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arXiv:1812.03924v2 [gr-qc] 23 Nov 2019

Pure and Applied Mathematics Quarterly

Volume 0, Number 0, 1, 2019

On the total mass of asymptotically hyperbolic

manifolds∗

Hamed Barzegar†, Piotr T. Chru´

sciel‡and Luc Nguyen

Abstract: Generalising a proof by Bartnik in the asymptotically

Euclidean case, we give an elementary proof of positivity of the

hyperbolic mass near the hyperbolic space.

It is a pleasure to dedicate this work to Robert Bartnik on the

occasion of his 60th birthday.

1. Introduction

The question of positivity of total energy in general relativity has turned out

to be a particularly challenging problem (cf. [14] and references therein),

with several open questions remaining. It therefore appears of interest to

provide simple proofs when available.

In his well-known paper on the mass of asymptotically Euclidean mani-

folds [2], Robert Bartnik gave an elementary proof of positivity of the ADM

mass near the Euclidean metric. Inspired by his work, we establish a similar

result for the hyperbolic mass near the hyperbolic metric. The argument

turns out to be somewhat more involved and calculation-intensive.

Indeed, we provide an elementary proof of positivity of the hyperbolic

mass, near the hyperbolic space, for metrics with scalar curvature bounded

below by that of the hyperbolic space. Namely, ignoring an overall dimension-

dependent constant, consider the usual deﬁnition (cf., e.g. [7]) of the mass

mof a metric gasymptotic to a metric gwith a static KID V(see below

∗Preprint UWThPh-2018-32

†The author is supported by the Austrian Science Fund (FWF) project No.

P29900-N27.

‡The research of the author is supported by the Austrian Science Fund (FWF),

Project P 29517-N27, and by the Polish National Center of Science (NCN) under

grant 2016/21/B/ST1/00940.

1

2 Hamed Barzegar et al.

for terminology):

m(V) = lim

R→∞ Zr=RV gmj giℓ Dmgjℓ −Dℓgjm

+ (gmj gki −gij gkm )(gjm −gjm)DkVdσi.(1.1)

We prove the following:

Theorem 1.1 For n≥3, let (M, g)be Rnequipped with the hyperbolic

metric,

g=dr2

1 + r2+r2dΩ2,(1.2)

where dΩ2is the canonical metric on the (n−1)-dimensional sphere Sn−1.

Let (A0,~

A)∈Rn+1 satisfy |~

A|:= p(A1)2+...+ (An)2≤A0and set

V=A0p1 + r2+X

i

Aixi.(1.3)

Let gbe a metric on Masymptotic to gwith well-deﬁned total mass m.

There exists δ > 0such that if

kg−gkL∞+kDgkL∞< δ ,

where Dis the covariant derivative operator of g, then gcan be put into the

gauge

ˇ

ψj:= Digij −1

2gjk gℓmDkgℓm = 0 (1.4)

in which we have

m(V)≥ZMhR−R+1

8n|Dg|2

giV dµg(1.5)

where, in local coordinates, dµg=√det g dnx.

It follows clearly from (1.5) that m(V)≥0 if

R≥R . (1.6)

Equivalently, if we set

m0:= m(V=p1 + r2), mi:= m(V=xi),(1.7)

On the total mass of asymptotically hyperbolic manifolds 3

then, under (1.6), the vector (mµ) is timelike future-pointing with respect

to the Lorentzian quadratic form m2

0−m2

1... −m2

n. The inequality (1.6)

holds of course for general relativistic initial data sets with vanishing trace

of extrinsic curvature and with matter ﬁelds satisfying the dominant energy

condition. Note that in vacuum, or in the presence of matter ﬁelds satisfying

well behaved equations, under suitable further smallness assumptions on the

extrinsic curvature of the initial data surface and on the matter ﬁelds, the

condition of vanishing of the trace of the extrinsic curvature can be enforced

by moving slightly the initial data hypersurface in space-time, after invoking

the implicit-function theorem.

Theorem 1.1 is, essentially, a special case of Theorem 3.1 below, with

the constants coming from (3.15). At the heart of its proof lies the identity,

which we derive below and which holds for any asymptotically hyperbolic

background (M, g) with a static KID V, under the usual conditions for

existence of the mass:

m=ZMh(R−R)V+n+ 2

8n|D φ|2

g+1

4|Dˆ

h|2

g

−1

2ˆ

hiℓˆ

hjmRℓmij −n+ 2

2nφˆ

hij Rij −n2−4

8n2λφ2

−1

2|ˇ

ψ|2

g−ˇ

ψiDiφ)V+hkiˇ

ψi+1

2φˇ

ψkDkV

+O|h|3

g+O|h|g|Dh|2

gV

+O|h|2

g|Dh|g|DV |gidµg; (1.8)

see (3.1)-(3.4) for notation. Throughout this work, the reader can assume

that indices are raised and lowered using the background metric g. We then

use a weighted Poincar´e inequality to control the non-obviously-positive

terms in (1.8).

The calculations leading to (1.8), presented in Section 4, are vaguely

reminiscent of those in [1], but the relation of the formulae presented there

to the hyperbolic mass is not clear.

We made an attempt to use similar ideas for perturbations of the Horowitz-

Myers instantons [8,11], with only partial results so far [3].

Remark 1.2 The Birmingham-Kottler [4,12] metrics with zero mass,

g=−r2

ℓ2+κdt2+dr2

r2

ℓ2+κ+r2hκ,(1.9)

4 Hamed Barzegar et al.

with constants ℓ > 0 and κ∈ {0,±1}, where (n−1

N, hκ) is an (n−1)-

dimensional space form with Ricci scalar equal to (n−1)(n−2)κ, are space

forms. Therefore all the calculations here apply verbatim to the case of

toroidal and hyperbolic conformal boundary at inﬁnity for such metrics.

There are, however, issues with the gauge, boundaries, and the weighted

Poincar´e inequality which would need to be addressed to be able to obtain

a positivity result:

1. In the κ= 0 case the associated manifold (0,∞)×n−1

Nis complete

with one locally asymptotically hyperbolic end, where r→ ∞, and

one cuspidal end, where r→0. Since the manifold is complete with-

out boundary, the proof of existence of the gauge should go through for

perturbations which vanish in the cuspidal end, but requires checking.

We note that positivity of the mass in the spin case has been estab-

lished in whole generality by Wang [15], using a variation of Witten’s

proof, and in [6] in dimension n≤7, but the non-spin higher dimen-

sional case remains open.

2. In the case κ=−1 the manifold of interest is [ℓ, ∞)×n−1

N, where

the boundary {ℓ} × n−1

Nsatisﬁes a mean-curvature inequality. If the

perturbations are not supported away from the boundary there will

be terms arising from integration by parts which are likely to destroy

positivity, since in this case there exist well behaved solutions with

negative mass.

2. Static KIDs

Let (M, g) be a smooth n-dimensional Riemannian manifold, n≥2 and let

Vbe a static KID on (M, g), i.e. a solution to

DiDjV=VRij −R

n−1gij.(2.1)

When ghas constant scalar curvature, an equivalent form is

∆gV+λV = 0 ,DiDjV=V(Rij −λgij),(2.2)

for some constant λ∈R. Here Rij denotes the Ricci tensor of the metric g,

Dthe Levi-Civita connection of g, and ∆g=DkDkis the Laplacian of g.

When λ < 0, rescaling gby a constant factor if necessary, when the

background metric has constant scalar curvature we can without loss of

On the total mass of asymptotically hyperbolic manifolds 5

generality assume that λ=−nso that

R:= gij Rij =λ(n−1) = −n(n−1) .

If gis an Einstein metric, namely Rij proportional to gij , using this last

scaling we obtain

Rij =−(n−1)gij , DiDjV=V gij .(2.3)

This implies

Di(|DV |2

g−V2) = 0 ,(2.4)

where | · |gdenotes the norm of a tensor with respect to a metric g. In

hyperbolic space, where the sectional curvatures are minus one, and when

Vtakes the form (1.3) in the coordinate system of (1.2), we have

|DV |2

g−V2=|~

A|2−(A0)2.(2.5)

3. The theorem

It is convenient to introduce some notation:

hij := gij −gij ,(3.1)

ψj:= Digij ⇐⇒ gij Dihjℓ =−gℓj ψj,(3.2)

φ:= gij hij =⇒φ:= gij hij =φ+O|h|2

g.(3.3)

We will denote by ˇ

h, respectively by ˆ

h, the g-trace-free, respectively the

g-trace-free, part of h:

ˇ

hij := hij −1

nφ gij ,ˆ

hij := hij −1

nφ gij .(3.4)

In order to address the question of gauge-freedom, we will apply a dif-

feomorphism to gso that

ˇ

ψi:= ψi+1

2gikDkφ(3.5)

vanishes. Note that the equation ˇ

ψi= 0 reduces to the harmonic-coordinates

condition in the case of a ﬂat background, where λ= 0.

We claim the following:

6 Hamed Barzegar et al.

Theorem 3.1 Let (M, g)asymptote to an asymptotically hyperbolic space-

form (M, g)and let Vbe a static KID of (M, g). Suppose that the usual

decay conditions needed for a well-deﬁned mass [7] are satisﬁed, namely, for

large r, in the coordinate system of (1.2),

hij =o(r−n/2), Dkhij =O(r−n/2),

V=O(r),|Dkhij |2

gV∈L1,(R−R)V∈L1.(3.6)

There exists δ > 0such that if

khkL∞+kDhkL∞< δ

and if |dV |g≤V, then we have

m≥ZMhR−R+n−2

8n|Dh|2

giV dµg

−1

2ZM(|ˇ

ψ|2

g−ˇ

ψiDiφ)V−2hkiˇ

ψi+φˇ

ψkDkVdµg.(3.7)

A sharper bound can be found in (3.15) below.

It follows from (2.5) that |dV |g≤Vholds for static KIDs as in the

statement of the theorem. It is well known (cf. e.g., the proof of [5, The-

orem 4.5]; compare [10,13]) that the gauge ˇ

ψk= 0 can always be realised

when gis close enough to the hyperbolic metric g. Hence Theorem 1.1 is

indeed a corollary of Theorem 3.1.

Proof: In Section 4we prove the identity

VR−R=D+Q − hkiˇ

ψi+1

2φˇ

ψkDkV , (3.8)

where

D:= DiV gmj giℓ Dmhjℓ −Dℓhj m+Dih(gmjgki −gij gkm)hjmDkVi

+1

2DihVgkℓ gjk Djgiℓ −gik Djgjℓ i

|{z }

(⋄)

+1

2Dih−3hiℓhk

ℓ+gik|h|2

gDkVi

+1

2Dihkiφ+1

4gkiφ2DkV(3.9)

On the total mass of asymptotically hyperbolic manifolds 7

is the sum of all divergence terms and Qis the sum of all quadratic or higher

order terms:

Q=−n+ 2

8n|D φ|2

g−1

4|Dˆ

h|2

g

+1

2ˆ

hiℓˆ

hjmRℓmij +n+ 2

2nφˆ

hij Rij +n2−4

8n2λφ2

+1

2|ˇ

ψ|2

g−ˇ

ψiDiφ) + O|h|3

g+O|h|g|Dh|2

gV

+O|h|2

g|Dh|g|DV |g.(3.10)

Here the Riemann tensor can be replaced by the Weyl tensor, and the Ricci-

tensor by its trace-free part.

We note that the term (⋄) in (3.9) is quadratic in (h, Dh):

gkℓ gjkDjgiℓ −gik Djgjℓ = (gkℓ −hkℓ)gjkDjgiℓ −gikDjgj ℓ

=−hkℓ gjkDjgiℓ −gik Djgjℓ .(3.11)

It is then easy to see that the integral of the divergence term Dgives the total

mass when integrated over the whole manifold, after taking into account the

fact that the boundary conditions needed for a well-deﬁned mass enforce a

vanishing contribution of higher-than-linear terms in the boundary integral.

This establishes (1.8).

We specialise now to the space-form version (3.10) of Q, which reads

Q=h−n+ 2

8n|D φ|2

g−1

4|Dˆ

h|2

g+1

2|ˆ

h|2

g−n2−4

8nφ2

+O|h|3

g+O(|h|g|Dh|2

g)iV

+1

2|ˇ

ψ|2

g−ˇ

ψiDiφ)V+O|h|2

g|Dh|g|DV |g.(3.12)

In order to absorb the undiﬀerentiated terms we use the weighted Poincar´e

inequality (A.8) below, namely

Z|ˆ

h|2

gV dµg≤1

nZh(|Dˆ

h|2

g− |Dˆ

h|2

g− |div ˆ

h−ˆ

hdV |2

g)V

+Dj(ˆ

hikDiVˆ

hjk )idµg.(3.13)

8 Hamed Barzegar et al.

with Ddeﬁned in (A.1). This leads to

ZQdµg≤Z−hn+ 2

8n|D φ|2

g+n−2

4n|Dˆ

h|2

g+n2−4

8nφ2

+1

2n|Dˆ

h|2

g+|div ˆ

h−ˆ

hdV |2

giV

+1

2nDj(ˆ

hikDiVˆ

hjk )+1

2|ˇ

ψ|2

g−ˇ

ψiDiφ)V

+O|h|3

gV+O(|h|g|Dh|2

gV) + O|h|2

g|Dh|g|DV |gdµg.(3.14)

Hence

m≥ZMhR−R+n+ 2

8n|D φ|2

g+n−2

4n|Dˆ

h|2

g+n2−4

8nφ2

+1

2n|Dˆ

h|2

g+|div ˆ

h−ˆ

hdV |2

giV dµg

−ZM 1

2|ˇ

ψ|2

g−ˇ

ψiDiφ)V−hkiˇ

ψi+1

2φˇ

ψkDkV

+O|h|3

gV+O(|h|g|Dh|2

gV) + O|h|2

g|Dh|g|DV |g!dµg.(3.15)

It is now clear that we can choose |h|g+|Dh|gsmall enough so that (3.7)

holds. ✷

4. The Ricci scalar of asymptotically anti de-Sitter

spacetimes

The aim of this section is to derive the the curvature identities (3.8) and

(3.12) needed in the proof of Theorem 3.1.

We consider the following metric

gij =gij +hij ,(4.1)

where gki is an anti de-Sitter metric. If we denote the connection of the

background metric by D, we have the relation

Dk≡Dk+δΓk,(4.2)

On the total mass of asymptotically hyperbolic manifolds 9

where δΓkis a (1,2) tensor equal to δΓ··k:= Γ··k−Γ··k. For example, applying

Dkon a vector component viwe get

Dkvi=Dkvi+δΓijk vj,(4.3)

where

δΓijk =1

2giℓ Djgkℓ +Dkgℓj −Dℓgjk

=1

2giℓ Djhkℓ +Dkhℓj −Dℓhjk .(4.4)

The Riemann tensors of the metrics gki and gki are related to each other via

the following equation

Rkimj =Rkimj +DmδΓkij −DjδΓkim +δΓkmℓδΓℓij −δΓkjℓδΓℓim .(4.5)

Contracting the ﬁrst and third indices, one obtains

Rij =Rij +DkδΓkij −DjδΓkik +δΓkkℓδΓℓij −δΓkj ℓδΓℓik .(4.6)

Inserting

δΓkik =1

2gkℓ Dihkℓ +Dkhiℓ −Dℓhki =1

2gkℓDihkℓ (4.7)

into (4.6), we obtain

Rij =Rij +1

2"Dkgkℓ Dihjℓ +Djhℓi −Dℓhj i

+gkℓ DkDihjℓ +DkDjhℓi −DkDℓhj i−DjgkℓDihkℓ

−gkℓDjDihkℓ +1

2gkpgℓqDℓhpk Dihj q +Djhiq −Dqhij

−1

2gkpgℓq Djhℓp +Dℓhpj −Dphj ℓDihkq +Dkhiq −Dqhki#.(4.8)

10 Hamed Barzegar et al.

And the Ricci scalar reads

R=gij Rij

=Rijgij +1

2gij "Dkgkℓ 2Dihjℓ −Dℓhj i

+2gkℓ DkDihjℓ −DkDℓhj i

−DjgkℓDihkℓ +1

2gkpgℓq Dℓhpk 2Dihjq −Dqhij

−1

2gkpgℓq Djhℓp +Dℓhpj −DphjℓDihkq +Dkhiq −Dqhki#.

(4.9)

Using

1

2gij gkpgℓq Djhℓp +Dℓhpj −DphjℓDihkq +Dkhiq −Dqhki

=1

2gij gkpgℓqDphj ℓ 2Dqhki −Dkhiq,(4.10)

this can be rewritten as

R=Rijgij +gij gkℓ DkDihjℓ −DkDℓhj i

+1

2gij hDkgkℓ 2Dihjℓ −Dℓhj i

−DjgkℓDihkℓ +1

2gkpgℓqhDℓhpk 2Dihj q −Dqhij

−Dphjℓ 2Dqhki −Dkhiq ii.(4.11)

In order to isolate the contribution of the mass we group all second-

derivative terms in (4.9) in a divergence with respect to the background metric

(similar to [7], except that there the divergence was taken with respect to

On the total mass of asymptotically hyperbolic manifolds 11

the physical metric):

R=Rijgij +Dkgij gkℓ Dihjℓ −Dℓhj i

−Dkgij gkℓDihjℓ −Dℓhji

+1

2gij hDkgkℓ 2Dihjℓ −Dℓhj i−DjgkℓDihkℓ

+1

2gkpgℓq Dℓhpk 2Dihjq −Dqhij −Dphjℓ 2Dqhki −Dkhiqi.

(4.12)

Note that

0 = Djδk

i=Dj(gkpgpi) = gpiDjgkp +gkpDjgpi ,

equivalently

Djgkp =−gℓk gipDjgiℓ =−gℓk gipDjhiℓ .(4.13)

This allows us to rewrite (4.12) as

R=Rijgij +Dkgij gkℓ Dihjℓ −Dℓhj i−Dkgijgkℓ Dihjℓ −Dℓhj i

+1

2gij h−gkpgℓqDkhpq 2Dihj ℓ −Dℓhji+gkpgℓq Djhpq Dihkℓ

+1

2gkpgℓq Dℓhpk 2Dihjq −Dqhij −Dphjℓ 2Dqhki −Dkhiqi

=Rijgij +Dkgij gkℓ Dihjℓ −Dℓhj i

+gkℓgip gjqDkhpq +gij gkpgℓq DkhpqDihjℓ −Dℓhji

+1

2gij gkpgℓqh−Dkhpq 2Dihjℓ −Dℓhji+DjhpqDihkℓ

+1

2Dℓhpk 2Dihjq −Dqhij −Dphjℓ 2Dqhki −Dkhiqi.(4.14)

After some simpliﬁcations one gets

R=Rijgij +Dkgij gkℓ Dihjℓ −Dℓhji+Q , (4.15)

where

Q:= 1

4gij gkpgℓq2DphjℓDqhki

|{z }

=:Q1

−DℓhkpDqhij −Dihpq Djhkℓ .(4.16)

12 Hamed Barzegar et al.

We note that

gij =gij −hij +χij ,(4.17)

where

hiℓ=gikhkℓ , hij =gikgj ℓhkℓ ,(4.18)

and

χij := gikgj ℓgmn hkmhnℓ +O(|h|3

g) = O(|h|2

g).(4.19)

In the notation of (3.2)-(3.5), the identity (4.15) becomes

−1

2DkgklDlφ=R−R+Rijhij −DkgkℓhjiDℓgij −ˇ

ψk−Q

|{z}

O(|h|2+|Dh|2)

|{z }

“higher order terms”

.(4.20)

If both gand gsatisfy the vacuum scalar constraint equation, so that

R=R, and in the gauge ˇ

ψi= 0, (4.20) takes the form

−1

2DkgklDlφ−R

nφ

=Rij ˆ

hij −R

n(φ−φ)−Dkgkℓhj iDℓgij +O(|h|2+|Dh|2)

|{z }

“higher order terms”

,(4.21)

which becomes an elliptic equation for φwhen all “higher order terms” are

thought to be negligible. Note that when gis a space-form the linear term

at the right-hand side vanishes, which implies that φitself is higher order.

However, this is not true in general, in particular one cannot assume that

φ= 0 for general perturbations of e.g. the Horowitz-Myers metrics.

We return to the calculation of the mass. Let Vbe a static KID as in

Section 2. Multiplying (4.15) by Vwe obtain

V R =VRij gij +Dkgij gkℓ Dihjℓ −Dℓhj i+Q

=VRij gij +Q+σ , (4.22)

where

σ:= VDkgij gkℓ Dihjℓ −Dℓhji

=DkV gij gkℓ Dihj ℓ −Dℓhji−gij gkℓ Dihjℓ −Dℓhj iDkV

|{z }

=:∗

.(4.23)

On the total mass of asymptotically hyperbolic manifolds 13

Then

∗=−Digij gkℓhjℓDkV+hj ℓDi(gij gkℓDkV)

+Dℓgij gkℓhjiDkV−hj iDℓgij gkℓDkV

=Dℓ(gij gkℓ −gℓj gki)hjiDkV+hjℓ(gij gkℓ −gℓj gki)DiDkV

|{z }

used in (4.36)

+hjℓDigij gkℓ −gℓj gkiDkV

|{z }

ﬁrst term in (4.29) .(4.24)

The last two terms in (4.16) are manifestly negative, which is the desired

sign for our purposes. The part Q1of Qrequires further manipulations, as

follows:

V Q1=1

2V gij gkp gℓqDphjℓDqhki

=1

2Vgij Dkhℓ

jDℓhki+O|h|g|Dh|2

gV

=1

2VgkℓDigjkDjgiℓ +O|h|g|Dh|2

gV

=1

2VnDihgkℓ gjkDjgiℓ −gik Djgjℓ i+gkℓDigik Djgjℓ

−gkℓgik Rmi gmℓ −Rℓmij gjm+O|h|g|Dh|2

go.(4.25)

In the notation of (4.17), Equation (4.25) becomes

V Q1=1

2VnDihgkℓ gjkDjgiℓ −gik Djgjℓ i+gkℓDigik Djgjℓ

−χijRij +hiℓhjmRℓmij +O|h|3

g+O|h|g|Dh|2

go

=1

2nDihVgkℓ gjk Djgiℓ −gik Djgjℓ i

−gkℓ gjkDjgiℓ −gik Djgjℓ DiV

|{z }

second term in (4.29)

+|ψ|2

g−χijRij +hiℓhjmRℓmij Vo

+O|h|3

gV+O|h|g|Dh|2

gV . (4.26)

In the special case where gis a (suitably normalised) hyperbolic space-form

14 Hamed Barzegar et al.

we have

Rijkℓ =R

n(n−1) δi

kgjℓ −δi

ℓgjk =−δi

kgjℓ −δi

ℓgjk ,(4.27)

and the relations in (2.3) are satisﬁed. In this case (4.26) becomes

V Q1=1

2nDihVgkℓ gjk Djgiℓ −gik Djgjℓ i

−gkℓ gjkDjgiℓ −gik Djgjℓ DiV

+|ψ|2

g−φ2+n|h|2

gVo+O|h|3

gV+O|h|g|Dh|2

gV. (4.28)

In order to simplify the expressions derived so far we consider similar

terms separately:

1. We wish to add the second term of (4.26) and the third term of (4.24):

hjℓDi(gij gkℓ −gℓj gki)DkV−1

2gkℓ gjkDjgiℓ −gik Djgjℓ DiV

=hjℓ ψjgkℓ +gijDigkℓ −Digℓj gki −gℓj ψkDkV

+1

2hkℓgj kDjgiℓ −hkℓgik ψℓDiV

=1

2hhkjψj+ 3hiℓDigkℓ +gkiDi|h|2

g−2φψk+O|h|2

g|Dh|giDkV

=1

2h−2hkiψi−2φψk

|{z }

=:Ak,taken care of in (4.34)

−3Dihiℓhk

ℓ+gkiDi|h|2

g

| {z }

=:Pk,taken care of in (4.31)

+O|h|2

g|Dh|giDkV , (4.29)

where we used

3hiℓDigkℓ =−3hiℓDihkℓ +O|h|2

g|Dh|g

=−3Dihiℓhk

ℓ+ 3Dihiℓhk

ℓ+O|h|2

g|Dh|g

=−3Dihiℓhk

ℓ−3hkiψi+O|h|2

g|Dh|g.(4.30)

We may rewrite the terms indicated by Pin terms of total divergences

On the total mass of asymptotically hyperbolic manifolds 15

as follows

1

2PkDkV=1

2Dih−3hiℓhk

ℓ+gik|h|2

gDkVi

+1

23hiℓhk

ℓ−gik|h|2

gVRik −λgik +O|h|2

g|Dh|g|DV |

=1

2Dih−3hiℓhk

ℓ+gik|h|2

gDkVi+1

2n3hiℓhk

ℓRik

−R+λ(3 −n)|h|2

g+O|h|3

goV+O|h|2

g|Dh|g|DV |.

(4.31)

When gis Einstein, the result simpliﬁes to:

1

2PkDkV=1

2Dih−3hiℓhk

ℓ+gik|h|2

gDkVi

+1

2(3 −n)|h|2

g+O|h|3

gV+O|h|2

g|Dh|g|DV |.(4.32)

Returning to the general case, in the notation of (4.29) we ﬁnd

AkDkV

≡ −hkiψi+φψkDkV

=−hkiˇ

ψi+φˇ

ψkDkV+1

2hkigiℓDℓφ+φgkℓDℓφDkV

=−hkiˇ

ψi+φˇ

ψkDkV

+1

2hkℓDℓφ+1

2gkℓDℓφ2+O|h|2

g|Dh|gDkV

=−hkiˇ

ψi+φˇ

ψkDkV+1

2(Dℓhkℓφ+1

2gkℓφ2DkV

+ (ψk−ˇ

ψk)φDkV

|{z }

=−1

4[Dℓ(gkℓφ2DkV)−gkℓ φ2DℓDkV+O(|h|2

g|Dh|g)|DV |g]

+ˇ

ψkφDkV

−hkℓφ+1

2gkℓφ2DkDℓV+O|h|2

g|Dh|g|DV |g).(4.33)

16 Hamed Barzegar et al.

Using (2.2), we thus obtain

AkDkV

=−hkiˇ

ψi+1

2φˇ

ψkDkV

| {z }

=:G

+1

2Dℓhkℓφ+1

4gkℓφ2DkV

+1

2n

4+ 1λφ2−1

4Rφ2−hkℓRkℓφV

+O|h|2

g|Dh|g|DV |g+O|h|3

gV . (4.34)

In the space-form case and using (2.3), (4.34) reads

AkDkV

=−hkiˇ

ψi+1

2φˇ

ψkDkV

|{z }

=G

+1

2Dℓhkℓφ+1

4gkℓφ2DkV

−1

2hn

4+ 1φ2V+O|h|2

g|Dh|g|DV |gi+O|h|3

gV . (4.35)

2. We can add the second term of (4.24) to the ﬁrst term of (4.22),

namely VRij gij , using (2.2). Thus, we have

VRijgij +hjℓ(gij gkℓ −gℓj gki)DiDkV

=VR−hij DiDjV+λgij V+Rij χij

+hjℓ gij gkℓ −gℓjgki + ˇχijklDiDkV

=VR+χij + ˆχij Rij −λˆχij gij

=VhR−hiℓhℓj Rij +hijRijφ+R|h|2

g

−λφ2+ (n−2)|h|2

g+O|h|3

gi,(4.36)

where

ˇχiℓjk := 2−gj[ihℓ]k−gk[ℓhi]j+hj[ihℓ]k+gj[iχℓ]k

+gk[ℓχi]j−hj[iχℓ]k−hk[ℓχi]j+χj[iχℓ]k

= 2 −gj[ihℓ]k−gk[ℓhi]j+O(|h|2),(4.37)

On the total mass of asymptotically hyperbolic manifolds 17

which possesses the algebraic symmetries of the Riemann tensor,

ˇχiℓjk =−ˇχℓijk =−ˇχiℓkj = ˇχj kiℓ ,(4.38)

and ˆχik := hjℓ ˇχiℓjk. Then, we have

ˆχikgik =hjℓ gjℓhik −gji hℓk −gkℓhij +gkihℓj +O|h|2

ggik

=φ2+ (n−2)|h|2

g+O|h|3

g.(4.39)

If gis space-form, using (2.3) and keeping in mind that λ=−n, (4.36)

becomes

VR+χij + ˆχij Rij −λˆχij gij

=VR+χijRij −(n−1 + λ)ˆχij gij

=VR+Vφ2− |h|2

g+O|h|3

g.(4.40)

Summarizing we obtain, quite generally,

VR−R=D+Q+G,(4.41)

where

D:= DiV gmj giℓ Dmhjℓ −Dℓhjm+ (gmj gki −gij gkm)hj mDkV

+1

2DihVgkℓ gjk Djgiℓ −gik Djgjℓ i

+1

2Dih−3hiℓhk

ℓ+gik|h|2

gDkVi

+1

2Dihkiφ+1

4gkiφ2DkV(4.42)

is the sum of all divergence terms, and where Gis the gauge-dependent

term deﬁned in (4.34), which has no obvious sign but which can be made to

vanish by a gauge transformation. Finally, Qis the sum of quadratic terms

18 Hamed Barzegar et al.

and error terms, in the general case given by

Q:= (−1

4gij gkpgℓq DℓhkpDqhij +Dihpq Djhkℓ

+1

2|ψ|2

g−χijRij +hiℓhjmRℓmij

+3

2hiℓhk

ℓRik −1

2R+λ(3 −n)|h|2

g

+1

2n

4+ 1λφ2−1

4Rφ2−hkℓRkℓφ

−hiℓhℓj Rij +hij Rijφ+R|h|2

g

−λφ2+ (n−2)|h|2

g+O|h|3

g+O|h|g|Dh|2

g)V

+O|h|2

g|Dh|g|DV |g

=n−1

4|Dφ|2

g−1

4|Dh|2

g+1

2|ψ|2

g+1

2hiℓhj mRℓmij

+1

2hij Rijφ+1

2R−λ(n−1)|h|2

g

+1

8(n−4) λ−Rφ2+O|h|3

g+O|h|g|Dh|2

goV

+O|h|2

g|Dh|g|DV |g.(4.43)

For space-forms this becomes

Q=h−1

4gij gkpgℓq DℓhkpDqhij +Dihpq Djhkℓ

+1

2|ψ|2

g−φ2+n|h|2

g+ (3 −n)|h|2

g+φ2− |h|2

g

−1

2n

4+ 1φ2+O|h|3

g+O|h|g|Dh|2

giV

=h−1

4|Dφ|2

g−1

4|Dh|2

g+1

2|ψ|2

g−n

8φ2+1

2|h|2

g

+O|h|3

g+O|h|g|Dh|2

giV . (4.44)

Using (3.4) and

|h|2

g=|ˆ

h|2

g+1

nφ2,|Dh|2

g=|Dˆ

h|2

g+1

n|D φ|2

g,(4.45)

On the total mass of asymptotically hyperbolic manifolds 19

we can rewrite Qof (4.43) in terms of the trace-free part of hand of ˇ

ψ:

Q=−n+ 2

8n|D φ|2

g−1

4|Dˆ

h|2

g+1

2ˆ

hiℓˆ

hjm Rℓmij +n+ 2

2nφˆ

hij Rij

+n2−4

8n2λφ2+1

2|ˇ

ψ|2

g−ˇ

ψiDiφ) + O|h|3

g+O|h|g|Dh|2

gV

+O|h|2

g|Dh|g|DV |g,(4.46)

where we used (2.2). Putting this into (4.41) we obtain (3.8). Also, when g

is a space-form metric, this gives

Q=h−n+ 2

8n|D φ|2

g−1

4|Dˆ

h|2

g+1

2|ˆ

h|2

g−n2−4

8nφ2

+O|h|3

g+O(|h|g|Dh|2

g)iV

+1

2|ˇ

ψ|2

g−ˇ

ψiDiφ)V+O|h|2

g|Dh|g|DV |g,(4.47)

which is precisely (3.12).

Appendix A. A weighted Poincar´e inequality

When ˇ

ψ= 0 all terms in (3.12) have the desired negative sign except for

those involving undiﬀerentiated occurrences of ˆ

h. To address this, some in-

tegral identities will be needed. Set

(Dˆ

h)ijk := 1

√2Diˆ

hjk −Djˆ

hik,(Lv)ij := 1

2Divj+Djvi,

(div ˆ

h)j:= −Diˆ

hij,(ˆ

hdV )i:= V−1ˆ

hij DjV , (A.1)

and note that L∗= div. For any symmetric tensor ˇ

hwe have (cf., e.g., [9,

Section 3])

(D∗D+L L ∗)ˇ

h= (D∗D+ Ric −Riem)ˇ

h , (A.2)

where

[(Ric −Riem)ˆ

h]ij =1

2(Rikˆ

hkj+Rjk ˆ

hki−2Rikjℓ ˆ

hkℓ).(A.3)

Assume, ﬁrst, that gis a space-form. Multiplying (A.2) by Vˆ

hand in-

20 Hamed Barzegar et al.

tegrating by parts, after some simple manipulations one obtains

Z|ˆ

h|2

gV dµg=1

n+ 1 Zh(|Dˆ

h|2

g− |Dˆ

h|2

g− |div ˆ

h|2

g)V

+Dj(ˆ

hikDiVˆ

hjk )−2ˆ

hikDiV Djˆ

hjk idµg

=1

n+ 1 Zh(|Dˆ

h|2

g− |Dˆ

h|2

g− |div ˆ

h−ˆ

hdV |2

g)V

+Dj(ˆ

hikDiVˆ

hjk ) + |ˆ

hdV |2

gVidµg.(A.4)

In a coordinate system in which the (suitably-normalised) anti-de Sitter

metric greads

g=−(r2+ 1)dt2+dr2

r2+ 1 +r2dΩ2(A.5)

we choose Vas in (1.3) with |~

A| ≤ A0so that

|dV |g< V =⇒ |ˆ

hdV |g≤ |ˆ

h|g.(A.6)

This gives

Z|ˆ

h|2

gV dµg≤1

nZh(|Dˆ

h|2

g− |Dˆ

h|2

g− |div ˆ

h−ˆ

hdV |2

g)V

+Dj(ˆ

hikDiVˆ

hjk )idµg.(A.7)

which provides the desired weighted Poincar´e inequality for space-forms

when the trace-free tensor ﬁeld ˆ

hdecays suﬃciently fast so that the di-

vergence term gives no contribution:

Z|ˆ

h|2

gV dµg≤1

nZ|Dˆ

h|2

gdµg.(A.8)

We now indicate how to adapt the above argument to the general case,

without assuming that the metric is a space form. For this, multiplying (A.2)

by Vˆ

hand integrating by parts we obtain

ZRikjℓˆ

hkℓ −Rik ˆ

hkjˆ

hij V dµg

=Zh(|Dˆ

h|2

g− |Dˆ

h|2

g− |div ˆ

h|2

g)V+Dj(ˆ

hikDiVˆ

hjk )

−2ˆ

hikDiV Djˆ

hjk −Rij ˆ

hjk ˆ

hik−λ|ˆ

h|2

gVidµg.(A.9)

On the total mass of asymptotically hyperbolic manifolds 21

To continue, it is convenient to introduce

ˆ

ψi:= −Djhji+1

2Diφ=−Djˆ

hji+n+ 2

2nDiφ(A.10)

(note that this diﬀers from ˇ

ψiby higher order terms). In this notation (A.9)

can be rewritten as

ZRikjℓˆ

hkℓˆ

hij −λ|ˆ

h|2

gV dµg

=Zh|Dˆ

h|2

g− |Dˆ

h|2

g− | ˆ

ψ−n+ 2

2nD φ|2

g

+2( ˆ

ψk−n+ 2

2nDkφ)(ˆ

hdV )kV+Dj(ˆ

hikDiVˆ

hjk )idµg.(A.11)

One should keep in mind that the divergence term at the right-hand side is

irrelevant for many purposes, in that it gives a vanishing contribution for

suitably decaying ﬁelds when the integral in (A.11) is taken over the whole

manifold.

Let γ > 0 be a constant, which might have to be chosen on a case-by-

case basis depending upon the background geometry at hand. The trivial

identity

−2Dkφ(ˆ

hdV )k=−|γ−1D φ +γˆ

hdV |2

g+|γ−1D φ|2

g+|γˆ

hdV |2

g(A.12)

leads to the following version of (A.11):

ZRikjℓˆ

hkℓˆ

hij −λ|ˆ

h|2

g−n+ 2

2n|γˆ

hdV |2

gV dµg

=Zh |Dˆ

h|2

g− |Dˆ

h|2

g− | ˆ

ψk−n+ 2

2nD φ|2

g+ 2 ˆ

ψk(ˆ

hdV )k

−n+ 2

2n|γ−1Dφ +γˆ

hdV |2

g+n+ 2

2n|γ−1D φ|2

gV

+Dj(ˆ

hikDiVˆ

hjk )idµg.(A.13)

Suppose that there exist constants c≥0 and ε > 0 such that for all φand

ˆ

hwe have

1

2ˆ

hiℓˆ

hjmRℓmij +n+ 2

2nφˆ

hij Rij +n2−4

8n2λφ2

≤cRikjℓˆ

hkℓˆ

hij −λ|ˆ

h|2

g−n+ 2

2n|γˆ

hdV |2

g−ε|ˆ

h|2

g.(A.14)

22 Hamed Barzegar et al.

Integrating (3.10) over the manifold in the gauge ˇ

ψ≡0, using (A.13)-(A.14)

and the decay conditions on hwe obtain

ZQdµg≤Znh−n+ 2

8n|D φ|2

g−1

4|Dˆ

h|2

g−ε|ˆ

h|2

g

+O|h|3

g+O|h|g|Dh|2

giV+O|h|2

g|Dh|g|DV |g

+c|Dˆ

h|2

g+n+ 2

2nγ−2−n+ 2

2n|Dφ|2

gVodµg.(A.15)

The right-hand side will be strictly negative, as desired, for all suﬃciently

small khkL∞and kDhkL∞, provided that V > 0, that V−1|DV |gis bounded,

and that

0< c < 1

4, cγ−2−n+ 2

2n<1

4.(A.16)

This reduces the positivity issue to the algebraic inequality (A.14), with γ

and csatisfying (A.16). The existence of c, and its value, has to be checked

on a case-by-case basis. We note that this strategy does not allow one to

conclude in the case of Horowitz-Myers instantons.

Acknowledgements: PTC wishes to thank IHES, Bures-sur-Yvette, for

hospitality and ﬁnancial support during part of work on this paper. Useful

comments from Erwann Delay are acknowledged.

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Hamed Barzegar

Faculty of Physics and Erwin Schr¨odinger Institute

University of Vienna

Boltzmanngasse 5

A 1090 Wien, Austria

E-mail: hamed.barzegar@univie.ac.at

24 Hamed Barzegar et al.

Piotr T. Chru´sciel

Faculty of Physics and Erwin Schr¨odinger Institute

University of Vienna

Boltzmanngasse 5

A 1090 Wien, Austria

E-mail: piotr.chrusciel@univie.ac.at

url: http://homepage.univie.ac.at/piotr.chrusciel

Luc Nguyen

Mathematical Institute and St Edmund Hall

University of Oxford

Andrew Wiles Building

Radcliﬀe Observatory Quarter

Woodstock Road, Oxford OX2 6GG, United Kingdom

E-mail: luc.nguyen@maths.ox.ac.uk