horizon the quantity

In the throat let's express

and,

At ρ = 0,

At the Rindler end of the throat where ρ = _{+} log (β/_{+}),

By following the trajectory of the infalling particle 2.15, and using 2.25 we find that

As the Rindler region is approached

We can only go so far in understanding the quantum mechanics of NERN black holes without having a concrete holographic system to analyze. That brings us to the well-studied SYK model. In this section the SYK/NERN dictionary is spelled out.

We'll begin with qualitative aspects of the SYK/NERN dictionary and then attempt to determine more precise numerical coefficients in the next subsection. The two-sided arrows in this subsection indicate qualitative correspondences.

The overall energy scale of the SYK model is called _{+}. In the SYK model acting with a fermion operator _{+}. Thus it makes sense to identify the process of dropping a particle from the black hole boundary, with acting with a single fermion operator.

A single boundary fermion operator in SYK has size 1 corresponding the assumption of [

Up to a numerical factor ≈1, the zero temperature extremal entropy of SYK is the number of fermion degrees of freedom

The 4-dimensional Newton constant can be obtained from the entropy formula,

Using 3.1 and 3.3 gives,

The SYK theory does not have sub-AdS locality (locality on scales smaller than _{+}). It is comparable to a string theory in which the string scale is of order _{+} or 1/

The black hole mass is _{+}/

Many of the detailed coefficients that appear in the subsequent formulas are dependent on

The literature on the bulk dual of SYK theory [

The dynamical boundary of SYK (described by the Schwarzian action) corresponds to the NERN black hole boundary, i.e., the top of the barrier where the throat meets the far region. The action governing the motion of the boundary is the Gibbons-Hawking-York boundary action.

The dilaton field ϕ in [

The time coordinate used in the SYK literature is called _{+}.

The time coordinate

For NERN black holes _{2r+} = 1/4, from which it follows that,

In some cases the numerical coefficients appearing in the various correspondences have been studied and allow more quantitative correspondences. I'll give some examples here, but I won't keep track of these coefficients in subsequent sections.

The specific heats of the SYK model and the NERN black hole can both be computed. On the NERN side the calculation is analytic and yields,

For SYK the calculation was done in [

where α_{s}(_{s} = 0.007 and for large ^{2}.

Setting 3.10 and 3.11 equal, we find the relation,

Let λ and

and

Plugging 3.13 and 3.14 into 3.12 gives one relation between

Another relation can be found by considering the entropy of SYK and the NERN black hole. On the NERN side we use the Bekenstein-Hawking formula which gives,

On the SYK side reference [

where

Combing 3.16 and 3.17 with 3.13 and 3.14 gives another equation for

The two relations 3.15 and 3.18 yield the following expressions for λ and

Thus, we find the following correspondences,

For _{s} and

giving,

and

Now let's return to the problem of a light particle dropped from the top of the potential 2.9 and estimate its energy ϵ. The height of the barrier is

We may compare this energy with the energy added to the SYK ground state by applying a single fermion operator

where the average 〈…〉 indicates disorder average. (The factor of 2 in the first term is present because of the SYK convention that ^{2} = 1/2).

Consider applying a single fermion operator at time

and becomes a superposition of many-fermion operators [

Tree-like operator growth. The size at any circuit-depth is the final number of fermions while the complexity is the number of vertices in the diagram. In this figure the size is 81 and the complexity is 40. The complexity at the next step would be 40 + 81 = 121. The time scale for a unit change in depth is Δ

It is similar to the evolution of a quantum circuit and it is natural to define a circuit depth. In general the circuit depth may not unfold uniformly with time. For example, if for some reason the computer runs at a variable time-dependent rate, the size will grow exponentially with depth but not necessarily with time. The time associated with a unit change in circuit depth is defined to be Δ

We can express this in terms of a rate of growth

The exponential growth as a function of circuit depth (for time less than the scrambling time) is the reason that size and complexity are proportional to each other. One may think of the size at a given depth as the number of “leaves” of the tree, and the complexity as the integrated number of vertices up to that point. Because the tree grows exponentially, the number of leaves and the number of vertices are proportional, and with some normalization (of complexity) the size and complexity can be set equal.

Roberts et al. [

Roberts, Stanford, and Streicher give a more detailed formula,

Apart from a brief transient the size grows exponentially. Dropping the 1 which is unimportant, the rate

which after a short time ^{−1} tends to

We may restate this in terms of Δ

At very low temperatures the pattern is quantitatively different. According to Qi and Streicher the size for low

Early on the rate is comparable to the infinite

but after a time β/2π (at which the infalling particle has reached the Rindler region) the rate has slowed to

Our interest will lie in the throat region during time period between

Let κ(ρ) be the surface gravity at position ρ,

and let

At the horizon the surface gravity is related to the temperature of the black hole,

and

The obvious guess for the interpolation between 4.9 and 4.10 is,

This is correct in the Rindler region but in the throat it is off by a factor of 2. Consistency between the Qi-Streicher formula and 2.28 requires,

or in terms of Δ

In [

(the factor of proportionality being _{0} a fixed distance from the boundary. If the temperature is sufficiently low the geometry between ρ = 0 and ρ = ρ_{0} is extremely insensitive to β and the growth up to that point should also be insensitive to β. But equation 5.1 implies that _{0}) blows up as

The formula used in [

From 5.2 one sees that complexity (or size) is not in one to one relationship with either position (ρ) or momentum (

Qi and Streicher [_{*}. Between _{*} Qi and Streicher find^{5}

Let us compare 2.28,

with the SYK calculation of Qi-Streicher. We first note from 4.15 that for _{+},

The first term in the Qi-Streicher formula 5.3 is unimportant. We may write,

and

Using 5.4 we find,

For _{+} <

Actually this is accurate for almost the entire passage through the throat. The ratio

is close to 1 as long as π

where Δρ is the length of the throat (see

There is a striking similarity between 5.3 and the infinite temperature formula 4.4 but quantitatively they are quite different. From 4.4 we see that at ^{Jt}. The quadratic growth only persists for a very short time of order 1/

By contrast, in the low

Now we come to the main point, the relation between the evolution of complexity and Newton's equations of motion. Let us compare 4.15,

and 5.2,

Eliminating ^{6}

between a dynamical quantity

The numerical constant relating the two sides of 6.1 is connected with the coefficient ϵ in the additional energy of applying a fermion operator the SYK low temperature state ground state.

Equation (6.1) resembles the ordinary non-relativistic relation between momentum and velocity. One might be tempted to think that

Nevertheless let's proceed to time-differentiate [6.1],

We next use the fact that the rate of change of momentum is the applied force,

In

Equation (6.3) looks temptingly like Newton's equation

Toy model involving a big and little ball. The big ball represents the boundary and little ball represents the particle. The big ball remains non-relativistic while the little ball quickly become relativistic.

One—the big-ball _{B} and the other—little-ball _{b}. Initially the two are attached and the combined system is at rest. At

It is evident from Newton's third law that both balls satisfy the equations,

but only

The connection between the toy model and the NERN system is clear: _{B}.

It is also worth noting that the heavy ball

These considerations, along with equation 6.3, lead to the conclusion that it is

Since

where ρ_{B} is the location of the boundary. If further follows that,

where ρ_{0} is constant. The obvious choice is for ρ_{0} to be the horizon location in which case

There are a number of ways of estimating the boundary mass _{B}. One way is to directly analyze the Schwarzian boundary term in the action. I will do something different making direct use of the complexity-volume (CV) correspondence [

The standard volume-complexity (CV) relation is,

The volume is the area of the throat times the length ρ,

where _{+}. One finds

or using the SYK/NERN dictionary,

From 6.3 we may write,

It follows that the mass of the boundary is,

This is to be compared with the energy of the infalling particle which is _{B} is of the same order as the mass of the NERN black hole.

If we now combine 6.12 and 6.13 with Equation (A.14) from the

for the motion of the boundary^{7}

The derivation in

On the other hand, the right side is just ^{2}^{2} which can be evaluated from the Qi-Streicher formula. In the throat region one finds the QS formula gives

Equating the right side of 6.16 to the right side of 6.17 determines the value of _{B},

consistent with 6.13.

There is also information in the Qi-Streicher formula about the relativistic motion of the light particle. For example consider the time that it takes, moving relativistically, for the particle to travel the distance Δρ = _{+} log β/_{+} from the boundary to the Rindler region. From 2.16 one sees that the time is β. Once the particle is in the Rindler region the size begins to grow exponentially with time. The Qi-Streicher formula 5.3 shows that this is indeed the case.

The basis for the derivation of Newton's equations in section 6 was the relation between momentum and the time derivative of complexity (Equation 6.1), which itself was based on the momentum-size correspondence of [^{8}

We begin by considering the approximate symmetries of matter in the background of a fixed, almost infinite, _{2} throat. The Penrose diagram for the throat is shown in

Penrose diagram for a two-sided non-dynamical background in the limit of low temperature and infinite throat length. Also shown are the matter generators ^{2}

The symmetry of infinite _{2} is the non-compact group _{2} but the left and right boundaries break the symmetry. As long as matter is far from the boundaries the symmetry will be respected.

_{0}, _{0}, _{0}, satisfying the algebra,

It is convenient to rescale

The commutation relations become,

Let's consider the generators one by one. The action of

The generator

Finally

The Rindler time is related to

so that

The orbits of the three generators are shown in

Orbits of the three generators.

The two-sided Penrose diagrams 7 and 8 represents two uncoupled but entangled SYK systems with Hamiltonians _{R} and _{L}. The generator

One might think that the global energy _{L} + _{R}]. However, there is no symmetry of _{2} generated by (_{L} + _{R}). Without going into details, Maldacena and Qi [_{int} that couples the left and right sides,

Using

and

we can write

In [_{iL} and _{iR}. Using our convention of calling size

where δ_{β} is a dimensionless normalization factor which normalizes the size of a single fermion to unity. This same operator appears in the interaction term _{int} in [

Combining 7.15 with 7.13 we find,

Thus, apart from the factor _{β}, β,

Again, the meaning of ≈ in 7.17 is: _{β} have an intricate mixed dependence [

It is known that the quantity μ is not independent of the other three parameters and that there is a relation between them. Zhao^{9}^{2}. It follows that,

Differentiating the Qi-Streicher formula also gives,

(In _{+} to the horizon at _{+}.)

It follows that the coefficient μ must satisfy,

so that 6.1 is satisfied. Equation (7.20) is non-trivial. On dimensional grounds the

That the product in 7.20 only depends on

The formal considerations of this section did not involve the momentum-size correspondence 5.2 postulated in [

We are almost where we want to be, but not quite because we have assumed the throat is infinite. If we make the throat finite by allowing

There is a formal way to restore the symmetry as a gauge symmetry [_{2} as illustrated in

Embedding a finite throated geometry in _{2}. Also shown are the three

The curved boundary separating the blue regions from rest of the diagram represents the Schwarzian boundary. The Penrose diagram can be conveniently parameterized by dimensionless coordinates −∞ < _{2}. This invariance allows us to move the geometry in various ways. In other words the representation of the finite throat in _{2} is redundant; the symmetry is a gauge symmetry. As such its generators should be set to zero. Denoting the gauge generators by tilde-symbols,

But the tilde generators are no longer the matter charges; they now include the charges of the boundary. In particular the spatial charge

Therefore, the gauge condition

is the Newtons third law of action and reaction, which tells us that the boundary recoils when the matter particle is emitted into the throat. Keeping track of the action=reaction condition seems to be the main point of the gauge symmetry. The un-tilded operators are the physical matter generators and their negatives are the generators that act on the boundary degrees of freedom.

There are a number of ways of insuring gauge invariance. One way is to construct manifestly gauge invariant objects and work with them. Lin et al. [

The embedding is not unique due to the _{2}. This invariance allows us to move the entire geometry—matter and boundary—in various ways by applying the three gauge generators.

The action of

Fixing a gauge.

We can use the gauge symmetries them to fix a convenient gauge:

The left black hole has a bifucate horizon. Using the Ẽ symmetry we can shift it to the

Next we can use _{0} in along the _{0}.

Finally we can fix the boost symmetry by assuming a particle is dropped from the right boundary at

That completely fixes the gauge. The resulting Penrose diagram is shown in 11.

Notice that in the limit that the temperature goes to zero that the bifurcate horizon moves all the way to the left boundary. The right Rindler patch becomes the Poincare patch, and the boosts become Poincare time translations. Again there is a one parameter family parameterized by _{0}. The boost operator

The gauge fixed Penrose diagram with the right boundary intersecting the

The transformations generated by _{0} parameter that move the right boundary. The momentum of the infalling particle that we called

Dropping the particle from the right-side boundary causes the boundary to recoil and move outward. That is indicated by the small separation shown as light blue. The effect is to change the right-side horizon (not shown) so that its bifurcate point is no longer on the

The time-slice shown as green is anchored on the boundaries at “boost time” _{R} at _{R} is a purely right-side operator it evolves according to,

Under this evolution _{R}(

The complexity of the evolving state can be determined from

Lin-Maldacena-Zhao argue that the generators can be decomposed into bulk matter, and gravitational (boundary) contributions. The bulk matter contribution to

The low energy ^{10}_{B} = _{B} = _{+}.

This agrees with the analysis in the previous section and provides a formal justification for it.

In addition there is a coupling between the matter and the boundary which has the form of a repulsive potential energy. As long as the particle is in the throat region the potential is linear in the distance between the infalling particle and the boundary. As shown in the

Susskind [

We could try modeling questions like this in AdS/CFT, but the tools I've used in this paper are special to SYK. Fortunately there is a simple case in which the question can be addressed. Anti de Sitter space has a gravitational field even in the AdS vacuum. The negative vacuum energy of AdS gravitates and attracts matter to the center. One does not need an additional mass.

The metric of AdS is,

Particles dropped from a distance experience an attractive radial gravitational force which behaves similarly to a harmonic oscillator force. A particle will move in a periodic orbit oscillating about the origin. There is no black hole, no horizon, no entropy.

Two dimensional AdS is not an exception, but engineering empty _{2} is subtle in the SYK system. Maldacena and Qi [_{2}. The geometry does not extend out to ^{11}_{2} except that the blue regions near the boundary have been excised.

Traversable wormhole with two boundaries.

In

and subsequently evolves to

In this case there is no black hole and the particle endlessly oscillates back and forth between the two boundaries.

A particle has been added to the right side of the traversable wormhole by acting with

The force on the particle is gravitational. From the bulk GR viewpoint it is produced by the vacuum energy in the region between the boundaries. The state without the particle (

When the particle is injected at _{R} the additional complexity of the state is initially very small. As the particle accelerates toward the center of AdS its momentum increases. The right boundary recoils so that the distance between the boundaries increases. According to CV duality, the complexity also increases.

Because the boundary is very heavy it moves non-relativistically which means its momentum and velocity are proportional to one another, and once again,

for both the boundary and for the particle.

The radial momentum reaches a maximum when the particle reaches the center of the diagram. It then switches sign. At the same time the complexity starts to decrease^{12}

The particle then gravitates back to the center and subsequently returns to the right boundary. The oscillating behavior of complexity may seem odd, but in fact it is generic for integrable systems. It is also characteristic of holographic systems below the black hole threshold [

To reiterate, the connection between gravitational attraction and complexity is not dependent on the presence of a black hole, or on the presence of a system with a large entropy. However without a black hole the system is integrable and the complexity oscillates. It should be pointed out that the complexity never get's very large during the oscillating behavior. At the maximum when the particle is at the center of the geometry the complexity is ~ β^{2}^{2} which is much less than

It would be interesting to confirm this behavior in the SYK theory using the methods of Qi and Streicher.

In this article, I have assembled further evidence that the holographic avatar of gravitational attraction is the growth of operator-size during the run-up to the scrambling time. During this period, size and complexity are indistinguishable, and one can say that gravitational attraction is an example of the tendency for complexity to increase. The presence of a massive object creates a kind of complexity-force, driving the system toward greater complexity in the same way that an ordinary force accelerates a particle toward lower potential energy. This conclusion was based on three things: the CV correspondence between complexity and volume; a duality between momentum and the time-derivative of complexity,

and the Qi-Streicher calculation of the time dependence of size at low temperature.

To test the duality, on the left side we used the standard relativistic classical theory of particle motion (in a gravitational field) to compute

One can object to such a connection (between momentum and complexity) on the grounds that it relates two fundamentally different kinds of quantities. Momentum is a linear quantum observable. Complexity is a nonlinear property of states; linear superpositions of states with the same complexity may have very different complexity. Thus, equating momentum and the time-derivative of complexity is inappropriately mixing concepts.

Similar things have been seen before. The Bekenstein formula and more recently, the Ryu-Takyanagi formula, equate area—a quantum observable—to entropy. This also seems inadmissible for similar reasons. A number of authors have written about this tension [see for example [

The same things should be true for complexity: in the small subspace of states encountered while a particle is falling toward the horizon of a black hole complexity and its derivative can behave like an observable, but beyond the scrambling time or when superpositions of classical states are considered the relation between complexity and observables must break down^{13}

On another point, E. Verlinde has also emphasized the need for a holographic explanation of gravitational attraction and has proposed an entropic mechanism [

In contrast to coarse-grained thermal entropy, complexity and operator size can oscillate, especially for non-chaotic or weakly chaotic systems. By the complexity-volume correspondence, the oscillating complexity may manifest itself as periodic motion. The motion of a particle in empty _{2}, discussed in section 8 is an example.

Returning to the case of a black hole, entropy approaches its maximum value well before the scrambling time, but as shown in [

It is quite possible that these remarks represent my own misunderstanding of Verlinde's theory.

Finally I would like to emphasize the importance of generalizing the concept of size to a wider class of gauge-gravity dualities. In a strongly coupled CFT it is not obvious what the fundamental constituents are, that are counted when we speak of size. I hope to come back to this issue in the future.

All datasets generated for this study are included in the article/

The author confirms being the sole contributor of this work and has approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

I thank Henry Lin and Ying Zhao for very helpful discussions about both the heuristic and formal arguments in this paper. The paper would not have been written without the many discussions I had with Alex Streicher, in which he explained his results on the growth of size in SYK, and related issues. This research was supported by NSF Award Number 1316699. The article has appeared as a preprint.

The Supplementary Material for this article can be found online at:

^{1}In the SYK literature the corresponding boundary condition is placed on the point where the dilaton achieves a certain value. In the correspondence between the dilaton theory and the NERN black hole the dilaton is simply the area of the local 2-sphere at a given radial location.

^{2}Frequently a radial proper coordinate is defined as the distance to the horizon. Note that in this paper ρ measures distance to the black hole boundary at _{+}, not to the horizon.

^{3}The idea of the boundary as a very massive particle was suggested by Kitaev, who developed this idea in [

^{4}In the sense that its momentum grows. Being relativistic the velocity is close to 1.

^{5}Qi and Streicher calculate the size but for reasons I have explained size and complexity are interchangeable for our purposes.

^{6}This relation was derived by Lin et al. by different arguments. See section 7 and Lin et al. [

^{7}It should be kept in mind that the

^{8}I am grateful to Henry Lin and Ying Zhao for explaining the argument to me.

^{9}Unpublished communication.

^{10}I am grateful to Herny Lin for a helpful discussion of this point.

^{11}AdS two is unique in having two disconnected boundaries.

^{12}This conclusion is based on the ability of the gravitational dressing to switch from the right to the left side. Such switching would be impossible without left-right coupling, but there is no obstruction to it when the Maldacena-Qi interaction is included.

^{13}I am grateful to Daniel Harlow for discussions about this point.