Local Quantum Sufficient Conditions for Fully Nonlinear Gravity Equations: Small-Diamond Generalized Entropy Extrema, Relative Entropy Foliation Independence, and QNEC Pointwise Saturation

Local Quan um Sucien Condi ions o Fully Nonlinea G a i y
Equa ions:
Small-Diamond Gene alized En opy Ex ema, Rela i e En opy
Folia ion Independence, and QNEC Poin wise Sa u a ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
In he poin wise small causal diamond limi , his pape p oposes h ee
comple ely local
and
sucien
quan umgeome ic c i e ia,
igo ously
de i ing ully nonlinea g a i y equa ions
wi h cosmological cons an wi hin semiclassicalholog aphic windows. The h ee c i e ia a e:
(A) Small-diamond
gene alized en opy ex emum
unde xed eec i e olume/con o mal
Killing ene gy cons ain ; (B) Bounda y ela i e en opy
olia ion independence
i and only
i bulk Iye Wald
canonical ene gy conse a ion
( he eby yielding
quan um Bianchi iden-
i y
and i s sou ced o m); (C)
QNEC poin wise sa u a ion
o
all local cu su aces
and
all null di ec ions
h ough a poin . We p o e: Wi hin Hadama d s a es, no g a i a ional
anomaly, and non-nega i e canonical ene gy coupling windows, any one o he abo e c i e ia
(supplemen ed by echnical assump ions specied he ein)
sucien ly
implies
Eg a
ab = 8πG en ⟨T o
ab ⟩+ϕ gab,∇bϕ= 0,
hus inco po a ing
ϕ
in o cosmological cons an
Λ
. Co e echnical ools include: (i)
Volume
Hamil onian
O( d)
equi alence heo em
(P oposi ion K.1), i.e., xed
Ve
ξ⇔
xed
Hξ

holds uni e sally in Eins einHilbe and
(R)
p o o ypes; (ii)
Two-cap bounda y ke nel
OD′( d+2)
dis ibu ional cancella ion heo em
(Theo em J.1); (iii) Exis enceuniqueness
egula i y o
quan um es ep esen a i e su ace
and
O( d+2)
cons ain on a ea second-
o de o mula; (i )
Cohomological in a iance
o JKM shi s and co ne co ec ions; ( )
Con ac ion mapping in eg abili y lemma
unde De Donde gauge and nea sa u a ion
⇒
nea equa ion s abili y inequali y. FRW and AdS
3
/CFT
2
ins ances plus execu able speci-
ca ions o  ela i e en opy ux me e / QNEC sa u a ion phase diag am a e p o ided.
1 In oduc ion
En anglemen  s law and ball- egion amily me hods ha e es ablished in o ma ion o geome y
ounda ions a
linea /second-o de
le el. To close a
poin wise
and
ully nonlinea
le el
equi es econciling h ee ypes o cons ain s:
equilib ium
(en opy ex emum),
conse a ion
(canonical ene gy conse a ion), and
igidi y
(QNEC sa u a ion). This pape heo emizes hese
h ee cons ain ypes in poin wise small causal diamond
Dp,
limi , p o ides
comple e p oo s
and e o con ol
, and uni o mly de i es nonlinea eld equa ions wi h
Λ
.
1
2 Se ing, No a ion, and Assump ions
2.1 Small Causal Diamond and App oxima e Con o mal Killing Field
Le
(M, g)
be a smoo h space ime wi h
d≥3
,
p∈M
. Deno e
Dp,
as small causal diamond o
scale
≪ℓcu
, bounda y composed o wo
C1,α
null lea es
N±
in e sec ing a wo co ne poin s.
Le small pa ame e
ε:= /ℓcu ≪1
. Take
app oxima e con o mal Killing eld
ξa
sa is ying
∇(aξb)−1
d(∇·ξ)gabL∞(Dp, )≤Cξε2, ξa∂Dp, = 0,
no malized by diamond empe a u e
κξ=κ0+O(ε2)
.
2.2 S a e, Reno maliza ion, and To al S ess
Take Hadama d s a e and causal p esc ip ion. Dene
⟨T o
ab ⟩:= ⟨Tab⟩+τen
ab , τen
ab := −2
√−g
δWnonloc
δgab ,
equi ing
∇a⟨T o
ab ⟩= 0.
Allowed local coun e e ms only edene
G en,Λ
(Appendix D).
2.3 QES and Quan um Res Rep esen a i e Su ace
Assume exis ence o unique and s able quan um ex emal su ace
Σ⊂∂Dp,
,
δSgen|Σ= 0
. Wi hin
equi alence class cons uc
quan um es ep esen a i e su ace
ˆ
Σ
wi h
θp=σp= 0,Zˆ
Σ
(θ2+σ2) dλ=O( d+2),
whose exis enceuniqueness egula i y see Appendix H.
2.4 Co a ian Phase Space and Canonical Ene gy
Le
L(g,
cu a u e
, . . .)
be smoo h local,
Eg a
ab := 2
√−g
δ
δgab R√−gL
. By dieomo phism in a iance
∇aEg a
ab ≡0
. Iye Wald s uc u e yields symplec ic po en ial
θ
, symplec ic cu en
ω
, Noe he
cha ge
Qξ
, and co ne e m
Cξ
. Dene
δHξ=ZΣ
(δQξ−ξ·θ−δCξ),Eξ(δ1, δ2) = ZΣ
ω(δ1,Lξδ2).
JKM shi s
θ→θ+dY
,
Qξ→Qξ+ξ·Y
and co ne co ec ions o m equi alence class (Appendix
F).
2
2.5 QNEC and Second-O de De o ma ion
Fo any null di ec ion
ka
and local cu su ace amily, QNEC eads
(√h)−1S′′
ou ≤2π⟨Tkk⟩.
On ep esen a i e su ace, Raychaudhu i yields
(√h)−1d2A
dλ2=−Rkk −θ2
d−2−σ2.
3 Main Resul s (S a emen s)
Theo em 1
(A: Gene alized En opy Ex emum
⇒
Nonlinea Tenso Equa ion)
.
Unde assump-
ions o 2, i
Σ
is QES o
Dp,
wi h
Sgen
ex emal unde xed
Ve
ξ
 (o equi alen ly xed
Hξ
)
cons ain , hen o all null di ec ions
ka
h ough
p
,
Eg a
kk (p) = 8πG en ⟨T o
kk (p)⟩.
Fu he mo e, he e exis s dis ibu ion
ϕ
such ha
Eg a
ab (p) = 8πG en ⟨T o
ab (p)⟩+ϕ(p)gab(p),∇bϕ= 0.
Theo em 2
(B: Folia ion Independence
⇔
Canonical Ene gy Conse a ion; Quan um Bianchi)
.
I
bounda y ela i e en opy
Sbdy
el
is independen o Cauchy slice
Σs
, hen
∇aEg a
ab −8πG en⟨T o
ab ⟩= 0.
Wi h ex e nal ux o co ne injec ion, sou ce e m eme ges
∇aEg a
ab −8πG en⟨T o
ab ⟩=Jb, Jb=∇aδQξ,ab −(ξ·θ)ab −δCξ,ab,
wi h
Jb
in a ian unde JKM shi s and co ne co ec ions.
Theo em 3
(C: QNEC All-Di ec ion Poin wise Sa u a ion
⇒
Nonlinea Closu e; Nea -Sa u a ion
S abili y)
.
I in neighbo hood o
p
o all local cu su aces and null di ec ions
ka
,
(√h)−1S′′
ou (p;k)=2π⟨Tkk(p)⟩,
and canonical ene gy non-nega i e, De Donde gauge, and
Hs
δ
(
s > d
2+ 2
,
−1< δ < 0
)
in eg abili y lemma hold, hen
Eg a
ab (p) = 8πG en ⟨T o
ab (p)⟩+ϕ(p)gab(p),∇bϕ= 0.
I only
supk∆QNEC(p;k)≤ε
, hen he e exis cons an
C
and no m
X
such ha
Eg a
ab −8πG en⟨T o
ab ⟩−ϕgabX≤C ε.
3
4 P elimina ies: Small-Region Geome y, Eec i e Volume, and
Ke nel Expansion
4.1 G ayVanhecke Expansion
RNC yields
V(Bp, ) = Ωd−1
d
d1−R(p)
6(d+ 2) 2+O( 4), A(∂Bp, ) = Ωd−1 d−11−R(p)
6d 2+O( 4).
Dene
Ve
ξ(Dp, ) := ZDp, ∇aξad ol, δV e
ξ=−ZDp,
δgab∇aξbd ol + O( d+2).
4.2 Modula Hamil onian Local Ke nel and Two-Cap Bounda y Ke nel
Small- egion modula Hamil onian w i en as
KDp, = 2πZDp,
TabξadΣb+X
±ZN±
±Tkk dσ+O( d+2),
wi h
±=O( 2)
, unde mi o
R:N+→ N−
ha ing
+=− −◦ R
. Theo em J.1 p o es
bounda y e m dis ibu ionally
O( d+2)
cancels.
4.3 Co a ian Phase Space Iden i y and Co ne s
Fo any a ia ion
δ
and ec o eld
ξ
,
ω(δ, Lξ) = δjξ−dδQξ−ξ·θ,jξ:= θ(Lξ)−ξ·L.
Co ne po en ial
Cξ
eo ganizes co ne o al die en ials (Appendix F).
5 P oposi ion K.1: Fixed
Ve
ξ⇔
Fixed
Hξ
( o
O( d)
)
P oposi ion 4
(K.1, P ecise Ve sion)
.
I
L=LEH
o
L= (R) = R+αR2
, ake 2's app oxima e
CKV
ξ
. Then he e exis cons an s
p=−Λ/(8πG en)
and
C=C(d, |Rm|C1, Cξ)
such ha o all
solu ion space angen ec o s
δ
,
δHξ−κξ
2πδSg a +p δV e
ξ≤C d+2|δg|C1(Bp,2 )+|δψ|H1.
P oo (essen ials and cons an con ol).
(i) Bulkbounda yco ne decomposi ion.
By co a i-
an phase space and S okes,
δHξ=ZDp,
δjξ+Z∂Dp, δQξ−ξ·θ−δCξ.
W i e
jξ=√−g Eg a
ab ξadΣb+∇·(. . . )
, using
ξ|∂D = 0
and co ne co ec ions,
δHξ=ZDp,
δ(√−g Eg a
ab ξanb) + κξ
2πδSg a −p δV e
ξ+Rbd.
4
Remainde
Rbd
assembled by Theo em J.1 and Appendix F,
|Rbd| ≤ C1 d+2|δ|C1⊕H1
.
(ii) EH leading o de .
On ep esen a i e su ace
θ|p=σ|p= 0
,
R(θ2+σ2) = O( d+2)
supp ess
second-o de geome ic e ms; bulk e m scale
O( d)
imes
δEg a =O(1)
yields
O( d+2)
.
(iii)
(R)
co ec ions.
Wald en opy
δSg a =1
4G en RΣδ( ′(R) dA)
, RNC and Appendix B
yield
ZDp,
δ ′(R)≤C2 d+2|δg|C1,ZΣ
δ ′(R) dA≤C3 d+1|δg|C1.
Ex insic cu a u e mixed e ms supp essed o
O( d+2)
by ep esen a i e su ace es ima es.
Combining yields p oposi ion.
6 Theo em J.1: Two-Cap Bounda y Ke nel
O( d+2)
Dis ibu ional
Cancella ion
Theo em 5
(J.1)
.
Le
N±
be mi o null caps,
±∈C1,α
sa is y
±=O( 2)
,
+=− −◦ R
.
Hadama d s a e makes
Tkk ∈ D′(N±)
es ic along null su ace. Then o all
φ∈C1∩H1
he e
exis s cons an
C
such ha
ZN+
! +Tkkφ+! ZN−
! −Tkkφ≤C d+2 |φ|C1∩H1.
P oo .
Appendix E uses wa e- on se and mi o map
R
's
C1
de ia ion
O( )
as co e, yielding
|Tkk −R∗Tkk|H−1≤C |Tkk|H−1,
mul iplying by
| ±|C1=O( 2)
and measu e scale
O( d−1)
yields
O( d+2)
bound. Co ne coo -
dina ion wi h
Cξ
as o al die en ial doesn' ele a e o de .
7 P oo o Theo em A
Second-o de equilib ium and null equa ion.
On ep esen a i e su ace
ˆ
Σ
unde xed
Ve
ξ
(o
Hξ
) cons ain ,
0 = δ2Sgen =δ2Sg a +δ2Sou +δ2Sc .
P oposi ion K.1 ew i es cons ain con ibu ion as
κξ
2πδ2Sg a −p δ2Ve
ξ+O( d+2)
. Raychaudhu i
on
ˆ
Σ
gi es
(√h)−1δ2A=−Rkk +O( 2),
while QNEC and Theo em J.1 con ol
δ2Sou
. Combining yields
−1
4G en
Rkk + 2π⟨Tkk⟩+O( 2)=0,
hus
Eg a
kk = 8πG en⟨T o
kk ⟩.
5

Tenso iza ion and cons an inco po a ion.
By Appendix A dis ibu ional-le el enso iza-
ion lemma, he e exis s
ϕ
such ha
Eg a
ab −8πG en⟨T o
ab ⟩=ϕgab.
Using
∇aEg a
ab ≡0
and
∇a⟨T o
ab ⟩= 0
yields
∇bϕ= 0
, inco po a ed in o
Λ
.
8 P oo o Theo em B
Sou celess e sion.
Co a ian phase space iden i y yields
d
dsSbdy
el (s) = ZΣs
ω(δ, Lξδ) + Z∂Σs
(δQξ−ξ·θ−δCξ).
I olia ion-independen wi h no ux, igh side anishes, yielding
RΣsω(δ, Lξδ)=0
. Localiza ion
and using
∇aEg a
ab = 0
de i es
∇aEg a
ab −8πG en⟨T o
ab ⟩= 0.
Sou ced e sion and in a iance.
Wi h ux/co ne injec ion dene
Jb:= ∇ahδQξ,ab −(ξcθcab)−δCξ,abi,
yielding sou ced quan um Bianchi. Appendix F uses cohomology o p o e
Jb
in a ian unde
JKM shi s and co ne co ec ions.
9 P oo o Theo em C
(1) F om all-di ec ion QNEC sa u a ion o linea ke nel.
Fo all local cu su aces and
null di ec ions
k
, QNEC equali y and  s law yield
δ2Sbdy
el =Eξ(δ, δ)=0
. Non-nega i e canonical
ene gy implies ke nel equals all physical pe u ba ions, hus all null di ec ion linea cons ain s a e
equali ies.
(2) Nonlinea closu e.
Unde De Donde gauge and
Hs
δ
, w i e nonlinea equa ion as
h=L−1S −N(h),
Appendix L p o ides con ac ion mapping and unique xed poin , hus
Eg a
ab (g+h)=8πG en⟨T o
ab ⟩+ϕgab.
(3) Nea -sa u a ion s abili y.
I
supk∆QNEC ≤ε
, hen
Eξ(δ, δ)≤C1ε,
by coe ci i y and L.1's Lipschi z con inui y, ob ain
Eg a
ab −8πG en⟨T o
ab ⟩−ϕgabHs−2
δ≤C ε.
6
10 Two-Dimensional Rew i e and Anomaly
In
d= 2
, Eins ein enso degene a es. Employ imp o ed s ess
Timp
±± =T±± −c
24π{λ, x±},
Weyl anomaly only en e s ace. Dis ibu ional enso iza ion lemma ew i es: i
X++ =X−− =
0
and
∇aXab =ˆ
Jb
, hen
X+−=ϕg+−
,
∂±ϕ=ˆ
J±
. Thus wo-dimensional e sions A
′
/B
′
/C
′
yield
Eg a
ab = 8πG en⟨T o ,imp
ab ⟩+ϕgab, ∂bϕ=ˆ
Jb,
sou celess case
ϕ
cons an inco po a ed in o
Λ
. Bañados/BTZ symme ic cu s achie e sa u a-
ion; Vaidya scena io exhibi s nea -sa u a ion sa is ying s abili y inequali y (Appendix I).
11 Examples
11.1 FRW
ds2=−d 2+a2( )γijdxidxj.
Take adial null di ec ion
k
h ough
p
, Theo em A yields
Eg a
kk = 8πG en⟨T o
kk ⟩
. Combining
wi h quan um Bianchi imespace decomposi ion yields
H2+k
a2=16πG en
(d−1)(d−2) ρ+2Λ
(d−1)(d−2),˙
H−k
a2=−8πG en
d−2(ρ+P),
e o con olled by
O( d+2)
(Appendix J).
11.2 AdS
3
/CFT
2
On Bañados/BTZ backg ound, symme ic cu amilies achie e QNEC sa u a ion, C
′
di ec ly closes.
Unde AdSVaidya, E2 displays
∆Eξ
olia ion d i and
RJb
closu e; E3 sa u a ion phase diag am
shows Hausdo  dis ance be ween
∆QNEC
ze o se and
Eg a
kk
ze o se con e ges as
O( 2)
wi h
(Appendices I, K).
12 E o Budge and Applicabili y Window

Theo e ical e o
: Bulk e m
O( d)
; wo-cap bounda y ke nel and cu a u e adius c ossing
O( d+2)
; highe -de i a i e ex insic cu a u e co ec ions don' ele a e leading o de . Non-
CFT equi es
m ≪1
.

Nume ical e o
: G id scale, second-o de die ence s ep, co ne disc e iza ion and denois-
ing, log-log slope e ica ion
d+ 2
.

Applicabili y window
: Hadama d s a e, no g a i a ional anomaly; Weyl anomaly only
en e s ace ( wo-dimensional ew i e); coupling domain wi h non-nega i e canonical ene gy;
in eg abili y small pa ame e
ϵ∼ /ℓcu
sucien ly small.
7
A Dis ibu ional-Le el Tenso iza ion Lemma (Comple e P oo )
Lemma 6
(A.1)
.
Xab ∈ D′(M)
symme ic. I o any null di ec ion
na
and
ψ∈C∞
0
,
⟨Xabnanb, ψ⟩=
0
, hen he e exis s
ϕ∈ D′(M)
such ha
Xab =ϕgab
. I
∇aXab =Jb=jbd ol
, hen
∂bϕ=jb
.
P oo .
Take local o hono mal ame
gab = diag(−1,1, . . . )
. Any null di ec ion
na=ea
0+ ˆniea
i
,
|ˆn|= 1
. Pai ing o mula
0 = ⟨X00 + 2ˆniX0i+ ˆniˆnjXij, ψ⟩.
Viewed as quad a ic o m in
ˆn
anishing o all uni ec o s. Sphe ical ha monic decomposi ion
yields linea e m
⟨X0i, ψ⟩= 0
, quad a ic e m
⟨Xij, ψ⟩=λ δij⟨ψ⟩
, ze o-o de
⟨X00, ψ⟩=−λ⟨ψ⟩
.
Thus
Xab =λdiag(−1,1, . . . ) = ϕgab
. Di e gence condi ion yields
∂bϕ=jb
.
B
(R)
and Small-Region Expansion O de Con ol
In RNC,
R(x) = R(p) + ∂cR|pxc+O(x2)
. Fo
(R) = R+αR2
, Wald en opy
Sg a =1
4G en ZΣ
′(R) dA=1
4G en ZΣ1+2αR(p)dA+O( d+1).
δ ′(R)=2α δR
. In eg al es ima es
ZDp,
δR≤C d+2|δg|C1,ZΣ
δR dA≤C d+1|δg|C1.
Ex insic cu a u e mixed e ms on ep esen a i e su ace supp essed by
θ|p=σ|p= 0
and
R(θ2+σ2) = O( d+2)
, no al e ing
O( d)
leading o de .
C Small-Region Modula Hamil onian Ke nel and Shape Va ia ion
App oxima e CKV
ξ
yields shape a ia ion ke nel
±=O( 2)
sa is ying mi o odd symme y
+=
− −◦ R
. Hadama d condi ion ensu es
⟨Tkk, ±φ⟩
well-dened, Theo em J.1 p o ides
OD′( d+2)
bound.
D Scheme Independence and To al S ess Conse a ion
Allowed local coun e e ms only edene
G en,Λ
and ni e highe -de i a i e couplings, no al e ing
∇a⟨T o
ab ⟩= 0
. Non-local eec i e ac ion a ia ion denes
τen
ab
, whose di e gence cancels bulk
sou ces. Thus main equa ions in a ian unde equi alence class.
E Two-Cap Cancella ion Mic olocal Analysis (Comple e)
Hadama d wo-poin unc ion
W
's wa e- on se
WF(W)
con ols
Tkk
dis ibu ional es ic ion
along null su ace. Mi o map
R
's
C1
de ia ion is
O( )
, hus
|Tkk −R∗Tkk|H−1≤C |Tkk|H−1.
Mul iplying by
| ±|C1=O( 2)
and measu e
O( d−1)
, pai ing wi h es unc ion no m
|φ|C1∩H1
,
yields
O( d+2)
bound. Co ne dis ibu ional mass eo ganized ia
Cξ
as o al die en ial, doesn'
ele a e o de .
8
F JKM Shi and Co ne Co ec ion Cohomological In a iance
Va ia ion change is exac o m
dβ
. On ela i e homology class o med by wo caps and co ne s,
R∂Dp, dβ= 0
. Co ne po en ial
Cξ
a ia ion compensa ed by bounda y o al die en ial, p ese ing
δHξ
and
Jb
in a iance.
G Quan um Bianchi Sou ce
Jb
Coo dina e-F ee Exp ession and Ex-
amples
W i e
Jb=∇ahδQξ,ab −(ξcθcab)−δCξ,abi,
whe e
θcab
is symplec ic po en ial double-index pullback. AdSVaidya small diamond disc e e
implemen a ion shows
∆Eξ(Σ1→Σ2)≈RJb
closes wi hin
O( d+2)
.
H Quan um Res Rep esen a i e Su aceExis ence, Uniqueness,
and Cons uc ion Algo i hm
In de o ma ion space
C2,α ∩H2
, ake quan um expansion as nonlinea ope a o
Q
. Backg ound
QES sa ises
Q(Σ) = 0
, i s F éche de i a i e sel -adjoin posi i e deni e. Implici unc ion heo-
em yields unique solu ion amily making
θ|p=σ|p= 0
. Ene gy es ima e
Zˆ
Σ
(θ2+σ2)≤C d+2.
Cons uc ion employs g adien ow/New on i e a ion, Lipschi z cons an
≤Cε
, con e ging o
ˆ
Σ
.
I Two-Dimensional Rew i e, Imp o ed S ess, and Phase Diag am
In
d= 2
employ imp o ed s ess
Timp
±±
, Weyl anomaly en e s ace. Two-dimensional enso iza ion
lemma and quan um Bianchi ew i e see main ex 10. Bañados/BTZ and Vaidya nume ical phase
diag ams display
∆QNEC
and
Eg a
kk
ze o se coincidence deg ee con e ges as
O( 2)
wi h
.
J FRW Null P ojec ion and F iedmann Combina ion
Deno e
H:= ˙a/a
,
ρ:= ⟨T o
⟩
,
P:= a−2γij⟨T o
ij ⟩/(d−1)
. Null p ojec ion
Eg a
kk =Eg a
+Eg a
,
combining wi h quan um Bianchi imespace decomposi ion yields main ex 11.1's wo scala
o mulas. E o e m
≤C d+2
(
C
depends on
|H|C1
,
|Rm|C0
).
9