Semi-Classical Black Hole Holography

Dissertation for the degree of Doctor of Philosophy

Lukas Schneiderbauer

School of Engineering and Natural Sciences Faculty of Physical Sciences Reykjavík, September 2020 A dissertation presented to the University of Iceland, School of Engineering and Natural Sciences, in candidacy for the degree of Doctor of Philosophy.

Doctoral committee Prof. Lárus Thorlacius, advisor Faculty of Physical Sciences, University of Iceland

Prof. Þórður Jónsson Faculty of Physical Sciences, University of Iceland

Prof. Valentina Giangreco M. Puletti Faculty of Physical Sciences, University of Iceland

Prof. Bo Sundborg Department of Physics, Stockholm University

Opponents Prof. Veronika Hubeny University of California, Davis

Prof. Mukund Rangamani University of California, Davis

Semi-classical Black Hole Holography Copyright © 2020 Lukas Schneiderbauer. ISBN: 978-9935-9452-9-7 Author ORCID: 0000-0002-0975-6803 Printed in Iceland by Háskólaprent, Reykjavík, Iceland, September 2020. Contents

Abstractv

Ágrip (in Icelandic) vii

Acknowledgementsx

1 Introduction1

2 Black Hole Model7 2.1 The Polyakov term ...... 8 2.2 The stress tensor ...... 11 2.3 Coherent matter states ...... 14 2.4 The RST term ...... 15 2.5 Equations of motion ...... 15 2.6 Black hole thermodynamics ...... 17

3 Page Curve 21 3.1 Generalized entropy ...... 24

4 Quantum Complexity 27 4.1 Complexity of black holes ...... 29 4.2 The geometric dual of complexity ...... 32

5 Conclusion 35

Bibliography 37

Articles 47 Article I ...... 49

iii Article II ...... 67 Article III ...... 99

iv Abstract

This thesis discusses two aspects of semi-classical black holes. First, a recently improved semi-classical formula for the entanglement entropy of black hole radiation is examined. This entropy is an indicator of information loss and determines whether black hole evaporation is an information preserving process or destroys quantum information. Assuming information conservation, Page expressed the entanglement entropy as a function of time, which is referred to as the “Page curve.” Using the improved formula for evaporating black hole solutions of a gravitational model introduced by Callan, Giddings, Harvey and Strominger (CGHS) and modified by Russo, Susskind and Thorlacius (RST), we find that the entanglement entropy follows the Page curve and thus is consistent with unitary evolution. Second, the notion of quantum complexity is explored in the context of black holes. The quantum complexity of a quantum state measures how many “simple operations” are required to create that state. Susskind conjectured that the quantum complexity of a black hole state corresponds to a certain volume inside the black hole. A modified conjecture equates the quantum complexity with the gravitational action evaluated for a certain region of spacetime which intersects the black hole interior. We test the complexity conjectures for semi-classical black hole solutions in the CGHS/RST model and find that both conjectures yield the expected behavior.

v vi Ágrip

Í ritgerðinni er fjallað um tvo eiginleika svarthola. Fyrst skoðum við nýja aðferð við að reikna flækjuóreiðu á milli svarthols og Hawk- ing geislunarinnar sem það sendir frá sér. Þessir útreikningar gefa til kynna hvort skammtaupplýsingar tapist við uppgufun svartholsins eður ei. Ef gert er ráð fyrir að upplýsingarnar varðveitist, þá fylgir flækjuóreiðan svonefndum Page ferli. Við reiknum flækjuóreiðuna, annars vegar í þyngdarfræðilíkani sem kennt er við Callan, Giddings, Harvey og Strominger (CGHS) og hins vegar í líkani kennt við Russo, Sus- skind og Thorlacius (RST), og sýnum fram á að hún fylgir Page ferlinum í báðum þessum líkönum. Niðurstaðan er í samræmi við það að uppgufun svarthols sé ferli þar sem skammtaupplýsingar tapast ekki. Við skoðum einnig hugtakið skammtafræðilegt flækjustig í tengslum við svarthol. Flækjustig skammtaástands segir til um hve margar einfaldar aðgerðir eða skref af ákveðinni gerð þarf til þess að mynda ástandið úr tilteknu viðmiðunarástandi. Susskind setti fram þá tilgátu að skammtafræðilegt flækjustig svarthols sé jafngilt tilteknu rúmmáli innan svartholsins. Önnur tilgáta tengir flækjustigið við þyngdarvirkni ákveðins hluta tímarúmsins sem skarast við svartholið. Við prófum þessar tilgátur fyrir svarthol sem gufa upp með Hawking útgeislun í CGHS og RST líkönunum og sýnum að báðar tilgáturnar gefa flækjustig í samræmi við skammtafræði.

vii viii Acknowledgements

First and foremost, I would like to express my gratitude to my supervisor Lárus Thorlacius as well as Valentina Puletti for their guidance and help throughout my years as a doctoral candidate. Special thanks go to the postdocs present during my stay at the University of Iceland, Daniel Fernandez, Friðrik Freyr Gautason, Napat Poovuttikul and Watse Sybesma for lively discussions (about topics not exclusively related to physics) and their valuable feedback. I am especially in debt to Friðrik who helped me translate the abstract into Icelandic in addition to giving me valuable feedback to various parts of this thesis, and Watse who provided his drawing skills to help create some of the figures. It was a delight to work with them as collaborators. I would also like to thank my doctoral siblings, Juan Angel-Ramelli and Aruna Rajagopal, for all our shared experiences. I wish them all the best for whatever the future may hold for them. For enduring me throughout as an office mate, I have to thank numerous people who came and went throughout my years in Iceland. I certainly enjoyed the en- lightening conversations with Daniel Amankwah, Arnbjörg Soffía Árnadóttir, Hjörtur Björnsson, Jóhann Haraldsson and Auðunn Skúta Snæbjarnarson. The running group and Bandí team I fortunately joined also deserve mentioning for accompanying me through most of my stay in Iceland. They provided me with a welcome diversion from academic work. Last but not least, I am grateful to my family and friends. I feel lucky being born to my parents, Grete and Helmut, who accepted and supported my major decisions throughout my life. I want to thank my siblings and their partners, Daniel, David, Eveline, Magdalena and Teresa. They kept my childhood interesting and turned every visit at home into a fun experience. A lot of joy is brought to me by my two nephews, Emil and Oscar, who managed to distract me even while writing these lines. I am especially thankful to three of my best friends, Dominik, Moritz and Philipp

ix with whom I spent the majority of my time during my studies. Every visit is a great pleasure to me, and I am lucky to have them in my life. This work was supported by the Icelandic Research Fund under grants 163422-053 and 195970-051, and by the University of Iceland Research Fund.

x 1

Introduction

Black holes are objects in space whose mass is so strongly concentrated that their gravitational attraction would not allow anything, not even light, to escape. They are, without doubt, among the most fascinating objects studied by physicists within the last century. Its original discovery as a solution to Einstein’s theory of general relativity (GR) by Schwarzschild [1] dates back to 1915. Although initially not taken seriously even by Einstein himself [2], and after a period of confusion about the nature of the event horizon of a black hole [3], there is no doubt to date that black holes are realistic solutions of GR. Not only was it recognized that black holes can be formed within ordinary circumstances, but it has also been realized that the appearance of black holes is actually hard to avoid in nature [4]. As of now, there is also a long list of observational evidence for the existence of black holes. Among the most prominent examples are the detection of gravitational waves [5], which was honored with a nobel prize in 2017, and the popular “Event Horizon Telescope” image [6]. GR is primarily concerned with describing physical phenomena at astronomical 11 scales, that is, distances of order & 10 cm. A complementary theory, quantum mechanics, was developed almost in parallel to describe the physical aspects concerning −8 atomic scales . 10 cm. With the advent of quantum mechanics into the second half of the 20th century its unification with special relativity resulted in what we know today as quantum field theory (QFT). Although it became clear at an early stage that QFT was the appropriate framework to correctly describe known fundamental particles and their interactions in a unified manner, it does not account for gravity.

1 2 Chapter 1. Introduction

This is not a problem for contemporary particle physicists, since gravity is a very weak force and its effect is negligible for usual particle experiments on our planet. However, the gravitational field can in principle become arbitrarily strong, for it only requires a high enough mass density. This leads us back to black holes which represent the archetype of objects with highly concentrated mass. If we want to understand the physics in the neighborhood of a black hole, which undoubtedly exist in nature, we need to understand how quantum mechanics and gravity function together. In other words, we need a unified theory, “quantum gravity.” While finding a full quantum theory of gravity has proven incredibly difficult, one can pursue a more humble path and content oneself with exploring situations where the gravitational effects are stronger than usual but not too strong. In such situations it is sometimes possible to ignore the quantum effects associated with the gravitational degrees of freedom but still incorporate quantum effects of normal matter and their effect on the spacetime geometry. It appears that already in this regime one stumbles across unexpected consequences and interesting puzzles arise. Around sixty years after the black hole’s first discovery, Hawking realized, taking into account quantum effects in the manner discussed above, that a black hole is in fact not black, but it radiates [7]. This radiation is of thermal nature and in line with earlier thermodynamic considerations regarding black holes by Bekenstein [8,9]. Shortly after, Hawking concluded that the scattering matrix is not unitary and information is lost [10]. At the time this came to a big surprise to the physics community, since unitarity, or loosely speaking, “conservation of information,” is a corner stone in the framework of quantum mechanics. Giving up such an important principle can not be easily accepted. This now famous problem was later dubbed the “black hole information paradox,” and was since then a primary drive for theoretical physicists to study black holes in all their varieties. Even though Hawking initially proposed to give up unitarity [11], major problems arose when trying to generalize quantum mechanics to incorporate non-unitary evolution [12, 13]. Strong hints were given by the discovered duality between Anti-de Sitter space (AdS) and conformal quantum field theory (CFT), in short AdS/CFT, in 1997 [14]. AdS/CFT describes a correspondence between two completely different theories. On one hand, we have a theory of gravity, GR including a negative cosmological constant, while, on the other hand, we have a usual quantum mechanical theory, a QFT. The AdS/CFT correspondence states that these theories are equivalent, and one can map a particular problem on one side to another problem on the other side. Now, assuming the AdS/CFT correspondence is correct, since the evolution of the QFT is 3 manifestly unitary, it also must hold true for the gravitational theory. Even though AdS/CFT states that black holes in AdS do not destroy information, that does not provide an answer to the question for what is wrong with Hawking’s calculation, even in an AdS/CFT setup, nor does it give definite answers about how the required information escapes the black hole. In the AdS/CFT context, a recent breakthrough was achieved by computing the Page curve [15, 16] using semi-classical methods, by coupling an asymptotically AdS black hole to a CFT reservoir [17, 18]. As we will discuss in more detail in chapter3, the Page curve quantifies the correlations between the black hole and its radiation and is a good indicator for unitarity. The result uses the Quantum Ryu-Takayanagi (QRT) formula [19–22] and the outcome exactly follows Page’s prediction assuming unitarity [15, 16]. It is the first time that it was accomplished to compute the Page curve by semi-classical methods alone. Although the QRT formula is motivated by AdS/CFT technology, there is evidence that it is actually valid even outside the AdS/CFT framework. It is claimed [23–25] that this formula can be derived using the Replica trick [26, 27] without making reference to a holographic dual system. The crucial ingredient is the inclusion of non-trivial topologies, i.e. Euclidean wormholes when evaluating the functional integral over Euclidean manifolds. On a practical level the QRT formula leads one to include the correlations from the radiation degrees of freedom in the interior of the Black Hole after the Page time. From this point of view, it is perhaps not surprising that this formula reproduces the Page curve, since the disappearance of information into the black hole interior was what caused the problem in the first place [7]. However, the point is that this instruction came from a semi- classical computation and the fact that the semi-classical geometry has information about these details is surprising. Even if this calculation still does not reveal how information escapes the black hole, it is at least another indicator that understanding the interior of the black hole is of major importance. Independently of the investigation of the Page curve, Susskind initiated a study of the black hole interior via the notion of quantum complexity [28– 30]. Quantum complexity defines a metric on the space of states, which in some sense captures an intuitive understanding of the difficulty to create a state by “simple operations” from another reference state. This is best illustrated by imagining a quantum computer consisting of a set of gates, which are modeled by unitary operators acting on a Hilbert space. These particular set of gates need a certain amount of applications to reach from an initial input state to a final output state. The number of (minimum) applications is what we call complexity, and it quantifies how “hard” it 4 Chapter 1. Introduction is for the computer to compute a certain state provided some input state. Susskind conjectured that the growth of complexity of a black hole state should be related to the volume growth of the interior of the black hole. We will review these arguments in chapter4. One of the initial motivations to study complexity was to argue against a “typical” occurrence of firewalls [31]. The term “firewall” [32, 33] refers to a hypo- thetical violent region at the event horizon of a black hole for an infalling observer, and thus violating the equivalence principle. It was proposed in [32] so to resolve an apparent paradox within black hole complementarity [34]. Susskind reasons [35] that a “typical” state has maximal complexity. Since the complexity of black hole states that are formed by collapse is increasing for an exponentially long time (in its entropy), they are not “typical,” and hence there is no reason to expect a firewall.

Outline

In this thesis we expand on the above ideas in the context of asymptotically flat black holes in a two dimensional dilaton gravity theory which we will describe in more detail in chapter2. Since the QRT formula was originally formulated in the context of AdS/CFT, it is an interesting question whether it also gives sensible results for asymptotically flat black holes. The two dimensional models we are considering incorporate back-reaction of the Hawking radiation to the geometry. This enables us, at least in principle, to directly compute the entanglement entropy which include effects such as the reduction of the black hole mass over time. In chapter3 we review Page’s argument which leads to the Page curve and elaborate on the proposed entropy formula. We evaluate the entanglement entropy on the semi-classical two-dimensional backgrounds in [36] and find perfect agreement with Page’s prediction. In chapter4 we revisit the notion of quantum complexity. Although complexity was originally studied for classical black holes in asymptotic AdS spacetime, we analyse Susskind’s ideas in the semi-classical regime of asymptotically flat black holes [37, 38]. A part of the problem is to precisely define the supposedly dual quantities. With our definitions we find positive evidence that supports his conjectures even in these cases. That such an identification is also possible in asymptotically flat spacetimes provides further evidence for black hole complementarity [34] which postulates that the black hole can be described by a quantum system with a finite number of degrees of freedom living near (and outside) the event horizon. 5

This thesis is based on three research articles,

I Holographic Complexity: Stretching the Horizon of an Evaporating Black Hole by Lukas Schneiderbauer, Watse Sybesma, and Lárus Thorlacius, published in the Journal of High Energy Physics, volume 3 (2020),

II Action Complexity for Semi-Classical Black Holes by Lukas Schneiderbauer, Watse Sybesma, and Lárus Thorlacius, published in the Journal of High En- ergy Physics, volume 7 (2020),

III Page Curve for an Evaporating Black Hole by Friðrik Freyr Gautason, Lukas Schneiderbauer, Watse Sybesma, and Lárus Thorlacius, published in the Journal of High Energy Physics, volume 5 (2020),

The articles are attached at the end of this thesis with copyright permission from JHEP. 6 Chapter 1. Introduction 2

Black Hole Model

In this thesis we primarily work with a two dimensional dilatonic gravity model introduced by Callan, Giddings, Harvey and Strominger (CGHS) [39] and a variant thereof by Russo, Susskind and Thorlacius (RST) [40]. One of the main advantages coming with the RST model is that the semi-classical equations can be solved explicitly as we show below. The CGHS model can be regarded as an eﬀective description for radial modes of near-extremal dilatonic magnetically charged black holes in four or higher dimensions [41]. While this aspect certainly provides an interesting point of view, in this work it will usually be suﬃcient to interpret the results from a purely two dimensional perspective.1 In this section we will review some aspects of the CGHS/RST models while trying to be complementary to the introductions in [36–38] and the already existing review by Thorlacius [42].

The classical CGHS action I0 is given by

I0 = Igrav + Imatter (2.1) with

1 Z √ I = d2x −g e−2φ R + 4 (∇φ)2 + 4λ2 , (2.2) grav 2π Z N 1 √ X 2 I = − d2x −g (∇f ) , (2.3) matter 4π i i=1 1This is going to be true apart from an isolated case, when we deﬁne the volume functional which represents complexity, see [37].

7 8 Chapter 2. Black Hole Model

and depends on the ﬁelds φ , the dilaton, the metric gµν , and the N free matter ﬁelds fi minimally coupled to the gravitational system. As usual, g := det gµν , and R is the Ricci scalar. A natural scale is provided by the parameter λ related to the magnetic charge of the higher dimensional theory. We take length to be measured in units of λ−1 and set λ = 1. 1 2φ The strength of the classical gravitational coupling is characterized by G(2) := 8 e which is allowed to vary over spacetime. If the gravitational coupling G(2) is small compared to N, and N much larger than 24,

N e−2φ 1, (2.4) 24 we can attempt a semi-classical treatment [39]. These conditions are necessary if we want to reliably ignore quantum effects induced by the dilaton and metric field since they will be dominated by the one-loop effects of N matter fields. Considering that the dilaton φ varies over spacetime, it is not clear a priori that these conditions can be satisfied and the solutions of the semi-classical equations of motion have to be checked for consistency. When discussing black hole solutions it turns out that the gravitational coupling G(2) goes to zero in the asymptotic region, but increases as one approaches the black hole singularity. One can, however, make the coupling arbitrary small at its event horizon by considering a black hole of large enough mass. One should then be able to trust the semi-classical solution at least everywhere outside the black hole as long as its mass is still large enough.2 We review the inclusion of quantum corrections in section 2.1. From the quantum effective action we derive an expression for the stress tensor in section 2.2 and discuss its properties. Further, in order to describe dynamical black hole creation we shortly examine the concept of coherent states in chapter 2.3. After inclusion of the RST term, discussed in 2.4, we consider the equations of motion corresponding to the quantum effective action in section 2.5. Finally, black hole thermodynamics in the context of the CGHS model is discussed in paragraph 2.6.

2.1 The Polyakov term

To calculate the correction I1 to the classical action induced by the matter ﬁelds fi, we can use a well known argument [43–45] which goes as follows. The crucial ingredient

2A classical black hole retains its initial mass for all times. However, quantum eﬀects lead to Hawking radiation and the black hole slowly evaporates. 2.1. The Polyakov term 9

is the fact that the free matter ﬁelds fi constitute a CFT on a ﬁxed background.

Formally, the correction I1 is given by the functional integral Z eiI1 := Df eiImatter , (2.5)

where we integrate over all matter ﬁelds fi over a two-dimensional manifold with ﬁxed metric gµν .

By the deﬁnition of the stress tensor Tµν , its expectation value is given by

4π δI hT i = −√ 1 . (2.6) µν −g δgµν

We will exploit this identity to deduce the form of I1. In a classical CFT the stress tensor is traceless. However, when quantizing the theory in a way compatible with conformal symmetry one encounters that the trace of the stress tensor operator acquires a non-zero expectation value, the so-called trace anomaly. In two dimensions its form is completely ﬁxed by the central charge c of the CFT (in the present case c = N) and is given by [46]

4π δI N hT µ i = −√ gµν 1 = R, (2.7) µ −g δgµν 12 with R being the Ricci scalar and N the number of scalar ﬁelds fi which equals the central charge c of the CFT. In conformal coordinates (y+, y−), in which the line element reads ds2 = −e2ρdy+dy−, (2.8)

−2ρ the Ricci scalar can be expressed as R = 8e ∂+∂−ρ, and so equation (2.7) becomes

δI N 1 = ∂ ∂ ρ. (2.9) δρ 6π + −

This equation is solved by3

N Z N Z √ 1 I = dx+dx−ρ∂ ∂ ρ = − d2x −gR R, (2.10) 1 12π + − 96π ∇2 where we write 1 Z R (x) := d2yp−g(y)G(x, y)R(y), (2.11) ∇2

3Given any solution we can of course add terms whose variation is zero as we vary the conformal factor ρ and the result still solves equation (2.9). We will come back to this issue later on. 10 Chapter 2. Black Hole Model

2 µ with G(x) being the Green’s function of the Laplacian ∇ = ∇µ∇ deﬁned by √ −g∇2G(x) = δ(x). (2.12)

It is a peculiarity of I1 that the covariant expression in (2.10) is obviously non- local, while in conformal gauge the term reduces to a single integral involving the conformal factor. For later convenience let us introduce a slightly shorter notation,

1 Z(x) := − R (x). (2.13) ∇2

Equation (2.12) does not define the Green’s function G uniquely. It is only determined up to a harmonic function. This is simply a manifestation of the fact that the Laplace operator ∇2 has a non-trivial kernel on non-compact manifolds. The undetermined harmonic function has to be fixed by imposing boundary conditions. Let us render these statements more explicit by choosing conformal gauge (2.8). In order to compare Green’s function with different boundary conditions, let us evaluate ± ± + − the field Z for a general Green’s function G(y ) = G0(y ) + h+(y ) + h−(y ). ± + − The function h(y ) = h+(y ) + h−(y ) is harmonic by construction and contains information about the boundary conditions of G. We have Z ± + − ± ± ± Z(y ) = −4 dz dz G(z , y )∂+∂−ρ(z ). (2.14)

Naively, one would perform repeated integration by parts and conclude that the harmonic function h has no inﬂuence on Z. However, this conclusion is not correct, since the function ρ has in general not the required falloﬀ behavior to justify integration by parts. In general, one has

Z(y±) = 2ρ(y±) + η(y±), (2.15)

± + − with η(y ) = η+(y ) + η−(y ) being another harmonic function, characterized by boundary conditions. Comparing this to the discussion before, especially equation (2.10), we conclude that the equal sign in (2.10) is only correct in a very speciﬁc conformal coordinate system, that is to say, in the coordinate system where h ≡ η ≡ 0. For reasons becoming transparent shortly, we will refer to this coordinate system as “vacuum coordinates.” One might ask, whether this is not at odds with our derivation of the 2.2. The stress tensor 11

Polyakov term (2.10) in the ﬁrst place, since nowhere did we assume a particular coordinate system (apart from using a conformal coordinate system). The answer is no, since when integration equation (2.9) we assumed there are no extra terms in I1 whose variation is zero. This assumption turns out to be simply incorrect when considering general boundary conditions and the missing piece is automatically restored, when writing the Polyakov term in a covariant way. To see this more explicitly let us consider the Polyakov term with general boundary conditions in conformal gauge (2.8),

N Z √ 1 N Z 1 − d2x −gR R = − dx+dx− ρ + η ∂ ∂ ρ. (2.16) 96π ∇2 12π 2 + −

The ﬁrst order variation of the second term is simply zero,

δ Z Z dx+dx−η∂ ∂ ρ = dx+dx−η∂ ∂ δ(x − y) = 0, (2.17) δρ(y) + − + − because of harmonicity of the function η. Therefore the more general expression (2.16) R + − still solves equation (2.9). Crucially, the term itself, dx dx η∂+∂−ρ, is not zero in general.

2.2 The stress tensor

To substantiate our conclusion from previous paragraph, it is instructive to derive the expectation value (2.6) for all components of the stress tensor Tµν directly from the

Polyakov term I1. There are two ways to do this. Either we vary the non-local term in (2.10) directly, or we introduce an auxiliary ﬁeld in order to render the non-local term eﬀectively local. Both lead to identical results, but the second way is technically simpler, so let us pursue this route. One replaces the Polyakov term by

N Z √ 1 d2x −g ZR − (∇Z)2 , (2.18) 48π 2 where the ﬁeld Z is now promoted to a dynamical variable. Extremizing this action with respect to Z yields the equation of motion

∇2Z + R = 0, (2.19) and its solution coincides with our deﬁnition of Z in (2.13). That is, by “integrating out” the ﬁeld Z one recovers the previous non-local Polyakov term (2.10). 12 Chapter 2. Black Hole Model

It is now possible to calculate the response of the action to the variation of the metric components gµν . One observes that by using the equation of motion (2.19) after carrying out the variation, that the result does not depend on the variation of Z itself, N Z √ 1 √ δI = d2x δ −gR Z − δ −ggαβ ∂ Z∂ Z , (2.20) 1 48π 2 α β √ so one only needs to be concerned about the variation of the metric determinant −g and the Ricci scalar R. These are well known [47], and the results we need can be written as √ √ σλ σ λ µν δ −gR = −g gµν g − δµδν ∇σ∇λδg , √ √ 1 (2.21) δ −ggαβ = −g − gαβg + δαδβ δgµν . 2 µν µ ν Thus the ﬁnal result in terms of the auxiliary ﬁeld Z is

4π δI hT i = −√ 1 µν −g δgµν (2.22) N 1 = − g (∇Z)2 + ∇ Z∇ Z − 2g ∇2Z + 2∇ ∇ Z . 24 2 µν µ ν µν µ ν

As a consistency check, we can calculate the trace to conﬁrm the conformal anomaly,

N N hT µ i = − ∇2Z = R, (2.23) µ 12 12 which is reproduced correctly as expected. More interestingly we can evaluate the diagonal components hT±±i of the stress tensor in conformal gauge (2.8), since they correspond to physical energy ﬂux. Using (2.22) and the general solution for Z in conformal gauge (2.15), we get

N hT i = 4∂2 ρ − 4 (∂ ρ)2 + 2∂2 η + (∂ η)2 . (2.24) ±± 24 ± ± ± ±

This is in line with an alternative derivation of this expression going back to [43], which 2 is also discussed in the original CGHS paper [39]. In [39, 40, 42, 43], the piece (∂±η) + 2 + − 2∂±η appears as integration “constants” t+(y ) and t−(y ) when integrating the conservation equations for the stress tensor Tµν . Again, these integration constants have to be ﬁxed by imposing boundary conditions. The above analysis shows how the functions t± are related to the function η originating from boundary conditions of the Green’s function G in (2.12).

We can now also justify why we called the coordinates with η = 0 the “vacuum 2.2. The stress tensor 13 coordinates.” In manifestly ﬂat spacetime the derivatives of the conformal factor are zero, ∂±ρ = 0, and the energy momentum ﬂux is purely given in terms of η,

N hT i = t ≡ (∂ η)2 + 2∂2 η , (2.25) ±± ± 24 ± ± so that if η = 0, hT±±i = 0, and no energy momentum ﬂux is detected by an inertial observer.

Another interpretation can be given to the function η by realizing that t± can be 0 2 f 00 1 f 00 written as a Schwarzian derivative {f(x); x} := f 0 − 2 f 0 ,

2 2 ± ± ± (∂±η) + 2∂±η = −2 z (x ); x , (2.26) with new coordinates z±(x±) given by

x± Z ± z±(x±) := dy±e−η±(y ), (2.27)

± ± ∂x (z ) ± ± so that ln ∂z± = η±(x (z )). It is then readily seen that under the coordinate transformation x± 7→ z±(x±) the conformal factor transforms as

1 ∂x+(z+) 1 ∂x−(z−) 1 ρ 7→ ρ + ln + ln = ρ + η. (2.28) 2 ∂z+ 2 ∂z− 2

As such, specifying η in some coordinates is equivalent to specifying the relation to the vacuum coordinates, i.e. the coordinates where the energy momentum ﬂux vanishes asymptotically. This also provides a direct interpretation for the scalar ﬁeld Z given in (2.15). It represents the value (more precisely, two times the value) of the conformal factor in vacuum coordinates. We already mentioned before, it is not necessary to introduce the auxiliary variable Z. One can also vary the non-local expression (2.10) directly. Following [48], one can use equation (2.12) to deduce the response of the Green’s function G to a change of the metric. Since the Dirac delta function does not depend on the metric, we have

√ δ −g∇2G = 0. (2.29)

After some algebraic manipulations of equation (2.29), one concludes that

δG(y, z) 1 = p−g(x) − gαβ(x)g (x) + δαδβ ∇xG(x, z)∇x G(y, x), (2.30) δgµν (x) 2 µν µ ν β α 14 Chapter 2. Black Hole Model and by using that

N Z Z δI = − d2x d2yp−g(y)R(y)G(x, y)δ p−g(x)R(x) 1 48π (2.31) N Z Z − d2x d2yp−g(x)R(x)p−g(y)R(y)δG(x, y), 96π one recovers the same result (2.22).

2.3 Coherent matter states

To dynamically create large macroscopic black holes, one needs a macroscopic particle beam consisting of a large number of f-quanta. To still have analytic control one models this particle beam as a coherent state built on top of the vacuum |0i corresponding to an inertial observer in ﬂat spacetime. Following [49], a left-moving coherent state can for example be written as

N ! i X Z |f ci ∝ : exp dσ+∂ f c(σ+)fˆ(σ+) : |0i, (2.32) π + i i i=1

± c where σ are the vacuum coordinates and fi is the classical left-moving matter proﬁle. ˆ The fis are the quantum ﬁeld operators corresponding to left-moving fi-quanta, and :: denotes normal ordering with respect to the vacuum state |0i. The important properties of such a state are

c ˆ c c hf |fi|f i = fi , (2.33) and in particular c ˆ c c ˆ hf | : Tµν : |f i = Tµν + h0| : Tµν : |0i, (2.34)

c c where Tµν is the classical energy momentum tensor calculated from the fields fi . If these quantities are sufficiently large, i.e. much larger than N, then one can easily incorporate the coherent state in our semi-classical analysis by perturbing around c a classical background field fi , which obeys the classical equations of motion. Ex- plicitly, the semi-classical quantum effective action is given by

c c I[g, φ, f ] = Igrav[g, φ] + Imatter[g, f ] + I1[g]. (2.35)

The coordinate dependence of the asymptotic energy ﬂux is still automatically taken 2.4. The RST term 15 care of in the way it was presented in section 2.2.

The form of the coherent state (2.32) is only meaningful if the scalar fields fi are free. However, even if the fields fi were not free, the quantum effective action (2.35) c still makes sense by perturbing around a field configuration fi which obeys the (in this case non-linear) classical equations of motion. When calculating the expectation value of the stress tensor hTµν i one could then still interpret the result in the spirit of (2.34) by identifying the first and second term with the leading order and next to leading order outcome in an expansion around Ne2φ → 0.

2.4 The RST term

Unfortunately, the equations of motion obtained by extremizing the quantum eﬀective action (2.35) are hard to solve and no general solution is known. However, numerical studies of this model exist [50–53]. Another way forward was to slightly modify the model, as for instance was done in [40, 54, 55]. Here we follow RST [40] who modify the gravitational sector of the theory by adding the term

N Z √ I = d2x −g φR (2.36) 2 48π to the eﬀective action. This term is of order N which is of the same order as the

Polyakov term I1 and therefore does not disturb the classical physics taking place when Ne2φ → 0. The eﬀect of this term is to restore a classical symmetry which allows to choose a coordinate system in which the conformal factor ρ equals the dilaton, φ = ρ. We will refer to this choice of coordinates as “Kruskal coordinates.” This feature allows the equations of motion to be solved explicitly.

2.5 Equations of motion

Finally, the full semi-classical action we are working with can be written down. It is given by c c I[g, φ, f ] = Igrav[g, φ] + I2[g, φ] + Imatter[g, f ] + I1[g]. (2.37)

By extremizing this action with respect to the metric components, the dilaton, and the matter ﬁelds we obtain the full set of equations of motion for the semi-classical 16 Chapter 2. Black Hole Model gravitational system,

N 2g ∇2 − 2∇ ∇ e−2φ + φ + µν µ ν 24 −2φ 2 2 c +4e 2∇µφ∇ν φ − gµν (∇φ) + λ = Tµν + hTµν i, (2.38) 1 ∇2φ − (∇φ)2 + λ2 + R = 0, (2.39) 4 2 c ∇ fi = 0, (2.40) where hTµν i is given by (2.22) and we temporarily reinstated the scale λ. To simplify the equations, one usually proceeds in conformal gauge (2.8) and further chooses Kruskal coordinates where the conformal factor ρ equals the dilaton φ [39, 40]. To show that this is possible, one takes the trace of equation (2.38),

N N 8e−2φ (∇φ)2 − 4e−2φ∇2φ + ∇2φ − 8e−2φλ2 = R, (2.41) 12 12 and combines it with the equation of motion for the dilaton (2.39) to get

N 1 e−2φ + ∇2φ + R = 0. (2.42) 24 2

−2φ N −2φ N Assuming e + 24 6= 0, which is always true in the semi-classical regime e 24 , the equation reads, in conformal gauge,

∂+∂− (φ − ρ) = 0, (2.43)

+ − + − + − and is solved by φ(y , y ) = ρ(y , y ) + ω+(y ) + ω−(y ). One then observes that the functions ω+ and ω− can always be eliminated by a conformal coordinate transformation y± 7→ x±(y±), so that in the coordinates (x+, x−) we are left with

φ(x+, x−) = ρ(x+, x−). (2.44)

This leads us to the simple equations of motion in Kruskal gauge,

N 1 ∂2 e−2φ + φ = T c + t , (2.45) ± 24 2 ±± ± N ∂ ∂ e−2φ + φ = −λ2, (2.46) + − 24 c ∂+∂−f = 0. (2.47) 2.6. Black hole thermodynamics 17

−2φ N These equations can be solved for the field Ω := e + 24 φ provided a general c classical matter profile T±± by specifying the asymptotic energy flux t± in Kruskal coordinates.

2.6 Black hole thermodynamics

In order to work out thermodynamic relations to leading order we will concentrate on equilibrium solutions in the classical limit Ne2φ → 0. It is also possible to calculate corrections in the RST model, but these will not be of major importance here.

We consider a canonical ensemble for the system of the gravitational field gµν , −1 the dilaton φ and the matter fields fi at temperature T ≡ β within a cavity of large size, represented by the variable V , and prescribed boundary conditions for the B B dilaton field φ bdry = φ and the matter fields fi bdry = fi . The partition function is then formally given by the Euclidean path integral [56] with Euclidean action IE, where the imaginary time coordinate is periodically identified with a period β, Z B B −IE [gµν ,φ,f] Z(β, V, φ , fi ) := Dgµν Dφ Df e . (2.48)

In the following we will only consider a cavity of inﬁnite size V → ∞. This step involves some subtleties which we will ignore for the sake of brevity. A more detailed analysis is for instance given in [57–59]. Let us from now on suppress the arguments B B but keep in mind that the expressions are parametrized by β, φ and fi . −2φ N In the classical approximation, valid for e 12 , the leading order result is just given by the saddle points obeying the given boundary conditions, that is

e−βF = Z ≈ e−IE [gcl,φcl,fcl] (2.49) where we introduced the free energy F . Using the free energy, one can then utilize the usual thermodynamic relations to calculate the entropy s,

2 s = β ∂βF, (2.50)

B B where the derivative is taken keeping the parameters V , φ and fi ﬁxed.

It is well known, that for arbitrary inverse temperature β, the metric gµν typically features conical singularities [60]. However, the conical singularities disappear for a special temperature, the Hawking temperature TH . When using equation (2.50) to compute the entropy of a black hole with Hawking temperature TH , it is obviously 18 Chapter 2. Black Hole Model

not enough to know F (β) only at β = βH , and so it is argued in [60] that one should still take contributions from conical singularities into account. We shall follow this approach here. A one-parameter family of static black hole solutions in the Euclidean classical CGHS model (in units such that λ = 1) is given by

ds2 = e2φdzdz,¯ (2.51) e−2φ = M + zz,¯ (2.52)

fi = 0, (2.53) with the complex coordinate z being related to asymptotic Euclidean time τ as z = 2πi τ re β , from which the periodic identiﬁcation τ → τ + β is obvious. The connection 2π to the Lorentzian solution is established by interpreting t = β iτ as a real-time coordinate. The ADM mass M of the black hole is related to the parameter M by M = πM [61]. Hence we can and will refer to M as the black hole mass. The ADM mass coincides with the usual Komar integral in Lorentzian signature,4

1 M = − lim ? dξ, (2.54) r→∞ 16πG(2) if one keeps in mind the relationship between the eﬀective coupling G(2) and the 2φ 2 2φ dilaton φ, 8G(2) = e . The one-form ξ = −r e dt is dual to a timelike Killing µ vector normalized such that ξ ξµ → −1 as r → ∞. In (r, τ) coordinates the metric reads

2π ds2 = e2φ dr2 + r2dτ 2 , (2.55) β which makes the appearance of a conical singularity for β 6= βH := 2π at r = 0 man- ifest. A peculiarity of the CGHS model becomes evident. The Hawking temperature does not depend on the mass M. This is atypical, as for a higher dimensional black hole the Hawking temperature always depends on its mass. A prominent example is the Hawking temperature of a four dimensional Schwarzschild black hole with mass −1 M which behaves like TH ∼ M . The fact that the Hawking temperature does not depend on the mass also implies that no smooth solutions (i.e. without conical

4In general the Komar integral is an integral over a co-dimension two manifold. In two dimension this is just a point and so no integral appears in formula (2.54). 2.6. Black hole thermodynamics 19 singularities) for temperatures other than the Hawking temperature exist.

To calculate the free energy F of this geometry, we need to evaluate the Euclidean on-shell action on a regularized manifold with boundary,

1 Z √ 1 Z √ − I = d2x g e−2φ R + 4 (∇φ)2 + 4 + dΣ h e−2φ (K − 2) , (2.56) E 2π π where the bulk action is supplemented by the Gibbons-Hawking-York term [56, 57] on the boundary to render the variational principle on a manifold with boundary well deﬁned. The determinant of the induced metric on the boundary parametrized by Σ is denoted by h while K is the extrinsic curvature of the boundary. The second term √ 2 R −2φ π dΣ he is needed to obtain a ﬁnite result.

The treatment of the conical singularity is simpliﬁed using the analysis in [62]. By regularizing the cone one can evaluate the Ricci scalar R, and after removing the regulator one obtains, in (r, τ) coordinates,

β R(r) = Rreg(r) + 2 1 − δ(r), (2.57) βH where Rreg(r) is the regular Ricci scalar of the geometry with β = βH .

One ﬁnds that the Euclidean action IE, given by (2.56), vanishes when evaluated B for the geometry with Ricci scalar Rreg, so that the free energy F (β, φ ) simply reads B 1 −2φ β F (β, φ ) = e − 1 . (2.58) π r=0 βH

Notice that we suppressed the dependence on the boundary condition for the matter fields f B due to the only nontrivial static case being f B = 0, c.p. (2.53). Further, due to (2.52), fixing the value of the dilaton φB at the boundary essentially fixes the mass M of the black hole solution.

Using (2.50), the thermodynamic entropy of the black hole at the Hawking tem- 1 perature TH = 2π yields

2 −2φ s = β ∂βF = 2e = 2M. (2.59) β=βH r=0

Remarkably, this result is in line with Bekenstein’s expression [8], s = A , where A 4G(2) 20 Chapter 2. Black Hole Model represents the area of the black hole horizon, since

1 2φ G(2) = e , (2.60) horizon 8 r=0 and A = 1. We also note that the vanishing of the free energy F (βH ) at the Hawking temperature is consistent with the thermodynamic relation F (T ) = U − T s, since the internal energy U is given by

1 −2φ U = (1 + β∂β) FM (β) = e = M/π = M (2.61) π βH r=0 and TH s = M/π. To conclude we collect our ﬁndings of the thermodynamics variables for the classical eternal CGHS black hole of mass M,

1 T = , (2.62) H 2π 1 U = M, (2.63) π s = 2M, (2.64)

with the Hawking temperature TH , the internal energy U and black hole entropy s. 3

Page Curve

A useful diagnostic to quantify the loss of information due to Hawking radiation turns out to be the entanglement entropy of the radiation Srad. To deﬁne this quantity, we imagine a total quantum system consisting of a black hole and its radiation, so that the Hilbert space H describing the total system is given by H = HBH ⊗ Hrad. We assume that such a product structure exists, at least approximately.1 Given a state ρ on H, the entanglement entropy of radiation can simply be deﬁned to be the von Neumann entropy of the reduced state ρrad = trBH (ρ),

Srad := −tr (ρrad ln (ρrad)) . (3.1)

Analogously we can deﬁne the von Neumann entropy of the black hole SBH. Perform- ing the calculation in the spirit of Hawking [7], one would ﬁnd that Srad is a growing and strictly monotonic function of time, at least for the time a semi-classical analysis can be trusted. This, however, appears to be in contradiction with unitarity, as was argued by Page [15, 65] as follows. Let us assume a unitary time evolution of the state ρ, so that an initial pure state stays pure for all times. In this case the two quantities Srad and SBH are equal to each other, Srad = SBH. Further, we assume that the initial

1That such an assumptions is justiﬁed, is not clear by any means. It is known that such a factorization typically fails for gauge theories, see for example [63]. Since gravity itself can be thought of as a gauge theory, it would not be very surprising if that were the case. However, we can still hope that the factorization holds in some approximate sense. One should remark that there are also arguments against the correctness of this assumption, see e.g. [64].

21 22 Chapter 3. Page Curve state (which we imagine to be shortly after a black hole has been formed), does not yet carry any radiation. This premise provides us with an estimate of the dimension of the accessible Hilbert space H if we accept the coarse grained entropy of the initial black hole to be one quarter of the initial area A0 of its event horizon in Planck units,

A0 A0/4 s0 = 4 [9]. The estimate thus is dim H ∼ e . It is then already easy to see that the entanglement entropy Srad must be bounded by the coarse grained black hole entropy, Srad ≤ s0. One gets a stronger bound by using Srad(t) = SBH(t) at each time and SBH(t) ≤ s(t), i.e.

Srad(t) ≤ s(t), (3.2) where s(t) represents the course grained black hole entropy at a time t. Notably, this bound is strongest, when the black hole entropy s attains its minimum, zero, which corresponds to a complete evaporation of the black hole, and weakest, when its maximal, s = s0, the initial moment after black hole creation. It is immediately apparent, that since s(t) → 0 as the evaporation progresses, sooner or later this will be in contradiction with Hawking’s result which states the entanglement entropy keeps growing. A similar but complementary bound, which is strongest at the initial moment and weakest at the evaporation end point, can be derived. It relies on an argument using thermodynamics of the radiated gas [66]. If we assume the radiation consists of massless particles at temperature T , the relation between it’s thermodynamic entropy srad and its internal energy U is given by

d U s = − , (3.3) rad d − 1 T where d is the spacetime dimension. If the black hole radiates particles of some energy ∆M, this will of course change the internal energy of the radiation gas accord- ingly ∆U = −∆M. By the ﬁrst law of thermodynamics, T ∆s = −∆M, and therefore d d ∆srad = − d−1 ∆s, or integrating, srad(t) = d−1 (s0 − s(t)). Since the entanglement entropy Srad should again be bounded by the coarse grained entropy srad, we have

d S (t) ≤ (s − s(t)) . (3.4) rad d − 1 0

The time when the two bounds for the coarse grained entropy of the radiation and the black hole meet each other is called the Page time tpage [16]. The two bounds are illustrated in Figure 3.1 on page 23 for a black hole emitting massless particles in four dimensions and a two dimensional black hole with a 23

4 3 s0

black hole

radiation Entropy 0 Time tPage

(a) Four dimensional black hole with dM ∼ − 1 .[16] dt M2

2s0

radiation

black hole Entropy 0 Time tPage

dM (b) Two dimensional black hole with dt ∼ −1.

Figure 3.1: The coarse grained (thermal) entropy of the black hole s(t) and radiation srad(t) respectively as a function of time shown in four and two dimensions. Their intersection determines the Page time tPage. 24 Chapter 3. Page Curve temperature independent of its mass. If the reduced density matrix of the radiation is thermal, then the von Neumann entropy Srad equals the coarse grained entropy srad. This is the behavior expected from Hawking’s result [7, 10]. However, we just reasoned that unitarity implies a turnover around the Page time, where the von Neumann entropy Srad should be bounded by the coarse grained black hole entropy s. Page showed [15], that the entanglement entropy of a typical state of a big enough system roughly assumes this maximum, after a time of order, say, half the evaporation time. That is to say, to good approximation, the entanglement entropy of the radiation Srad(t) should follow the minimum of the two values srad(t) and s(t) if unitarity applies.

3.1 Generalized entropy

Up until very recently, it was not known how to “repair” the computation of Hawking in order to reproduce the Page curve, i.e. in order to restore unitarity, by a semi- classical computation. So far it was not even clear, it can be restored by a semi- classical computation and it was suspected that the full theory of quantum gravity has to be formulated in order to get the correct behavior. Quite the opposite is claimed by [17, 18, 67] where a unitary Page curve is obtained only by using semi-classical gravity. Motivated by AdS/CFT and the QRT formula [19–22], it was proposed that the correct semi-classical formula for the entanglement entropy Srad is given by

Area Srad(A) = min ext (I) + SBulk(SAI ) . (3.5) I 4G

To explain this formula it will be convenient to concentrate on a Penrose diagram of an asymptotically ﬂat evaporating black hole in Figure 3.2, page 25. Let us imagine dividing spacetime into a region which includes the black hole and its complement which potentially only includes radiation. This is indicated by the blue line in Figure 3.2. This line (which in four dimensions would be a large sphere at each instant, S2 × R) should be placed in a region where gravity is very weak. We would then attribute the degrees of freedom carried by the outside region to the Hilbert space Hrad and the rest being described by HBH, such that the total Hilbert space is H = HBH ⊗ Hrad. We are interested in calculating the entanglement entropy Srad associated to the inﬁnite region outside A, that is the von Neumann entropy of a state in Hrad. The prescription (3.5) then tells us to consider all possible spacetime 3.1. Generalized entropy 25

singularity A S I S A

matter

horizon

Figure 3.2: Penrose diagram of an asymptotically flat evaporating black hole. The green region represents an infalling matter beam. The event horizon is shown by the dashed line. The blue line separates the inner black hole region from the outer region where the outgoing Hawking radiation is collected. A point on the blue line A corresponds to a particular time for an observer far away. As time moves forward, the point A moves upwards on the blue line. The pink dot I indicates the appearance of an island. 26 Chapter 3. Page Curve points I to the left of the dividing blue line and extremize the sum of the area divided by Newton’s constant G at the point I and the entanglement entropy SBulk corresponding to a matter state defined on the subregion SAI connecting the point I with A. If multiple extrema exist one is instructed to take the minimum. When we apply this prescription to a semi-classical black hole one expects to find two extrema of the expression in formula (3.5). One of them will be found at the (yellow) boundary far inside the black hole while the second one will be found near the horizon, indicated in Figure 3.2 by the pink dot. At early times (meaning, up to the Page time tPage after the black hole was formed) the minimum of these two extrema is given by the first extremum at the boundary while after the Page time the second extremum at I will overtake. Notice that the first minimum instructs us to evaluate the bulk entanglement entropy Sbulk from the anchor point A on the whole spatial slice going inside the black hole where the area term is zero. If one were to consider a pure state on the whole spatial slice divided into two complementary regions RL and RR, this would just coincide with the naive statement that SL = SR. At this point formula (3.5) is telling us nothing new. We get the usual result obtained in the spirit of Hawking (see the blue lines in Figure 3.1), which closely tracks the coarse grained entropy of the radiation. The novel feature enters at the Page time, when the second extremum located near the horizon will actually overtake. Then (3.5) will be dominated by the area term which near the horizon is nothing else than the coarse grained black hole entropy s and we recover a Page curve that is consistent with unitarity, Figure 3.1. It is also quite interesting to observe that SBulk only has to be evaluated on a region between a point on the horizon I and the anchor A. In other words we are explicitly instructed to exclude the interior of the black hole in the calculation. In [36] we explicitly evaluate the generalized entropy (3.5) for an evaporating black hole in the RST model described in chapter2. We indeed find the exact behavior just described and one recovers the entanglement entropy depicted in Figure 3.1b at leading order. The crucial difference to the original derivation of entanglement entropy curve by Page is that in Page’s derivation unitarity was assumed. In [36], however, we just perform the calculation of (3.5) in semi-classical gravity without explicitly assuming unitarity. 4

Quantum Complexity

We can give a precise definition of quantum complexity in the context of Quantum Information Theory (QIT) [68, 69]. In QIT one models a quantum computer in terms of a finite set of gates which are unitary operators acting on a finite dimensional Hilbert space H. In order for the gate set to be useful in practice the gates should be universal, i.e. able to produce any quantum state in an approximate sense. However, at the same time, the gate set should not be too large since they represent the building blocks of the quantum computer. Let H be a finite dimensional Hilbert space. Consider a finite set of unitary operators on H,

G := {Ui : Ui ∈ U(H), i = 1,...,M} (4.1) for M being a positive integer. Further consider the group generated by the set G via matrix multiplication, denoted by hGi. A typical element SG ∈ G is of the form k1 k2 kM ki SG = U1 U2 ...UM , with Ui 6= I, or permutations thereof. Deﬁnition. (Universal Gate Set) A gate set G is called universal iﬀ for any U ∈ U(H) and > 0 there exists an element SG ∈ hGi such that

kU − SGk < , (4.2) with the matrix norm being deﬁned as usual,

kUk := sup kUψk . (4.3) ψ∈H,kψk=1

27 28 Chapter 4. Quantum Complexity

In other words, the set hGi ⊂ U(H) is dense in U(H) with respect to the matrix norm.

Equivalently, given a reference state ψ0 ∈ H with kψ0k = 1, any state ψ1 ∈ H with kψ1k = 1 can be constructed to arbitrary accuracy by acting on ψ0 with gates in G. That is, given an > 0, there exists a sequence of gates SG ∈ hGi with

kSGψ0 − ψ1k < . (4.4)

It is perhaps not obvious that this “factorization problem” has a solution, i.e. that a universal gate set exists. But it does, and the rate of convergence in the number of required gates is even remarkably fast [70]. Typically there exist more than one such sequence SG. Let us denote the set of all such sequences for a given ψ0, ψ1 and as c(ψ0, ψ1; ). As just stated, this set is always non-empty, c(ψ0, ψ1; ) 6= ∅.

Within QIT the element SG ∈ hGi represents a particular computer program. The number of steps of a computer program SG is then given by the sum of exponents

#SG := k1 + k2 + ··· + kM . Intuitively, the complexity of a computer program should be the number of its steps. However, it could occur, that there is “better” program (i.e. a program which requires less steps), which performs the same task. In the language of above, this would be represented by another element in c(ψ0, ψ1; ) for the same ψ0, ψ1 and . In this case it is clear, that the actual task was really not as “complex” as suggested by the original computer program. Therefore, we require complexity to be the number of steps of the “best” computer program performing a particular computation.

Deﬁnition. (Quantum Complexity) For a ﬁxed > 0, ψ0 ∈ H, ψ1 ∈ H, Quantum

Complexity CG is deﬁned to be the minimum number of steps of all elements in c(ψ0, ψ1; ),

CG(ψ0, ψ1; ) := min #SG. (4.5) SG ∈c(ψ0,ψ1;)

As such, for a fixed > 0, CG can be regarded as a function H×H → R+. It would 1 ∼ d−1 be tempting to view CG as a metric on the projective Hilbert space P (H) = CP [30], where d is the dimension of the Hilbert space H. However, strictly speaking, there is a slight subtly. Since there exist two states ψ0, and ψ1, which are not equal but within the same -ball, kψ0 − ψ1k < , we have CG(ψ0, ψ1; ) = 0 and thus the function CG is not positive definite on P (H). This is easily fixed by defining our metric

1 A metric on H is here understood to be a symmetric, positive deﬁnite function H × H → R+, which satisﬁes the triangle inequality. 4.1. Complexity of black holes 29

1 to assume a value between zero and one, say 2 , if evaluated on states ψ0 6= ψ1 with kψ0 − ψ1k < . Some elements of c(ψ0, ψ1; ) are illustrated in Figure 4.1.

ψ1 -- H

ψ 0 --

Figure 4.1: Illustration of the Hilbert space H and two states ψ0 ∈ H, ψ1 ∈ H surrounded by -balls related by diﬀerent sequences of gates G. Each of the dashed lines represents an element of c(ψ0, ψ1; ). Complexity C is determined by the shortest path.

Note, that equivalently we could’ve deﬁned the complexity metric on the set of unitary operators U(H), since U(H) acts transitively on P (H). We are then asking the equivalent question, what the minimum number of gates is to construct a particular unitary matrix U provided an accuracy . Such a geometric approach to complexity appears to be fruitful and is studied for example in [71].

4.1 Complexity of black holes

Motivated by Bekenstein’s black hole entropy formula [9], the idea of the holographic principle [72–74] was developed, which roughly states that a gravitational system in d spacetime dimensions is described by a quantum system in d − 1 dimensions. More speciﬁcally, a black hole can be described by a quantum system with its number of degrees of freedom dictated by the Bekenstein entropy. This ﬁts nicely with the principle of black hole complementarity [34], which replaces the black hole interior for an asymptotic observer by a quantum system living on the “stretched horizon”, a surface with an area of one unit more than the area of the event horizon in Planck units. 30 Chapter 4. Quantum Complexity

The precise dynamics of the quantum system living on the stretched horizon is not known. However, some necessary features of the quantum features can be de- duced. One example is the “fast scrambling” property [75, 76], which refers to a short thermalization time scale. Thinking of the quantum system as a system consisting of a ﬁnite number of qubits [75], this property implies that the interaction between diﬀerent qubits are “k-local” and “all-to-all” [30]. That means in particular that the interaction is not local if one thinks of the qubits to be distributed on the horizon surface, but that each interaction involves k qubits or fewer. Further, “all-to-all” implies that every subset of k qubits interact. In the language of QIT, the universal gate set G includes gates that act on at most k qubits. Given any k qubits, there exists a k-local gate in G that acts on them. As an example, a small quantum circuit is shown in Figure 4.2.

5 τ = 0 τ = 1 τ = 2

6 Figure 4.2: Sketch of a small quantum circuit SG with #SG = 3 × 3 and dim H = 2 . Each connected brown component represents a (3-local) gate. Each black line corresponds to a qubit, while the circles correspond to the action of gates. This circuit only includes three steps while in principle an arbitrary number is possible.

One can now heuristically argue, that the complexity growth for early times should be linear in time [29], where “time” is proportional to the number of steps in the quantum circuit. Let’s consider an initial state ψ0. If we have k-local gates and at each time step all qubits are involved, then the number of gates acting at each time step is given by s #S ∼ , (4.6) G k where s is the number of total qubits in H. At this point the state ψ0 evolved to the new state ψ1 via the application of those gates. If the total space of states is large enough (this also requires that the regulator is chosen small enough), then it is likely 4.1. Complexity of black holes 31

that the minimal amount of gates required to go from ψ0 to ψ1 is the same as those which we just acted with, s C(ψ , ψ ; ) ∼ . (4.7) 0 1 k This behavior is expected to roughly last until the space of states is exhausted, so that the minimal amount of gates is not given anymore by the number of gates we acted with. The time when this happens is argued to be exponential in the total number of s qubits, τ ∼ e [30]. Changing notation a little bit, C(τ) := C(ψ0, ψτ ; ), we have

dC(τ) s ∼ for τ es. (4.8) dτ k

Now, we shall interpret this result as a result for a quantum system living on the stretched horizon of a black hole, describing its interior. The number of total qubits A is dictated by the Bekenstein entropy s = 4 of the black hole, where A is the area of its event horizon in Planck units. Further, we identify τ with the proper time on the stretched horizon. The relation to asymptotic Schwarzschild time t is then simply given by τ ∼ TH t, where TH is the Hawking temperature associated to the black hole. Keeping k constant, we can conclude that [35]

dC(t) es ∼ TH s for t . (4.9) dt TH

Let us remark, that for asymptotically ﬂat black holes, the evaporation time is at worst polynomial in s, so the behavior (4.9) is expected to be relevant for pretty much the whole existence of the black hole. Of course, for an evaporating black hole, the Hawking temperature TH and the coarse grained entropy s typically depend on time, so that in general we expect a departure from constant growth. For example, 1 a four dimensional Schwarzschild black hole has a Hawking temperature of TH ∼ M while the Bekenstein entropy is s ∼ M 2, so that one expects [30]

dC(t) 1/3 ∼ M(t) ∼ M 3 − 3t , (4.10) dt 0

dM 1 assuming the black hole radiates only massless particles, dt ∼ − M 2 . On the other hand, in this thesis, we are primarily interested in two dimensional black holes with dM constant Hawking temperature TH ∼ 1 and entropy s ∼ M with dt ∼ −1. This leads to an expected complexity growth of

dC(t) ∼ M(t) ∼ M − t. (4.11) dt 0 32 Chapter 4. Quantum Complexity

4.2 The geometric dual of complexity

Black hole complementarity posits that the full quantum gravitational system should be accurately described by a quantum system on the stretched horizon which in eﬀect replaces the interior of the black hole. However, in the semi-classical limit, the dynamics of the quantum system should also be reﬂected by the original black hole interior geometry. This begs the question, what the corresponding geometric quantity of complexity is in the black hole interior. Obviously, whatever this quantity is, it should obey equation (4.9). The original conjecture made by Susskind [29] was based on exactly that point. He realized that the growth of the volume of an Einstein- Rosen bridge is in accordance with equation (4.9). Susskind then conjectured that the quantum complexity of a general black hole is given by its volume. In short, Complexity equals Volume (CV). A Penrose diagram of the Einstein-Rosen bridge, including their growing volumes, can be seen in Figure 4.3.

black hole singularity

horizon

white hole singularity

Figure 4.3: Penrose diagram of an Einstein-Rosen bridge including two volume slices (in green) of its interior at diﬀerent times t = t1 and t = t2.

Some month later, a refined quantity that also behaves like (4.9) was discussed [77, 78]. This quantity is the action I which describes the gravitational model. The claim is, that the action I evaluated on a certain bounded region of the black hole spacetime, called the Wheeler-DeWitt (WdW) patch, yields complexity. We refer to this proposal as Complexity equals Action (CA). For the Einstein-Rosen bridge, we can define the WdW patch by considering the union of all spacelike surfaces connecting two anchor points on each side of the wormhole, see Figure 4.4. An obvious difference between those two proposals is, that while CV can be defined inherently for any black hole solution, CA requires this black hole to be a solution 4.2. The geometric dual of complexity 33

black hole singularity

A2 A2

horizon

horizon A1 A1

white hole singularity Figure 4.4: Penrose diagram of an Einstein-Rosen bridge including the WdW patch anchored at symmetrical anchor points A1 and A2. of equations of motion derived by an action principle with a given action I. Both these quantities have been extensively studied in literature,2 and most results show consistent growth behaviors for both of them. To my knowledge, the precise relation between CA and CV is not yet completely understood. In [37, 38] we precisely deﬁne the notion of volume and action respectively for the two dimensional dilaton gravity model discussed in chapter2. We ﬁnd that the results are consistent with the growth in (4.11) which we inferred from the more general formula (4.9).

2To name a few inﬂuential studies in this direction: [79–85]. 34 Chapter 4. Quantum Complexity 5

Conclusion

In this dissertation we applied a QRT formula to asymptotically flat semi-classical black holes of a two-dimensional dilaton gravity model to compute the entanglement entropy between the black hole and its radiation. Waiting for a scrambling time tscr ∼ log(s) after the black hole has been formed, one finds two extrema of the generalized entropy (3.5). The first one is located at the spacetime boundary “inside” the black hole, while a second one is located close to the horizon. At early times, the first extremum is minimal and the entanglement entropy follows the coarse grained entropy of radiation. This behavior persists until the Page time which in the present model is one third of the black hole lifetime. At that time, the second extremum starts to take over. Since this extremum is located near the horizon, its value is dominated by the area term in (3.5) which is nothing else than the coarse grained entropy of the black hole. As a result, we recover the Page curve in Figure 3.1b on page 23, consistent with unitarity. Although originally the QRT formula is motivated by AdS/CFT, this result provides evidence that the applicability of this formula extends beyond asymptotically AdS spacetimes. This claim is substantiated by [86] who provide a derivation of the QRT formula in the CGHS/RST model using the Replica trick which does not rely on any holographic dual theory. It should be noted that computing a Page curve by semi-classical methods is only a first step in solving Hawking’s black hole information paradox in the context of the presently discussed model. It is clear that there is still a need to understand the detailed mechanisms involved that allow information to escape the black hole.

35 36 Chapter 5. Conclusion

How precisely information is encoded in the Hawking radiation is not evident and an answer requires further research. In addition to the Page curve, we investigated the CV and CA proposals as poten- tial dual candidates of quantum complexity. The CV proposal instructs us to evaluate the volume of extremal volume slices anchored at certain points which are labeled by time. Although we work with a theory in two dimensions, we were led to consider a four-dimensional volume functional which upon dimensional reduction reduces to a two-dimensional integral weighted with the inverse of the two dimensional gravitational coupling e−2φ. A volume in two dimensions is a curve and curves extremizing their length are simply geodesics. It turned out that geodesics do not even have the right qualitative behavior to be consistent with the behavior of quantum complexity, (4.9). To be able to reproduce the correct time-dependence of quantum complexity, it was further necessary to place the anchor points on the stretched horizon [34]. This supports the interpretation of the black hole interior being described by an ordinary quantum system residing at the stretched horizon, whose chaotic dynamics generates the growth in complexity. To inspect the CA proposal we had to evaluate the gravitational action on a Wheeler-DeWitt patch. To ensure a well-posed variational principle for a gravitational action, the action has to be supplemented by appropriate boundary terms. These boundary terms are generally not unique which leaves an ambiguity in the CA prescription. The late-time limit of complexity growth is usually not sensitive to these ambiguities. However, since we are dealing with evaporating black holes a late-time limit is not useful, and we have to ﬁnd a way to resolve these ambiguities. To do that we proposed to restrict the possible boundary terms in the action by a symmetry principle. Using this prescription we recovered a growth rate that is consistent with (4.9). In contrast to the CV computation where we had to rely on numerical methods to compute the volume functional in the semi-classical black hole geometry, the CA calculation provides explicit results without having to resort to numerics. Bibliography

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Article I

JHEP03(2020)069 Springer March 12, 2020 : December 2, 2019 February 10, 2020 : : Published Received Accepted [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP03(2020)069 [email protected] , . 3 1911.06800 The Authors. c 2D Gravity, Black Holes, Classical Theories of Gravity

We obtain the holographic complexity of an evaporating black hole in the , [email protected] Science Institute, University ofDunhaga Iceland, 3, Reykjav´ık107, Iceland E-mail: Open Access Article funded by SCOAP where the rate ofhole growth evaporates. tracks the shrinking area of theKeywords: stretched horizon as the black ArXiv ePrint: semi-classical RST model ofthat two-dimensional dilaton takes gravity, into using account aholes, the volume we prescription higher-dimensional recover origin thearise of expected at the the late semi-classical model. time level.the behaviour evolving By black For considering of hole, classical the we thewith volume obtain black complexity, an inside sensible early but the results onset stretched for new of horizon the order features rate of the of black growth hole of scrambling the time complexity, followed by an extended period Abstract: evaporating black hole Lukas Schneiderbauer, Watse Sybesma and LárusThorlacius Holographic complexity: stretching the horizon of an JHEP03(2020)069 9 7 ]. In the present paper, we explore 3 2 3 – 1 – ]. In order to reproduce black hole thermodynamics, 1 12 1 ]. 2 The dilaton gravity model has explicit classical solutions describing black hole forma- Quantum complexity has in recent years emerged as an important entry in the holo- 2.1 Complexity of2.2 a two-sided black Complexity hole of a black hole formed by gravitational collapse tion from arbitrary incoming matter energy flux. We focus, for simplicity, on black holes evaporate due to thecalculations emission in of a two-dimensional Hawking dilaton radiation.classical gravity model effects, Our that is allows main that analyticshrinking result, study the area of based rate semi- of on of the explicit growthhorizon stretched of is horizon holographic taken as to complexity theorder be precisely black one a tracks hole in the membrane evaporates, the with where appropriate an the units area of stretched larger the than model. that of the event horizon by graphic dictionary for black holes following Susskind’svolume conjecture of that the the expanding spatial Einstein-Rosening bridge complexity of of a athe two-sided relation corresponding eternal between quantum black quantum state hole complexityof and [ reflects semi-classical interior the black black grow- hole holes geometry that in are the context formed by gravitational collapse and subsequently entropy and the dynamics must bea sufficiently relatively chaotic to short scramble timescale, quantumdynamics but, information on and beyond the that, holographic thefollows, encoding precise we of nature will the of not black thecan hole make stretched interior any be horizon remain specific useful elusive. assumptions tosimple In about keep model what the in [ scrambling mind dynamics a but collection it of qubits undergoing k-local interactions as a The interior of aapproach black to hole quantum is gravity. the Theinterior archetype principle geometry of of and an black hole any emergent complementarityfinite matter spacetime posits number that in of that quantum enters the mechanical the degrees a holographic oflocated black freedom associated hole outside with can the a stretched be event horizon the described horizon number in [ of terms stretched horizon of degrees a of freedom should match the Bekenstein-Hawking 4 Conclusion and outlook 1 Introduction 2 Complexity of classical CGHS black holes 3 The RST model: complexity in a semi-classical black hole Contents 1 Introduction JHEP03(2020)069 , 9 (2.1) , ]. We will be # 14 2 ) – i f 11 ∇ ( N =1 i X 1 2 − 2 λ ]. These are simple toy theories for + 4 7 2 ) φ ∇ + 4( – 2 – R φ 2 − e " g − ] is also of interest for these dilaton gravity models, but it is √ 5 x , 2 4 d M Z = ]. CGHS 8 S ]. 6 The CGHS model and related semi-classical models were studied extensivelyThe in classical the GGHS action, The original CGHS model can be viewed as a spherical reduction of a four-dimensional The transitory nature of semi-classical black holes highlights certain technical aspects ]. The two-dimensional theory captures the low-energy dynamics of radial modes in the reflected in our calculations below. early 1990’s and severalbrief reviews and were only written introduce at the that minimal time, ingredients including needed [ for the purposes of this paper. dilaton gravity theory in10 a near-extremal magnetically chargednear-horizon black hole region background of [ higher-dimensionalas geometry. the quantum The complexity volume istheory, that that rather is of than a to the spacelike be three-dimensional length identified surface of in the a original spacelike curve in two-dimensions, and this will be black holes formed by the gravitational collapsefields of in matter fields. a black The quantization holegeometry of background matter causes leads the to Hawkingsimple black radiation hole to and to track its in back-reaction evaporate.Susskind, on a and the The variant Thorlacius subsequent ofanalytically (RST), evolution [ the where is semi-classical the particularly model semi-classical that field was equations introduced can by be Russo, solved 2 Complexity of classicalWe CGHS work black holes within aCallan, Giddings, class Harvey, of and two-dimensionalblack Strominger hole (CGHS) dilaton physics [ that gravity can theoriesclassical be systematically first solutions studied that introduced at the describe by an semi-classical black event level. horizon. hole They The geometries have black with holes a include spacelike static singularity two-sided black inside holes and also dynamical tation of the semi-classical blackaddress. hole The complexity, where alternative these formulationa issues of Wheeler-DeWitt are holographic patch relatively complexity [ easy inmore to terms subtle to of implement the at the actionpaper semi-classical on [ level, and we postpone this to a forthcoming well. We then consider amain semi-classical analytically extension soluble of and the numerically model evaluateblack the where hole volume the background. functional field in equations an re- evaporating of the identification between complexitysidering and classical volume, black that holes. can In often the be present ignored paper, when we con- only consider the volume represen- formed by an infalling thin shellture and to start this off context. by adaptinglinear A the growth complexity suitably with as defined time volume volume at conjec- horizon functional late of exhibits times the and precisely dynamically by the formed restricting expected black to hole the one volume finds inside reasonable the early-time stretched behaviour as JHEP03(2020)069 , ) N M 2.4 (2.4) (2.5) (2.6) (2.2) (2.3) = . These − from the φ x 0) via the + λ = x < ρ − x , for black holes in V f ±± C T 0 and − > = . + φ = 1. 2 x λ = 0 − e i , 2 , ± − ν ˙ y x , µ + − ˙ matter fields. y , f x M i , ∂ dx µν 4 ]. In the case of the Jackiw-Teitelboim f − − g 1 + x = 0 and there are two asymptotic regions, 15 + − and thus set p M dx x − φ ρ 1 2 = 2 x = − − − e φ + – 3 – λ ρ 2 x − 2 − M ds e e ∇ = = − 2 2 Z ∂ ρ − 2 + ds = − = . The two-dimensional theory inherits a scale e i V f R = has constant area and the calculation of the complexity re- . The corresponding coordinate transformation on the left φ σ , ∂ 2 0, the curvature is singular on the spacelike curves 2 + S − t e ± = 0 e i M > ± , which is proportional to the area of the local transverse two-sphere f φ − 2 = ∂ − + e ± ∂ ) in the outside region on the right (where x , where the curvature goes to zero. One can introduce Schwarzschild-like t, σ ) is a spacelike curve in the two-dimensional spacetime and the integrand in- is the energy-momentum tensor of the s ( → ∞ f geodesics. µ µν , is identical to the one obtained for a Schwarzschild black hole in 3+1 dimensional y T − 1 x In order to obtain the holographic complexity of a classical CGHS black hole, we We find it convenient to work in a conformal gauge, not of the near-extremal dilaton black hole (in Einstein frame) in the higher-dimensional + x 2 coordinates ( transformation it is apparent that, for corresponding to the white holefigure and black hole singularity. TheEinstein Penrose gravity. diagram, shown The in event horizon− is at The first solution we consider describes a two-sided eternal black hole, From the Ricci scalar, two-dimensional Jackiw-Teitelboim gravity inblack [ hole, the transverse duces to calculating aother two-dimensional hand, the geodesic transverse length. areaare depends on For spatial a location CGHS and curves black that hole, maximize ( on2.1 the Complexity of a two-sided black hole where introduce a volume functional where cludes a factor of S parent theory. A corresponding factor was included when defining and use a residual conformalare referred reparametrisation to to as choose Kruskal coordinatescoordinate coordinates where system, for reasons the that classical will equations become of apparent motion below. and In constraints this reduce to involves the two-dimensional metric,minimally a coupled scalar scalar dilaton fields field,parent theory and set matter by the inwe magnetic the take charge of length form the to of near-extremal be black measured hole. in In units the of following, JHEP03(2020)069 on ) = (2.9) (2.7) (2.8) − a ). We ∞ , x + 1 + a x → ( φ )) with fixed 0 σ s i ( − ) , y R ) s , t + − ( a I I ). σ + ( y R Points t Anchor ( R τ L t + − = , i i ) τ − , ) of a two-sided CGHS black hole y L + y , t µ . With these conventions, the metric M R σ − = t , ( + − τ t , V − M a ∓ ( C y + φ e Event Horizons 2 − y ˙ ∓ − y − e e e Singularity + White Hole = ˙ y – 4 – 7→ 7→ ± − x log + − p y y 1 2 = 0 ds propagates to the future in the ‘upward’ direction on τ = , with t a a Z σ σ = = L − + i i τ V σ = 0) is given by τ on the left and right anchor curves. This amounts to maximizing > ) is divergent for spacelike curves that extend all the way to spatial ) R L − Black Hole Singularity t − 2.4 , t x I a σ ( and L 0 and t 0 i < . This cartoon depicts the Penrose diagram of a two-sided eternal CGHS black hole. + or Anchor Curve Stretched Horizon x ) coordinates approaches the two-dimensional Minkowski metric as In order to proceed, we note that the functional is invariant under the transformations Our prescription for the volume complexity The volume in ( t, σ , providing a coordinate invariant notion of spatial position outside the black hole, and Figure 1 a the volume functional, and evaluating the resulting maximal volume. anchor curves outside the blackfind hole it where the convenient to volume integral useφ is anchor cut curves off on (see which figure place the dilaton the field curves is symmetrically constant, anchor curves on take the a particularly left-they simple and are form curves right-hand in of the side constant Schwarzschild-like coordinates, of where the black hole. The where the anchor curve on the left (right) isis parametrised then by given by the maximalendpoints volume at on the set of spacelike curves ( (where in ( both sides of theboth black sides. hole and infinity. In order to obtain a finite expression for the complexity, we introduce timelike JHEP03(2020)069 0 at (2.17) (2.13) (2.14) (2.16) (2.10) (2.11) (2.15) (2.12) < run into L τ > M . | E . | σ , . 4 ) ) e L ) = 0 L τ . τ and obtain the follow- L , and the time relations 2 + 4 τ 1 − ) σ ) coordinates, ) with − E L σ . − µ + 2 µ τ 2 t µ ) where the curve meets the − t, σ ± − µ − . 2.13 ) but these labels are in one- e ) R τ Me 2 µ t sinh( R − ± τ sinh( − ) sinh( − ˙ R = y 2.14 0 + 4 = . y τ L t + sinh( τ 2 2 2 and R ± y σ 2 τ − 0 in ( − 2 E sinh( y 2 t 2 L e e t − dt 0 dσ − E sinh( t p √ ) runs from a negative value e = + − − E y 2 = − sinh( L − t + M + and L 2 2.14 ˙ 1 M y = t σ + 2 E 0 − s t R ) = in ( − t 2 , e µ y = 0, where the curve enters the black hole from – 5 – e dt − dσ τ Me ) ˙ + y τ σ − µ y ) e + , e + 4 τ , y ˙ R 2 + 2 y R − 0 − τ t τ − , while curves satisfying ( 2 − − 2 R y τ E E M M , and can be parametrised as e + sinh = . We illustrate this setup in figure sinh( sinh( s = 0 E t R = 2 R My log e τ sinh( µ = τ ) is satisfied by imposing 2 L 0 1 2 t t = 2 E + 4 √ − + τ 2 R e sinh 2.15 t = σ e + + − y M < E < M = 2 = y and tanh − 0 = a R t t 2 the curve exits the black hole to the right and reaches the endpoint on σ 2 0 2 2 t E e − e µ e − − = t 2 τ M √ . The corresponding conserved quantity is given by R = ∈ The maximal volume curve is labelled by hand a pair of relations involving the Schwarzschild times at the anchor points, The second equation in ( can then be re-expressed as to-one correspondence with theanchor Schwarzschild times curves. To seecurve this, and consider the the anchor intersection curves. points On between the the one maximal hand we volume have relating curve parameters to the spatial location of the anchor curves, and on the other with the curvature singularity. The parameter the endpoint on the leftthe anchor left. curve to At the right anchor curve at Maximal volume curves that extendcharge between in the the anchor range curves correspond to a conserved We then convert back to Kruskaling coordinates equation by for using a maximal volume curve, which is valid over the entire extended spacetime, for We construct the correspondingregion maximum on volume the curve right by and first rewriting focusing the conserved on charge the in outside the ( This integrates to JHEP03(2020)069 (2.24) (2.21) (2.22) (2.23) (2.18) (2.19) (2.20) , 2 , . M ) ) to give . , as 2 L + t ) L a t + µ M σ 2.17 R 2 t − + 2( a and τ − Me σ 2 e limit, where the anchor R t )+2 O . sinh( Me L ) t + ) µ → ∞ √ . µ + ) − R a ) + 2 ) L − t t 2 σ L µ ( R M + 1 − t τ 2 τ ) and doing some algebra one even- R − + t ( M R log τ 1 2 t , cosh ( − 2.17 − cosh ( 2 a e sinh (2 a σ µ τ − 4 σ 2 cosh ( e + 1 = cosh = τ ) at the event horizon, + q a . EH cosh − σ φ – 6 – L 1 + 2 4 2 )+ t µ + tanh e 2.20 cosh dτ − L + t e − µ µ q R 2 0 + t M R dτ τ , y = Z τ R 2 R M t 2 ) + ) to obtain an exact, if somewhat unwieldy, formula for + τ L SH L τ a t φ = = sinh Z σ 2 2 2 ) is easily evaluated in this parametrisation, M + 2.20 − e cosh( R e EH √ ). t a = = ) 2.8 V σ L 2 t ) + ) can be combined with second equation in ( ) can then be re-expressed in terms of V e L − 2.18 t R cosh( t + ( 2.15 a 2.14 1 2 1 R σ M t e 2 ( ) with the second equation in ( e = 2 − M + 2.15 = y = ) given by ( L V t = arccosh + R µ t ( − The expression for the volume simplifies enormously at late times, The volume functional ( R τ 2 which can be shown to growcurves. at the same rate Later at on, latecollapse times when as of the we full matter, volume consider betweencurve. we anchor dynamical will In black the see holes case thatblack at formed hole, the hand, by with we stretched an the define horizon area the gravitational that is stretched is horizon a one to natural unit be larger choice a than membrane of the outside anchor area the of the event horizon, the location of thetaken to anchor only curves. include the Theto volume cutting volume inside off prescription the the event for volume horizon complexity integration of is in the ( sometimes black hole. This amounts with a leading term that growson linearly the with location time, of followed the by anchor aat curve, constant and term late subsequent that terms times. depends that arecurves As exponentially suppressed are expected, moved the off volume to diverges spatial in infinity, the but the late time rate of growth is unaffected by which can then be insertedthe in maximal ( volume as a function of The first equation in ( By combining ( tually arrives at The parametrisation ( with JHEP03(2020)069 . The (2.25) (2.26) (2.27) M ). = 2 , independent π BH 1 2.27 2 S ], for instance, it ]. = 3 16 , . H ) is thus proportional + 0 + 0 T x 2.22 < x ≥ , mediated by matter fields + + . x x + 0 proposal [ x 2 if if V = = , + , we sketch the setup. In contrast M ) ) in the one-sided black hole back- x C , 2 + 0 + x 2.4 = 0 − + 0 M − x + ˙ x x = 0. Expressing the corresponding condition + ( + δ x r − – 7 – + 0 x − − M x x + = 0. In figure + + ˙ = x x x − + ). The result differs from the volume inside the event − − x f x ) = 0 at ++ r    − T ( 0 x 2.22 t = ρ 2 − e ], and the Bekenstein-Hawking entropy is given by = = 0 in ( 7 φ a 2 σ − ) coordinate system and the volume inside the stretched horizon can be e ) is a delta function. For the geometry this implies t, σ + 0 x − , which is precisely in line with the original + = 0. An alternative prescription is to consider all locally extremal curves orig- H x ( T − δ x = 0 in the ( Curves that maximise the volume functional ( Just as in the eternal black hole geometry, we need to introduce an anchor curve outside We note that the Hawking temperature of a CGHS black hole is + BH x S , into a CGHS vacuum and creating a black hole. This amounts to SH i ground are obtained by patching together maximal curves across the infalling shockwave. at inating from the anchorthe point, black and hole. selecting theprescription This curve leads computation that to maximizes can exactly the the be same volume done curves inside and, as the interestingly, boundary it conditions turns ( out that this to the eternal black holecurve where connecting we these had two twohave anchor points, to points there and supply therefore is a awas prescription now unique argued for only extremal that additional one boundary in anchorboundary conditions. order point conditions to should and In obtain be [ thereforein a we our smooth setup volume in at Kruskal coordinates the one radial finds origin, the the relation additional The resulting black hole is one-sided, as illustrated inthe figure black hole to obtainthe a anchor finite curve volume. to The a extremal point curve on now reaches from a point on where horizon by only a smalltwo-sided amount black and hole the the location late ofin time the is volume anchor the growth curve late is is time the unimportantblack rate same. if holes of all that For growth we a the of are static advantageapparent. interested the of complexity. using It the is stretched only horizon when as we the consider anchor dynamical curveof becomes the black hole mass [ late time rate of growth ofto the complexity given by the volume in ( 2.2 Complexity ofNext, a we black consider hole collapsing formed af by thin gravitational shell collapse of matter at This is a curveour of anchor constant curves dilaton above.σ field Indeed, outside with the this black definition,obtained hole, the by which stretched setting is horizon how is we located defined at JHEP03(2020)069 1 t ) ) 2 = t t 2.13 2.27 = (2.28) (2.29) (2.31) (2.30) Curve t Anchor

+ x 0

+ = x , and vice versa. Curves 0 Extremal t Horizon Stretched and , . E + 0 + 0 + − x x > x < x . + + + 0 . y y M x − in terms of ˙ β + y + − β + y − y x 2 + ). 0 ˙ 0 y t = i and − 2 2 M + x − 4 = 0 in the inside region match onto curves − α y e + α β p + y + – 8 – be continuous across the shock. Those matching I 0 = t + 0 ds − 2 M 2 x M y − 4 , y 2 e Z + − + i + , the geometry is that of a static black hole and the I x x − = , the geometry is flat and the volume functional takes + 0 − and + + 0 ) in shifted Kruskal coordinates, = V + 0 y + M M 2 x y + > x 2.8 ) This is a Kruskal diagram of the Penrose diagram on the left. y < x are real valued parameters. The boundary condition (    + + x − 0 = . We use Weierstraß-Erdmann conditions to patch across the x − Right 0 − i t y q < x = 0, which are simply straight lines emanating from the origin β and Black Hole Singularity β < E ) This cartoon depicts the Penrose diagram of a one-sided CGHS black hole formed + Horizon − on the outside leaving us with a one-parameter family Stretched i I 0 and Left M = 0 in Kruskal coordinates (see figure + .( I = − α > x Event E Horizon There is again a two-parameter family of maximal volume curves satisfying ( In the outside region, = 0 i + Matter Fields that the momenta conjugate to conditions uniquely determine the parameter One finds, in particular,with that curves with and labelled by shockwave. First of all,from the viewing curve the itself integral should in be the continuous. volume A functional second as condition a comes Lagrange density and requiring In the inside region,the 0 simple form, where selects curves with x volume functional reduces to ( Figure 2 by gravitational collapse.The ( color coding coincides. The gray line is an equal time curve. One obtains a two-parameter family of maximal curves, JHEP03(2020)069 t on t (2.33) (2.34) (2.32) . This 2 ], which is 8 ). One is now . 2.32 + 0 + 0 > x < x , , ) and ( 2 2 + + t y 2.31 ≥ t < t M y , , − 1 1 1 M t . For this reason, we prefer to use the + 0 M ≥ x ). 3 O t < t , but disagrees strongly before that time. − a + 2.4 2 φ 2 − , we define the anchor curves, parametrised by combining ( y − M t − t = ) 1 2 y 0 t 2.1 φ + + – 9 – can be uniquely related to the tortoise time t > t    y y and 0 and it would imply that complexity starts growing t − − 0 cosh( t ) = 2    t , which is exactly half of the eternal black hole result, , ( 2 MM t a = 2 1 2 1 σ 0 AC    V M t > t − . As in section ) = a t φ 2 ( 2 for 2 corresponds to the fact that in the current case we only consider 0 − SH e 2 1 V M/ = = a ]. Here we adopt a particular modification due to RST [ 0 σ 2 V e 18 , The result agrees at times 17 1 is the time when the anchor curve crosses the shockwave line, see figure is the moment the black hole is formed. We conclude that the volume growth sets ). This factor 1 2 t t , so that the dilaton field is constant 2.23 a We can compare the complexity growth of the gravitational collapse model to the com- In contrast to the aforementioned, we could consider the total volume up to an arbitrary The growth of the volume inside the stretched horizon as a function of tortoise time In fact, as we move the anchor curve infinitely far away, the time difference of black hole creation and In particular, we could imagine this anchor line to be asymptotically far away, in analogy to computa- σ 2 1 also goes to infinity. 1 R t 3 The RST model:The complexity CGHS in model a cananalytically semi-classical be [ black extended hole such that one cantions study done a in the semi-classical AdS/CFT setup. black hole plexity growth of the eternalcollapse black model hole, is both at latesee time. ( The result forhalf the of gravitational the volumeeternal slice black (since hole case we (which only is have two-sided). a one-sided black hole), as compared to the where is long before the blackat hole the is moment created the shockwave isstretched released, horizon see as figure the anchorThe curve volume and integral can only be consider cut the off volume at inside either the the black event hole. horizon or at the stretched horizon. anchor line. We have is given by where in essentially at the time the black hole is formed, which is consistent with causality. Subsequently, the remaining parameter the anchor curve, see figure by We now obtain a relationin between a position to evaluate the volume functional ( JHEP03(2020)069 0 AC V (3.6) (3.1) (3.2) (3.3) (3.4) (3.5) . 0 AC V . and # i 0 f SH − V ∂ i of the form 2 f t + 0 + represents the leading = x ∂ t q κρ . which is much earlier than S = N =1 . i 1 X + χ t + 1 2 x φ , which is put to unity. The 2 κ ~ + at . − Ω) ρ . . ct ρ . φ M − 2 ) − S − χ ( ∂ − + 0 e κ + + x ρ∂ 2 R √ t q + e − S := x φ∂ + x ∂ + 2 2 x , to be large, then κχ t Time Time i d ( Ω + d f δ √ − M M and as a result Ω = + 0 ∂ – 10 – Z M CGHS Z x Ω ρ S κ κ + φ , = = ∂ − − = 2 κ φ + = = f ++ + χ q T ct RST φ − S 2 S S 1 − t χ∂ e = + t ∂ − Ω := " κ x 2 √ d 0 M/2 M

Z ouegrowth Volume , the number of matter fields 12 can be thought of as playing the role of N = 2 N/ the black hole is created. 2 . This graph demonstrates the difference between the volume growth t := RST S κ The solutions of the equations of motion can be written in compact form if one defines Again, we study an incoming shockwave of energy new field variables, The RST model continues tochoose enjoy Kruskal the coordinates, symmetry setting of the classical theory that allowed us to Here additional RST counter term has the following form in conformal gauge, This term is allowed by thefield symmetries equations of the take model a and particularly when simple it is form added and the are semi-classical easily solved analytically. Using the new fields, the semi-classical action reads If one takes order quantum correction due to matter, which in conformal gauge reads Figure 3 starts growing at thethe time time the shockwave crosses the anchor curve given by JHEP03(2020)069 crit (3.8) (3.7)

+ ) = Ω − S x 0 . Horizon

= Stretched + , x E − 2 + S

t x t x x = + = t x Curves t Extremal − ln are considered physical. 4 κ − − 4 κ + x ) x ) for which Ω( + 0 − S x log , x ν − ˙ y − + S and an evaporating black hole for + µ x 1 ˙ x y + 0 κ µν x Θ( 4 g √ + 0 ≤ M 0 p x i φ := 2 + + − x x crit – 11 – , the curve ( Ω − ds e + 0 + + 0 ≥ I Z x ) > x − = + + − , x V x I − + x . It is important to note that in the RST model only those x + + 4 i x − ) Depicts Kruskal diagram of the scenario illustrated in the left figure. crit ) = − Right Fields Matter , x Ω = Ω + Horizon ) Cartoon of the Penrose diagram of the life cycle of an evaporating black hole x ]. Stretched Black Hole Singularity − i 12 Ω( Left κ crit √ .( , as shown in figure + 0 Event Horizon Ω = Ω Although we have closed expressions for the solutions of the RST model, we were not Since the extremization of > x + be found in [ able to solve the extremizationcheck problem whether of the a complexity spacelike asexpectations volume volume we analytically prescription had and gives to in results resort order consistent to to with numerical general methods. a timelike curve that candimensional be parent interpreted theory. as For theturns origin spacelike in and spherical defines coordinatesblack in the hole evaporates a location and higher- eventually of thethe the singularity solution terminates black at can hole an be endpoint, singularity.again after extended timelike. which The into A semi-classical a more detailed late description time of flat the region semi-classical geometry where can the for physical instance boundary is Figure 4 formed by collapse. ( The color coding coincides. The semi-classical collapse solution of interest, given in Kruskal coordinates, is The solution describesx flat spacetime for regions of spacetime whereIn Ω( the flat spacetime region inside the infalling shell, the boundary of the physical region is JHEP03(2020)069 2 . 1 . We . We ≈ 5 5 . ) is not particularly t 1 2 ( 0 − V 4 κ − are plotted in figure ). We do not expect that φ ), which is interpreted as the 2 − e 3.8 M/κ M/κ in ( φ 2 = 1 yields a coupling strength of − the numerical result is consistent with e = log(4 S t M/κ 1 which demonstrates that we should be able M/κ . 0 – 12 – ≈ . We find that the slope of crit that is determined by the choice of anchor curve in the and then use a shooting algorithm to obtain the corre- ) for different values of Ω t t ( 0 − crit V on the stretched horizon at the scrambling time is approximately . For instance, a ratio 1 2 3 ) inside the stretched horizon provides us with the volume complexity which barely exhibits a linear growth rate. On the other hand a ratio − ) at Ω = Ω S 3.8 5a t which shows a linearly falling growth rate over a long period of time. 4 κ 2.27 − 5d M and 5c = 100 yields a coupling strength of ) showing the decreasing growth rate after a scrambling time has passed from the Results for the functions Further, it could be reasoned that the factor Note that, like in the case of classical collapse, it is not a priori clear what the appro- ) at a value of tortoise time In this model the gravitational coupling is given by the value of t 3 ( the coupling strength given by figure 4 Conclusion and outlook In this paper webridges have inside computed black holographic holes complexity in as two-dimensional dilaton the gravity volume models of where Einstein-Rosen we have explic- evaluation is reliable (see below). observe that for reasonablya high linear values decrease of the after volume growth the follows scrambling theenough, linear time so trend that forever, quantum corrections since become eventually, strong the on black the hole stretched is horizon. small In this model, which indicates we should notreflected trust in figure the result atM/κ all for this choiceto of trust parameters. the This solution is for quite some time after the scrambling time. This is confirmed in sponding maximal volume curve.to We apply patch across the the Weierstraß-Erdmann matching shockwave conditions and then continue thearea numerical of evaluation outwards. the transverse two-spherethe in quantum the corrected higher-dimensional area theory,sensitive Ω should to be this replaced by replacement, at least not in the parameter range where our numerical note, however, that thegrowth choice rate of of anchor the curveM/κ complexity, only but affects not the theonset onset slope of time complexity of growth. and the the curve early (at leading orderpriate boundary for conditions large at the originanalogous are. to We ( have chosen to apply boundary conditions V same way as for athe classical black dynamical hole, black hole. thenshockwave reaches the If the we complexity stretched use horizon. begins achor If to curve curve, we of grow then instead constant use very the Ω the complexity early,and far stretched growth long horizon this from turns as before prescription on our the essentially is an- when used incoming the in black obtaining hole the is numerical formed results presented in figure can be performed by solvinglem corresponding is simply Euler-Lagrange to equations, solve the aconditions, numerical non-linear prob- whose ordinary differential solutions equation provide with appropriate acally boundary integrating parametrisation ( of the extremized volume. Numeri- JHEP03(2020)069 E E t t = = t t . The blue E t = t = 5. = 100. S t . The black hole creation M/κ = M/κ t (b) M/κ ← (d) S t 2 t = t = 2 t t ← = t 0 0 ) ) t t ( ( 0 0 V V – 13 – E E . t t S t = = t t S t while the evaporation process is completed at time = t 2 t = 1. = 50. = t M/κ M/κ 2 (a) t (c) S = t t = t . Numerical results of volume growth for different values of 2 ← t Second, when considering dynamical black holes formed by gravitational collapse, we First, in order to obtain sensible results, we have to calculate extremal volumes in = t 0 0 ) ) t t ( ( on abruptly. If, on therange, other then hand, complexity we growth use turnsof the on Hawking stretched emission smoothly horizon at at to the a delimit semi-classical time the level. that integration coincides with the onset find it natural to cutrather off than the extending volume the integrationunimportant integration at range if the to all stretched a we horizonfor distant are of anchor a the interested curve. classical black in black This hole prescription hole is distinction extends but is the to it late a does time distantas affect anchor rate soon the curve of the onset then growth infalling the of shockwave of complexity complexity passes already the growth. the starts complexity If anchor growing the point volume and the complexity growth turns reduction. Thethe appropriately classical defined CGHS volume modeltime functional and at can it late exhibits be times. thegeodesics, explicitly expected that At evaluated will the linear otherwise in growth sameof arise, with time complexity. does Schwarzschild it not is lend easy itself to to a check direct that interpretation the in length terms of the spacelike itly known semi-classical black holeof the solutions. complexity This of allows aevaporates us black by to hole emitting that follow Hawking is the radiation.statements. formed time in Our evolution gravitational main collapse results and can subsequently be summarized inthe three four-dimensional parent theory rather than lengths of geodesics in the two-dimensional Figure 5 time is indicated by curve depicts the numerical result whileof the the dashed curve orange around line the is scrambling obtained by time a linear extrapolation 0 0 V V JHEP03(2020)069 at that time. In the T ], that the rate of complexity 2 in these models. M – 14 – should decrease linearly in time. This precisely the and the Hawking temperature S ST ), which permits any use, distribution and reproduction in ˙ C ∝ the number of matter channels available for Hawking emission. This CC-BY 4.0 N This article is distributed under the terms of the Creative Commons ], where we find that the numerical results obtained in the present paper for the ]. The scrambling dynamics that generates the growth in complexity takes place 6 2 Our results also support the notion that the holographic complexity corresponds to Our results fit very well with Susskind’s argument [ Holographic complexity can also be calculated in these models using the Wheeler- Third, using numerical methods, we obtain the complexity of an evaporating black Attribution License ( any medium, provided the original author(s) and source are credited. It is a pleasure to thankwas Shira Chapman supported and Nick by Poovuttikul the forand discussions. Icelandic by This Research the research Fund University under of grants Iceland 185371-051 Research Fund. and 195970-051, Open Access. degrees of freedom that haveblack been hole emitted has from the disappearedstate black and of hole. high, all At but that the no end is longer of left growing, the complexity. is day, the a long trainAcknowledgments of outgoing radiation in a the scrambling time, which is logarithmic in the quantum complexity of theation combined [ system of blackat hole the and stretched emitted horizon Hawking andThe radi- the outgoing growth rate radiation is is reduced free as the streaming area and of no the horizon further shrinks. complexity is generated by the models we are considering,is the proportional to Hawking black temperaturerate hole remains determined mass. by constant It and followstranslates the that into entropy a the growth black rate holeof of loses complexity the area that black is at hole initiallyevaporated. a proportional and constant In to then the other drops initial words, behaviour mass linearly we with see time in until our the numerical calculations, black following hole an has initial completely onset period of order paper [ volume complexity are confirmed by actioneven calculations at that can the be semi-classical carried level. out analytically growth for anproduct evaporating of black the hole black hole should entropy at any given time be proportional to the horizon. The growth rateblack then hole reliably evaporation. tracks Towards the thecorrections area end are of expected of the to the become horizon blackbe important as hole trusted. and it lifetime, semi-classical shrinks higher calculations can due order no to quantum longer DeWitt action formalism. We will present our results on that in a forthcoming companion hole as a functionfind of that time after using the black the holethe volume is scrambling prescription created, time inside the to complexity the growth settle stretched needs to horizon. a a time We rate period of of order growth proportional to the area of the stretched JHEP03(2020)069 ]. , ]. , ]. , (1992) SPIRE (2016) B 394 ]. IN SPIRE 64 IN SPIRE ][ (1992) 278 [ D 46 Complexity IN ][ SPIRE -dimensions ]. IN ]. ]. B 289 NATO Advanced ][ Nucl. Phys. Summer School in , (1 + 1) SPIRE Phys. Rev. SPIRE SPIRE , IN IN Fortsch. Phys. 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Giddings and A. Strominger, L. Schneiderbauer, W. Sybesma and L. Thorlacius, C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, J.G. Russo, L. Susskind and L. Thorlacius, L. Susskind, L. Susskind, A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y.A.R. Zhao, Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, L. Susskind, L. Thorlacius and J. Uglum, S. Chapman, H. Marrochio and R.C. Myers, A. Bilal and C.G. Callan Jr., S.P. de Alwis, S.B. Giddings, A.R. Brown, H. Gharibyan, H.W. Lin, L. Susskind, L. Thorlacius and Y. Zhao, L. Thorlacius, A. Strominger, T. Banks, A. Dabholkar, M.R. Douglas and M. O’Loughlin, J.A. Harvey and A. Strominger, [9] [6] [7] [8] [2] [3] [4] [5] [1] [16] [17] [18] [14] [15] [12] [13] [10] [11] References Article II

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t JHEP07(2020)173 Springer July 23, 2020 June 10, 2020 : : February 4, 2020 : Published Accepted Received [email protected] , Published for SISSA by https://doi.org/10.1007/JHEP07(2020)173 [email protected] , . 3 2001.06453 The Authors. c 2D Gravity, Black Holes

We adapt the complexity as action prescription (CA) to a semi-classical model , [email protected] Science Institute, University ofDunhaga Iceland, 3, Reykjav´ık107, Iceland E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: of two-dimensional dilaton gravity andplexity determine for the an evaporating rate black ofcal hole. increase results The of for results holographic semi-classical are com- black consistentsame hole with complexity model, our using previous but a numeri- volume thetest prescription CA for (CV) the calculation in holographic the is representation fully of analytic the black and hole provides interior. a non-trivial positive Abstract: Action complexity for semi-classical black holes Lukas Schneiderbauer, Watse Sybesma and LárusThorlacius JHEP07(2020)173 ], the 2 24 7 ]. 8 1 15 9 In line with black hole complementarity [ – 1 – 4 1 11 ]. A refined version of the conjecture instead relates 3 11 15 11 22 15 21 6 ]. 5 , 4 1 3 2.2.4 Complete action 2.2.1 Gibbons-Hawking-York2.2.2 boundary terms Time-/spacelike2.2.3 joint contributions Null boundary contributions The volume (CV) and action (CA) complexity conjectures have been explored for For a recent review of complexity and black holes, see [ 4.1 Evaporating black4.2 hole Eternal black hole 3.1 Gravitational collapse 3.2 Eternal black hole 2.1 A well-posed2.2 variational principle RST symmetry 1 bridge of a two-sided eternalthe black corresponding hole is quantum conjectured statethe to [ reflect complexity the to growing the complexityblack gravitational of hole spacetime, action referred evaluated to onblack as hole a the interior Wheeler-DeWitt specific [ (WdW) bounded patch, that region intersects of the the a variety of black hole geometries in Einstein gravity as well as extended to black hole degrees of freedom cangates acting be on usefully a finite modelledqubits number in can of qubits. terms be of takenhorizon to a and be quantum the located circuit black atprovides with hole (or a k-local interior near) dual is a geometric thendegrees stretched description viewed horizon of of as just freedom. the an outside quantum emergent the dynamics In spacetime event of particular, region the that the stretched expanding horizon spatial volume of the Einstein-Rosen 1 Introduction The quantum complexity of atum black hole mechanical is degrees generated of by freedom the that scrambling are dynamics enumerated of by quan- the black hole entropy. These 5 Discussion and outlook A Stokes’ theorem in two dimensions with null boundaries 3 Classical black hole complexity 4 Semi-classical black hole complexity 2 RST model Contents 1 Introduction JHEP07(2020)173 (1.1) and the 2 ], that arise from 12 ]. , 1 11 ST, is the proper time of a distant fiducial observer ∝ t – 2 – V C dt d ]. Here 1 ]). Black holes are stable in classical gravity and at late times, long after 9 – 6 ] we initiated the study of holographic complexity in semi-classical gravity with ] we found non-trivial agreement between the volume of certain extremal surfaces ], but in this case the two-dimensional spacetime is asymptotically AdS ) holds both for two-sided eternal black holes and at late times for dynamical black 10 10 14 , 1.1 In [ In [ 13 approaches zero atobtained the analytically endpoint for classical of CGHS evaporation. holes, we had While to the rely on extremal numerical volume evaluation for could be evaporating RST black holes, respectively.ity We restricted (CV), our using attention a to suitablythe volume defined complex- higher volume dimensional functional that parenttheory. corresponds theory to rather For spatial classical than volume CGHS geodesic in which black length is holes, in proportional the the volumetemperature, two-dimensional to complexity the grows product at a of constant Bekenstein-Hawking rate entropy and the Hawking as is expected on general groundsand [ ( holes formed by gravitationalhand, collapse. complexity growth For slows semi-classical down RST as black the holes, black hole on evaporates the and other the rate of growth make precise statements about the complexity ofof an two-dimensional evaporating black black holes hole. has The beenin complexity studied [ previously in the Jackiw-Teitelboimblack model holes do not evaporate. and the expected behaviour of holographic complexity of classical CGHS black holes and toy models, the Callan,and Giddings, Thorlacius Harvey, (RST) and models Stromingerthe (CGHS) of near-horizon and two-dimensional limit dilaton Russo, of gravity Susskind, aThe [ near-extremal RST charged model dilaton black isblack hole hole particularly in by well higher gravitational dimensions. suited collapse,spacetime, and for is its our described subsequent by purposes. evaporation an in analytic The asymptotically solution flat formation of the of field an equations RST and this allows us to the steady reduction ofoutgoing train the of black Hawking radiation holethe but area quantum no complexity and black of hole. endsno this For in longer a final growing a large state as initial final will the black state be Hawking hole radiation mass, very where free large, there streamsthe but is outwards finite, aim [ an of and presumably testingblack the hole geometric evaporation. representation of In quantum order complexity to in have the analytical context control, of we considered semi-classical resemble that of athe stationary holographic one. complexity (both Similarly, CV into and the that CA) late of of time a aa limit, classical stationary factor dynamical the black of black rate two hole hole of accountingsolution). reduces of growth for the of the This same two-sided changes, nature massasymptotically however, of flat and when the spacetime, other semi-classical maximally black conserved effects extended hole charges are stationary evaporation (up taken due to into to account. Hawking emission In results in solutions in other gravitational theories.on Much stationary of black the attention hole andholes solutions effort (see while has e.g. there been [ have focused beenits formation fewer by studies gravitational of collapse, dynamical the black geometry of a dynamical black hole will closely JHEP07(2020)173 (2.1) ]. The numer- . We end with a 16 , 4 ], it has the distinct 15 ] are consistent with 10 10 , followed by the corresponding 3 , , we carefully develop the boundary ct 2 S + q S + ) carries over to the semi-classical theory with 0 – 3 – S 1.1 = ]. Our numerical results confirmed that at leading ], we are able to eliminate a certain ambiguity in the 10 bulk 12 S ]. By working in so-called Kruskal gauge and arranging 18 , 17 given to leading order by the time-dependent area of the black hole. t for most of the black hole lifetime. The structure of the paper is as follows. In section A Wheeler-DeWitt patch is bounded by co-dimension one and co-dimension two sur- In the present paper we extend this work by evaluating the holographic complexity CGHS black hole formed in gravitational collapsesemi-classical in section calculation for anbrief evaporating discussion RST and black outlook hole for in future section work. 2 RST model The action of the semi-classical RST model consists of three terms, the boundary terms infield the equations action of to the respect RSTholographic model action the complexity [ same and symmetry obtain that a simplifies remarkably simple the end bulk terms result. needed togravity have models. a well-posed We variational then problem present the for holographic the action two-dimensional complexity dilaton (CA) for a classical faces in spacetime. The associatedder gravitational to action make must the include variational boundaryone problem terms boundaries in well-posed. or- For the time-liketained appropriate and space-like from boundary co-dimension the terms standardtheory, in while Gibbons-Hawking-York contributions the term from two-dimensional in nulla boundaries theory the more and higher-dimensional are co-dimension careful parent ob- two treatment boundaries [ require growth of complexity expectedzon, for provides a the non-trivial quantum positive testduality dynamics for between of black the hole stretched an complementarity horizon and evolving andports the the stretched the holographic black validity hori- hole of interior. the At holographicregime. the complexity same conjectures time, themselves in it sup- a new dynamical the entropy at time The next-to-leading order (logarithmic) termbe in read the off Bekenstein-Hawking from entropywith can our previous also analytic semi-classical expression entropy forical calculations the results in complexity for the growth volume RSTthe and complexity model new the obtained [ analytic in result results our agrees calculations for previous for action work a complexity. [ semi-classical The black hole fact geometry that reproduce our the holographic time-dependent complexity rate of of semi-classical black holesthis is in more terms technically involvedadvantage of than that an the the volume action computations entireyields on in explicit semi-classical a [ results calculation Wheeler-DeWitt for canthroughout the patch. rate its be of evolution. carried While complexity The out growth relation of analytically ( an and evaporating RST black hole semi-classical RST black holesorder in in a [ semi-classical expansiona the function rate of of the growth of properST the time complexity, of when expressed a as distant fiducial observer, is proportional to the product JHEP07(2020)173 (2.4) (2.2) (2.3) # 2 generated by the ) . It enters at the i 0 κ f along with a dilaton S with respect to a flat ∇ will be directly related ( µν µν i is retained in the action g κ g φ = N i 2 X ~ − 1 2 e , − is set by the magnetic charge of 2 1 R λ − 2 1 λ ∇ ] and captures the one-loop correction + 4 gφR , R 2 − 11 ) φ √ g y ∇ ], in order to have a definite prescription for − 2 d √ 18 y – 4 – Z + 4( 2 d 2 κ R ]. This term is allowed by general covariance and − Z φ 2 4 κ 12 = . The length scale − i − e ct f " S ] were enough to ensure a unique answer for the late-time = g and thus any expression involving q 18 − ]. However, it is well known that the action associated to a κ S is the conformal factor of the metric √ = 1 throughout but note that when 5 ρ , y 2 ], where a particular set of co-dimension one boundary terms and 2 4 ~ e d 18 M Z and serves to preserve the classical symmetry of = q ), where 0 S φ S scalar matter fields − 12, was introduced by Callan et al. in [ N ρ ( N/ µ this term dominates over one-loop effects coming from the dilaton gravity sector ∂ = and N κ φ = 1. The second term, patch while leaving thefurther variational criteria, beyond principle those well-posed. considered in One [ therefore has to introduce co-dimension two joint terms were proposed.the In general, requirements these imposed terms are in still [ complexity not growth unique, but rate in well-knownhowever, classical for black hole the geometries. semi-classicalcomplexity This model is growth not we for the consider dynamical case, that solutions below. certain that boundary Indeed, describe terms can evaporating when be black we added holes, calculate that we change the the find value of the action on the WdW not change the equations ofTo motion restrict but the will set in of general possiblethat change actions, the the the value variational CA of principle proposal themotion on action comes the should itself. with the WdW follow further patch from requirement conditions should a on be the stationary well-posed. boundary action The offor principle Einstein-gravity the equations in assuming WdW of [ patch. appropriate A boundary solution to this problem was presented to quantum corrections in the semi-classical theory. 2.1 A well-posedThe variational CA principle proposal instructsa us so-called to WdW evaluate the patchgiven on-shell [ set action of of equations the of model motion in is question not on unique. For instance, adding boundary terms does was introduced by Russodoes et not al. disrupt in thesame [ classical order physics as obtainedcurrent in the limit reference metric. We set it accompanies the prefactor with to the quantum effectivelarge action due tothat the can conformal therefore anomaly be of ignored. the The matter third term fields. in the For semi-classical action, where field the higher-dimensional near extremal blackλ hole and from now on we work in units where is the classical CGHS action involving a two-dimensional metric JHEP07(2020)173 ] , λ κ by 12. 18 0 ]. In (2.6) (2.7) (2.8) (2.5) N/ λ ], . The λ = 19 14 ]. 7→ , κ λ 12 13 # can a priori 2 l ) | i l f λ ∂ ∇ | ( Here log 3 N =1 i X ˜ χ λ 1 2 parametrized by − N 2 dλ ∂ . The terms on the second will return the original non- ) . The integration variable S N Z Z Z N ∇ does not vanish anywhere. ( N l . σ 4 κ λ ) ∂ N N ∈ Z − β X φ − N ∈ 2 k − φ κ ( e + 2 to be an affine parameter is measured by 2 Σ κ = κ β λ − ˜ k χ . + 1 – 5 – in the bulk action, φ α ˜ χKd j 2 2 | dλ ) a ∇ R − h j | φ α e

N k ˜ χ Z ∇ p j ( ) is not invariant under reparametrizations := S N σ ˜ Z χ σ 2.7 S N + 4 σ joints X ˜ X ∈ χ S . The failure of N ∈ j R X N S∈ " . One way to remedy this problem is to introduce an auxiliary ) being coordinates of the null curve q + 2 + 2 g λ 2 ( S ] gives the following boundary terms involving the combination of − = 2 α is the extrinsic curvature of each time/space-like boundary component γ √ 18 y 2 K d and is the determinant of the induced metric on M boundary and write the action in terms of integrals over local quantities only [ α Z h S ∂λ ∂γ Z ) accompany null boundary components = ] for a detailed analysis. := 2.7 and that multiplies the Ricci scalar 18 bulk α χ S k S Considering a region with piecewise smooth space-like, time-like or null boundaries, To obtain a well-posed variational problem, we adapt the procedure proposed by [ See [ Unfortunately, conventions dictate using the Greek letter kappa both in this context and as 3 2 We’ve opted for using boldface for one of them to reduce the scope for confusion. with first term on theitself second and line the of second ( termbe is an added arbitrary to function offset of this any pathological scalar feature. field, provided The first term onYork term, the where right handS side ∈ is theline analogue of of ( the familiarparametrizes the Gibbons-Hawking- null line defined by the equation scalar field As one can easilylocal check, integrating action, out up the to auxiliary boundary field terms. the prescription of [ fields ˜ explained in detail below, afull sufficient criterion action, for including the boundary situationfield terms, at equations hand the is of same to the symmetry impose RST that on model the allowed to the be semi-classical analyticallyto solved a in two-dimensional the dilaton-gravity firstby theory. place the in However, non-local [ a term direct application is obstructed with the holographic action complexity. Ato similar evaluate issue the arises complexity whenthis the of CA case, two-dimensional prescription Jackiw-Teitelboim the is black rate used terms holes of [ and complexity additional growth is physical found input to is be required crucially to influenced by fully boundary determine the complexity. As JHEP07(2020)173 φ S ]. 18 (2.9) (2.14) (2.13) (2.10) (2.11) is the unit µ n ) depends on an that are sensitive j 2.7 σ that points outwards which depends on the 1 j S respectively (and corre- and a , ¯ λ φ N 2 σ e φ 4 , κ 2 and S e − σ λ 1 , | . µ 2 , − Σ (2.12) n =

| d µ µ ) µ ) dy Z ¯ 1 k n + to vanish at the boundary. However, it is p µ µ k k φ, Z dy Z | + RST ( 1 2 is a tangent vector of ρ δ

2 g 1 1 | e 1 2 n – 6 – p h ( | and − | = , we have to add a term = log p φ j = ρ = log and a ). Adding a boundary term does not alter the equa- S 2 ), was introduced to preserve the symmetry generated i a Z = log S ds RST 2.4 φ, Z δ a ( reads g = a ) are accompanied by signs φ ), 2.7 α ] ¯ RST k , which again influences the holographic complexity. The above ). The corresponding infinitesimal transformation of the fields δ l , given by ( φ 20 , ct and − S 12 ρ α ( k µ ∂ , that are separately either spacelike or timelike, one finds corresponds to the null boundary in the way explained above and are unit normal vectors to 2 µ i S k n are given by [ ρ In the following, we work in conformal gauge, where the line element takes the form The various terms in ( Now, consider adding to the action a boundary term of the form In case of a joint of two null-lines parametrized by Finally, for each non-smooth joint and and the variation of the scalar fields 1 while the matter fields do not transform. Recall that the term by the current and 2.2 RST symmetry In order to overcomepropose the to troublesome restrict arbitrariness theunder in allowed the the terms symmetry choice by which of guided an the boundary invariance definition requirement terms, of of we the the RST total model action in the first place. tions of motion and a termif of this we particular impose form will Dirichleth not boundary influence the conditions, variational principle i.e.easy take to the see variation of thatin the such our induced a setup. metric term Furthermore, canundetermined considering drastically function regions change with the null-boundaries,prescription result ( thus of needs holographic to be complexity supplemented by additional restrictions, as discussed below. to the conventions adapted in the procedure. A coherent set of rules is presented in [ involving an arbitrary function where from the region of interest. sponding vectors while in casewe of have a joint between a null andwhere a space- ornormal timelike vector associated boundary to component, the space- or timelike boundary. type and position ofS the joint. More explicitly, in the case of joints formed by two curves JHEP07(2020)173 . η + (2.20) (2.17) (2.18) (2.19) (2.15) (2.16) ρ boundary ) vanishes = 2 Z # 2.17 2 ) i , it is convenient f ∇ including ( , field via S N =1 i X Z . boundary 1 2 S S ρ + − ]. µ ∂ 10 Ω) µ bulk − n χ | S ( h 2 κ | , e , 2 φ , , p g 2 − κ Σ i − 2 + ) . = − η √ µ 0 φ , ), it is evident that, generically, the RST up to a boundary term that will have to n χ ˜ ∇ χKd 2 κ | + κρ µ = 0 η can easily be evaluated and yield ∂ h ( 2 | | + + S 2.12 0 RST χ RST ∇ δ ) of the form φ φ Ω) – 7 – p h δ | 2 2 8 κ S ∇ − − RST ( 2.7 p Z e e δ Ω = 1 κ S ) + = σ ρ := 2 µ − ∇ χ RST Ω := n 2 = 0, obtained from the auxiliary δ χ ) µ (˜ χ η 2 |∇ ) can be written as ∇ ∇ ), the following identity h h ( ∇ | g 2.5 1 is a unit normal vector to the surface κ p − 2.13 " µ g n √ y − ) is invariant under 2 d √ y 4 2.5 2 Z where d 2 µ Z − n = 0. It is now evident that the RST variation of the first line of ( µ = η ∇ is a harmonic field, = bulk RST η δ S K In conformal gauge ( The variations of the new fields Ω and In order to work out the RST variation of the action We now impose the additional requirement that the total action Note that this definition differs slightly from the one employed in [ 4 with RST variation of the firstthe total Gibbons-Hawking-York derivative (GHY) term cancels term against that a we contribution consider coming next. from 2.2.1 Gibbons-Hawking-York boundary terms Let us now consider the GHY term in ( where while and also the variation of the last term on the second line. The remaining non-vanishing to define the fields The bulk action ( be cancelled. Goingvariation back of to such the a term exampleadditional will ( boundary not vanish. term We soaction. can that use this its to variation our cancels advantage and against choose the the variation of the bulk for which the bulk action ( terms, remains invariant under the RST transformation, JHEP07(2020)173 (2.27) (2.23) (2.25) (2.26) (2.21) (2.22) (2.24) ). This 2.17 , . Σ ). This term does 6= 0 , ρ d µ ) Σ ∂ ρ µ 2.21 d µ ) ∇ 0 0 χ n ˜ χn (˜ K ( | | ρ 0 h ∇ e | . h g | | p = µ − ) , p S j 2 j √ Φ Z S a y n ρ j Z ( 2 S ∂ µ d S σ ) Φ σ σ j 1 Φ σ 4 M p µ ∂ Z − ∂ S µ 0 + X σρ 0 Σ + 2 n = g j 1 ρ d . It follows that the inner product is inde- 4 n 1 − 0 p ( e j p − | Σ = 2 ρ a e j ˜ – 8 – = χK =

| ρ d log 0 ˜ = µ χ µ j ! j h , its RST variation is easily evaluated,

S Z = S 2 RST σ δ 2 ds Since the term is proportional to ˜ locally. Similarly, out joint terms of thisnot form. influence This the is variational possible, principle because, in as two dimensions. we have just seen, such terms do and the same ispendent true of for the the conformal tangent factor vector which does not vanish in general. The RST symmetry is easily enforced by simply leaving where Φ is temporarily introducedS as a scalar field whose contour lines describe the surface It turns out thatwell-posed in variational two principle. dimensions,not The these depend reason terms on is are the thatvector actually conformal as the factor not argument at necessary of all. to the This obtain logarithm is a does easily seen by writing the unit normal This variation can be cancelled by introducing a suitably2.2.2 chosen additional boundary term. Time-/spacelike jointStill contributions assuming no null boundaries,to this the leaves us action with the analysis of the joint contributions which precisely cancels theleaves first us only term with innot the the contribute term second when involving line we the obtainreference flat of metric field reference the is equations metric bulk fixed). using in action field However, ( its ( variations RST (under variation which does the not flat vanish in general, holds, where quantities with subscript 0metric are to be evaluated with respect to the flat reference and furthermore, due to Stokes’boundaries), theorem the (assuming second for term the can moment be a written region as without null JHEP07(2020)173 := 0 λ (2.31) (2.32) (2.33) (2.34) (2.29) (2.30) (2.35) (2.28) 7→ ), so that λ is fixed by | N ρ , l 2.28 λ σ λ ∂ | ˜ χ∂ dλ . log l λ N ν Z ∂ ˜ χ∂ l . µ 0 , σ N

, N λ ∂ σ dλ ∇ | l µ σ − N ν − l λ k N k 7→ k λ ∂ µ + 4 Z | , ∂ µ − λ l m l k N ν ν ˜ σ χ log ∂ l -dependent term in ( ∂ A 2 l µ

σ ˜ µ χ κ σ ∂ dλ ∂ λ − ∂ , ∇ ν − σ − | N log 2 1 σ − ν − k k µ

Z | k ˜ k χ µ − k l dλ ∂ µ − k σ µ λ N k ˜ ∂ N σ χ | 2 Z = 2 + A m dλ | | ) the full expression is also reparametrization N − log l κ σ – 9 – N 2 1 λ ˜ χ

Z | ∂ log = | l 2.31 N | ˜ + 2 λ N χ l σ ∂ σ λ σ | κ log if joint lies in the futureif of joint lies in the past of 2 2 ∂ ˜ χ | ˜ χ cancels the original = 2 − log σ | dλ l κ N N 2 log = ˜ χ λ σ σ N λ l ∂ − N | Z ν − + ∂ σ ) ∂ l ( N µ depends on conventions, but the relative sign to σ log is the null vector associated with the null boundary in question, σ ∂ = ∇ ˜ 2 σ χ 2.32 α σ − ν ( − ) = 2 λ σ . When the original joint terms are combined with the new terms k k k α µ − k k dλ∂ 2.28 β ( ˜ χ − N e Z dλ N 7→ N σ is future directed. This implies, that for the total joint contributions on either ) corresponding to null boundaries, Z α 2 µ k N 2.7 k is a constant, σ is a vector that depends on nature of the joint, which we will be agnostic about for 2 A µ − m We have now successfully rewritten the terms corresponding to a null boundary and its , since λ β which is also manifestly invariant under reparametrization attached joints in a manifestlythe reparametrization RST variation, invariant form. it will Next,on again in the be order conformal convenient to to factor split obtain and those parts terms which into do parts which not. depend We have assuming side of the null boundary, we obtain of the form where and this argument. The sign and using that The second term is nowe manifestly invariant under a changeobtained of from parametrization integration byinvariant. parts Independent in of ( the precise nature of the joints, the original contribution will be in a way that is manifestlysecond reparametrization term invariant. by This parts is which achieved gives by integrating the we note, that the termin involving total we have 2.2.3 Null boundaryLet contributions us now includeterms null in boundaries ( in our discussion. It will be beneﬁcial to rewrite the JHEP07(2020)173 , . ρ ν 1 k ) is

µ µ ¯ k (2.37) (2.36) (2.38) ). k µ l 2.37 µν and the λ η ¯ 1 m 0 ,

ρ ∂ ˜ 2.17 ρ χ , 2 ) λ µ λ e − ρ k e µ l ∇ = 7→ ¯ λ A ¯ χ m ν

∂ (˜ dλρ∂ ρ λ k − ∇ µ N e log g ¯ k Z 1 ¯ )). In case of a joint − A µν ˜ χ N

g √ σ N y 2 2.25 σ 2 log d − 1 ˜ χ ρ + 2 M λ 2 N Z

σ ˜ µ χ∂ k in case of a joint with a space- or µ l + 2 dλ ρ λ 2 Σ = 2 m N

∂ ρ . Importantly, the ﬁrst term in the Z µ ρ − ρ d k 5 e N µ µ l ∂ σ λ A 1 µ m

∂

ρ +2 µ − ˜ χn l k | e log λ µ h 2 – 10 – ∂ | A ¯ ˜ m χ

p ¯ N A . S ), we can write them as

σ log ρ Z 2 2 . Hence, in all cases, the resulting joint terms will be S ˜ χ log does not depend on − ν 2.34 σ 1 N k µ )) ˜ χ S µ σ is given by a normal vector, see ( ρ m 2 ¯ k e N X ρ µ S∈ σ µν − − m η e l log( ν + 2 + 2 χ ∂ l ρ 2 (˜ µ σ

λ ∂ λ ∂ µ l ν − σ − ˜ k χ∂ λ k k µ ∂ dλ∂ µ − dλ for a justiﬁcation of the formula when including null boundaries. m k N N ˜ A χ Z A Z

N dλ N σ log σ N 2 2 Z − . ) does not vanish, but since it is reparametrization invariant and the variation of ˜ N χ ρ N X N = σ N ∈ Since the RST variation of the remaining bulk contribution vanishes, the RST vari- Combining the resulting terms, and using the product rule, we obtain Considering the joint terms ( σ 2 2.37 See appendix 2 2 5 − − well-posed variational problem andAs we these leave them terms out areoverall to manifestly reparametrization implement reparametrization invariance. the invariant, The RST removingof RST symmetry. them ( variation will of not thevanishes spoil at last the term boundary in (therea are the boundary no last term. derivatives involved), line it can be cancelled by adding in order to cancel with the total derivative contributionation coming from of the the bulk null ( independent contributions of has the to conformal vanish factor, as these well. terms are Because not the necessary first in line order of to ( ensure a where each termfirst is line now manifestly issecond invariant independent line under combines of with reparametrization the the correspondingcomponents, terms conformal arising in from conjunction factor space- with or timelike Stokes’ boundary theorem, One can easily check that timelike curve (since then between two null curves, thebut argument of now the logarithm the isthe above given resulting by procedure argument is is independent performed of the twice conformal (once factor for each null surface), so that where the first term isfactor reparametrization invariant and does not depend on the conformal JHEP07(2020)173 (3.1) (2.41) (2.39) . This φ . In this χ # 2 ) i f ∇ ( . # N =1 η i X − 1 2 in the CGHS model. The , implying Ω = η∂ φ − + M ∂ = Ω) is given by 2 κ − ρ f χ , ( + 2 ) κ i = 0. Note that the action does not e + 0 f g x − S ∂ − 2 i − Ω (2.40) with mass ) where f 2 √ RST + − + δ f ∇ x ∂ + ( , x δ = 2 + N =1 + 0 i x X – 11 – χ Ω) M x is subject to a remarkable simplification, 2 − ∇ ∇ S ( = 2 . " 1 κ f )] − ++ − η T 2 dx ∇ ) + η χ ( dx ∇ ∇ ( [ M 1 κ g The equation of motion Z " − g = √ , in conformal gauge, − y S S 2 √ d y 2 Z d 4 κ M Z − anymore. Further, as a side note, the value of holographic complexity on the l = S As a result, the expression for the holographic complexity does not involve an arbitrary 3.1 Gravitational collapse Before discussing the semi-classicalof case, let an us infinitely analyse thinenergy the shell momentum classical tensor of gravitational associated collapse incoming to matter the matter field In particular, in this form theis, action of has course, no somewhat explicit misleading, dependencewill for on the indeed the shape depend dilaton of field on the the WdW patch spacetime in metric Kruskal and coordinates therefore the3 dilaton as well. Classical black hole complexity allows us to choosecoordinate Kruskal system, coordinates the ( on-shell action WdW patch can also bespace- obtained by or a timelike limiting surfaces procedure, only.obtained regulating by The the the WdW resulting above patch limit with RST is symmetric finite, prescription and for itThe null agrees on-shell boundaries. with action. the result region bounded byunder spacelike, RST timelike, transformations or ininvolve null the any boundaries. boundary sense or that joint Additionally,or terms canceled anymore, it since against is they total were invariant derivative either contributions consistently from removed, the originalfunction bulk action. 2.2.4 Complete action Let us now collectfor the the results total of action the above considerations. We obtain a simple expression This action has the properties that the variational principle is well-posed on any spacetime JHEP07(2020)173 = 0 (3.2) (3.3) t

+ x 0 ], it seems

+ = 10 x Horizon Stretched

. The holographic Singularity , ). Furthermore, since + 0 + 0 2 x < x ≥ 2.41 + + + − x x x x . if if + 1 M to zero in ( M + κ is defined as the union of all spacelike 0 i + 0 t M x + 1 = this implies + + EH I ρ − φ x – 12 – 2 − − x ], we take the WdW patch to be anchored at the . + e t − i + + I 10 x x = − − SH    φ 2 = − e ρ 2 − − i e is then given by the action evaluated on the WdW patch. The = t φ and conformal factor 2 Black Hole Singularity − φ e + Horizon − Stretched i at time I C curve and extending towards the black hole, see figure + I t outside the black hole. If we instead anchor the WdW patch on a curve far outside . Left panel: Penrose diagram of one-sided CGHS black hole formed by gravitational φ , which depicts a Penrose and Kruskal diagram on the left and right, respectively. Event Horizon 1 0 The WdW patch at a given tortoise time In line with our previous paper [ i Matter Fields complexity classical action can be obtained by formally setting more physical to placecomplexity the growth coincide, anchor at curve least(here at approximately, defined with the as the stretched the time tortoise horizon of time and black at hole have which formation the thesurfaces shock onset originating wave of meets from the theconstant stretched point horizon). on the anchor curve that intersects the appropriate In the classical collapse solution considereddilaton here, the stretched horizon is athe curve of black constant hole, thetime, main to difference the isthe time to anchor shift in curve the tortoise on onset coordinates its of way when complexity to the growth forming forward infalling the in black shock hole. wave As passes was through discussed in [ and for the dilaton in Kruskal coordinates.figure The infalling shell creates a black holestretched horizon, singularity, defined as as shown aunit in membrane larger outside than the the black area hole, of with the an event horizon, area that is one Figure 1 collapse. Right panel:line the denotes corresponding a Kruskal curve diagram of equal with tortoise the time same color coding. The gray JHEP07(2020)173 ) − A dx + )) (3.6) (3.4) (3.5) − A − A , x dx + A + dx ) − A − A

+ x = 0

( + A t , x − S + is the tortoise x

+ A 0 ( x t , x x

( = + A + x Horizon Stretched dx . + 0 + = 0, the matter term + A ), which are related to x , where ( . To this end, we denote f C t, x dt > x − 1 d )) ∂ − A + that the change of the area x ), while the singularity curve , x ( − A 2 − S A + − , x , x . x x + A + A x patch x − A ( WdW =: 2 . x − dx + , t + A dx x − x ) , and after, e + + − A t 0 + − e − dx dx + A − = = < x − A dx + + 0 – 13 – + − A , x WdW x M x x x + A Z − + dx

−

+ + = 0 = 0 = 2

x t + A x It is apparent from ﬁgure ) cl x

= A

( + − A S x d , x = Horizon + A Stretched x ) by the equations C ( −

, before the shockwave arrives, is given by Singularity , x A ). It is practical to consider two separate cases: the WdW patch + − S x , x + S x − + x x patch . WdW 1 can be interpreted as the reference metric “area” of the WdW patch drawn in . Evolution of the WdW patch for classical gravitational collapse. Color coding coincides ) does not contribute to the action. This is true independent of the precise form of A An asymptotic observer would use the tortoise coordinates ( 2.41 time associated to the anchor pointthe Kruskal of coordinates the describing WdW the patch, anchor seeis point figure as described ( by ( anchored before the shock wave arrives, Before incoming shockwave. in Kruskal coordinates, where Kruskal coordinates. Kruskal coordinates ( We are interested in the growth rate of holographic complexity the collapse solution onlyin involves ( infalling matter, forthe matter which profile. Inmatter particular, flux, we instead could of have started athat with sharp the a shockwave, on-shell and smoothly action this varying reduces incoming conclusion to would still hold. It follows, Figure 2 with figure JHEP07(2020)173 (3.7) (3.8) (3.9) (3.12) (3.13) (3.14) (3.10) (3.11) of the model, λ = 0 corresponds t , or dt t and remains constant 0 + Me x = = , so that Mdt , . t + A − A = x , d dx + A + A . + A , , , dx dx x + 0 + 0 + A + 0 − M x M x = 1 . < x + 0 = 0, the black hole creation time. The + A > x M x = + A t = + dx + + 0 x + A ), which we identify with the anchor curve, + − A dx x M x at − A − S + 0 dx dx 3.3 for M x + x x is continuous at for M + A ˙ , are consistent with the expectation that the − A x C − – 14 – = t + 0 x + S ) 3 M x − x + A A M Me + A d x + A = ( ) + x − S dx = 2 + A = 2 x ˙ − A M − x C ˙ The analogous calculation can be done for times after the x ( C , if we shift the time variable − S = − dt x + A A d x = = 0. One easily finds A + A d t > dx . + 0 ) in conjunction with the defining relation for the black hole singularity, 3.8 > x + Our findings, summarized in figure x complexity of the quantumproportional state to corresponding the black to hole a entropy black times hole its temperature. should grow with a rate or so that the holographic complexity growth for one obtains After incoming shockwave. black hole creation, Using ( We observe an exponential onsettime towards scale 2 of thewhich exponential we have growth set is to given 1. by the characteristic scale so that Furthermore, since to where the shock wave meets the stretched horizon, then From the definition of the stretchedwe horizon obtain ( the equivalent relation which immediately implies that JHEP07(2020)173 of , so + 0 (4.1) R x t (3.15) (3.16) . As is − 4 = = , using the t + L x t at . For simplicity, t 4 M ]. The result for complexity 23 , , 22 − . x ) + + 0 ) are the tortoise times w.r.t. the right x x L t − − ( + R M t x M, ( ˙ 0 C δ = = 0 100 t ρ = 4 + 0 M is easily performed and indeed provides the ex- – 15 – 2 M 2 ˙ x C − where A e . In this case, we expect the complexity growth to = L R t t = − f φ ++ = 2 R T t t − e of holographic complexity as a function of tortoise time ˙ C . − S x ]. The black and white hole singularities are located on the curves defined + S 21 . Growth rate x = The variation of the Kruskal area A Kruskal diagram including the WdW patch for late times is given in figure M 4 Semi-classical black hole4.1 complexity Evaporating black hole Again, we study anthe form incoming leftmoving shockwave pulse of energy have twice the contribution of a one-sided black hole. pected holographic complexity growth in the late time limit. usual for the CAa prescription second in anchor the point context ongrowth of will a two-sided then ‘left’ be black anchor a holes, curve, function(left) we of see side have e.g. associated to [ to provide we the take respective the anchor two anchor pointthat points position, the to see result move symmetrically figure only as depends time on progresses, i.e. eternal black hole for late time.is given Its by solution in terms of the dilaton, in Kruskal coordinates, see e.g. [ by Figure 3 CA prescription, for classicalcomplexity gravitational grows linearly collapse. with time. Following an exponential onset, holographic 3.2 Eternal black hole For completeness we should mention that our prescription also works for the classical JHEP07(2020)173 ) = (4.5) (4.2) (4.3) (4.4) ± . The x φ ( ] ). Since ± = t 24 ρ , , 2.41 11 − ), using Kruskal x ] to be + , stating that there x ), as [ 2.16 11 − − , which fixes the form I R 2.13 2 κ ln , 4 κ = ) i − , cf. ( ± µ ) ± y f µ ( + 0 y R T , t x ± h + t ) η , x − − ] 2 ± + f x µν + ∂ ρ T 19 + x 2 ± x ∂ + 2 can be determined [ Θ( − η − + 0 ± . ± M x ρ x − ± η∂ I + Patch WdW ± – 16 – . In this form, the energy momentum ‘tensor’, ρ∂ x ∂ ) = log( − ± (at least in Kruskal coordinates), it can be viewed and the metric via its conformal factor − ) and reads [ I − ∂ 1 4 φ ± , x + 0 t κ 4.4 x + = White Hole Singularity Black Hole Singularity − x − ( ± x t + η L = t i − x f ±± + ] T x h 12 − is fixed by ( η in Kruskal coordinates ) = − ± t )), needed for the holographic complexity computation, is related to the , x + x 2.17 Ω( determines the functions . The field 2 η . Kruskal diagram of eternal black hole. A symmetric WdW patch is presented. Color ) ± (see ( x η The form of (2 / 1 which is needed forthe the field evaluation of holographic complexity, see equation ( to a specific coordinatethe system. metric For is the manifestly case Minkowskian at near hand, this is− the coordinate system where is determined by boundary conditionsshould imposed be at no past outgoing nullincluding infinity its energy quantum flux corrections, at isfact, actually that not the a quantum tensor corrections anymore. depend This on is a related choice to of the vacuum, which makes reference The semi-classical collapse solution, incoordinates, terms is of given the by field [ Ω defined in ( field conformal anomaly of the energyof momentum tensor the energy momentum tensor in light cone coordinates where Figure 4 coding agrees with previous figures. which in turn determines the dilaton JHEP07(2020)173 + + 0 ). x I (4.9) (4.6) (4.7) (4.8) ≤ 2.17 + x . A useful (0)) repre- + 0 − S , x . This is in line > x i (0) f + + S T x h x , (1)) describes the point ) for     − S 4 κ , x ln + 1 − +1 (1) u u + S defines the boundary of physical . − (1 x − , u 4 κ 1 + 0 u 4 κ 1). The point ( M κ − 1 , 4 M = κ u 4 e ] and take the stretched horizon of an = ), describes flat spacetime for , < x − e M κ 2 4 κ 4 4 κ u crit + e 4.5 x + 0 M κ with outgoing Hawking radiation towards 4 = M x M κ ) = e 4 u − B + 0 for ( + ) = Ω x 0 + This reflects a breakdown of the semi-classical κ − S x − S – 17 – + B − A M κ 4 4 κ x 6 4 x x > x ) , x − ln u + S − ( + + 0 x + A is the interval (0 − x + S x x x 1 u     − − ) together with ( 4 κ = . The location of the black hole singularity is determined by 4.2 = ! ) ) 5 only appears within a total derivative term in the action ( u u η ( ( crit + S − S implies, that an asymptotic observer near future null infinity x x i

) = Ω f ++ − B T ) which satisfies Ω( , see figure h , x − S + + B I , x x ] for more details. + S x 21 As in the classical theory, we take the anchor curve for our WdW patch to be the Altogether, the solution ( The form of See e.g. [ 6 horizon, with an areaevent of horizon. order For 1, technicalRST simplicity, in we black Planck hole follow formed units, [ by larger shockwavethe collapse than period to coincide of the with evaporation. area the This apparent of horizon determines the during the black anchor hole curve as in Kruskal coordinates. stretched horizon of the black hole, defined as a membrane outside the black hole event where the black holeproperty has entirely evaporated. This parametrization has the convenient The curve Ω( spacetime before the matter shockwave arrives, and is given by similar limitations as the black hole mass is depleted. and an evaporatingfuture black infinity hole for the curve ( where the range ofsents the the parameter formation of the black hole singularity, while ( on with an exponentialthe onset black as hole the has been blackmodel depleted. hole and is The even endpoint formed requires of andemanating a Hawking turns from emission small off the is adjustment abrupt when black in indescription hole the the the when endpoint. mass form RST the of of remaining a blacka hole negative reminder mass energy that approaches shock our the wave semi-classical Planck calculation scale of and holographic serves as complexity will be subject to parametrization of this curve, which we employ at a later stage, is given by as encoding the boundary conditionswith the of fact the that energy the field momentum tensor will see a non-vanishing outgoing matter energy flux, i.e. Hawking radiation, which turns JHEP07(2020)173

(4.12) (4.11) (4.10)

= 0 t + is completely x 0 A

= . + Horizon Stretched x + 0 . , − > x − ), can be formulated as − A x dx + x 4.5 dx + + x ) dx − A x ( − + x + B WdW x x Z given by ( − 2 κ η , and after, . + 0 + A ) + x − ) in Kruskal coordinates, the semi-classical − . For future evaluation purposes, we note x + < x B x − − dx A 2 κ + + A + 0 x i . We have . dx dx – 18 – 6 6 = log( = log( . − A t + − x WdW σ σ + Z − The evaluation of the change of area I ) + A := x − ( B I − B 2 κ ), together with the field x + + i 2.41 = A A Matter Fields crit d = 2 C Ω = Ω Black Hole Singularity − i crit . Left panel: Penrose diagram depicting the life cycle of evaporating black hole formed by Event Horizon Horizon Ω = Ω Stretched The on-shell action ( To evaluate holographic complexity we againin consider two the cases: region the before WdW the patch shock anchored waveBefore arrives, incoming shockwave. analogous to the classical case, see figure units there. However, the error isto negligible as the long Planck as scale, the black i.e.first hole remains whenever place. large the compared With semi-classical these approximation ingredients,fashion can it as be is in now relied the possible on classical to in case, define the see the WdW figure patch in a similar In addition to theholographic ‘flat’ complexity reference acquires metric a area correction that the correctionKruskal term coordinates can to their also logarithm, be given an ‘area’ interpretation, by changing from Figure 5 collapse. Right panel:line the denotes corresponding a Kruskal curve diagram of equal with tortoise the time same coloras coding. usual, in The gray Kruskalvanishes coordinates. at With the this evaporation convention, end the point, area whereas of it the should stretched strictly horizon speaking be 1 in Planck JHEP07(2020)173 (4.13) (4.15) (4.16) (4.14)

x 0 = 0

+ = t x , 0 . Horizon ) Stretched − A , t < x S t − . x − log( S + t x d before the incoming shock- . − ˙ t C ) − A dt . dt . for for x + A ( x t + B 1 x − ), to give + t − M e 4.9 , but gives a non-negligible contribution κ log M κ M O 4 − + ) O + A ) and ( 4 κ → −∞ + x – 19 – log t t 0. − = 0) of the form 4.8 t t + log( 2 κ t < d Me M κ + 4 ) for

x 0 + A = 0 = B ≈ −

7 t = x + − A d ( A x x 2 κ − B d Horizon . x 3 1 + log Stretched 5 t e 2 κ − κ M is the scrambling time. A graphical representation of this tiny on-set x − + = log t x M κ B 4 O d Me 2 κ    = 0 of = log . Kruskal diagrams of evolution of a WdW patch of an evaporating black hole. Color ˙ C S t t < Combining the results leads to a total complexity growth Similarly, . κ where behaviour can be seen in figure wave The result is exponentially suppressedfor as times near the black hole creation ( Figure 6 coding coincides with figure which can be evaluated, making use of ( The result deviates fromto the classical calculation by a constant contribution proportional JHEP07(2020)173 S E t t (4.22) (4.23) (4.17) (4.20) (4.21) (4.18) (4.19) is related to u = 0. t . ) . − A x . − A t log( , − , dx d e B . ) κ u , M − A ) ), we can express the result as before the scrambling time 1 M κ x − A . The parameter 4 ( ! S x + A B 4.6 − O − t ( + B x t e u x κ M is linear after the scrambling time. + B dt + − 4 x t e t . A short time before the lifetime , but contributes near the black hole − κ M ˙ O C S 4 + + A t t κ M + x 4 − − log M κ e 4 u dt + 1 + before the scrambling time to provide linear + log − − 4 + κ M κ t ) , can be expressed as 4 + A A = 0. log S + A e with time − M κ t . The plot confirms the continuity and linear t 4 κ M x 2 dx κ – 20 – 4 = dt A 7 κ M even before the scrambling time. The final exact ) 4 − A 1 log( 1 u x − + 1 + log d ≈ − − log t − B ≈ − ) t M Replacing the spacetime boundary by the singularity ) d S e ) (1 t − 2 + A κ 2 κ + A 1 x M x − A & ( ( −

κ M x t − S gives the correct change of Kruskal area, = − S 4 ( is continuous at the black hole creation time ), using the parametrization ( x x u + 0 M M is continuous at A A dt 4.9 d d = B dt = 2 > x d A ) = 2 + t d ( x = log ˙ C ) and ( B d = 0, 4.8 t via t The total result is plotted in figure It is interesting to observe, that the contribution of That leaves us with the evaluation of the logarithmic area be negative. However, atPlanck scale, that and, time, as the stated mass above, of our the result is black not hole trustworthy has anymore. already attained the unreliable when the remaining mass is of order thefalloff Planck of scale. the growth rateof of the holographic black complexity holesubsequently has becomes been slightly reached, negative. thereason value This to of is believe, the that potentially complexity the problematic, growth complexity since growth rate rate there hits of is zero an no and evaporating black hole should ever This result holds until the black hole has fully evaporated, but, as stated above, it becomes which shows that also conspires with the non-linear contribution of growth up to correctionsresult of for order the complexity growth rate after the black hole creation is given by The result is suppressedcreation after time the scrambling time In addition to ( which, for times after the scrambling time Notably, the growth of theMoreover, Kruskal the area result for After incoming shockwave. curve in the region where corrections are suppressed aftertime a scrambling time JHEP07(2020)173 t M The E (4.26) (4.24) (4.25) t , using 7 t ] 21 = 100. . M/κ ). The Kruskal diagram for 4 κ (b) S t = 3.15 t log 4 κ ← ˙ 00 0 C − 200 , i.e. on curves satisfying M 4 κ 2 crit + M, M Mdt , = + − S is only sensitive to the location of the singularity = 2 – 21 – x − t A x + S A x + d x E − , goes rapidly to zero asymptotically. t φ e ) = − of holographic complexity as a function of tortoise time ˙ , x C + x Ω( = 10. , but the physics described by the semi-classical solution is somewhat 4 M/κ proportional to the ADM mass of the black hole. Since the semi-classical (a) is characteristic of the black hole size and therefore we will continue to refer not M S . The growth rate t ) is ˙ 0 C 20 Since the semi-classical Kruskal diagram is unchanged compared to the diagram of a 4.24 The infinite train of radiation does not lead to a catastrophic back-reaction on the geometry because M 7 2 the gravitational coupling, governed by to it as the ‘mass’ of the black hole. classical black hole, and the Kruskal area and not the detailed form of the dilaton field, it agrees with the classical calculation, hole, shown in figure different. In contrastin ( to all othersolution describes solutions a considered blackvanishing in hole energy this in density work, equilibrium inparameter the with the parameter a asymptotic heat region bath and at the infinity there ADM is mass non- diverges. infinity, with a temperature equalbath to the provides Hawking a temperature of steadyradiating the black incoming black hole. hole. energy The The flux solution, heat which in matches Kruskal coordinates, the is outgoing given flux by [ from the The spacetime curvature is singular where Ω = Ω describing a black holesingularities and of the white classical hole eternal black singularity.a hole semi-classical described eternal These by black are hole ( solution the is thus same identical to curves that as of a for classical eternal the black the CA prescription, forat evaporating the black creation time holesevaporated. of of the different black initial hole mass. is followed The by exponential a linear onset falloff4.2 period until the black Eternal hole black has hole We can also study a semi-classical eternal black hole by including a heat bath at spatial Figure 7 JHEP07(2020)173 M (5.1) ), has (4.27) = 2 0 , already S 4.23 S t = 0 in Kruskal coordinates and t . = 0, the time of black hole forma- 2 κ t ± t − = 0. It follows that the semi-classical M η ), ) that M, ) = 2 t ( 3.16 4.24 = 4 – 22 – ˙ with time after the scrambling time M C ˙ C ] where we numerically evaluated the semi-classical 10 ) = 2 t ( 0 S ] using a CV prescription. In fact, up to small corrections, 10 ]. In the context of semi-classical black holes, the finite black hole 18 ) immediately implies that 4.4 ) exhibits linear behaviour already from vanishes. B 4.23 4, so that κ/ First, it confirms linear falloff of In order to ensure a well-posed variational principle for the action on a Wheeler- We conclude that the complexity growth of the semi-classical eternal black hole for observed (numerically) in [ equation ( tion, in contrast to CVthe where scrambling time. the numerics The indicates linearentropy falloff an of is initial important, the adjustment as period evaporating itin black of captures order hole. the this time model evolution Classical of and blackrate the hole at entropy the is semi-classical given level by the black hole radiates mass at a constant terms needs to bebe addressed achieved in in a ordersemi-classical natural to way bulk have in theory a the tocomplexity RST well-defined growth the model CA rate by boundary prescription. during extendingseveral terms the a interesting This as evaporation symmetry features. can well. process, of the presented original The in formula final ( analytic result for DeWitt patch, it is necessaryboundary to terms are include not appropriate unique, boundaryblack something terms holes that is in in true the classical for action. Einsteinincrease CA gravity of in These the complexity general, [ ambiguity but does forlifetime a not range limits affect of the the late ability time to rate take of a late time limit and the ambiguity involving boundary complexity using a CV prescription forhere the provide same a model. much Theevaporates analytic compared more to CA detailed the results previous presented picture numericalthe CV of semi-classical evaluation. approximation how For can parameter be complexity values trusted, where starting evolves the from two as a approaches scrambling are the time in after black goodof the agreement, hole the black hole black is hole formed lifetime. and for most of the remainder 5 Discussion and outlook In this paper wein have a investigated the toy holographic modelfor complexity where of black the evaporating hole semi-classical black complexityexpressions holes geometry can for is the be known increase adapted explicitly.This in to extends complexity The our this over CA previous model the proposal work lifetime and in of we [ a have semi-classical obtained black analytic hole. late times agrees with the classical result ( and does not receive semi-classical corrections. for late times.then Further, equation it ( followscorrection from ( JHEP07(2020)173 ) (5.4) (5.5) (5.2) (5.3) If we 4.23 and a 8 crit φ = φ Both the Kruskal 9 ), describing an eter- over time. With this ˙ C ], it was shown that the 4.24 16 , , 15 4 κ , 2 κ log − 4 κ + κ M T, 2 κ 4 ) t − ( M. h S φ log = 2 will have minuscule values, which do not change 2 ∝ κ expansion for large initial black hole mass. – 23 – 2 κ ) for the static solution ( ) S B t + ( − ˙ h 5.3 C φ κ/M 2 M − ), even including the subleading logarithmic term. e = 2 5.2 S = 2 ) given by the semi-classical entropy. S t ( S ) for an eternal RST black hole and we once again find that the 4.27 ) holds with ) shows that the rate of complexity growth at the onset of black hole evaporation is the value of the dilaton field at the horizon. When evaluated for a dynamical 5.2 h ]. Interestingly, the corresponding cancellation also takes place in the rate of com- and the semi-classical correction 4.23 φ 15 A One may wonder how to interpret our formulas after the black hole has evaporated. Second, the subleading logarithmic term in the rate of complexity increase in ( Due to the slowThe evolution same of is the of logarithm, course this true remains for true a for WdW the patch bulk at of very the early black times, hole long lifetime. before the black hole is formed. 8 9 but having a vanishingly smalllarge absolute WdW complexity patch that action wascarried built at up in late during the times the outgoing does evaporationof train of not the the of reflect black WdW Hawking the hole patch radiation. and very actionway is For around continues a this indefinitely is classical and to black this onlyand hole, use issue define the the does the action growth not prescription absolute arise. to holographic calculate An complexity the obvious as change in the complexity integral of One can still defineis a one stretched unit larger horizonWdW than as patch zero. anchored the on timelike This itarea curve only curve covers is where a very the microscopic close spacetime transversewith region. to time. area the This boundary is at consistent with zero growth of the holographic complexity at late times, The cancellation of thereaction semi-classical on the corrections spacetime canflux geometry [ ultimately due be to tracedplexity the to increase matched the ( ingoing back- relation and ( outgoing radiation immediately after the black holeing is to formed ( and zero atis the consistent evaporation with endpoint. the relation Compar- instead ( evaluate the entropy formula ( nal black hole in equilibriumvalue, with a heat bath, we find that the entropy takes its classical can also be given anleading order interpretation quantum-corrected in entropy terms for a ofa semi-classical entropy. thermal black In heat hole [ bath, in equilibrium is with given by where solution of the RST modelgives describing a black hole formed by an incoming shock wave, this The Hawking temperature is independent of black hole mass in this model so the relation is seen to hold at leading order in a JHEP07(2020)173 ) at | A.1 h (A.1) (A.2) | = const. , that we − 0 y ∂M = − y is spacelike (timelike) ∂M ... + µ are degenerate in this case. Since j µ , µ n n | − h | dy ) does not make sense for null boundaries, + p dy A.1 ρ ∂M 2 Z e – 24 – − = and the determinant of the induced metric µ | = j g ], or the semi-classically corrected Jackiw-Teitelboim 2 | µ ds 26 |∇ , ) and consider a null boundary component, g | − 25 p and described by a null curve of the form , y Thus Stokes’ theorem also has to be valid for manifolds with + M y M Z 10 and the unit normal vector ], since black holes in those two-dimensional models have varying h 27 . . The result for piecewise smooth boundaries is obtained by summing over ∂M ∂M is a inwards (outwards) pointing unit normal vector if µ n We work in conformal gauge, The expression on the right hand side of ( The constant Hawking temperature of CGHS and RST black holes simplifies calcu- The need for a metric only arises if one wants to integrate scalar functions over a manifold in a 10 coordinate-invariant way. with light-like coordinates ( take to lie in the future of Stokes’ theorem in its general formto is a a metric statement involving on differential a forms,null it manifold. boundaries. is oblivious Our goalfor in null this boundary appendix components is ina to a limiting find two-dimensional procedure a context. where simpletimelike the This expression or null is to spacelike boundary easily curves. replace curve achieved using is ( approximated by a family of either Stokes’ theorem in thewith context spacelike of or Lorentzian timelike boundaries. manifoldscomponent is For simplicity, usually we focus presented on for aindividual single manifolds smooth boundary boundary components. The theorem states that where and the. . . indicatesrally contributions involve from the other metric boundary determinant the components. boundary The integrals natu- as the induced metric It is a pleasure to thankis Shira Chapman supported and by Nick the Poovuttikul for Icelandicby discussions. Research the Fund This under University research grants of 185371-051 Iceland and Research 195970-051, Fund. and A Stokes’ theorem in two dimensions with null boundaries the CGHS black holes, see(JT) e.g. model, [ see e.g.temperature. [ Acknowledgments prescription the complexity isdiscontinuously a adjusted monotonically to growing a function near-zero of value time at and the does endpointlations not of but get the is evaporation process. rather unphysical. It would be interesting to study the charged version of JHEP07(2020)173 ]. only ). (A.7) (A.8) (A.3) (A.4) (A.5) (A.6) . 0 M (2016) λ = const. A.6 SPIRE + 64 IN 7→ [ y λ ]. SPIRE ) is given by IN Fortsch. Phys. ][ , A.3 µ ]. . M j = const., is introduced, such arXiv:1810.11563 + M µ N j , , by a family of spacelike curves ∓ + y − SPIRE 0 , dλ k y dy − IN ) N + 0 ][ = ∂ Z y ∂M − Z √ N , − y hep-th/9306069 σ − + √ + [ . The general result, for the case of ρ y + N = e ( ∂ M µ X = j N ∈ The Stretched horizon and black hole 1 | µ √ + – 25 – lies in the past of lies in the future of h = n | | − − 0 µ y h N N = const., and not making reference to a particular | j p (1993) 3743 ρ µ arXiv:1403.5695 = ∓ p − 1 y ), the null curve is approached by a family of timelike e − |∇ 48 1 − − y g 0 ∂M | = ), which permits any use, distribution and reproduction in ( Z , y p n 0 + 0 = y 0. → M lim N Z (2016) 44] [ σ → are signs determined by 64 , tangential to the null boundary N µ σ Phys. Rev. D CC-BY 4.0 k , This article is distributed under the terms of the Creative Commons Three Lectures on Complexity and Black Holes Computational Complexity and Black Hole Horizons , and µ ∂ µ k 0, in the limit Addendum ibid. = [ λ > ∂ 24 complementarity Furthermore, a similar procedure can be applied for null curves defined by The outward-directed unit normal to the timelike curve in ( L. Susskind, L. Susskind, L. Susskind, L. Thorlacius and J. Uglum, [3] [1] [2] References In this form, the expression is manifestlyOpen invariant under Access. reparametrizations Attribution License ( any medium, provided the original author(s) and source are credited. where the sum runsdirected null over vector all piecewisethat smooth null boundary components. The future- and for null boundaries that liehaving in null the boundaries past of definedcoordinate by system, can be written as thus giving We could also have approximatedand the considered null the curve past directed normal vector. 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JHEP05(2020)091 Springer , May 20, 2020 April 22, 2020 April 29, 2020 : : : b Received Published Accepted [email protected] , Watse Sybesma Published for SISSA by b https://doi.org/10.1007/JHEP05(2020)091 [email protected] , Lukas Schneiderbauer, a,b . 3 b 2004.00598 The Authors. c 2D Gravity, Black Holes, AdS-CFT Correspondence

A Page curve for an evaporating black hole in asymptotically flat spacetime , [email protected] Instituut voor Theoretische Fysica, KUCelestijnenlaan Leuven, 200D, 3001 Leuven, Belgium Science Institute University ofDunhaga Iceland, 3, 107 Reykjavík, Iceland E-mail: [email protected] b a Open Access Article funded by SCOAP Hawking evaporation is balanced by an incoming energy flux. Keywords: ArXiv ePrint: Abstract: is computed by adapting thesolvable Quantum semi-classical Ryu-Takayanagi (QRT) two-dimensional proposal dilaton tobe gravity an theory. analytically one The third Page ofPage time curve the is is black also found obtained hole to for lifetime, a at semi-classical eternal leading black order hole, where in energy semi-classical loss corrections. due to A Friðrik Freyr Gautason, and Lárus Thorlacius Page curve for an evaporating black hole JHEP05(2020)091 ]. 3 ]. Early on, the entanglement 2 , 1 ]. A version of the Page curve can 11 The result hinges on the use of the Quantum – 1 – 1 ]. 18 5 , ] and the existence of extremal hypersurfaces termi- 4 16 10 – 8 7 15 3 6 12 11 1 12 5 19 ] for further details on such a setup. 6 5.2.2 No-island configuration 5.2.1 Island configuration In a recent breakthrough, a Page curve was computed using semi-classical methods by See e.g. [ 5.1 Eternal black5.2 hole Dynamical black hole 3.1 Coupling to3.2 matter Semi-classical black holes 1 important step towards resolving Hawking’s black hole information paradoxstudying [ black holes informal asymptotically field anti-de theory Sitter (CFT) (AdS)Ryu-Takayanagi reservoir (QRT) spacetimes formula [ coupled [ tonating a on con- so-called islandsalso behind the be event obtained horizon for [ eternal AdS black holes, but in this case the islands extend outside grained thermal entropy ofchanges the when radiation the coarse thattropy grained has associated entropy been to of the emittedlimited the remaining up radiation by black to the exceeds hole, that the black atwhen point. hole coarse which the point grained entropy entanglement This the and en- entropy entanglementthe transitions becomes entropy from Page a is increasing time. decreasing to function decreasing Reproducing of is the time. referred Page to The curve as without time explicitly assuming unitarity is an 1 Introduction If black hole evaporationgoing is radiation a and unitary the process,to quantum the follow state entanglement the associated entropy so-called between to Pageentropy the the is curve out- remaining then as black a a hole monotonically function is increasing of expected function time of [ time which closely tracks the coarse 6 Discussion 4 Generalized entropy 5 Page curves 2 Page curve from QRT 3 The model Contents 1 Introduction JHEP05(2020)091 ], 15 lifetime t ] for a discussion of islands in , but rather than working with a 13

1 oisland no island c ]. More specifically, the dilaton gravity 14 ], but see [ – 2 – 11 , 5 Page t . Page curve for an evaporating RST black hole. Figure 1 ]. Explicit computations have for the most part been restricted to two- 0 12 0 init init ] in the context of two-dimensional AdS gravity, and use a three-dimensional grav- S , where the entanglement entropy of the outgoing Hawking radiation that has passed S 2 11 1 We work with a two-dimensional dilaton gravity model of the type introduced by Callan, In the present paper we demonstrate that the QRT formula can also be applied in the future null infinity.bath This is is essential in for contrast letting to the the radiation escape. AdS setup, Our where computation the in coupling asymptotically flat to a heat ter is holographic. Thiset allows al. us [ to take advantageitational of dual an description insight to put calculateto forward the the by generalized contribution Almheiri entropy of in the thefrom two-dimensional QRT collapsing bulk formula. matter CFT In this matteranalytically. model, into the The vacuum formation Hawking of and radiation a its black emitted hole subsequent in evaporation this can process be naturally propagates studied towards Giddings, Harvey, and Stromingersector (CGHS) is in that [ ofwhich the remains analytically model solvable at introduced the by semi-classicala level. Russo, two-dimensional For CFT Susskind, the matter with and sector, alarge Thorlacius we take large number (RST) of central in free charge scalar [ fields as in the CGHS-model, we assume that the conformal mat- context of an evaporatingclassical black order hole in the intime model asymptotically is that flat found we spacetime. to use,figure be and At for one leading a third semi- largebeyond of initial a the black distant black hole spatial holereference mass, reference lifetime. point. the point This Page is main plotted result as is a presented function in of time registered at the the horizon [ dimensional Jackiw-Teitelboim gravity [ higher dimensional AdS black hole spacetimes. JHEP05(2020)091 , ]. rad , at , or . In 24 (2.1) ¯ ] and H A ) takes CFT 11 5.2 = , H 2.1 ¯ 6 and A , 4 H A is dominated by ) above is taken ], which uses the and , where Hawking 5 2.1 gen , rad 4 S H CFT in ( H A = A ] and the study of chaos [ H . 23 ] . AI which lies just inside the horizon, I S [ I ∪ ). At early times the minimum value A Bulk 2.1 we review the main result of [ S 2 + ) , for which the generalized entropy ( I A – 3 – ( N and the generalized entropy G and for a dynamical black hole in section 4 ∅ Area = 5.1 = I ] and the effect of a quench protocol on the Page curve 21 gen S we derive the QRT formula for our model. Results for eternal ]. Furthermore, the Page curve has been computed from bound- , homologous to 4 I 20 of the spatial manifold on which the CFT in question is defined. – A so that the dual geometry is an asymptotically AdS black hole. The 16 . The second term is the von Neumann entropy of bulk quantum fields T ]. Outside the direct scope of black hole physics, Page curves are found to A 22 we introduce the two-dimensional dilaton gravity model and establish notation. 3 ] the system consists of a standard holographic CFT, with Hilbert space we present some conclusions and outlook. 4 denotes a codimension two region that penetrates into the dual bulk spacetime and 6 I In [ The paper is organised as follows. In section The QRT prescription can be motivated via a replica trick involving Euclidean worm- the von Neumann entropy ofEventually the another Hawking radiation extremum, which involving grows an monotonically with “island” time. to be the entireand one boundary looks where for the regions extremal CFT values. is defined, Thein i.e. QRT this prescription case for theis entanglement entanglement then entropy entropy between given between the bycorresponds black the to hole smallest the and “empty” extremal the surface value Hawking of radiation, ( is homologous to with support inside a spacelike region bounded by finite temperature CFT is assumed toradiation be emanating coupled from the to black an hole auxiliary is system, collected. denoted The by region Our goal is towill compute do Page the curves computation forspacetime both black for and holes an for in eternal ainto asymptotically black dynamical a hole flat black linear in spacetime. dilaton hole an vacuum.quantum formed We asymptotically Ryu-Takayanagi We linear by (QRT) follow formula dilaton the the gravitational holographic collapse approach of of [ matter The first term onto the a right given hand subregion sideHere is the standard Ryu-Takayanagi entropy associated in section Following that, in section black holes are presentedsection in section 2 Page curve from QRT holes, as explored in [ ary conformal field theorieswas in studied [ in [ be connected to the eigenstate thermalization hypothesis [ argue for the validity of the QRT formula in our asymptotically flat spacetime model. Then spacetime thus gives rise toconceptual a rather point clean of physical picture, view,asymptotically both AdS and from spacetimes. a offers computational evidence and that the QRT prescription applies beyond JHEP05(2020)091 (2.2) (2.3) ) and is 2.1 ] for details). ]. In this paper, 26 ]. However, this 11 [ . To simplify this 27 I I . Hence, we do not need 2 ]. The dual field theory in this 25 , ) , BH 3 S ,S × rad φ S ( is dominated by the area term in ( R ) for any given trial island itself as the island. – 4 – I × duality to compute the von Neumann entropy of 2.1 min gen 5 , 2 S 1 = R gen /CFT S 3 . gen S supersymmetric Yang-Mills theory in six dimensions, which does not ] and use AdS 1) 11 , -exact solution of heterotic string theory [ 0 α For this latter extremum, . = (1 2 denotes the direction along which the string theory dilaton is linear. This back- ]. The dilaton gravity models studied in the present paper are in fact closely 4 φ 26 N R The holographic dictionary for such linear dilaton backgrounds works in a similar way In this paper we study black hole geometries in 1+1 dimensional dilaton gravity, which More precisely, the island refers to the internal entanglement wedge defined by 2 we will often neglect this distinction and refer to that it follows a Pageremain curve stationary as with a respect function to the ofis black time hole. the experienced evaluation The by challenging of asymptotic aspecttask, the observers of we who the second computation follow term [ inthe ( bulk fields usingin a section standard Ryu-Takayanagi prescription. We will come back to this to artificially split ourwhere system into the a Hawking QFT quantataken dual care are to of the collected in black a ascontaining natural hole in way the plus by black an an the hole anchor auxiliary AdSWe curve and system background. will dividing an the compute “outside” system the Instead into region entanglementthe an the entropy containing anchor “inside” between curve split outgoing and the part all Hawking is radiation that radiation. that remains inside, has including passed the black through hole itself, and see explicitly as in standard AdS/CFT,dilaton except region the instead dual ofIn variables the our are AdS computation region defined we in inoutside will conventional the the holography black place asymptotic hole. (see a linear [ Inblack timelike hole the “anchor will gravitational pass theory curve” through the at the Hawking anchor a radiation curve emitted as fixed from depicted radial in the figure position far case is flow to a conventional QFTtheory in [ the UV butrelated rather to the to above a fivebranewill non-local background, not theory as called play explained little an forspacetimes string example important can in role serve [ in as our holographic backgrounds discussion for beyond a exemplifying class that of linear non-conformal dilaton theories. some non-conformal theories.NSNS A fivebranes, prominent which example is is a given spacetime by of the the form near-horizon limit of where ground is an and a Page timesmallest extremal defined value as of the time when theare asymptotically two flat with extrema an asymptotically tradeare linear places dilaton familiar field. providing from Linear the dilaton constructions spacetimes in string theory where they arise as holographic duals to takes over. therefore given approximately byend the result Bekenstein-Hawking is entropy the of Page the curve, black hole. The JHEP05(2020)091 λ = 1 (3.3) (3.1) (3.2) λ ,

2 λ + 4 2 ], ) σ , , φ 14 − − ∇ . The two-dimensional model can 0 x φ d + −∞ + 4( = x d R Curve ) Anchor − φ is proportional to the area of the transverse , φ ,x 2 + − φ − x 2 e ( – 5 – σ tends to ρ − d g 2 e e σ + − σ − √ d x = − 2 is a characteristic length scale that can be set to 2 d s = λ d 2 Z s π d 1 2 . The strength of the gravitational coupling is controlled by ) Island = − σ − grav + I σ ( 1 2 = σ is an arbitrary constant that can be absorbed by a constant shift of the spatial . Penrose diagram of an evaporating RST black hole formed from collapsing matter is the dilaton field and 0 φ φ We will find it useful to employ so-called Kruskal coordinates, for which the metric in the dilaton field andbe becomes viewed large as a as is spherical inherited reduction from of the a2-sphere parent four-dimensional in theory four-dimensional theory. and Planck In units. this case, the scale conformal gauge takes the form We start with the classical CGHS dilaton gravity action [ spacetime with a linear dilaton profile, where coordinate Figure 2 (green). A timelike anchor curveevolves separates along the spacetime this into curve, interiorfuture and more null exterior infinity. and regions. The more As island time Hawking moves radiation with time has along passed the through purple3 it curve inside on the its event way The horizon. to model where by a rescaling of the two-dimensional coordinates. The vacuum solution is given by flat JHEP05(2020)091 (3.6) (3.7) (3.8) (3.9) (3.4) (3.5) . The ), writ- 3.2 = 0 ) is that while vu , a rescaling of 0 3.9 M > coordinates, is given by . The temperature of a λ ± . x . = 0 φ M 2 so that semi-classical corrections λ π − 2 e 24 , 2 − = , ∂ − x . In this coordinate system the equations u , = M + vu φ . d N φ x 2 v − π π = λ we get back the vacuum solution ( = 2 d − − 2 λM 1 e ρ , gives the well-known two-dimensional “cigar” . 2 + − = = M N – 6 – ∂ = 0 and a bifurcate event horizon at = G Mu T = gravitational dual. This is an important technical 4 the solution exhibits a naked singularity analogous M 2 ], Horizon √ φ / M 3

s 2 0 φ d = 1 − = 2 29 + 1 = , e − − φ Area 28 2 ], the dilaton gravity sector is coupled to matter in the x vu M < − = ), and this must be taken into account when evaluating the 14 e = 2e S − ) reduce to and 3.1 S ∂ + 3.1 ∂ Mv is proportional to the black hole mass, √ that has an AdS M c = + x minimally coupled free scalars, with an integration constant. For N M large central charge assumption which allows us to simply evaluate the von Neumann entropy of the CFT fields 3.1 Coupling toIn matter the original CGHSform model of [ are dominated by one-loopa effects strongly due coupled to matter the sector matter described fields. by Here a we holographic will two-dimensional instead CFT assume with A purely two-dimensional argument leading tothe the area dilaton dependence of in thedilaton ( as horizon is is apparent unity, fromBekenstein-Hawking the ( entropy gravitational coupling constant is controlled by the CGHS black hole is independent of its mass, The Bekenstein-Hawking entropy, given by one quarterthe of original the four horizon area dimensional in theory, Planck canat units be the in expressed horizon, in terms of the dilaton field evaluated to the negative mass Schwarzschildthe solution coordinates, in four dimensions. For with a curvature singularityintegration constant at where we have temporarily restored the characteristic mass scale In the absence of mattersolution fields, to the the dilaton above gravity field theory equations, is up non-dynamical to and constant shifts the of general the where ten in Kruskal coordinates. For solution in Lorentzian signature [ with the conformal factor equal to theof dilaton motion obtained from ( JHEP05(2020)091 , ) − x ] by a ( (3.11) (3.12) (3.13) (3.14) (3.15) (3.10) 14 −− T . 12 and c/ ) = + κ x large compared to corresponding to a ( . + M and ) ++ , x , but to get started it c T + x µν , = ( T , ) N ++ + = ±± T x i ( T 1 2 ), G = + 1 + φ ) = 3.3 2 2 + − , ) − x x . e φ ( 0 µν 2 1 ± . ∇ + δS ∂ ( δg x 2 3 g (3) − − π − L − 4 G 3 φ ) √ M 2 c ,G + , – 7 – − ) 48 x = ( + = µν c x ≡ F = 1 ] we make the identification g ( µν φ = T − 2 15 , ++ ρ − φ ] except we are dealing with Lorentzian CFT. In particular, the T 2 e 14 ν limit where semi-classical effects are turned off. In this − + − 30 ∇ x ∂ 0 µ 1 2 4 + = e ∂ ∇ → φ h below, we can arrange our computation of the von Neumann 2 − φ ) = − 2 + e − 5.2 x ( 4e 0 F 3 spacetime. We are interested in semi-classical black holes with initial mass Through the holographic dictionary, the two-dimensional central charge is related to 3 When comparing to blackOur holes conventions in match [ those of [ 3 4 and the normalization forfactor of the 2. energy-momentum tensor therefore differs from the one used in [ thin shell of infalling matter energy incident on the linear dilaton vacuum. For our purposes, classical energy-momentum tensor is defined as where We take the energy-momentum tensor to have compact support in The CFT energy-momentum tensor has twoeach non-trivial components of which only dependsa particularly on simple one form of in the the light Kruskal cone coordinates coordinates. ( The field equations take and the response to arbitrary incoming matter energy flux is easily obtained, is useful to considerlimit, the the gravitational fieldtwo-dimensional matter equation CFT, is sourced by the energy-momentum tensor of the while the equation of motion of the dilaton field is unaffected by the coupling to matter. three-dimensional gravitational quantities via the Brown-Henneaux formula, AdS the scale set by the centralparameter charge given of by the matter CFT, for which there is a natural expansion In most of what follows we work to leading non-trivial order in on a spacelike section butstill does has not solutions affect the describingmomentum. gravitational dynamical sector. black In holes particular, formed the from theory incoming matter energy- As discussed in section entropy of the matter fields in such a way that we only have to deal with pure gravity in JHEP05(2020)091 (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.16) (3.17) , . + 0 + 0 , ± . , < x > x t 1 1 + + − x x 2 ) if if v > v < ρ if if ± ∂ region outside the shock wave, + 0 . 2( M x , while a set of coordinates, for 1 , − − σ + − u , ρ + 1)) − ω v > − e 2 ± + 0 u ) = 0 − . x ∂ ( M x e − − ) 2 v − 1 − + = , ω R, − − σ x − vu c = + x 12 − c 12 − (1 + ω + − = ( x = ( + ρ = i − M σ – 8 – i = ( µ v , x    µ ) 1 2 ±± , u + 0 T T = + , u v,u h h x = ) is not needed, only the behaviour at early and late ( ω ) t φ + e − 2 = σ ,x e − 3.14 = + ρ , e + are sensitive to the choice of coordinate system. This is x x v = − ( 1 ∂ ρ ± M v 2 t + − ∂ = c 6 ) = e − v,u ] leading to the following expressions in conformal coordinates, ) ( = − ρ 31 2 i ,x − + − e x + ( φ T 2 h − are functions of integration determined by physical boundary conditions that e linear dilaton region, the change of coordinates, ) 1 ± ) as input [ x is the Ricci scalar of the background metric. In two spacetime dimensions the ( ± R v < t 3.22 The boundary functions to be expected since notions of positive frequency and normal ordering depend on the only ( where reflect the matter quantum state. On a curved spacetime background,longer the traceless energy-momentum due tensor to of the the matter conformal CFT anomaly, is no where continuity equation expressing energy-momentum conservation can be integrated using Time as measured by asymptotic observers at rest with respect to the black hole is 3.2 Semi-classical black holes brings the metric intowhich manifestly the flat metric is form, asymptotically Minkowskianis in given the by brings the metric and dilaton into the following simple form, In the the detailed form of thetimes, solution and ( we canare therefore patched consider together an across idealised an solution infinitely where thin two null static shock configurations wave, A rescaling of the coordinates, JHEP05(2020)091 ), 3.28 (3.29) (3.27) (3.28) (3.24) (3.25) (3.26) ). At ) a non- . Upon at leading 3.18 3.1 / . ), we have a , ) = 0 1 in ( 2 2 − ) ) 0 = 1 00 ω y ( y grav ( → ∞ ( − v < I t 3 2 + . σ − lifetime t 0 2 should be dominant com- 000 . ) y . y Q − ) is only valid on timescales } − − I σ y x = ± 1 d d } ], 3.14 , x ρ , + + ± − then 14 y x (1 + e y y, x d d { ρ∂ { 24 ), in which the metric is manifestly − gφR , + ). ∂ 1 − log , ) + − 3.20 1 2 } √ ± c x 3.23 x x ± ) and it follows that d c ( 2 − 24 + d in ( ± , x ) ] and add the following term, t x ± − ± 3.19 related to the old one via d Z t y ) = 15 – 9 – ) { , x − π ) = Z ± + c c σ y ± 12 If we take x ( 48 π ( y ( c − ( 5 ± ρ − t 12 t = ± ) t c − 12 ± 2 ) for the energy-momentum tensor in the new coordinates, ) = x = − RST ( − is not immediately apparent in the conformal gauge expression ( I Q ± coordinate system ( ± , y Q 3.23 I ±± I ) = ) + T dy dx − y − ( σ ρ ( , σ ) = ) this leads to the usual anomalous transformation of the energy- + ± −− σ y coordinate system in ( ( T ( 3.23 ) − ±± T , ω 2 + , the conformal factor of the metric transforms as ) ω ( ± ± ± x ( dy dx ± ) has a small but finite value, the classical solution ( y ] this expression was interpreted as the energy flux of Hawking radiation from the 14 While the non-local nature of 3.11 → 5 ± it enters the formalism via the boundary functions pared to semi-classical contributions fromto the the dilaton theory gravity sector. areof Further needed modifications motion. in order We will to follow find the analytic approach solutions of to [ the semi-classical equations order. The semi-classical back-reaction onmatter the spacetime can, geometry however, due be to accounted Hawkinglocal for emission by Polyakov adding term to induced the by classical matter action quantum effects [ written here in conformal coordinates. In [ black hole as observed by anhole asymptotic emitting observer. Hawking radiation Energy loses conservation mass.in implies ( When that the a semi-classical black expansion parameter that are short compared to the lifetime of the black hole, which is linear dilaton vacuum andflat vanishing in energy-momentum tensor. the transforming The to metric the is manifestly asymptotically flat, one finds non-vanishing outgoing energy flux at we effectively obtain a new function As an example, considerearly a advanced time black before hole formed the by arrival gravitational of collapse the as collapsing in matter ( (i.e. choice of time variable. Underx a conformal reparametrization of the light-cone coordinates, When inserted in ( momentum tensor involving a Schwarzian derivative, In order to preserve the form ( JHEP05(2020)091 and c (3.35) (3.31) (3.32) (3.33) (3.34) (3.30) ). The solution , , where it takes 1 ) ), where the field as expected. 3.32 v > 3.3 ]. Here we are mainly Mvu 15 − ) = 0 involves a factor of ± σ log( ( , ± , RST ± t . t ) I − Q , − c I 24 , x − the gravitational coupling becomes σ + + 1) x crit − crit e u − Ω = φ , even if this is the vacuum solution but ( Ω − 2 ± v c 24 ∂ ) is that semi-classical solutions of the full ). Here we find that in Kruskal coordinates = ) + Ω log( → − − 6= 0 + 1 vu c Ω 3.6 ) 48 φ 3.29 2 , ± − − M − – 10 – x ( (1 , u − ± x t + M renders the functions Ω = σ + ]. This has a suggestive physical interpretation, where Ω = e e x ± 15 − σ = , and when Ω + 1 = 0 Ω = u , v − d c ∂ 48 ). v Ω = + d ∂ φ log . This corresponds to a flat space energy-momentum tensor 3.30 can be expressed in Kruskal coordinates ( 2 1 2 e − 1 M RST − I ) = ) limit of the theory. The RST term c 48 0 ± = but if we transform to coordinates for which the metric is manifestly + which is the temperature of the eternal black hole. The outgoing energy σ 2 = ( which is exactly the energy-momentum tensor of a thermal gas of tem- Q s → π I 1 ± d 2 related via ( t c crit 24 + Ω Ω = ) = 0 represents a boundary of spacetime, analogous to the boundary at the origin of ra- ± ≥ T the same boundary functions as before. The linear dilaton vacuum remains an x grav ) = ( I and Ω crit ± ± ± t t Ω φ σ Finally, consider the formation and subsequent evaporation of a dynamical black hole. A two-sided eternal black hole solution is given by One benefit of including the RST term ( The resulting semi-classical field equations simplify dramatically when a new field vari- ( → ±± with As in the classicalof case matter without energy back-reaction, is we injecteddescribing imagine into the a a full situation linear evolution where dilaton ofinterested a vacuum in such described short a the by burst geometry black ( outside holethe the can form collapsing be matter found shell, in i.e. [ for we find that T perature flux carried by the Hawkingat radiation the is same matched temperature by as an the incoming Hawking flux temperature of thermal of radiation the black hole. where we have rescaled the coordinatesthat as in ( asymptotically flat, exact solution of the semi-classical equations and takes the form in the new fieldthis variable. is Notice because that themanifestly metric flat is coordinate not system manifestly flat in Kruskal coordinates. Transforming to a able is introduced, dial coordinates in the higher-dimensional theory from which the CGHS model istheory descended. equations reduce to with to the semi-classical action,the which classical is ( allowedtherefore by enters general at covariance the and same does order as not the disturb Polyakov term but there are subtletiesbelow, involved. Instrong particular, in the the semi-classical theory new [ Ω field variable is bounded from JHEP05(2020)091 so AI and S does (4.5) (4.3) (4.4) (4.1) (4.2) M as the . This A 3 ) I ) vanish. Bulk ( φ S A 2 ) a curve of . Then both − 3.23 on a timelike ρ e 2 2 e A 3.33 at one end and a Ω = Ω A ) yields 3.9 curve with , Ω ] , . AI . the drift is extremely slow and can − , S crit y [ in the manifestly asymptotically flat d − Ω M (3) y + A − d Bulk y G ) is universal and is the main focus of this σ ) d + S 4 is instead bounded by Length I y = 4.1 A − + d ρ Ω( Ω ' 2 ) σ AI 2 ] z I e S and at the other end by a point ( d N – 11 – − AI ). The spatial location of the anchor curve drifts I are zero. The second step is to perform a Weyl G = 2 S 2 3 2 [ 4 at the other. = ) ) z L Area I 2 ± 3.35 ( s N y crit Bulk ) but for = ( = d G S 2 3 ± 4 t s Area gen 3.20 d S Ω = Ω term in ( is chosen as long as it is a spacelike surface that connects log ). For an evaporating black hole, the corresponding statement is no AI are embedded in the dual three-dimensional spacetime, S I 3.34 ], we compute the von Neumann entropy holographically by passing to and 11 spacetime. In Poincare coordinates the metric is A 3 is at a fixed spatial coordinate, Ω and the gravitational contribution the energy-momentum tensor in ( ) ± Following [ The second term on the right hand side in ( y ( . In the absence of an island, the surface ± where the integration functions rescaling of the two-dimensional metrict that strips off theIn conformal this factor case, the matteris CFT empty is AdS in a vacuum state and the dual three-dimensional geometry points where The calculation is simplified ifcan we be arrange achieved the in embedding two geometry steps. to The be first pure step AdS is to identify a set of light-cone coordinates be ignored on timenot scales depend of on order which theI black hole lifetime. Thepoint final on answer the for boundary curve a three-dimensional gravitational theory and evaluating the geodesic length between the anchor curve. We takethat the it anchor is curve located toconstant be well outside a constant the blackcoordinate system hole. ( For anlonger eternal exact black due to holein the ( the asymptotic coordinates ( this expression, which gives zero in the absence of an island, is given by section. It is the vonthat Neumann is entropy bounded of at the one CFT end matter by fields the on island a spacelike surface 4 Generalized entropy In order to derive afor Page the curve for generalized these entropy, semi-classical black holes, we adapt the expression to the two-dimensional settingthe at area of hand. theabsence transverse The of two-sphere an first evaluated island. locally term Comparingarea at on with contribution an of the the island, an black right and hole island hand entropy in gives in the zero side ( classical in involves limit. the The natural semi-classical extension of JHEP05(2020)091 (5.1) (4.6) (4.7) in the is a UV I δ and A , . It was also noted that is the appropriate set of π 1 ) =0 2 ± t

= v, u i ( ) T I ( ρ e ) A , ( , ρ ) e − c 2 24 ,y ) ). The relevant matter CFT vacuum state + . We have dropped the UV cutoff from the y ( ) = 4.7 ρ A, I ( ± geodesics then leads to the following result for − d e σ – 12 – 3 in ( h ( ) = 0 δ ± = . Then each anchor point has a mirror anchor point y ±± log ( T Bulk z 3 c 6 ± S t ] = and the subscript is a reminder that the formula should be AI S [ − y d Bulk + S y d and the thermal gas must be at the same temperature. As was noted − for an eternal black hole and therefore π 1 2 = = is the two-dimensional distance measured between the points 2 flat ) = 0 T s ) u ), the energy flux outside an eternal semi-classical RST black hole is d ( − t A, I ( 3.33 d ) = The eternal black hole is two sided and we place a timelike anchor curve in each A standard calculation involving AdS v ( + asymptotic region. For simplicity,the we anchor assume curves, that as our shown anchor in figure points lie symmetrically on momentum tensor of at thermal CFT at acoordinates temperature to of use when evaluating is the Hartle-Hawking stateto where time positive in frequency Kruskal coordinates modes rather are than determined asymptotic with Minkowski time. respect temperature below ( when evaluated in manifestly asymptotically flat coordinates and this is precisely the energy- flat spacetime which turns out to be considerably5.1 simpler than the Eternal evaporating black case. hole A semi-classical eternal black hole in asymptoticallygas flat of spacetime incoming is supported radiation by that ato maintains thermal the Hawking mass radiation. of the black The hole against two-dimensional the black energy loss holes studied in this paper all have 5 Page curves We now have everything inthe place QRT prescription. to calculate Our primary generalizedhole goal entropy that is in has to the a obtain finite RST the lifetime Pageclassical model but curve eternal using of first black hole. an we carry evaporating This black out provides the a corresponding first test calculation involving for a a black semi- hole in asymptotically that depends on dynamical inputcutoff from parameter. the two-dimensional matter theory. Here the holographic entropy, where flat metric evaluated in coordinates forformula which as it just contributes an additive constant. and the geodesics areformation semi-circles centered in on step the twodimensions holographic above which boundary. can maps The be the Weyl implemented regulated trans- as holographic boundary a to coordinate a transformation surface, in three JHEP05(2020)091 ) 3 4.7 (5.5) (5.2) (5.3) (5.4) , i ) m A ,u m A v ( ρ 2 e ) , is empty. The von Neumann A , and the anchor curves to be A I 1 ,u t . A c 3 v σ ( ρ ≈ + coordinates via, . With these assumptions in place 2 t e ) M . 2 − c 4 M ) 2 e ) 48 ) , A A − vu v, u m u 3 u ( = 1 A A − = u v A − 1 Ω A − − v ≈ A ( (1 ) – 13 – u ( , u 2 v,u ( ) log σ ρ + m 2 t c e A 12 e v = = − v are related to the A v ) bulk ( S h ) in the other exterior region, which is related to the original , as indicated in figure t, σ m . We also take the black hole mass to be large compared to the ( m ) log A ], we now perform two calculations: one with no islands, and one c 12 u, v 11 , = 4 = ( denotes an anchor point on the curve on the right in the figure. The anchor m bulk ) ) S A replaced by v, u , u is asymptotic time, measured by an observer on the anchor curve, and the asymp- I ( . A Penrose diagram of an eternal black hole. A pair of timelike anchor curves (blue A v A ( t Inspired by [ The bulk entropy then takes a simple form, where totically flat coordinates where curves are assumed to bewell located approximated by well its outside classical the value, black hole where the conformal factor is point by scale set by the matter central charge, so that with a single islandarea on term of each the side. generalizedentropy entropy Consider of is the first by bulk definition the zero, fieldsthat no-island as is connects scenario. non-vanishing the and two In given mirroredbut by anchor this with the points. case, length This the of means the that geodesic we in can directly AdS apply ( Figure 3 curves) separates the spacetimeintersect into the an anchor interior curves anddominated at two by different exteriors. the times. area The On term two associated the spatial to late hypersurfaces the time islands surface denoted the(denoted by generalized by purple entropy dots. superscript is located in the linearthe dilaton semi-classical field region, variables so areregions well that of approximated by interest their and classical our counterparts calculations in simplify. all JHEP05(2020)091 ) ) ) A σ and 5.7 5.3 2 (5.7) (5.8) (5.9) (5.6) ) − (5.10) . I e 4 ] for the , u or I 32 v A t 2 − = ( the generalized e I that connect the . Both the anchor 1 4 3 , , we obtain 1 ) crit 2 + ) gives I ) Ω I u A I 5.6 u − v ) − I > σ − A Ω( into ( u A term accounts for the time it ( t )(1 I 2 A ) A u I u σ . v A .... v A , − and 4 u + − . A I M c v v A ( A (1 σ 12 ≈ − M Ω c 3 I = log r + c 6 1 4 4 , v ) shows that for – 14 – M ≈ A = A ) + . In this case, the area term in the generalized gravity, the island and its mirror were found to be 5.4 v I u I = 4 2 3 u d I = Page v ), as promised. For another perspective on the location t I I − v u and working to leading order in island gen 5.6 ) S (1 I , or both. Our computation includes, by construction, the M ]. This remains true here as well. The island saddle point ( , u ) is precisely twice the rate that was obtained in [ I v 12 ( 5.4 = 4 ) for the semi-classical area function 3.33 island gen S , as indicated in figure ) I ) is non-vanishing and bulk term involves geodesics in AdS , v 4.1 I u For a two-sided black hole in AdS We now repeat the calculation with symmetrically placed islands at = ( m tion for the conformal factorof in the ( island, consider an observer sitting on the anchor curve who sends an ingoing light is outside thebetween event island horizon and but event horizon inside given the by a stretched tiny horizon, number, with the proper distance The fact that the island is close to the event horizon justifies using the classical approxima- for this, we obtain for the Page time of an eternal RST blackoutside hole. the event The horizon Page [ curve is drawn in figure Inserting the leading order saddle point values for Comparing to the no-islandentropy result is dominated in by ( takes the for the island Hawking radiation configuration. to travel from The the black hole to the anchor curve. Correcting island. Extremizing over This is a saddlefind point all and extrema and not select a the minimum. one that However, gives the the QRT lowest prescription value instructs for the us generalized entropy. entanglement entropy of radiation emitted to one side. I entropy ( anchor point and correspondingfrom island the on two sides each of side the of black the hole black are identical hole. and add The up contributions to where we have used ( point and the island arecan assumed be to used lie for inoutside a the a region conformal large where factor. mass the classical black This approximation hole is ( and automatically we satisfied will for check an ex anchor post point facto that it also holds for the Corrections to this result areor either exponentially subleading suppressed in (by factors powersentropy of of of the radiationentropy growth emitted rate on in both ( sides of the black hole and we note that the JHEP05(2020)091 ) ) ]. 12 4.7 3.19 (5.11) the time 2 system in ( . The graph plots ) /c − , the signal must be A BH ) we are instructed to t , ω S + 4.7 ω ( = 6 Page t , s t , such that + obs A t A island σ contribution to the two-dimensional matter = 2 ± t – 15 – obs A t − A t

Page

t no-island is the scrambling time. This is in line with a similar result in [ , that corresponds to an anchor point at time explicitly depends on the location of the anchor curve. ) ) I 1 4 obs A , u t 0 I v − ( as a function of retarded time on the anchor curve. A t 0 = log ( . Page curve for the eternal RST black hole with BH A BH σ s S S t c 3 2 − In the holographic evaluation of the bulk entropy term in ( gen when calculating the generalizedwhere entropy the on CFT a is trial manifestly surface in in its the vacuum linear state dilaton and vacuum the three-dimensional holographic semi-classical black hole solution but the remaining bulk termidentify requires light-cone more coordinates work. whereenergy the momentum tensorwhere is the zero. metric is manifestlycollapses The flat in to correct the form initial choice the linear black is dilaton hole. region the before These the coordinates matter are shell suitable for the evaluation of ( where 5.2 Dynamical black hole We now turn our attentionhole to that dynamical is black formed holes by and gravitationalHawking collapse compute emission. of a matter The Page and steps curve then inof for gradually the a the evaporates calculation due black generalized are to entropy theminimum with same value. and as The without before, area i.e. an term to island in find the and extrema generalized determine entropy which can one be gives read off the directly from the S signal to the island.an island A at straightforward calculationemitted shows from that the anchor in curve order at to an earlier be time received at However, because our black holedifference is in asymptotically flat spacetime and not AdS Figure 4 JHEP05(2020)091 2 (5.12) /CFT 3 , # )) I u I u (1 + I I v v . The final result is the − 5 (1 )) ], we take the incoming matter A u 32 A u ) (1 + A I A v u v I − Mv (1 − 2 – 16 – I A log ( u u − log ) I I A ) to calculate the bulk term in the generalized entropy, u v v 4.7 ], the von Neumann entropy of such a state is identical to log (1 + 32 I v " − log 1 c 12 coordinates. M ) . Of course, a dynamical black hole is not the linear dilaton vacuum and + − 3 = 2 , ω + ω ( island gen . Penrose diagram of a dynamical RST black hole with two spacelike hypersurfaces S is non-vanishing due to the incoming energy flux that forms the black hole. There in ) ) + The generalized entropy is to be computed for the two competing configurations, with ω ( A, I ( + 5.2.1 Island configuration Let us start by determining the generalized entropy for an island configuration, provided we use theThis coordinate means system in that particular, corresponds thatd to we are the instructed CFT to in calculate its the vacuum two-dimensional state. distance and without an island, indicatedone in that the gives Penrose a diagram smaller in value figure for the entropy. dual is pure AdS t is, however, a simple way aroundto this be problem. described Following [ bynull a coherent infinity. state As builtthe on shown the von in vacuum Neumann [ state entropy ofRyu-Takayanagi of prescription inertial the ( observers vacuum at state. past As a result, we can use the AdS Figure 5 indicated, one before theconfigurations, Page respectively. time and the other after, corresponding to the no-island and island JHEP05(2020)091 ) and 3.27 0 (5.15) (5.16) (5.17) (5.18) (5.19) (5.13) (5.14) ≈ A σ . , − ) I A u I t A v v . and this turns out c 1 I A log u u > I v = 4(1 + corresponds to an island I c 6 v ) . log I − I A and the coordinate distance I u , A u t u I u ) 1 v 6 + c A crit 24 σ − (1 + . e . I Ω log ) we find I ) v + ) coordinates. We are assuming that u 2 = 2 c − 1 ) )) ( ) ( A 24 5.15 I I O O u and + v, u I ( + + u )) I 2(Ω( I + log I (1 + 3 v , v v ) we have u I A A v 1 + = 0 − t σ – 17 – 2 − 1 − 1 + 3.20 I A (1 + v − − ≈ σ I (1 e ) + . The expression for the outgoing energy flux ( v yields the following two conditions, = I = ) I − u v ) I − I A A c / I v 1 v u u 12 (1 − , u I I (1 + v = I log(1 u ) + c ( log v I A ) 12 & u u ) for the area term has been expressed in A + − σ I − ) over 3.35 (1 + (1 ω − Mv M ∆ − A 5.13 2 2 t + 2 − − ω )) I ∆ u − 0 = 0 = , which allows us to drop the last term on the right in the top equation, and √ 1 (1 + I ) v ) = ( I A v Extremizing ( v A, I ( reveals that the first Hawking radiationthe passes island through solution the is anchor curve already at valid within a time of order the scrambling time after that. so the simplifying assumption thatWe we also used assumed to in obtain the theto island calculation be solution that valid is the when indeed island justified. is located at In terms of the asymptotic coordinates ( The above relations imply that in the near horizon region, The other solution has the islandthe near island the black solution hole into singularity the and is remaining unphysical. equation Inserting in ( log( later on we verifyrearranged the as self-consistency of this assumption. The resulting equations can be One of the two solutions of the quadratic equation for In order to solve for the location of the island we make the simplifying assumption factor of the dynamicalthe black black hole hole metric. where Thevalid the for anchor the classical point island approximation is forout by is much with assumption always of a far valid. the large outside lifetime enoughhole It mass of has turns but an evaporated it out evaporating down will black to to fail hole a also towards provided the small be it end size. starts of the lifetime when the black where we have usedd ( the island is locatedand outside the island the lie infalling in shell a region of where matter a and classical approximation that can both be used the for anchor the point conformal JHEP05(2020)091 ) ) 0 , u 5.19 0 (5.21) (5.20) v ( ) gives in ( the black A s 5.13 v t and into ( with # I is contained in the I v s u )) 0 t v A + u and A A σ I linear dilaton region inside u v (1 + A . The generalized entropy is 1 , the signal must be emitted A v 1 = 2 ) v I ..., < − , u v < obs A 0 extends from the anchor curve to I t v (1 ) + v ( A 2 − end at a fixed reference point AI σ S A 0 t A AI − u u S A t ( for some log c 0 ..., 24 0 A – 18 – v v . The gravitational coupling becomes strong at − /v ) + − crit A log M , such that σ = obs A " − 0 = 2 t u A , compared to the leading order area term, it is important t log Ω = Ω ( c c island gen 12 12 S = = no-island gen ) on asymptotic time and this will not be greatly affected by the detailed S 4.7 In the following we let the spacelike surface Finally, inserting the leading order saddle point values for We can again probe the location of the island by considering an observer sitting on the sub-leading terms that we havethe dropped. scrambling This time expression and is onwards. valid for retarded time of order up to logarithmic correction terms. In particular, all dependence on on the boundary curveadjustment for of all the anchoranchor endpoint points. point. at In We the take otherthe semi-classical the matter words, boundary reference shock we point in wave, will togiven i.e. response simply by be with to ignore in any changing the the higher dimensional perspective thistransverse has two-sphere a going simple to interpretation zero.bulk in entropy terms We ( of are the primarilyconditions area interested in of imposed the the at dependence thethe of origin. strong the This coupling can regionchange be and the seen leading then by order checking result adopting that at a a late simple change times prescription on in for the the anchor prescription curve. does not a no-island configuration that we now turn our5.2.2 attention to. No-island configuration In the absence ofthe an semi-classical island, boundary the at the spacelike semi-classical surface boundary andthe it semi-classical is black hole not solution a indeed priori breaks clear down how near to the proceed. boundary but The from validity of a hole scrambling time. Earlier, wehorizon found of the an same eternal resultvery for black close an hole. to island just the Here outside event the the horizon. island event is inside the black hole but still located as a functioncontribution of is the initially of retarded order to time keep in at mind that the thisto contribution anchor the grows leading with curve. order time, result. anda eventually Although This becomes time expression comparable the of for the time-dependent order generalizedinto the the entropy, which outside scrambling is region time valid beyond when has the anchor passed curve, after is to the be first compared Hawking to radiation the contribution emerges from anchor curve who sendsimplies an ingoing that light in signal. orderfrom The to the relation be anchor between received curve at at an a island time at JHEP05(2020)091 (6.1) M/T (5.22) ∆ − = BH S ∆ ), respectively, shows 5.21 black holes but our computation ) and ( 2 . ]. Radiating into a cold surrounding 5.20 33 M is the energy of the emitted gas, which c . . 16 1 U . U ∆ = T 2 BH 1 3 S = – 19 – S = 2∆ where − Page = the generalized entropy will be dominated by the island t U/T 1 3 gas S > ∆ = 2∆ A σ gas − S A t ∆ computation. It is furthermore convenient that in two dimensions there 2 , and we get M CFT ∆ is the mass lost by the black hole in a given time interval. The entropy increase / 3 M ∆ Our results rely on two dimensional conformal methods. On a technical level this is For the evaporating black hole the Page time comes out to be 1/3 of the black hole life- For the eternal black hole we confirm the appearance of an island outside the horizon, Comparing the island and no-island results in ( via a AdS are no grey body factors. On a more fundamental level, we are able to account for radiation the entropy lost by thewhere black hole. The blackof hole the satisfies gas the radiated firstmust is law equal reflected in the fact that we can conveniently obtain the bulk contribution to the entropy ration rate of thebetween black internal hole. energy and This entropy can of be the gas seen emitted more by explicitly a by black considering body the in two relation dimensions, We can compute the ratio of the entropy of the gas that the black hole emits compared to end of evaporation theat island the appears black to hole melt endpoint.as together the Our with black semi-classical hole both calculation mass is,matter singularity remains however, CFT. and large only horizon compared reliable to as the long scale set by thetime. central This charge fits of the withspace the is following an simple reasoning irreversible process [ and the entropy of the radiation grows at twice the evapo- trivial Ryu-Takayanagi area termThis associated is with consistent an withoffers island earlier a near results rather obtained the clean for physical blackbeyond picture AdS asymptotically hole and AdS horizon. offers spacetimes. evidence that the QRT prescription applies whereas for an evaporating black hole the island is always inside the horizon. Towards the black holes in asymptoticallyThis flat includes both spacetime an inthe eternal a outgoing black Hawking two-dimensional flux, hole, dilaton and supported aevaporates. gravity by black model. In hole an formed both incoming by cases, energy gravitationalvon the collapse flux Neumann that generalized matching gradually entropy entropy of is thecrosses minimised two-dimensional at over matter early to CFT times but a by at the configuration a bulk where Page time the the minimum system generalized entropy includes a non- that for retarded time The corresponding Page curve is drawn in figure 6 Discussion By assuming a QRT formula we have explicitly obtained Page curves for semi-classical configuration. The Page time forlifetime, a dynamical RST black hole is one third of the black hole JHEP05(2020)091 , ]. 08 SPIRE D 14 , IN JHEP ][ , ]. (2013) 028 Phys. Rev. , 09 (2019) 063 (1993) 3743 ]. SPIRE IN 12 71 ][ ]. 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IN IN (2006) 181602 [ ][ IN ][ CC-BY 4.0 JHEP ][ 96 , Breakdown of Predictability in Gravitational Collapse SPIRE Entanglement Wedge Reconstruction and the Information Paradox This article is distributed under the terms of the Creative Commons IN Evaporation of large black holes in AdS: Coupling to the evaporon arXiv:0804.0055 Information in black hole radiation Time Dependence of Hawking Radiation Entropy [ [ arXiv:1905.08762 arXiv:1301.4995 hep-th/9306083 Entropy beyond the Classical Regime Phys. Rev. Lett. entropy proposal entanglement entropy arXiv:1905.08255 and the entanglement wedge[ of an evaporating black hole (2008) 075 [ (1976) 2460 [ N. Engelhardt and A.C. Wall, S. Ryu and T. Takayanagi, V.E. Hubeny, M. Rangamani and T. Takayanagi, T. Faulkner, A. Lewkowycz and J. Maldacena, A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, J.V. Rocha, D.N. Page, S.W. Hawking, G. Penington, D.N. Page, [7] [8] [9] [5] [6] [2] [3] [4] [1] [10] Attribution License ( any medium, provided the original author(s) and source are credited. References ing the completionResearch of Fund this under work. grants 185371-051search and This Fund, 195970-051, research and by was the byFFG is University supported the a of in Postdoctoral National Iceland Fellow part Re- ofsupported Science the by by Foundation Research the the Foundation under KU — Icelandic Leuven Flanders grant C1 (FWO). NSF grant FFGOpen ZKD1118 PHY-1748958. is C16/16/005. also Access. theory with a higher-dimensional AdS dual. Acknowledgments We thank Valentina Giangrecothank M. the Puletti Kavli Institute for for stimulating Theoretical discussions. 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