12
Variational principle and one-point functions in three-dimensional flat space Einstein gravity Stephane Detournay, 1,* Daniel Grumiller, 2,Friedrich Schöller, 2,and Joan Simón 31 Physique Théorique et Mathématique, Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium 2 Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/136, A-1040 Vienna, Austria 3 School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Kings Buildings, Edinburgh EH9 3JZ, United Kingdom (Received 27 February 2014; published 18 April 2014) We provide a well-defined variational principle for three-dimensional flat space Einstein gravity by adding one-half of the Gibbons-Hawking-York boundary term to the bulk action. We check the zero-point function, recovering consistency with thermodynamics of flat space cosmologies. We then apply our result to calculate the one-point functions in flat space Einstein gravity for the vacuum and all flat space cosmologies. The results are compatible with the ones for the zero-mode charges obtained by canonical analysis. DOI: 10.1103/PhysRevD.89.084061 PACS numbers: 04.20.Fy, 04.20.Ha, 04.60.Kz, 11.25.Tq I. INTRODUCTION Given some bulk action I ½Φ and some boundary conditions on the fields Φ, it is important to check whether or not there is a well-defined variational principle, δI ¼ 0 on shell, for all variations δΦ that preserve the boundary conditions. Physically, the importance of δI ¼ 0 for sol- utions of the classical equations of motion (EOM) is evident: Only then solutions of the classical EOM are actually classical solutions, in the sense that they stabilize the action and allow for a meaningful (semi-)classical approximation to the path integral. Gravity theories are notorious in this regard, since the simplest way to resolve these issues is not accessible: Imposing natural boundary conditions on the metric, g μν 0 would be most unnatural, as the metric must be nondegenerate near the boundary. For any given boundary value problem (for instance, a Dirichlet, Neumann or Robin boundary value problem) it is thus important to check whether or not there is a well-defined variational principle. If there is no well-defined variational principle, i.e., the first variation of the action does not vanish on shell, then suitable boundary terms must be added to the bulk action to achieve a vanishing first variation. A textbook example is provided by the pure Einstein-Hilbert (EH) action, whose variation yields a boundary term involving the metric variation, but also its first derivatives in the direction normal to the boundary. An additional termthe Gibbons- Hawking-York (GHY) termcan then be added to the original action to compensate these unwanted terms and make the variational principle well defined for variations such that δg μν ¼ 0 at the boundary (Dirichlet problem) (see [1], Appendix E). Another archetypical situation is provided by antide Sitter (AdS) gravity, where after the addition of the Gibbons-Hawking-York term and appropriate counter- terms, the Dirichlet variational principle is well defined for variations δg ð0Þij keeping the boundary metric fixed [2] (note that other variational problems can be posed for AdS 3 gravity, see [3] for Neumann conditions and [4,5] for mixed conditions). In general however, if achieving a vanishing first variation turns out to be impossible, then the particular set of boundary conditions/boundary value problem must be discarded as unphysical. Insisting on a well-defined variational principle is not only a matter of internal consistency, but has physical consequences. For example, the free energy as derived from the Euclidean path integral approach can only be correct if the action obeys a well-defined variational principle. Also holographic response functions are sensitive to the boundary terms that are added to the bulk action. The main purpose of the present work is to establish a well-defined variational principle for flat space Einstein gravity in three dimensions and to calculate the zero- and one-point functions, in particular for the flat space vacuum and for flat space cosmology solutions. This paper is organized as follows. In Sec. II we review the situation for Einstein gravity with negative cosmologi- cal constant. In Sec. III we discuss flat space Einstein gravity, provide a well-defined variational principle, check the free energy and calculate the holographic response functions. In Sec. IV we point to some loose ends in flat space holography. Before starting we mention some of our conventions. We work exclusively in Euclidean signature. Our sign * [email protected] [email protected] [email protected] § [email protected] PHYSICAL REVIEW D 89, 084061 (2014) 1550-7998=2014=89(8)=084061(12) 084061-1 © 2014 American Physical Society

Variational principle and one-point functions in three-dimensional flat space Einstein gravity

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Page 1: Variational principle and one-point functions in three-dimensional flat space Einstein gravity

Variational principle and one-point functions in three-dimensional flat spaceEinstein gravity

Stephane Detournay,1,* Daniel Grumiller,2,† Friedrich Schöller,2,‡ and Joan Simón3,§1Physique Théorique et Mathématique, Université Libre de Bruxelles and International Solvay Institutes,

Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium2Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/136,

A-1040 Vienna, Austria3School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh,

King’s Buildings, Edinburgh EH9 3JZ, United Kingdom(Received 27 February 2014; published 18 April 2014)

We provide a well-defined variational principle for three-dimensional flat space Einstein gravity byadding one-half of the Gibbons-Hawking-York boundary term to the bulk action. We check the zero-pointfunction, recovering consistency with thermodynamics of flat space cosmologies. We then apply our resultto calculate the one-point functions in flat space Einstein gravity for the vacuum and all flat spacecosmologies. The results are compatible with the ones for the zero-mode charges obtained by canonicalanalysis.

DOI: 10.1103/PhysRevD.89.084061 PACS numbers: 04.20.Fy, 04.20.Ha, 04.60.Kz, 11.25.Tq

I. INTRODUCTION

Given some bulk action I½Φ� and some boundaryconditions on the fields Φ, it is important to check whetheror not there is a well-defined variational principle, δI ¼ 0on shell, for all variations δΦ that preserve the boundaryconditions. Physically, the importance of δI ¼ 0 for sol-utions of the classical equations of motion (EOM) isevident: Only then solutions of the classical EOM areactually classical solutions, in the sense that they stabilizethe action and allow for a meaningful (semi-)classicalapproximation to the path integral.Gravity theories are notorious in this regard, since the

simplest way to resolve these issues is not accessible:Imposing natural boundary conditions on the metric,gμν → 0 would be most unnatural, as the metric must benondegenerate near the boundary. For any given boundaryvalue problem (for instance, a Dirichlet, Neumann or Robinboundary value problem) it is thus important to checkwhether or not there is a well-defined variational principle.If there is no well-defined variational principle, i.e., the

first variation of the action does not vanish on shell, thensuitable boundary terms must be added to the bulk action toachieve a vanishing first variation. A textbook example isprovided by the pure Einstein-Hilbert (EH) action, whosevariation yields a boundary term involving the metricvariation, but also its first derivatives in the direction normalto the boundary. An additional term—the Gibbons-Hawking-York (GHY) term—can then be added to theoriginal action to compensate these unwanted terms and

make the variational principle well defined for variationssuch that δgμν ¼ 0 at the boundary (Dirichlet problem) (see[1], Appendix E). Another archetypical situation is providedby anti–de Sitter (AdS) gravity, where after the addition ofthe Gibbons-Hawking-York term and appropriate counter-terms, the Dirichlet variational principle is well defined forvariations δgð0Þij keeping the boundary metric fixed [2](note that other variational problems can be posed for AdS3gravity, see [3] for Neumann conditions and [4,5] for mixedconditions). In general however, if achieving a vanishingfirst variation turns out to be impossible, then the particularset of boundary conditions/boundary value problem must bediscarded as unphysical.Insisting on a well-defined variational principle is not

only a matter of internal consistency, but has physicalconsequences. For example, the free energy as derived fromthe Euclidean path integral approach can only be correctif the action obeys a well-defined variational principle.Also holographic response functions are sensitive to theboundary terms that are added to the bulk action.The main purpose of the present work is to establish a

well-defined variational principle for flat space Einsteingravity in three dimensions and to calculate the zero- andone-point functions, in particular for the flat space vacuumand for flat space cosmology solutions.This paper is organized as follows. In Sec. II we review

the situation for Einstein gravity with negative cosmologi-cal constant. In Sec. III we discuss flat space Einsteingravity, provide a well-defined variational principle, checkthe free energy and calculate the holographic responsefunctions. In Sec. IV we point to some loose ends in flatspace holography.Before starting we mention some of our conventions.

We work exclusively in Euclidean signature. Our sign

*[email protected][email protected][email protected]§[email protected]

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conventions for the Ricci tensor are fixed by Rμν ¼∂αΓαμν þ � � �. We always consider three-dimensional

manifolds M with boundary ∂M.

II. EINSTEIN GRAVITY WITH NEGATIVECOSMOLOGICAL CONSTANT

Before studying the variational principle in three-dimensional flat space, we revisit it in AdS3 to understandwhether there exists a smooth limit to the case of vanishingcosmological constant. None of the results in this sectionare new, but they provide a useful starting point for thenovel results in Sec. III.In Sec. II A we recall the Brown-Henneaux boundary

conditions. In Sec. II B we review the variational principlefor various boundary value problems. In Sec. II C werecover the zero-point function and Bañados-Teitelboim-Zanelli (BTZ) black hole thermodynamics. In Sec. II D wereconsider one-point functions.

A. Euclidean anti–de Sitter boundary conditions

Asymptotically AdS3 metrics satisfying Brown-Henneaux boundary conditions are defined by [6]

grr ¼l2

r2þ hrrðt;φÞl4=r4 þ oð1=r4Þ grt ¼ Oð1=r3Þ

gtt ¼r2

l2þ httðt;φÞ þ oð1Þ grφ ¼ Oð1=r3Þ

gφφ ¼ r2 þ hφφðt;φÞl2 þ oð1Þgtφ ¼ htφðt;φÞlþOð1=rÞ: (1)

The notation Oðr−nÞ [oðr−nÞ] means that the quantityscales like r−n or smaller (smaller than r−n). We considerthe following set of variations preserving these boundaryconditions (for convenience we scale all fluctuations δhμνwith appropriate factors of the AdS radius l so that they aredimensionless quantities)

δgrr ¼ δhrrðt;φÞl4=r4 þ oð1=r4Þ δgrt ¼ Oð1=r3Þδgtt ¼ δhttðt;φÞ þ oð1Þ δgrφ ¼ Oð1=r3Þδgφφ ¼ δhφφðt;φÞl2 þ oð1Þ δgtφ ¼ Oð1Þ: (2)

While more general boundary conditions compatiblewith asymptotic AdS behavior are possible, for three-dimensional Einstein gravity we do not need more generalones. All interesting physical solutions/states/fluctuationsof Einstein gravity are already allowed by the boundaryconditions above.For later purposes we perform a 2þ 1 split of the metric,

gab ¼ γab þ nanb, where γab is the boundary metric and na

is the outward pointing unit normal vector. We always takean r ¼ const hypersurface as boundary. Extrinsic curvatureis given by the standard expression Kab ¼ γacγbd∇cnd.

For a collection of useful formulas for variations andboundary quantities see [7].The boundary conditions (2) yield the following

identities:

na ¼ δralrþOð1=r3Þ (3a)

ffiffiffiγ

p ¼ r2

lþOð1Þ (3b)

K ¼ 2

l− ðhtt þ hφφ þ hrrÞ

lr2

þOð1=r4Þ (3c)

γabδgab ¼ ðδhφφ þ δhttÞl2

r2þ oð1=r2Þ (3d)

Kabδgab ¼ ðδhφφ þ δhttÞlr2

þ oð1=r2Þ (3e)

nanbδgab ¼ δhrrl2

r2þ oð1=r2Þ (3f)

γabnc∇cδgab ¼ −2ðδhφφ þ δhttÞlr2

þ oð1=r2Þ: (3g)

B. Variational principle

We consider the EH action including a cosmologicalconstant with a GHY [8,9] and a cosmological boundaryterm with arbitrary dimensionless coefficients α and β,respectively,

Γðα;βÞ ¼ −1

16πG

ZMd3x

ffiffiffig

p ðR − 2ΛÞ

−1

8πG

Z∂M

d2xffiffiffiγ

p �αK þ β

l

�; (4)

where Λ ¼ − 1l2. The first variation of the bulk action (4)

equals

δΓðα;βÞ ¼1

16πG

ZMd3x

ffiffiffig

p ðGab þ ΛgabÞδgab

þ 1

16πG

Z∂M

d2xffiffiffiγ

p �Kab −

�αK þ β

l

�γab

þ ðα − 1ÞKnanb�δgab

þ 1 − α

16πG

Z∂M

d2xffiffiffiγ

pγabnc∇cδgab

þ 2α − 1

16πG

Z∂2M

dxffiffiffiffiγ0

pn0anbδgab: (5)

Primed quantities are corner contributions that arise in caseof nonsmooth boundaries, see e.g. [10–14]. We assume for

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the remainder of our AdS3 review that there are nocorner terms.The identities (A3) allow us to write the variation

δΓðα;βÞ as

δΓðα;βÞ ¼1

16πG

Z∂M

d2xð2ðα − 1Þδhrr− ðβ þ 1Þðδhtt þ δhφφÞÞ þ oð1Þ: (6)

Since we are interested in a well-defined variationalprinciple, we study the situations under which this variationvanishes.The first observation is that if we choose

α ¼ 1 β ¼ −1 (7)

no matter whether the variations δhab are on shell or not,the first variation (6) vanishes, leading to a well-definedvariational principle for the well-known action [15–17]

Γ ¼ −1

16πG

ZMd3x

ffiffiffig

p �Rþ 2

l2

−1

8πG

Z∂M

d2xffiffiffiγ

p �K −

1

l

�: (8)

To explore whether there exist more general situations,we study the linearized EOM satisfied by the variationsδhab ¼ gab − gab, where gab stands for the AdS3 metric.These are given by [18]

Gð1Þμν ¼ Rð1Þ

μν −1

2gμνRð1Þ − 2Λδhμν ¼ 0 (9)

Rð1Þμν ¼ 1

2ð−∇2δhμν − ∇μ∇νδhþ ∇σ∇νδhσμ þ ∇σ∇μδhσνÞ

(10)

Rð1Þ ¼ −∇2δhþ ∇μ∇νδhμν − 2Λδh; (11)

where all barred quantities are computed using the AdS3metric. If we take the trace of the linearized Einsteinequation, we learn that Rð1Þ ¼ 0.In transverse gauge for the fluctuations,

∇μðδhμν − gμνgρσδhρσÞ ¼ 0, the linearized EOM implytracelessness (see e.g. [19]).

δhtt þ δhφφ þ δhrr ¼ 0 (12)

Inserting this asymptotic on-shell constraint into the varia-tion (6), one obtains

δΓðα;βÞjEOM ¼ 1

16πG

Z∂M

d2xδhrrð2αþ β − 1Þ þ oð1Þ:(13)

Thus, for variations preserving the boundary conditions (2)and tracelessness (12), there is a one-parameter family of

actions (4) with a well-defined variational principle, deter-mined by the condition

2α ¼ 1 − β: (14)

This result is expected on general grounds, see e.g. [20].Their key observation is that asymptotically there is nodifference between an extrinsic curvature term and aboundary cosmological constant, see (3c). Thus only thevalue of 2αþ β is of relevance, this being fixed to unity bythe condition (14).Note that it is rather unusual to use the linearized

equations of motion for the fluctuations—here, relation(12)—in order for the variational principle to be welldefined. Indeed, one would normally expect to haveδΓ ¼ 0 for all δgμν allowed by the boundary conditionsdefining the theory. In the AdS3 case considered here, thetracelessness condition (12) could be interpreted as restrict-ing the Brown-Henneaux boundary conditions (2). Onemight worry whether this is too restrictive to yield asensible theory. However, the Virasoro descendants ofthe BTZ black holes satisfy the tracelessness condition.Also, one can check that this condition is invariant underthe action of the Brown-Henneaux generators. Therefore,this restricted set of boundary conditions not only containsthe physically most relevant solutions but still displays thetwo Virasoro algebras characteristic of two-dimensionalconformal symmetry.Besides the special choice (7) that leads to the standard

situation of a Dirichlet problem for the metric, a GHY termwith the usual normalization and the well-known holo-graphic counterterm, there is another case that stands out.Namely, for the choice1

α ¼ 1

2β ¼ 0 (15)

the boundary action is independent of the cosmologicalconstant [and the corner term in (5) vanishes]. In this casethere is no holographic counterterm, and the GHY termdoes not have its usual normalization, but has an additionalfactor of 1

2. The independence from the cosmological

constant makes this choice most suitable for an l → ∞limit.

C. Zero-point function

Evaluating the action (4) with the condition (14) onclassical solutions leads to the free energy, F ¼ TΓðα;βÞ,where T is the temperature, determined by the inverse of theperiodicity of Euclidean time.

1A similar choice was studied in arbitrary dimensions in [21],see also [22,23].

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F ¼ limrc→∞

1

4G

�r2c − r2þ

l2− ð2αþ βÞ rc

lffiffiffiffiffigtt

p �(16)

The first term comes from the bulk action, integrated fromthe center or outer horizon, r ¼ rþ, to the asymptoticboundary, which we regulate by a cutoff r ¼ rc that wesend to infinity at the end of the calculation. The secondterm comes from the boundary action evaluated at the samevalue of the cutoff. Note that the result for the free energy isuniversal and finite as long as α and β satisfy the condition(14) and the metric obeys the boundary conditions (1).For instance, taking BTZ black holes [24,25]

ds2 ¼ −ðr2 − r2þÞðr2 − r2−Þ

l2r2dt2 þ l2r2

ðr2 − r2þÞðr2 − r2−Þdr2

þ r2�dφ −

rþr−lr2

dt

�2

(17)

and inserting their Euclidean continuation into the formulasabove leads to the free energy [T ¼ ðr2þ − r2−Þ=ð2πrþl2Þ,Ω ¼ r−=ðrþlÞ]

FðT;ΩÞ ¼ −r2þ − r2−8Gl2

¼ −π2T2l2

2Gð1 −Ω2l2Þ (18)

independently from α and β, provided they obey therelation (14).The result (18) leads to the correct mass,

M ¼ ðr2þ þ r2−Þ=ð8Gl2Þ, and entropy, S ¼ 2πrþ=ð4GÞ,of BTZ black holes, and is consistent with the firstlaw of thermodynamics, dF ¼ −SdT − JdΩ, whereJ ¼ rþr−=ð4GlÞ. We stress that this would no longer bethe case if α and β violate the condition (14). Thus, theBTZ free energy emerges correctly if and only if we have awell-defined variational principle.

D. One-point functions

We focus first on the standard case α ¼ 1, β ¼ −1. Tocompute the one-point functions, we consider the sameasymptotic metric as in (1), but we allow a different set ofmetric fluctuations to accommodate for sources

δgrr ¼ irrelevant δgrt ¼ irrelevant

δgtt ¼ δhð0Þtt ðt;φÞr2=l2 þ δhttðt;φÞ þ oð1Þδgφφ ¼ δhð0Þφφðt;φÞr2 þ δhφφðt;φÞl2 þ oð1Þδgrφ ¼ irrelevant δgtφ ¼ δhð0Þtφ ðt;φÞr2=lþOð1Þ: (19)

Notice that these fluctuations do not respect the Brown-Henneaux boundary conditions (2) in general. The leadingcontribution, in an Oð1=rÞ expansion, of the on-shellvariation of the classical action equals

δΓjEOM ¼ 1

16πG

Z∂M

d2x

��hrr2

þ hφφ

�δhð0Þtt

− 2htφδhð0Þtφ þ

�hrr2

þ htt

�δhð0Þφφ

�: (20)

The fluctuations (19) are non-normalizable and are inter-preted as sources for the corresponding operators in thedual conformal field theory (CFT) [26].Let us evaluate this variation for the (Euclidean version

of the) BTZ black hole (17). The nontrivial functions habare identified to be

hrr ¼ −htt ¼r2þ þ r2−

l2htφ ¼ i

rþr−l2

; (21)

giving rise to an on-shell variation (20)

δΓjEOM ¼ 1

32πGl2

Z∂M

d2xððr2þ þ r2−Þðδhð0Þtt − δhð0ÞφφÞ

− 4irþr−δhð0Þtφ Þ: (22)

This is the usual “holographic” structure [26] ofδΓjEOM ∼ ðvevÞδðsourceÞ, where “vev” here refers to theholographically renormalized Brown-York stress tensor,Tab, and “source” to the non-normalizable fluctuations,hð0Þab , that we switched on in order to get a finite response.Evaluating the holographically renormalized Brown-Yorkstress tensor

δΓjEOM ¼ −1

2

Zd2x

ffiffiffiffiffiffiffigð0Þ

qTabδhð0Þab (23)

on the (Euclidean version of) BTZ solutions (17) using ourresult (22) yields (here we set l ¼ 1 and provide onlyabsolute values for easier comparison with Minkowskianresults)

jTttj ¼ jTφφj ¼r2þ þ r2−16πG

¼ M2π

jTtφj ¼jrþr−j8πG

¼ J2π

:

(24)

The expressions (24) are well known and reproduceprecisely previous calculations, see e.g. [16,27] and refer-ences therein.For other values of α and β, subject to the condition (14),

we obtain the same response functions, as expected.

III. FLAT SPACE EINSTEIN GRAVITY

Equipped with the explicit results for Euclidean AdS3from the previous section, we address in this section thesituation in flat space Einstein gravity.In Sec. III A we formulate a specific set of flat space

boundary conditions. In Sec. III B we provide the varia-tional principle for Einstein gravity compatible with these

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boundary conditions. In Sec. III C we check the zero-pointfunction and flat space cosmology thermodynamics. InSec. III D we calculate one-point functions.

A. Euclidean flat space boundary conditions

The flat space boundary conditions are usually providedin Eddington-Finkelstein (EF) gauge adapted to nullinfinity [28].2 However, for Euclidean purposes this gaugeis inaccessible since there is no Euclidean analog of a nullvector. Thus, our first task is to translate these boundaryconditions into a more suitable gauge.We start with the set of boundary conditions provided in

[31] (these are looser boundary conditions than the ones byBarnich and Compère [28]).3

guu ¼ huu þOð1=rÞ guφ ¼ huφ þOð1=rÞgur ¼ −1þ hur=rþOð1=r2Þ grφ ¼ h1ðφÞ þOð1=rÞgφφ ¼ r2 þ ðh2ðφÞ þ uh3ðφÞÞrþOð1Þ grr ¼ Oðr−2Þ:

(25)

The Minkowski background is given by huu ¼ −1 with theremaining hμν ¼ hi ¼ 0 and all subleading terms set tozero, ds2 ¼ −du2 − 2dudrþ r2dφ2.The asymptotic symmetry group is generated by vector

fields

ξL ¼ ξLðφÞ∂φþξ0LðφÞðu∂u− r∂rÞ−ξ00LðφÞur∂φþ��� (26)

ξM ¼ ξMðφÞ∂u þOð1=rÞ∂u

þ ðuf1ðφÞ þ f2ðφÞ þOð1=rÞÞ∂r

þ ðf3ðφÞ=rþOð1=r2ÞÞ∂φ; (27)

where the dots refer to subleading terms generating trivialgauge transformations which are modded out in theasymptotic symmetry group.The equations of motion impose additional relations

between the free functions appearing in (25):

∂uhuu ¼ 0 (28)

∂uhrr ¼ −2hur (29)

∂uhrφ ¼ u∂φhuu þ ∂φhur − 2huφ þ h4ðφÞ: (30)

Note that the integration function h4ðφÞ erroneously wasset to zero in the equation displayed in the text belowEq. (7b) in [31]. While this omission is not relevant for flatspace chiral gravity, it is important to take into account thisfunction in Einstein gravity.It is of interest to derive the centrally extended BMS3

algebra [28,32,33] as the asymptotic symmetry algebra dueto the boundary conditions (25). Since the canonicalcharges QM associated with the super-translation gener-ators M [31] are QM ∼

RdφξMðφÞðhuu þ h3Þ, we can

answer this question by computing the variation of thestate dependent function huu þ h3. Using the on-shellrelation ∂uhuu ¼ 0, we find from the Lie derivativesalong ξL;M

δξMhuu ¼ 0þ� � � δξLhuu ¼ h0uuξLþ 2huuξ0Lþ� � � (31)

δξMh3 ¼ 0þ��� δξLh3¼ h03ξLþ2h3ξ0L−2ξ000L þ�� � : (32)

From the left-hand equations we can read off the vanishingcommutator between the super-translation generators,while the right-hand equations yield the expectedSchwarzian derivative for the combination huu þ h3,including the anomalous term. With the appropriate nor-malization of the charges QM, the correct central chargecM ¼ 3=G is reproduced [28] and the asymptotic symmetryalgebra matches the centrally extended BMS3 algebra,which is isomorphic to a two-dimensional Galilean con-formal algebra [34].We convert now the boundary conditions from EF gauge

(25) into diagonal gauge, starting from the line element

ds2 ¼ −fðφÞdu2 − 2du drþ r2dφ2 þ 2gðφÞdu dφþ h1ðφÞdr dφþ � � � ; (33)

where the ellipsis denotes subleading terms. We definediagonal gauge as one in which there is no dr dt term,where t is the time coordinate replacing the retarded time u.

u ¼ tþ Kðr;φÞ ⇒ du ¼ dtþ ∂rKdrþ ∂φKdφ: (34)

Absence of dr dt terms yields the partial differentialequation

∂rK ¼ −1

f⇒ Kðr;φÞ ¼ −

rfðφÞ þ K0ðφÞ: (35)

The transformed line element is then

ds2 ¼−fðφÞdt2þ dr2

fðφÞþAdφ2þ 2Bdtdφþ 2Cdrdφþ �� �(36)

with the functions

2There is earlier work where asymptotic flatness was definedin space polar coordinates and time, for example see [29,30].

3In particular, the function h3 does not appear in [28], so thecentral charge term does not emerge from (32), but instead fromthe (modified) transformation behavior of huu. Their boundaryconditions are preserved by Lie variations along vector fields(26), (27), provided a specific trivial contribution (27) withf1 ¼ ξ000L is added to the generators.

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A ¼ r2 − fð∂φKÞ2 þ 2g∂φK

¼ r2�1 −

ðf0Þ2f3

�þ 2r

f0

f2ðg − fK0

0Þ þ K00ð2g − fK0

0Þ(37)

B ¼ −f∂φK þ g ¼ −rf0

f− fK0

0 þ g (38)

C ¼ −f∂φK∂rK − ∂φK þ g∂rK ¼ h1 −gf: (39)

From the expression for the function A we see that onlyzero-mode solutions, f0 ¼ 0, have the usual r2dφ2 term indiagonal gauge. Restricting to constant f also eliminatesthe term linear in r in the function B. Since it is not obviousto us how to interpret the asymptotic line element fornonconstant f we restrict from now on to constant f andpostpone comments on the general case to Sec. IV.Converting the time t to Euclidean time τ, we shall study

background metrics of the form

ds2 ¼ hττðφÞdτ2 þ hrrðφÞdr2 þ r2dφ2; (40)

with hrr ¼ 1=hττ and fluctuations satisfying

δgφφ ¼ OðrÞ δgφτ ¼ Oð1Þ δgττ ∼ δgrr ¼ Oð1Þδgrφ ¼ OðrÞ δgrτ ¼ Oð1=rÞ δðgrrgττÞ ¼ Oð1Þ;

(41)

where to highest order δgrr, δgrτ, and δgττ only depend onφ. We stress that our subset of fluctuations δgττ, δgrr in (41)differs from the ones considered in [29,30]. In that respect,the theories studied in these references are different fromours. The conditions on the fluctuations δgrr and δgττ implythat we can parametrize them as

δgττ ¼ δhττ þOð1=rÞ δgrr ¼ δhrr þOð1=rÞδðhrrhττÞ ¼ 0: (42)

In summary, the boundary conditions above are adiagonal gauge version of the flat space boundary con-ditions imposed in Einstein gravity [28] or flat space chiralgravity [31]. They can be transformed into each other forzero-mode solutions [meaning that the φ-dependent func-tions in (40) actually are constant] using the standardcoordinate transformation between EF and Schwarzschildcoordinates. We do not address nonzero mode solutions inthis work.4

For later purposes we collect here identities analog to(A3) for an r ¼ const boundary.

na ¼ δraffiffiffiffiffiffihrr

pþOð1=rÞ (43a)

ffiffiffiγ

p ¼ rffiffiffiffiffiffihττ

pþOð1Þ (43b)

K ¼ 1

rffiffiffiffiffiffihrr

p þOð1=r2Þ (43c)

gabδgab ¼δhrrhrr

þ δhττhττ

þOð1=rÞ (43d)

Kabδgab ¼ Oð1=r2Þ (43e)

γabnc∇cδgab ¼ Oð1=r2Þ (43f)

B. Variational principle

We establish now that the EH action with the addition ofone-half of the GHY boundary term,

Γ ¼ −1

16πG

ZMd3x

ffiffiffig

pR −

1

16πG

Z∂M

d2xffiffiffiγ

pK; (44)

gives rise to a well-defined variational principle for flatspace boundary conditions (40), (41).To prove this statement, we consider a one-parameter

family of actions given by

ΓðαÞ ¼ −1

16πG

ZMd3x

ffiffiffig

pR −

1

8πG

Z∂M

d2xffiffiffiγ

pαK: (45)

Note that (45) is the l → ∞ limit of the action (4). Its onefree parameter α will be determined by requiring the firstvariation

δΓðαÞ ¼1

16πG

ZMd3x

ffiffiffig

pGabδgab

þ 1

16πG

Z∂M

d2xffiffiffiγ

p ðKab − αKgab

þ ð2α − 1ÞKnanbÞδgabþ 1 − α

16πG

Z∂M

d2xffiffiffiγ

pγabnc∇cδgab

þ 2α − 1

16πG

Z∂2M

dxffiffiffiffiγ0

pn0anbδgab (46)

to vanish on shell. Notice we used gab ¼ γab þ nanb whenwriting the second line in (46).The identities (44) allow us to prove the relationsZ

∂Md2x

ffiffiffiγ

pKabδgab ∼Oð1=rÞ (47)4We stress that this restriction already covers physically

interesting configurations as conical defects and flat cosmologies.

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Z∂M

d2xffiffiffiγ

pKgabδgab ¼

Z∂M

d2x

ffiffiffiffiffiffihττhrr

s �δhrrhrr

þ δhττhττ

þOð1=rÞ (48)

Z∂M

d2xffiffiffiγ

pKnanbδgab ¼

Z∂M

d2x

ffiffiffiffiffiffihττh3rr

sδhrr (49)

Z∂M

d2xffiffiffiγ

pγabnc∇cδgab ∼Oð1=rÞ: (50)

Thus, the first variation of the action (44) yields on-shell

δΓðαÞjEOM ¼ −α

16πG

Z∂M

d2x

ffiffiffiffiffiffihττhrr

sδðhrrhττÞhrrhττ

þOð1=rÞ

þ 2α − 1

16πG

�Z∂M

d2x

ffiffiffiffiffiffihττh3rr

sδhrr

þZ∂2M

dxffiffiffiffiγ0

pn0anbδgab

�: (51)

The first line in (51) vanishes, for any value of α, as r goesto infinity due to (42). The point we want to stress is thateven in the absence of corner terms, the second line in (51)does not vanish for all fluctuations δhrr in (42).5 Thus,requiring the existence of a well-defined variational prin-ciple for the set of boundary conditions (41) fixes the valueof α ¼ 1

2. Notice that this value would also ensure the

vanishing of the corner terms, if these were to exist. Thus,

δΓð12ÞjEOM ¼ Oð1=rÞ: (52)

The standard GHY boundary term corresponds to thechoice α ¼ 1. Our analysis in (51) proves the lack of awell-defined variational principle for this boundary term forthe flat space boundary conditions (40), (41). It is interest-ing to point out that our answer α ¼ 1

2is compatible with the

consistency relation (14) together with the fact that in flatspace β is effectively zero.In conclusion, the action (44), which contains one-half of

the usual GHY boundary term, with flat space boundaryconditions (40), (41) leads to a well-defined variationalprinciple. The action (44) arises as a smooth l → ∞ limitof the AdS action (4) with the choice (15).

C. Zero-point function

Like in the AdS case, the action (44) can be used todetermine the free energy of specific classical solutions.

FðαÞ ¼ limrc→∞

−α

4Gffiffiffiγ

pKjr¼rc ¼ −

α

4G

ffiffiffiffiffiffihττhrr

s ����r→∞

(53)

Our result for free energy is finite, but does depend on thechoice of α.The solutions of interest here are the flat vacuum and flat

space cosmologies [35–37]. Their Euclidean version isgiven by the line element [38]

ds2 ¼ r2þ

�1 −

r20r2

�dτ2 þ dr2

r2þð1 − r20

r2Þþ r2

�dφ −

rþr0r2

�2

(54)

which depends on the mass parameter rþ and the rotationparameter r0. The action (44) evaluated on flat spacecosmologies (54) leads to the free energy [T ¼ r2þ=ð2πr0Þ, Ω ¼ rþ=r0]

FðT;ΩÞ ¼ Fð12Þ ¼ −

r2þ8G

¼ −π2T2

2GΩ2(55)

in accordance with [38]. The entropy derived from the freeenergy (55) is consistent with independent derivations ofentropy, S ¼ 2πr0=ð4GÞ, using either the Bekenstein-Hawking relation SBH ¼ Ah=ð4GÞ or a generalization ofthe Cardy formula for Galilean (or ultrarelativistic) con-formal algebras [39,40] (see [41] for a recent generalizationthat takes into account logarithmic corrections). We stressthat consistency with the first law of thermodynamics,dF ¼ −SdT − JdΩ [where J ¼ −rþr0=ð4GÞ], is onlyachieved for the choice α ¼ 1

2in (45). For later comparison

we note that the mass (as defined through the canonicalcharges) is given by M ¼ −F ¼ r2þ=ð8GÞ.Comparing the free energy (55) of flat space cosmolo-

gies with the free energy of “hot flat space,” F ¼ −1=ð8GÞit was shown in [38] that there is a phase transition betweenthese two spacetimes, similar to the Hawking-Page phasetransition [42], with a critical temperature Tc ¼ 1=ð2πr0Þ.Note that the factor 1

2in the boundary term in (44) is

crucial to obtain the correct normalization of the freeenergy. While the importance of this factor should beevident on general grounds and from our AdS discussion inSec. II C, we illustrate this point now by explicitlyevaluating the thermodynamics of flat space cosmologieswhen considering instead the standard Einstein-Hilbertaction with the usual GHY boundary term. Since the bulkaction vanishes on shell, the only change as compared tobefore is that the free energy gets multiplied by a factor of 2relative to the result (55).

5Notice that for the boundary conditions considered in [30],this last term would also be subleading, since those boundaryconditions impose stronger falloff behavior on the fluctuationsthan ours. Thus, we reproduce the claim in [30] that there exists awell-defined variational principle for α ¼ 1. In fact, we showedabove that this claim is true for any α.

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Fð1ÞðT;ΩÞ ¼ −π2T2

GΩ2(56)

Assuming now the validity of the first law, we then obtainan entropy

Sð1ÞðT;ΩÞ ¼ −∂Fð1Þ∂T

����Ω¼const

¼ 2π2TGΩ2

¼ πr0G

¼ 2Ah

4G¼ 2SBH (57)

that differs from the expected Bekenstein-Hawkingentropy SBH by a factor of 2. If we instead insist on thevalidity of the Bekenstein-Hawking entropy (which isconfirmed independently through a Galilean CFT calcu-lation [39,40]) then we violate instead the first law,dFð1Þ ≠ −SBHdT − JdΩ. Thus, as in our AdS discussion,the existence of a well-defined variational principle iscrucial to reproduce the thermodynamics of flat spacecosmologies.

D. One-point functions

Now that we have established the existence of a well-defined variational principle for the action (44), we

compute the one-point functions by turning on sourcesin the bulk. As usual in holographic correspondences, thesources correspond to non-normalizable modes thatsolve the linearized EOM [26]. We call any mode “non-normalizable” whenever it violates our Euclidean flat spaceboundary conditions (40), (41).In the Appendix we provide the most general solution ψ

to the linearized EOM in a suitable gauge and discuss theconditions for normalizability. We find

ψττ ¼ −ψ rr ¼ 2ξ1 þOð1=rÞ (58a)

ψτφ ¼ r2∂τξ0φ þOð1Þ (58b)

ψφφ ¼ 2r2∂φξ0φ þOðτrÞ (58c)

ψ rτ ¼ ψ rφ ¼ 0: (58d)

In order to switch on suitable non-normalizable modes thatcan act as sources we switch on the function ξ0φ in (59).Having done so, we consider now the first variation (46)

with α ¼ 12for the class of metrics

grr ¼ hrrðφÞ þ hð1Þrr ðτ;φÞ=rþOð1=r2Þ grτ ¼ hrτðφÞ=rþOð1=r2Þgττ ¼ hττðφÞ þ hð1Þττ ðτ;φÞ=rþOð1=r2Þ gφφ ¼ r2 þ hð1Þφφðτ;φÞrþOð1Þgττ ¼ hττðφÞ þ hð1Þττ ðτ;φÞ=rþOð1=r2Þ gφφ ¼ r2 þ hð1Þφφðτ;φÞrþOð1Þ; (59)

with variations of the form [again δðhττhrrÞ ¼ 0]

δgrr ¼ δhrrðφÞ þOð1=rÞ δgrτ ¼ Oð1=rÞδgττ ¼ δhττðφÞ þOð1=rÞ δgφφ ¼ δhð0Þφφðτ;φÞr2 þOðrÞδgrφ ¼ Oð1Þ δgτφ ¼ δhð0Þτφ ðτ;φÞr2 þOð1Þ: (60)

These variations by construction are compatible with (59),appropriately generalized to allow also small gauge fluc-tuations that lead away from the gauge choice imposed inthe Appendix. The fluctuations with superscript (0) corre-spond to non-normalizable contributions, i.e., sources.In this case the identities (44) together with on-shell

relations give rise to

δΓjEOM¼ 1

32πG

Z∂M

d2x�ð2rchrφþOð1ÞÞ∂τδh

ð0Þτφ þhττδh

ð0Þφφ

þ�hð1Þττ ∂τhrφ

hττþhrτ∂φ lnhττ−2∂φhrτ−2hτφ

þhrφ∂τhð1Þφφ

�δhð0Þτφ

�þOð1=rcÞ: (61)

Requiring a finite response when the radial cutoff issent to infinity, rc → ∞, restricts the sources to satisfy∂τδh

ð0Þτφ ¼ 0. This means the function ξð0Þφ appearing in (59)

can only be linear in τ,

ξð0Þφ ¼ ξJðφÞτ þ1

2

Zφdφ0ξMðφ0Þ (62)

and thus depends on two free functions of φ that we callξM;J. [Note that adding a constant to ξð0Þφ does not changethe sources since the quantity ξð0Þφ appears only with firstderivatives in (59), so the arbitrary integration constant inthe second term in (62) is irrelevant.] The correspondingsources are then parametrized by these two free functionsand generate two independent response functions:

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δhð0Þφφ ¼ δξM þ 2τδξ0J δhð0Þτφ ¼ δξJ: (63)

In terms of these two functions the variation (61) (aftertaking the limit rc → ∞) reads

δΓjEOM ¼ 1

32πG

Z∂M

d2x

�hττδð2ξ0Jτ þ ξMÞ

þ�hð1Þττ ∂τhrφ

hττþ hrτ∂φ ln hττ − 2∂φhrτ

− 2hτφ þ hrφ∂τhð1Þφφ

�δξJ

�: (64)

For reasons explained already, our focus is exclusivelyon zero-mode solutions. In that case we can drop allderivative terms in the response functions and the sourcefunctions ξL;M. Then the variation (64) establishes ourmain result

δΓjEOM ¼ 1

32πG

Z∂M

d2xðhττδξM − 2hτφδξJÞ: (65)

We give the two response functions appearing in thevariation (65) suggestive names

M ¼ hττ8G

J ¼ hτφ4G

(66)

in terms of which we can rewrite our main result (65) as

δΓjEOM ¼ 1

2

Z∂M

d2x

�M2π

δξM −J2π

δξJ

�: (67)

The overall factor 12is the same as in the definition of the

stress tensor (23). The factors 2π in the denominators areexactly as in (24). The signs are adjusted suitably forEuclidean signature.We evaluate now the response functions for flat space

ds2 ¼ dτ2 þ dr2 þ r2dφ2 (68)

and find

Mflat ¼1

8GJflat ¼ 0: (69)

The expressions (69) coincide precisely with mass andangular momentum of flat space [28,31].Similarly, we calculate the response functions for flat

space cosmologies

ds2 ¼ r2þ

�1 −

r20r2

�dτ2 þ dr2

r2þð1 − r20

r2Þþ r2

�dφ −

rþr0r2

�2

(70)

and find

MFSC ¼ r2þ8G

JFSC ¼ −rþr04G

: (71)

Again the expressions (71) coincide precisely with massand angular momentum of flat space cosmologies [28,31],as well as with the thermodynamical expressions wederived in Sec. III C.In conclusion, we recover as one-point functions for

zero-mode solutions precisely the response functionsexpected from previous canonical and thermodynamicalanalyses. This further confirms the validity of the action(44) with one-half of the usual GHY boundary term.

IV. OUTLOOK

After reviewing how to obtain a well-defined variationalprinciple for Einstein gravity with (Euclidean) AdS3boundary conditions and recovering known results forzero- and one-point functions we applied the same methodsto flat space boundary conditions. One of our main resultsis that the bulk action has to be supplemented by one-halfof the GHY boundary term. We exploited this result torecover thermodynamics of flat space cosmologies and tocalculate the one-point functions for zero-mode solutions.We stress that the factor 1

2in the boundary term was crucial

to obtain the correct free energy [38], since without thisfactor either the first law does not hold or the Bekenstein-Hawking relation is violated. Recent work that overlapswith ours [43,44] does not use this boundary term butinstead takes the AdS results for one-point functions andperforms a suitable limit of vanishing cosmological con-stant. The results of these papers for the response functionsagree with our corresponding results, which were directlycalculated in flat space. Thus, our results provide a proof ofthe smoothness of zero- and one-point functions in the limitof AdS with vanishing cosmological constant.A technical reason for our restriction to zero-mode

solutions was that the transformation between EF anddiagonal gauge only works straightforwardly for suchsolutions. It would be interesting to generalize the dis-cussion and to describe nonzero mode solutions inEuclidean signature. While our Euclidean boundary con-ditions allow also φ-dependent functions in (40) consistentwith a variational principle, the relation of such nonzeromode configurations to configurations in EF gauge is notclear to us. We leave this issue as an open problem, and justmention that its resolution probably requires us to set upboundary conditions that are valid both at lightlike infinityand at spatial infinity. Physically, the restriction to zero-mode solutions was sufficient for our purposes, namely tocheck the consistency of the free energy and the one-pointfunctions for flat space cosmologies.

ACKNOWLEDGMENTS

We are grateful to Arjun Bagchi for a pleasant andfruitful collaboration on related topics. We thank Tomás

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Andrade, Glenn Barnich, Geoffrey Compère, RezaFareghbal, Don Marolf, Ali Naseh, IoannisPapadimitriou, Alfredo Perez, David Tempo and RicardoTroncoso for discussions. S. D. is a Research Associate ofthe Fonds de la Recherche Scientifique F. R. S.-FNRS(Belgium). D. G. and F. S. were supported by theSTART project Y 435-N16 of the Austrian Science Fund(FWF) and the FWF projects I 952-N16 and I 1030-N27.The work of J. S. was partially supported by theEngineering and Physical Sciences Research Council(EPSRC) (Grant No. EP/G007985/1) and the Scienceand Technology Facilities Council (STFC) (GrantNo. ST/J000329/1).

APPENDIX: LINEARIZED SOLUTIONS INFLAT SPACE EINSTEIN GRAVITY

Any solution to the linearized Einstein equations in threedimensions must be locally pure gauge.

ψμν ¼ ∇μξν þ∇νξμ ¼ ∂μξν þ ∂νξμ − 2Γαμνξα (A1)

For a Euclidean background in polar coordinates theonly nontrivial Christoffel symbols are Γr

φφ ¼ −r andΓφ

rφ ¼ Γφφr ¼ 1=r. Since transverse-traceless gauge does

not lead to sensible results in flat space [31] we imposeinstead axial gauge on the fluctuations.

ψ rτ ¼ ψ rφ ¼ 0 ψ rr ¼ −ψττ (A2)

The last condition makes sure that the relation between ψ rrand ψττ required by our boundary conditions (42) remainsintact. Globally, linearized solutions of the form (A1) arenot necessarily pure gauge, since they can have nontrivialcanonical boundary charges. Fluctuations ψ that are puregauge even globally will be referred to as “small gaugetransformations.”To construct the most general solution to the linearized

EOM, we are thus left to find the most general vector field ξpreserving the axial gauge (A2). This gives rise to threeconditions that can be fully integrated. Preserving ψ rr þψττ ¼ 0 gives rise to

∂rξr þ ∂τξτ ¼ 0: (A3)

Preserving ψ rτ ¼ 0 requires

∂rξτ þ ∂τξr ¼ 0: (A4)

Both conditions together imply

ξτ ¼ ξ0τðφÞ þ ξþðrþ τ;φÞ þ ξ−ðr − τ;φÞ (A5)

ξr ¼ ξ0rðφÞ − ξþðrþ τ;φÞ þ ξ−ðr − τ;φÞ: (A6)

Finally, preservation of the gauge condition ψ rφ ¼ 0 yields

∂rξφ þ ∂φξr −2

rξφ ¼ 0: (A7)

This differential equation is solved by

ξφ ¼ r2ξ0φðτ;φÞ þ ~ξφ; (A8)

where ~ξφ is a particular solution of (A7), whose preciseform depends on the functions ξ� and ξ0r . The functionξ0φðτ;φÞ arises as a homogeneous solution and will play animportant role for turning on sources.Any solution for the vector field ξ above then leads to a

solution for the linearized fluctuations ψ :

ψττ ¼ −ψ rr ¼ 2∂τξτ (A9a)

ψτφ ¼ ∂τξφ þ ∂φξτ (A9b)

ψφφ ¼ 2ð∂φξφ þ rξrÞ (A9c)

ψ rτ ¼ ψ rφ ¼ 0: (A9d)

Let us consider now solutions to the linearized EOMcompatible with our boundary conditions (41). Then thefunctions ξ� above can be at most linear in r� τ for largevalues of r.

ξτ ¼ ξ0τðφÞ þ rξ0ðφÞ þ τξ1ðφÞ þOð1=rÞ (A10)

ξr ¼ ξ0rðφÞ − rξ1ðφÞ − τξ0ðφÞ þOð1=rÞ (A11)

Solving the differential equation (A7) yields

ξφ ¼ r2ξ0φðτ;φÞ þ r2 ln rðξ1Þ0 − τrðξ0Þ0 þ rðξ0rÞ0 þOð1Þ:(A12)

Compatibility with periodicity in φ and the boundaryconditions (41) requires ∂τξ

0φ ¼ 0, ∂φξ

0φ ¼ ξ1 and

ðξ1Þ0 ¼ 0, so with no loss of generality we set ξ0φ ¼ φξ1

(we set to zero an additive constant to ξ0φ since it does notcontribute to the metric). There are no further conditionsemerging from (41).Thus, the most general result for the vector field

compatible with periodicity, gauge and boundary condi-tions of the metric is given by

ξτ ¼ ξ0τðφÞ þ rξ0ðφÞ þ τξ1 þOð1=rÞ (A13)

ξr ¼ ξ0rðφÞ − rξ1 − τξ0ðφÞ þOð1=rÞ (A14)

ξφ ¼ φr2ξ1 − τrðξ0Þ0 þ rðξ0rÞ0 þOð1Þ: (A15)

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Inserting this result into (9) establishes the normalizablesolutions6 to the linearized EOM:

ψττ ¼ −ψ rr ¼ 2ξ1 þOð1=rÞ (A16a)

ψτφ ¼ Oð1Þ (A16b)

ψφφ¼−2τrðξ0þðξ0Þ00Þþ2rðξ0rþðξ0rÞ00ÞþOð1Þ (A16c)

ψ rτ ¼ ψ rφ ¼ 0: (A16d)

If we preserve the linearized on-shell property but violateour boundary conditions, we can generate non-normal-izable modes. In particular, if we further turn on thehomogeneous solution ξ0φðτ;φÞ in (A8), vector fields ofthe form

ξτ ¼ OðnormalizableÞ (A17)

ξr ¼ OðnormalizableÞ (A18)

ξφ ¼ r2ξ0φ þOðnormalizableÞ (A19)

lead to specific non-normalizable solutions:

ψττ ¼ −ψ rr ¼ 2ξ1 þOð1=rÞ (A20a)

ψτφ ¼ r2∂τξ0φ þOð1Þ (A20b)

ψφφ ¼ 2r2∂φξ0φ þOðτrÞ (A20c)

ψ rτ ¼ ψ rφ ¼ 0: (A20d)

In Sec. III D we show that the non-normalizable sol-utions (20) appear as sources for the correspondingoperators in the one-point functions.

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6As stressed in Sec. III D, in this work we defined normal-izable modes to mean solutions to the linearized EOM compatiblewith our boundary conditions. Normalizability also has a defi-nition in terms of the symplectic norm of the correspondinglinearized solution, defined as

ðψ ;ψÞ ¼ −iZΣt

nαωαðψ ;ψ�; gÞ;

where Σt is a spacelike surface (here t ¼ const), nα its normal,and ωα the symplectic current (see e.g. [45]). From the asymp-totic behavior (16), one can check that the contribution to thenorm from the r ¼ ∞ boundary is finite (actually, vanishes),which is expected for linearized perturbations satisfying theboundary conditions (42).

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