Radiation in AdS and Flat Holography

Gabriel Arenas-Henriquez
Yau Mathematical Sciences Center, Tsinghua University
In coll bor tion with Felipe Di z (ITMP, Moscow), Weizhen Ji (CUHK, Chin ), nd D vid River -Bet ncour (ITMP, Moscow)
rxiv:2505.xxxx

The University of Edinburgh

May 5th, 2025

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 Motivation
• AdS/CFT correspondence is well-est blished p r digm th t rel tes gr vity in n nti-de
Sitter sp ce in d + 1 dimensions with conform l ield theory t its d− dimension l
bound ry Maldacena 97; Gubser, Klebanov, Polyakov 98; Witten 98

• It provides hologr phic du lity: bulk dyn mics is fully c ptured by qu ntum ield theory
without gr vity on the bound ry.

Successes:

• Non-perturb tive de inition of qu ntum gr vity in AdS.

• Re lis tions of strongly coupled QFTs, bl ck hole thermodyn mics, ent nglement,
tr nsport, etc.

C n we formul te hologr phic du lity in sp cetimes th t re not symptotic lly AdS?

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 • Fl t Hologr phy: Aims to gener lise the hologr phic principle to symptotic lly l t
sp cetimes.

Key ch llenges:

• No timelike bound ry where is the du l theory pl ced?

• No conform l bound ry metric like in AdS wh t repl ces the role of the CFT?

• Absence of r di l hologr phic direction hologr phic RG low interpret tion?

Luckily, some of these questions h ve lre dy been nswered (or p rti lly nswered)
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 Current ppro ches:

• C rrolli n Hologr phy: The bound ry of symptotic lly l t sp ces is the null in inity - its
geometry is given by degener te metric -> C rrolli n m nifold
• [Barnich-Troessaert 10, Barnich-González 13, Hartong 16, …]

• Exploit the properties of its symptotic symmetry group: Bondi-v n der Burg-Metzner-S chs
(BMS) group

4 ∼ℭ 3
[Duval-Gibbons-Horvathy 14]

• T king the ultr rel tivistic limit c → 0 of CFT le ds to C rrolli n ield theory
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 Celesti l Hologr phy: Du lity between the gr vit tion l S-m trix of four-dimension l
symptotic lly l t sp ce nd correl tors in conform l ield theory living on the celesti l
2
sphere S loc ted t the sp ti l sections of the null in inity [Strominger 14]
[Pasterski-Pate-Raclariu 21]
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 • These ppro ches re rel ted s the sourced W rd identities of conform l C rrolli n ield
theory living t null in inity reproduce the BMS lux-b l nce l ws
[Donnay-Fiorucci-Herfray-Ruzziconi 22]
[Bagchi-Banerjee-Basu-Dutta 22]

• Correspondence between l t limit in AdS nd ultr rel tivistic limit t the bound ry
[Barnich-Gomberoff-Gonzalez 12]
ℓ→∞↔c→0 [Bagchi-Detournay-Fareghbal-Simon 12]
[Ciambelli-Marteau-Petkou-Petropoulos-Siampos 18]
[Campoleoni-Delfante-Peka-Petropoulos-Rivera Betancour-
Vilatte 23]
[Alday-Nocchi-Ruzziconi-Yelleshpur Srikant 24]

• Still lot to underst nd on the Fl t/C rrolli n-Celesti l limit from AdS/CFT correspondence
[de Boer-Hartong-Obers-Sybesma-Vandoren 23]
[Poulias, Vandoren 25]
[Lipstein-Ruzziconi-Yelleshpur Srikant 25]
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 Gr vit tion l r di tion pl ys n import nt role in l t hologr phy!

C rrolli n: It rel tes to non-equilibrium phenomen in the bound ry C rrolli n luid

[Ciambelli-Marteau-Petkou-Petropoulos-Siampos 18]
[Fiorucci-Pekar-Petropoulos-Vilatte 25]
Celesti l: Soft theorems, memory e ects, nd Celesti l W rd Identities

Two guiding questions:

• How do we ch r cterise r di tion in AdS?

• Is the l t sp ce limit of ‘’r di tion`` in Ads comp tible with l t hologr phy?

-> C n we recover celesti l oper tors nd C rrolli n dyn mics from AdS?
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 Radiation in Asymptotically Flat Spacetimes

• In l t sp ce, gr vit tion l w ves re un mbiguously de ined s the

r di tive modes t null in inity

• To study r di tion system tic lly, we use the Bondi g uge

( )( )
2 β V 2 2β A 1 A B 1 B
ds = e du − 2e dudr + gAB dΘ + U du dΘ + U du
r 2 2

• Null geodesics re gener ted long the ine p r meter r

• Extr g uge condition ∂rdet ( 2 ) = 0

gAB
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• Asymptotic lly, for l rge r, we h ve the following f llo conditions

2 2 2
2m B 2 2 z
ds = − du − 2dudr + 2r γzz̄dzdz̄ + du + rCzzdz + D Czzdudz
r

r [3 ]
1 4 1
( Nz + u∂zmB) − ∂z(CzzC ) + cc + …
zz
+
4

mB = mB(u, z, z̄) : Bondi M ss

Czz = Czz(u, z, z̄) : She r

Nz = Nz(u, z, z̄) : Angul r Momentum Aspect

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 +
The she r tensor CAB is free d t t ℐ , nd it cont ins inform tion bout r di tion
through the Bondi news tensor

Nzz = ∂uCzz

• C n be interpreted s the gr vit tion l n logue of the ield strength Fuz = ∂u Az

+
• The tot l energy lux cross ℐ is

dE 1
32πG ∫S2
AB
= NABN dΩ
du

• Non-zero news ⇒ non-zero gr vit tion l r di tion

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 Gr vit tion l w ves c use tr nsverse tid l deform tions,
governed by the geodesic devi tion equ tion:
2 A
d ξ 1 A B
= − NB ξ
du 2 2
Wh t detector sees:

• If NAB ≠ 0, detector records oscill tory stretching nd

squeezing of the sp cetime —this is the physic l imprint of
gr vit tion l w ve.
• The integr ted e ect of burst of r di tion gives the displ cement memory
— gr vit tion l memory e ect
uf

∫u
ΔCAB = duNAB(u)
i [Braginsky-Thorne 87, Christodoulou 91, …]

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 These me surements re directly tied to symptotic ch rges nd symmetries in the
BMS fr mework:

• The gr vit tion l memory e ect corresponds to l rge supertr nsl tion
rel ting the v cuum before nd fter r di tion p sses.

• The ch nge corresponds to the soft sector of gr vity — encoded hologr phic lly
in Celesti l mplitudes nd memory observ bles. [Strominger-Zhiboedov 14]

Direct me surement of the gr vit tion l memory effect m y be possible in the coming dec des
[Van Haasteren-Levin 10, Lasky-Thrane-Levin-Blackman-Chen 16, Einstein Telescope Collaboration 25]
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 A novel characterisation of radiation
[Fern ndez-Álv rez nd Senovill 19,20,21,22] proposed geometric ch r cteris tion of gr vit tion
r di tion for symptotic lly l t nd de Sitter sp cetimes.

This novel propos l provides criterion b sed on geometric construction th t utilises conform l
2
comp cti ic tion techniques, introducing non-physic l metric g ̂ =Ωg

Using this metric, we c n construct resc led version of the Bel-Robinson tensor

̂ = ŵ β α β α
μνρσ ρμαŵ νσβ + *ŵ ρμα * ŵ νσβ ,

β
where ŵ ρμα ̂ β−1
= Ω W ρμα is resc led version of the Weyl tensor
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 From here, we de ine the super Poynting vector s

̂μ=− ̂μ n ̂ α β γ
n ̂ n ̂
αβγ
μℬ
where nμ̂ = ∂μΩ is norm l to the bound ry, s tisfying nμ̂ n ̂ = − Λ/3

• In bscence of m tter, the super Poynting vector is conserved t the bound ry ∇̂ μ ̂ = 0

μℬ

Wh t inform tion is encoded in this qu ntity?

• In l t sp ce, using Bondi coordin tes, one c n check th t the le ding order is

̂ r = 2∂ N ∂ N AB ̂ u = 2D N C D N AB ̂ A = − 4∂ N ABD N C
u AB u C A B u C B

NAB = 0 ⇒ ̂ = 0
ℐ μ ℐ
• Therefore, in l t sp ce
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 Criterion:

Absence of gr vit tion l r di tion t ℬ if nd only if

̂ =0
μ ℬ

• This criterion h s been tested for known solutions in l t sp ce (within Petrov

cl ssi ic tion) greeing with previous results.

C n we extend this de inition to AdS?

If so, wh t inform tion is c ptured by this qu ntity?

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 Radiation in AdS
[Ci mbelli-P sterski-T bor 24] proposed to extend the fr mework to neg tive cosmologic l
const nt
Using Fe erm n-Gr h m exp nsion, we c n rewrite the super Poynting in terms of hologr phic
d t

̂ = − 4C i T jk = ϵ ijk[ , T]
i ℬ
jk jk

Bound ry Cotton tensor Hologr phic stress tensor

• For non-trivi l Poynting vector in AdS, we need non-conform l bound ry!

• Addition lly, the Cotton nd hologr phic stress tensors need to be line rly independent
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 Ex mple A: T ub-NUT-AdS
2 2 2 2
dsbdy = − (dt + 2n(1 − cos θ)dϕ) + ℓ dΩ
[Kalamakis-Leigh-Petkou 20]

ℓ ( ℓ )
2
n 4n
( )
(0)
Non-conform lly l t ij = 1 + 3ui uj + g With u i = δti
• 4 2 ij

M
( )
(0)
• Hologr phic stress tensor describes perfect luid ij = 3uiuj + gij
8πGℓ 2

• Energy-momentum/Cotton du lity ij ∝ ij ⇒ ̂ =0
iℬ
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 Ex mple B: Robinson-Tr utm n AdS

2 i j 2 2 dζd ζ̄
dsbdy = hijdx dx = − dt + 2ℓ
P 2

• Non-conform lly l t provided non-const nt G ussi n curv ture

( ij )
(0)
• Hologr phic stress tensor h s dissip tive corrections ij = ε 3ui uj + g + Π ij
[de Freitas-Reall 14]
[Bakas-Skenderis 14]
[Ciambelli-Petkou-Petropoulos-Siampos 17]

ℬ
• Non-v nishing Super Poynting ̂ i
≠0

Are dissipative e ects in uid/gravity related to radiation?

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 Fluid/gravity perspective
Fluid/Gr vity correspondence connects the dyn mics of long-w velength perturb tions
in AdS sp ce to the one of rel tivistic conform l luid de ined t the timelike bound ry
[Bh tt ch ryy -Hubeny-Minw ll -R ng m ni 07]

(0)
The conform l luid dyn mics is given by its b ckground geometry gij nd the stress
ij ij
tensor th t s tis ies ∇i =0

The luid energy-momentum tensor is decomposed long the velocity ield s

ε : Energy Density
( )
2 (0) 2 2
ij = ε + p ℓ u u
i j + pgij + τij + ℓ uiqj + ℓ ujqi
ij
τ : Viscous Stress
Conform l inv ri nce requires the energy momentum to be i
q : Heat Current
tr celess, hence implying the conditions 1
(0) i j
gij u u =− 2
ε = 2p , and τii = 0 ℓ
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 • The Cotton-York tensor c n lso be decomposed long the timelike congruence

2( )
c 1 (0)
ij = 3ℓuiuj + gij − ℓcij + ℓuicj + ℓujci
ℓ

• Here, c, ci nd cij correspond to Cotton density, Cotton current nd Cotton stress tensor,
respectively

Just s the bound ry energy-momentum tensor, the Cotton-York tensor ij is tr celess nd

cov ri ntly conserved, n mely
ij (0) ij
g(0) ij =0 ∇i =0
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Let us focus on the lgebr ic lly speci l solutions of the Petrov type.

The bound ry geometry s tis ies the following identities

1 ℓ2
σij = 0 , qi = *ci , and τij = − *cij .
8πG 8πG

[Ciambelli-Marteau-Petkou-Petropoulos-Siampos 18]

Using bulk reconstruction techniques, the line element written in the Newm n-Unti g uge is

( )
4
ℓ 8πGεr + cγ
( )
2 2 i 2 (0) 2 4 i j
ds = 2ℓ uidx dr + r gij + 2rℓ u(i Aj) + ℓ Sij + uiuj dx dx
r2 + γ2

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 We c n obt in the super Poynting vector in terms of luid v ri bles

1
( k )
̂ i = 3ℓ 2 (8πGεq i − c *q i) − 16πGℓ 2 2τ ijq + u i (τ klτ − ℓ 2q kq )
j kl
16πG
i ij
Cle rly, it v nishes when the luid is perfect, i.e., q = 0 nd τ = 0

In lgebr ic lly speci l solutions, dissip tive processes re m pped to the bulk sp cetime s
gr vit tion l r di tion.
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For sp cetimes outside Petrov cl ssi ic tion, the super Poynting is

1
( j) ( j k)
̂ i = 3ℓ 2 (ε *c i − c *q i) + 2ℓ 2 ℓ 2 *c ijq + *τ ijc + 2ℓ 3η ijk c jlτ + c q
j j lk
16πG

Even in the bsence of dissip tion, the super Poynting is non-v nishing

1 ̂ i = 3ℓ 2ε *c i .
16πG

According to this, the minimum necess ry condition for the solution to r di te is non-v nishing
bound ry Cotton current (with non-zero energy density).
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The condition of the Poynting to be divergence-free t the bound ry implies

̂ ̂
∇i = 0 i ℬ

[2 ]
1 3 4 1
(0) i
∇i S = (0)
∇i ε u + τ qj − u (τ − ℓ q ) +
2 i ij i 2 2 2 j
c *q − εT
εT 3 16πG

There is bound ry entropy production due to bulk r di tion

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Flat Limit
• In order to comp re with the l t sp ce result, we c n consider the l t limit of the super
Poynting of AdS. To this end, we will m ke use of the Λ− BMS g uge
[Compère-Fiorucci-Ruzziconi 19]

V
ds = e du − 2e dudr + gAB (dΘ − U du) (dΘ − U du)
2 β 2 β A A B B
r

( )
2 1 1 1 1
gAB = r qAB + CAB + 2 DAB + 3 EAB + 4 FAB + …
r r r r
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Solving Einstein’s equ tions order by order in the symptotic exp nsion

A 1 AB
β = β0(u, x ) − C CAB + …
32r 2

(2 )
A A B 2 2β0 A 1 2β0 1 AB AB
U = U0 (u, x ) + e ∂ β0 − 2 e DBC + C ∂B β0
r r

[ ( 3 ) ]
2 2β A 1 AB C 1 AB 3 CD A
− e 0 N − C D CBC + ∂B β0 + DB − CCDC ∂ β0 + …
3 2 16

V Λ 2β 2
= e r − r (DAU0 + ℓ)
0 A ̂
r 3

[2 ]
2β0 1 Λ AB A A 2m
−e R[q] + CABC + 2DA∂ β0 + 4∂Aβ0∂ β0 + +…
16 r
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The tr celess symmetric tensor becomes free in the l t limit

Λ
[( 0) A]
CAB = e −2β0
∂ − ℓ ̂ − D U C
q + D U 0
+ D U 0
u C AB A B B
3

The bound ry structure

(3 )
2 Λ 4β0 A 0 2 0 A A B
ds(0) = e + U0 UA du − 2UAdudx + qABdx dx ,

0 ∘
In the l t limit (Λ → 0), the bound ry g uge freedom U = 0 , β0 = 0 , q= q
renders the line element degener te.

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 There is di eomorphism order by order in the hologr phic coordin te to comp re with the
Fe erm n—Gr h m g uge

4 (Λ) 2 (Λ)
3|Λ| −3m − 3 NB
Tij = ,
16πG 2 (Λ)
− 3 NB 2 (Λ)
m qAB + JAB
Λ

1
( )
(Λ)
mB = mB + ∂ + ℓ ̂ C C AB
AdS Bondi M ss
u AB
16

( ) 4 (Λ )
(Λ) 3 B 1 ̂ 3 1 3 CD AdS ngul r
NA = NA − D NAB − ℓCAB − ∂A R[q] − C CCD
2Λ 2 8 momentum spect

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 Then, we c n use the hologr phic stress tensor expressed in terms of Λ− BMS qu ntities nd
exp nd in powers of the cosmologic l const nt before t king the l t limit

(Λ) (Λ) (Λ)

3 2 2
3 3 3
̂u= ̂u (Λ ) , ̂A= (Λ )
AB −1 AB C −1
2∂uN ∂uNAB + (−2) + 2∂uN DC N B +

[Ciambelli-Pasterski-Tabor 24]

Λ→0 ( 3 )
0 0
(0 − 16πG ∂uNAB)
|Λ| 2
lim Tij = 1

1
0 0
Λ→0 ( 3 )
ij = ( C ) (0 − 2 NAB)
|Λ| 2
0 0
lim lim ij = 1
0 −ϵAC∂uN B Λ→0

NAB = 0 ⇒ ̂ = 0
ℐ μ ℐ
We c n see th t it is consistent with the l t sp ce de inition!
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 Carrollian luid structure

Consider the P p petrou-R nders p r meteris tion

( )
(0) i j 1 A 2 A B
gij dx dx = − 2 Ωdu − bAdx + aABdx dx
ℓ
• Under C rrolli n di eomorphisms, Ω tr nsforms s sc l r, bA tr nsforms s connection,
nd aAB tr nsforms s r nk two tensor

i 1 i 1
• We lso consider the bound ry luid velocity u = ∂ u to be t rest, s tisfying u ui = −
Ω ℓ2
2 A B
ds = aABdx dx : Degener te metric
After t king the l t limit Carroll
1
υ = ∂u : C rroll vector
Ω
A
μ = − Ωdu + bAdx : clock-form with bA being the Ehresm nn conn.
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 • We w nt to obt in the C rrolli n exp nsion of the super Poynting for lgebr ic lly speci l
solutions.

• Exp nding for sm ll cosmologic l const nt (l rge ℓ), we ind the following exp nsion of the
he t current nd viscous stress

A A 1 A AB 2 AB AB
q = Q + 2π , τ =−ℓ Σ −Ξ
ℓ

1 1
With the C rroll he t currents given by QA = *χ A , πA = *ψ A
8πG 8πG

AB 1 AB AB 1 AB
And C rroll viscous stress Σ = *X , and Ξ = *Ψ
8πG 8πG

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 Exp nding the super Poynting round the l t limit

1 ̂ 4 ̂ (−4) 2 ̂ (−2) ̂ (0) 1 ̂ (2)

u = ℓ u + ℓ u + u + u
Ω ℓ 2

̂ A = ℓ4 ̂ A 2 ̂A + ̂A + 1 ̂A
(−4) + ℓ (−2) (0) (2)
ℓ 2

We ind the le ding terms to be

̂ (−4) = 16πGΣABΣ , ̂ (−2) = 16πG (2Σ ΞAB − Q AQ )

u AB u AB A

̂A = 32πGΣ Q AB
(−4) B

The le ding beh viour is rel ted to the C rrolli n energy lux -> non-conserved qu ntities
[Ciambelli-Marteau-Petkou-Petropoulos-Siampos 18]
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
𝒫
32 [Campoleoni-Delfante-Peka-Petropoulos-Rivera Betancour-Vilatte 23]
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Comp ring with the result in the Λ -BMS g uge, we propose the following C rrolli n version of
the Bondi news tensor

AB 1 ¯ AB
Σ ∝ u
Ω
𝒟
𝒩
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 AdS C-Metric
As n ex mple, we c n consider n cceler ting bl ck hole in AdS
[Kinnersley-Walker 70]
[Plebanski-Demianski 76]
[Hong-Teo 03]
¯
( Σ̄(x̄) )
2 2
2 1 ¯ 2 dr̄ 2 d x̄ 2
ds = − f(r̄)dt̄ + + r̄ + Σ̄(x̄)dȳ
Ω̄(r̄, x̄)2 ¯
f(r̄)

Using the Newm n-Penrose form lism, it h s been

Ω̄(r̄, x̄) ≡ 1 + Ar̄x̄
shown th t it is r di tive solution for ny v lue of
the cosmologic l const nt
2
Σ̄(x̄) ≡ (1 + 2mAx̄)(1 − x̄ ) […, Bicak-Krtous 02, Podolsky-Ortaggio-Krtous 03, Fernández-Álvarez-
Podolsky-Senovilla 24]

(3 )
2m Λ Slow cceler tion regime Aℓ < 1 , only one bh horizon
¯f(r̄) ≡ 1 − − r̄ 2 2 2
+ A + 2mA r̄
r̄
R pidly cceler ting regime Aℓ > 1 bh horizon + cc horizon
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 In order to obt in hologr phic qu ntities, we put the metric in the Weyl—Fe erm n—Gr h m
g uge [AH-Cistern -Di z-Gregory 23]
[AH-Di z-River -Bet ncour 24]

( z̄ )
2
1 i n 2 2 2 dz̄ i i j
r̄ ∑
= ζ(n)(x̄ )z̄ ds = ℓ H − kid x̄ + hijd x̄ d x̄
n≥0

2 2
ℓ Ω f 1
(2Aδi − z̄ ∂iα)
−2 i n x̄ 2
(z̄ + αz̄ ) ∑
H = ζ(n)(x̄ )z̄ , ki =
2 2
n≥0
2z̄(1 + αz̄)

(0)
ki = αi Weyl Connection

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 The bound ry is non-conform lly l t nd the stress tensor is non-perfect

( i j γij ) diag (0, −Σ )

2 4
(0) ϵ (0) 3mA ℓ 1 2
Tij = Tij + Πij = 2
3u u + + 16πG
,
1 − A 2ℓ 2Σ
Perfect Fluid part Dissipative corrections

We ind th t the Poynting vector is non-v nishing for ny v lue of the cceler tion p r meter

( ) ( )
2 2 2 2 2
9Am Σ 1 − A ℓ Σ 1 − 2A ℓ Σ
̂ i = − 4C i T jk = δ i
jk x̄
2πG
• There is no bound ry entropy production, in greement with [Ci mbelli-Petkou-Petropoulos-
Si mpos 17]
(0) i
∇i S =0
𝒫
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To c st the metric in the Λ-BMS g uge, we propose the following di eomorphism which llows to
solve the coe icients order by order ne r the bound ry

A −n
∑
t̄ = U(n)(u, x )r
n≥0
1 A −n
∑
= r̃ = R(L)r + R(n)(u, x )r
z̄ n≥0

A A A −n
∑
x̄ = X(n)(u, x )r
n≥0

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 (r ) (r )
−4 −3
With these functions, we ind th t grr = grA =

(r )
∘
detgAB = detqAB + −3
β0 = 0 = ℓ̂

Addition lly, we obt in

·2
(Λ) 3Υ
m = mB+ ·
32Υ 2
(Λ + 3A 2Υ
)

(Λ) 3 B
NA = − D NAB + NA
2Λ

3
(Λ )
−2
JAB = − ∂uNAB +
Λ
𝒳
𝒪
𝒪
𝒪
𝒪
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As expected from r di tive solution, in the l t limit, we ind non-trivi l she r nd news tensor

[ ]
·
( )
1 2
Υ Υ∂ x − 1 ∂x
CAB = ·
Υ2
2A ∂x 1

This llows us to construct the Celesti l oper tors ssoci ted with the C-metric

One c n comp re with l t sp ce comput tion by [Strominger nd Zhiboedov 16], where they
provide physic l interpret tion for superrot tions in the presence of topologic l defects.
𝒴
𝒳
𝒴
𝒴
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 Final remarks
Gr vit tion l r di tion in AdS sp cetimes o ers powerful window into the dyn mics of
hologr phy beyond equilibrium.

Cruci lly, underst nding r di tion in AdS m y sh rpen our intuition nd strengthen the
conceptu l bridge to l t sp ce hologr phy

T ke- w y mess ges:

1. The Super-Poynting vector o ers consistent nd cov ri nt criterion for identifying gr vit tion l r di tion in AdS
sp cetimes.

2. This notion is supported by well-known ex mples, including lgebr ic lly speci l solutions nd the AdS C-metric.

3. 3. Dissip tive corrections in the bound ry luid encode r di tive bulk degrees of freedom

4. The l t limit of r di tive AdS solutions n tur lly connects to celesti l nd C rrolli n hologr phy, suggesting
uni ied picture of r di tion cross symptotics.

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 Future Work

• Poynting vector through n improved ction principle?

• A more robust criterion for r di tion in AdS?

• Check more gener l b ckgrounds with bound ry entropy production

• Rel te the lgebr nd OPEs of Energy Detector Oper tors with contr ctions of the AdS
symptotic symmetry group

• A generic m p between NU nd Λ-BMS g uge

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Thanks for listening!

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