Necessity and Extensions of Gibbons--Hawking--York Boundary Terms: Variational Well-Posedness, Corners and Null Boundaries, and Closure to Quasilocal Energy and Thermodynamics

Necessi y and Ex ensions o GibbonsHawkingYo k Bounda y
Te ms:
Va ia ional Well-Posedness, Co ne s and Null Bounda ies, and
Closu e o Quasilocal Ene gy and The modynamics
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Ve sion: 1.7
Abs ac
On pseudo-Riemannian mani olds wi h (possibly non-smoo h) bounda ies, he a ia ion o
he Eins einHilbe bulk ac ion con ains no mal de i a i e- ype bounda y uxes; xing only
Di ichle da a o he induced me ic
hab
does no suce o well-posedness. In he amewo k
o Le iCi i a connec ion and ex insic cu a u e, his pape igo ously p o es ha adding he
GibbonsHawkingYo k (GHY) e m wi h o ien a ion ac o
ε:= nµnµ∈ {±1}
a non-null
bounda ies cancels all no mal de i a i e con ibu ions, he eby es ablishing a s a iona i y p in-
ciple o a ia ions xing
hab
. Fo piecewise bounda ies, we p o ide a unied dic iona y o join
(co ne ) e ms and p o e ac ion addi i i y; o null segmen s, we cons uc a null bounda y e m
wi h expansion
θ
and su ace g a i y
κ
ha is in a ian unde cons an escaling, elucida ing
he endpoin and di e gence con ibu ions in oduced by non-cons an escaling and ans e se
supe ansla ions espec i ely, along wi h hei compensa ions. Subsequen ly, we es ablish in
ADM/ReggeTei elboim canonical decomposi ion and co a ian phase space (Iye Wald, Wald
Zoupas) ha he GHY/join s uc u e ende s he Hamil onian die en iable, wi h bounda y
gene a o s consis en wi h B ownYo k quasilocal s ess; compa ibili y wi h co a ian cha ges
is achie ed wi hin he same bounda y condi ion class and ep esen a i e. Fo
(R)
and Lo e-
lock (including GaussBonne ) heo ies, we cons uc bounda yco ne unc ionals ma ching
Di ichle da a and p o ide addi i i y p oposi ions o piecewise non-smoo h cases. Finally, in
Euclidean black hole geome ies, explici compu a ion wi h
K
and e e ence
K0
, oge he wi h
necessa y join and (AAdS case) coun e e ms, yields consis en ee ene gy, ene gy, and en-
opy. Appendices p o ide s ep-by-s ep ep oducible de i a ions, o ien a ionsign dic iona ies,
and wo ked examples in co a ian phase space.
MSC
: 83C05; 83C57; 58A10; 49S05
Keywo ds
: GibbonsHawkingYo k bounda y e m; a ia ional well-posedness; co ne s
and join s; null bounda ies; B ownYo k quasilocal ene gy; co a ian phase space;
(R)
g a i y;
Lo elock/GaussBonne g a i y; Euclidean black holes; he modynamics
1 No a ion, O ien a ion, and Da a Classes

Space ime and cu a u e
:
(M, gµν)
is a ou -dimensional o ien able pseudo-Riemannian
mani old wi h signa u e
(−,+,+,+)
. The Riemann enso is
Rρσµν =∂µΓρσν −∂νΓρσµ + ΓρλµΓλσν −ΓρλνΓλσµ,
1
wi h
Rµν =Rρµρν
,
R=gµνRµν
, and
Gµν =Rµν −1
2Rgµν
.

Non-null bounda y geome y
: On a bounda y segmen
B
, ake uni no mal
nµ
wi h
ε:= nµnµ∈ {±1}
. The induced me ic and ex insic cu a u e a e
hµν =gµν −ε nµnν, Kµν =hµαhνβ∇αnβ, K =hµνKµν.

Null bounda y geome y
: On
N
, ake null ec o
ℓµ
and auxilia y ec o
kµ
wi h
ℓ·k=−1
.
The ans e se wo-dimensional me ic is
γAB
. The shape ope a o and expansion a e
WAB := γAµγBν∇µℓν, θ := γABWAB,
wi h indices aised/lowe ed by
γAB
; ans e se co a ian de i a i e
DA
and Háji£ek one- o m
ωA:= −kµ∇Aℓµ
a e induced by he igging connec ion.

Ane pa ame e and su ace g a i y
: Le
λ
be an ane pa ame e along he gene a o
ℓ
, wi h
∂λ:= ℓµ∇µ.
Unde he no maliza ion
ℓ·k=−1
, su ace g a i y is dened as
κ:= −kµℓν∇νℓµ,
yielding
ℓν∇νℓµ=κ ℓµ
.
This deni ion is compa ible wi h he escaling laws in 4: when
ℓ→eαℓ
and
k→e−αk
,
θ→eαθ
and
κ→eα(κ+∂λα).

Piecewise bounda ies and join s
:
∂M=SiBi
, wi h
Cij =Bi∩Bj
allowing signa u e ips
o con aining null segmen s.

Bounda y da a (Di ichle class)
: Non-null segmen s x
hab
; null segmen s x he Ca oll
s uc u e
(γAB,[ℓ])
, whe e
[ℓ]
is an equi alence class unde cons an escaling
ℓ→eαℓ
; each
join xes an angle ( he
η
in 3 and loga i hmic angle
a
in 4).

Measu es
: Bulk
√−gd4x
; non-null bounda y
p|h|d3x
; null bounda y
√γdλd2x
; join s
√σd2x
.
2 Va ia ion o EH Bulk Ac ion and Bounda y Flux
SEH =1
16πG ZM
√−g R d4x.
The  s a ia ion is
δ(√−gR) = √−g Gµνδgµν +∂µh√−ggαβδΓµαβ −gµαδΓβαβi,
whe e
2
δΓρµν =1
2gρσ∇µδgσν +∇νδgσµ −∇σδgµν.
A e angen /no mal decomposi ion, he bounda y e m con ains an i educible p incipal e m
nµ∇µδgαβ
;
SEH
alone is ill-posed unde Di ichle da a.
3 GHY Cancella ion and Va ia ional Well-Posedness
SGHY[g] = ε
8πG Z∂Mp|h|Kd3x
Va ia ional se up (xed embedding, uni no mal gauge)
: The bounda y geome ic lo-
ca ion is held xed; only he me ic a ies. Thus
δ(nµnµ)=0, δnµ=1
2ε nµnαnβδgαβ .
This se up is compa ible wi h Di ichle da a (xing
hab
) and makes
SGHY
and join e ms cancel
bounda y uxes e m-by- e m.
Theo em 1
(GHY Cancella ion)
.
Fo a ia ions xing
δhab = 0
,
δ(SEH +SGHY) = 1
16πG ZM
√−g Gµν δgµν d4x.
P oo .
See Appendix B o e m-by- e m ma ching.
Sel -check hin
: Align he p incipal e m
nρhµαhνβ∇ρδgαβ
om Appendix A wi h he
∇δg
e ms in
δKab
a ising om
δnµ=1
2εnµnαnβδgαβ
in Appendix B; di ec e m-by- e m e ica ion
yields cancella ion.
4 Piecewise Bounda ies, Signa u e Flips, and Co ne Addi i i y
Non-nullnon-null join angle dic iona y
: Le wo segmen s ha e uni ou wa d no mals
n1, n2
wi h causal ypes ma ked by
εi:= n2
i∈ {±1}
. The join angle
η
is dened as
η=






a ccosh −n1·n2, ε1=ε2=−1 (
bo h spacelike, no mals imelike
),
a ccos n1·n2, ε1=ε2= +1 (
bo h imelike, no mals spacelike
),
a csinh nT·nS, ε1ε2=−1 (
mixed causal;
n2
T=−1, n2
S= +1).
The co ne e m is
S(nn)
co ne =1
8πG ZC
√σ η d2x ,
wi h o ien a ion and sign die ences uni o mly xed by he mas e o mula and o ien a ion
ables.
Nullnon-null and nullnull join s
: Loga i hmic angles
a(nℓ)= ln |−ℓ·n|, a(ℓℓ)= ln −1
2ℓ1·ℓ2,
wi h join e ms
1
8πG RC√σ a d2x
.
3
Theo em 2
(Addi i i y and Necessi y)
.
Unde bounda y da a xing he espec i e angles (
η
o
a
),
SEH +SGHY +Sco ne /join
is a ia ionally well-posed and sa ises addi i i y
S[M1∪ΣM2] = S[M1] + S[M2].
Join e ms a e in a ian unde any
C1
egula iza ion limi , independen o egula ize de ails.
5 Null Bounda ies:
θ+κ
S uc u e, Rescaling, and Endpoin Com-
pensa ion
SN=1
8πG ZN
√γ(θ+κ) dλd2x
Theo em 3
(Null Well-Posedness)
.
Fixing
(γAB,[ℓ])
,
δ(SEH +SN)
con ains no no mal de i a i e
esiduals.
Pu e escaling
(p ese ing
ℓ·k=−1
, no ans e se componen s):
ℓ→eαℓ, k →e−αk⇒WAB →eαWAB, θ →eαθ, κ →eα(κ+∂λα).
When
α=
cons ,
RN√γ(θ+κ) dλd2x
plus join e ms is in a ian ; when
α=α(λ)
, endpoin o-
al a ia ions a e p oduced, abso bable by loga i hmic angle coun e e ms (see Appendix D; pa h B
akes
ln(ℓc|Θ|)
equi ing
Θ
sign-deni e; i
Θ
c osses ze o, use pa h A endpoin /join compensa ion).
T ans e se supe ansla ion/c oss-sec ion epa ame iza ion
:
ℓ→eα(ℓ+ AeA)⇒θ→eα(θ+DA A),
belonging o c oss-sec ion edeni ion eec s, ea ed sepa a ely om pu e escaling abo e.
Dimensional no e
: In
D
dimensions, ans e se space dimension is
D−2
; co esponding di e gence
s uc u e gene alizes s aigh o wa dly by dimension.
Null B ownYo k s ess
:
TABN=−1
8πGWAB−θ δAB,
sa is ying ans e se conse a ion dened by he igging connec ion, compa ible wi h null Wald
Zoupas cha ges wi hin he same bounda y condi ion class.
6 Canonical Fo malism: Die en iable Hamil onian and Quasilocal
Ene gy
In
3+1
decomposi ion, wi h
SEH
alone he Hamil onian unc ional is non-die en iable; adding
SGHY
wi h necessa y join /null e ms yields:
Theo em 4
(Die en iabili y and Bounda y Gene a o s)
.
Unde Di ichle da a and he o ien a-
ion/ egula i y assump ions o his pape , aking he ac ion
S=SEH +SGHY +Sjoin (+SN)
4
wi hou in oducing any in insic bounda y unc ional depending solely on he bound-
a y in insic me ic
hab
, he Hamil onian
Hξ
is F éche die en iable on phase space, wi h bound-
a y gene a o
uniquely
gi en by
Tab
BY =1
8πG(Kab −Khab)
I in insic e ms (such as
Sc
in 9 o e e ence e m
S e
) a e added/sub ac ed wi hin he same
bounda y condi ion class,
Hξ
emains die en iable wi h bounda y gene a o modied o
Tab
BY, en =Tab
BY +Tab
c −Tab
e ,
consis en wi h co a ian phase space analysis in 6 and eno maliza ion coun e e ms in 9.
The ene gy on a spacelike slice
S
is
EBY =ZS
√σ uaubTab
BY d2x
which in he asymp o ically a limi app oaches he ADM mass.
7 Co a ian Phase Space and Rep esen a i e Independence
δL=E·δϕ + dΘ(ϕ, δϕ),Jξ=Θ(ϕ, Lξϕ)−ξ·L= dQξ.
I
L→L+ dB
, hen
Θ→Θ+δB
and
Qξ→Qξ+ξ·B
. Wi hin he same bounda y condi ion
class and he same (o gauge-equi alen ) ep esen a i e, mass, angula momen um, and ho izon
en opy a e in a ian ; ux bounda ies employ WaldZoupas co ec ions o ensu e in eg abili y.
Skele on o mula (loca ing die en iabili y sou ce)
: In he ReggeTei elboim amewo k,
δHξ=ZΣ
(
cons ain s
·δϕ) d3x+I∂ΣΠabδhab +···d2x.
Wi h bulk e m alone, bounda y a ia ion con ains
Πabδhab
and no mal de i a i e e ms, non-
die en iable; adding
SGHY
(and join /null e ms) ans o ms bounda y a ia ion in o BY su ace
gene a o s, ende ing
Hξ
die en iable.
Wo ked Example ( ep esen a i e independence compu a ional chain)
: Take a s a ic
black hole wi h Killing eld
ξ=∂
, a inni y
I
and ho izon
H
:
δHξ=ZS∞δQξ−ξ·Θ−ZSHδQξ−ξ·Θ.
I
L7→ L+ dB
, hen
Θ7→ Θ+δB,Qξ7→ Qξ+ξ·B,
wi h
δ(ξ·B) = ξ·δB
, so inc emen s a bo h ends anish,
δHξ
in a ian ; i ux bounda ies exis ,
apply WaldZoupas co ec ion making endpoin die ence ze o, es o ing in eg abili y.
Reno malized BY su ace s ess
:
Tab
BY, en =2
p|h|
δSGHY +Sjoin +Sc −S e 
δhab
=Tab
BY +Tab
c −Tab
e ,
whe e
Tab
c := 2
p|h|
δSc
δhab
and
Tab
e := 2
p|h|
δS e
δhab
.
Minimal coun e e ms o ou -dimensional AAdS appea in 9.
5

8
(R)
G a i y: Di ichle -Compa ible Bounda yJoin s
Using he scala  enso equi alence
Φ = ′(R)
,
S=1
16πG ZM
√−g(ΦR−V(Φ)) d4x.
Unde Di ichle da a xing
(hab,Φ)
,
S (R)
bdy =1
8πG Z∂M
εp|h|ΦKd3x, S (R)
join =1
8πG X
CZC
√σΦ (
angle
) d2x.
I ins ead xing
(hab, nµ∇µΦ)
as Robin- ype da a, compensa ion e ms
∝p|h|nµ∇µΦ
mus be
added a he bounda y, wi h co espondingly weigh ed join e ms (Appendix G).
9 Lo elock (GaussBonne ) G a i y and Piecewise Non-Smoo h
Addi i i y
Fo GaussBonne (GB) e m in
D≥5
,
SGB =α
16πG ZM
√−gRµνρσRµνρσ −4RµνRµν +R2dDx,
he Di ichle -compa ible Mye s- ype bounda y e m is
SGB
bdy =α
8πG Z∂M
εp|h|2b
GabKab +JdD−1x,
whe e
b
Gab
is he Eins ein enso o
hab
,
Jab =1
32KKacKcb+KcdKcdKab −2KacKcdKdb −K2Kab, J =habJab.
P oposi ion 5
(GB Addi i i y, Piecewise Non-Smoo h)
.
Taking he abo e bounda y e m and
adding co esponding GB join polynomials (quad a ic combina ions o angles
η
/loga i hmic angles
a
wi h
(K, b
R)
), unde xed Di ichle da a
SGB[M1∪ΣM2] = SGB[M1] + SGB[M2].
P oo ske ch.
In eg a e by pa s on each piece; a join s appea esiduals
∝δ(
angle
)
; chosen GB
join polynomials' a ia ion exac ly cancels hese esiduals. Rep esen a i e: De uelleMe inoOlea
(2018).
10 Non-Compac Bounda ies and AAdS Coun e e ms (Fou -Dimensional
Minimal Rep esen a i e)
Sc =1
8πG Z∂Mp|h|2
L+L
2b
Rd3x
whe e
L
is he AdS cu a u e adius and
b
R
is he bounda y in insic Ricci scala . This ep e-
sen a i e is equi alen o koun e e ms/holog aphic eno maliza ion in ou dimensions o yielding
he same ni e s ess and con o mal-in a ian e ms; highe dimensions equi e addi ional highe -
cu a u e coun e e ms.
6
11 Dis ibu ional Cu a u e, Thin Shells, and Ze o-Measu e Bound-
a y o Bounda y
I
Kab
exhibi s jumps ac oss a hype su ace, bulk cu a u e de elops
δ
- ype dis ibu ions; hei
con ibu ion o he ac ion is abso bed by join / hin shell e ms. Timelike/spacelike hin shells
sa is y Is ael junc ion condi ions
[Kab −Khab] = −8πG Sab
; null hin shells sa is y Ba abèsIs ael
condi ions. The join and null ules o his pape a e compa ible he ewi h.
12 Euclidean Black Holes:
K
,
K0
, F ee Ene gy, and En opy
Fo Schwa zschild Euclidean geome y
ds2= ( ) dτ2+ ( )−1d 2+ 2dΩ2
2, ( ) = 1 −2M
,
unca ed a
=R
,
τ∈[0, β]
. Wi h ou wa d uni no mal
nµ=√ δµ
,
K(R) = 2p (R)
R+ ′(R)
2p (R), K0(R) = 2
R.
To al ac ion
IE=IEH +IGHY[K] + Ijoin −I e [K0].
Remo ing conical deci
β= 8πM
and aking
R→ ∞
ni e pa yields
F=IE
β=M
2, E =∂β(βF) = M, S =β(E−F) = A
4G.
Pe iodici y iden ica ion no double-coun ing
: Due o
τ∼τ+β
, la e al edge co ne s
a
=R
a
(R, 0)
and
(R, β)
a e equi alen ; in eg a ion by pa s on in e al
[0, β]
yields co ne
con ibu ions a wo ends whose sum equals he con ibu ion o a single co ne on he pe iodic
mani old, no double-coun ing occu s.
13 Va ia ional Well-Posedness s PDE/F edholm
This pape es ablishes closu e o ac ion  s a ia ion on gi en bounda y da a se s; PDE well-
posedness and F edholm p ope ies equi e unc ional space and bounda y- alue ope a o analysis.
On compac bounda ies, pu e Di ichle /Neumann maps a e gene ally non-F edholm; na u al mixed
da a (e.g.,
([γ], H)
o Ba nik da a) a e mo e sui able. This wo k is conned o he a ia ional
well-posedness le el; Appendix L p o ides illus a i e examples.
Appendices: Numbe ed De i a ions, Dic iona ies, and Examples
Unied no e
: All in eg als explici ly w i e measu es
dnx
; se no a ion unied as
{±1}
; mas e
o mula
SGHY = (8πG)−1εRp|h|Kd3x
wi h o ien a ion able uniquely xes sign die ences.
7
Appendix A: EH Ac ion Bounda y Flux (Te m-by-Te m Decomposi ion)
A.1
δSEH = (16πG)−1RMδ(√−g)R+√−g δRd4x
,
δ√−g=−1
2√−g gµνδgµν
.
A.2
δR =Rµνδgµν +∇µgαβδΓµαβ −gµαδΓβαβ
.
A.3
S okes o mula yields bounda y e m
(16πG)−1R∂Mp|h|nµ(···)d3x
.
A.4
P ojec ion
hµν=δµν−εnµnν
w i es bounda y ux as
Z∂Mp|h|hΠabδhab +nρhµαhνβ∇ρδgαβ +···id3x.
whe e
Πab := Kab −Khab
.
Appendix B: GHY Cancella ion and Example O ien a ion Table
B.1
δ(p|h|K) = p|h|δK +1
2K habδhab
,
δK =habδKab −Kabδhab
, whe e
δKab =haµhbν∇µδnν+δΓρµνnρ.
B.2
Subs i u ing uni no mal gauge
δnµ=1
2ε nµnαnβδgαβ ,
he
∇δg
in
δK
cancels e m-by- e m wi h Appendix A p incipal e m, while
Πabδhab
mu ually
cancel, yielding Theo em 2.1.
B.3 Example o ien a ion able
Segmen Causal ype
n2=ε
Ou wa d no mal GHY weigh
Ini ial/nal slices Spacelike
−1
Fu u e/pas
−Rp|h|Kd3x
La e al edge Timelike
+1
Ou wa d
+Rp|h|Kd3x
Euclidean bounda y Riemannian
+1
Ou wa d
+Rp|h|Kd3x
(This able is o eading guidance only; ac ual compu a ions uni o mly use he mas e o mula.)
Appendix C: Th ee Types o Join s and Addi i i y (Dic iona y and P oo Ou -
line)

Non-nullnon-null:
η
dened piecewise by causal ype (see 3); co ne e m
1
8πG R√σ η d2x
.

Nullnon-null:
a= ln |−ℓ·n|
.

Nullnull:
a= ln |− 1
2ℓ1·ℓ2|
.

Piecewise GHY in eg a ion by pa s lea es only endpoin e ms
∝δ(
angle
)
, canceled by join
e ms; ac ion addi i e; esul independen o join egula ize de ails.
8
Appendix D: Null Rescaling, Endpoin Compensa ion, and Supe ansla ion
D.1 Pu e escaling
ℓ→eα(λ)ℓ, k →e−α(λ)k
:
θ→eαθ
,
κ→eα(κ+∂λα)
. In a ian unde
cons an
α
; non-cons an p oduces endpoin o al a ia ions.
D.2 Pa h A (LMPS endpoin /join compensa ion)
:
Send =1
8πG X
endpoin s
Z√σ α d2x.
D.3 Pa h B (loga i hmic coun e e m)
:
S epa am =1
8πG ZN
√γΘ ln ℓc|Θ|dλd2x, Θ := θ.
No e: Pa h B equi es
Θ
sign-deni e on each gene a o ; i
Θ
c osses ze o (as a oci), ea
ze o-c ossing poin s as join s and handle pe D.2 endpoin /join compensa ion, o use pa h A.
D.4 T ans e se supe ansla ion
ℓ→eα(ℓ+ AeA)
in oduces
DA A
, classied as c oss-
sec ion edeni ion.
Appendix E: ReggeTei elboim Die en iabili y and BY Gene a o s
δHξ=ZΣ
(N δH+NiδHi) d3x+Z∂Σ
√σε δN +jiδNi+Tab
BYδhabd2x.
whe e
ε:= uaubTab
BY
,
ji:= −σiaubTab
BY
,
σab =hab +uaub
.
Adding GHY/join s ende s
Hξ
die en iable and gene a es co ec e olu ion; asymp o ically
a
EBY →MADM
.
Appendix F: Co a ian Phase SpaceRep esen a i e F eedom and Wo ked Ex-
ample
F.1 Rep esen a i e eedom
:
L→L+ dB⇒Θ→Θ+δB
,
Qξ→Qξ+ξ·B
. Cha ge elemen
kξ:= δQξ−ξ·Θ
emains in a ian .
F.2 Wo ked example (s a ic black hole)
: Main ex 6 al eady p o ides wo-end cancella ion
chain; ux bounda ies es o ed o in eg abili y ia WaldZoupas co ec ion, yielding  s law and
S=A/(4G)
.
Appendix G:
(R)
/Lo elock Bounda yJoin Co espondence
G.1
(R)
: Di ichle :
S (R)
bdy = (8πG)−1Rεp|h|ΦKd3x
, join
∝Φ (η
o
a)
. Robin: add
∝
p|h|nµ∇µΦ
wi h dual join e ms.
G.2 GaussBonne (
D≥5
)
:
SGB
bdy = (8πG)−1αRεp|h|(2 b
GabKab +J) dD−1x
; piecewise
non-smoo h GB join polynomials ensu e P oposi ion 8.1 addi i i y (coecien s xed by Che n
Weil/ ansg ession; see De uelleMe inoOlea, 2018).
Appendix H: Schwa zschild Euclidean Ac ion (Including
K
and
K0
)
ds2= dτ2+ −1d 2+ 2dΩ2
2
,
= 1 −2M/
.
K(R) = 2p (R)
R+ ′(R)
2p (R)
,
K0(R) = 2
R
.
IE=IEH +IGHY[K]−I e [K0] + Ijoin ⇒F=M/2, E =M, S =A/(4G)
.
Pe iodici y iden ica ion no double-coun ing explained in main ex 11.
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