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On the total mass of asymptotically hyperbolic manifolds

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Abstract

Generalising a similar proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of hyperbolic mass near hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.
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´
scieland 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 DmgjDgjm
+ (gmj gki gij gkm )(gjm gjm)DkVi.(1.1)
We prove the following:
Theorem 1.1 For n3, let (M, g)be Rnequipped with the hyperbolic
metric,
g=dr2
1 + r2+r2d2,(1.2)
where d2is the canonical metric on the (n1)-dimensional sphere Sn1.
Let (A0,~
A)Rn+1 satisfy |~
A|:= p(A1)2+...+ (An)2A0and 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
kggkL+kDgkL< δ ,
where Dis the covariant derivative operator of g, then gcan be put into the
gauge
ˇ
ψj:= Digij 1
2gjk gℓmDkgm = 0 (1.4)
in which we have
m(V)ZMhRR+1
8n|Dg|2
giV dµg(1.5)
where, in local coordinates, g=det g dnx.
It follows clearly from (1.5) that m(V)0 if
RR . (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
0m2
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(RR)V+n+ 2
8n|D φ|2
g+1
4|Dˆ
h|2
g
1
2ˆ
hiℓˆ
hjmRℓmij n+ 2
2nφˆ
hij Rij n24
8n2λφ2
1
2|ˇ
ψ|2
gˇ
ψiDiφ)V+hkiˇ
ψi+1
2φˇ
ψkDkV
+O|h|3
g+O|h|g|Dh|2
gV
+O|h|2
g|Dh|g|DV |gig; (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
2+κdt2+dr2
r2
2+κ+r2hκ,(1.9)
4 Hamed Barzegar et al.
with constants ℓ > 0 and κ∈ {0,±1}, where (n1
N, hκ) is an (n1)-
dimensional space form with Ricci scalar equal to (n1)(n2)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,)×n1
Nis complete
with one locally asymptotically hyperbolic end, where r→ ∞, and
one cuspidal end, where r0. 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 n7, but the non-spin higher dimen-
sional case remains open.
2. In the case κ=1 the manifold of interest is [ℓ, )×n1
N, where
the boundary {} × n1
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, n2 and let
Vbe a static KID on (M, g), i.e. a solution to
DiDjV=VRij R
n1gij.(2.1)
When ghas constant scalar curvature, an equivalent form is
gV+λV = 0 ,DiDjV=V(Rij λgij),(2.2)
for some constant λR. 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(n1) = n(n1) .
If gis an Einstein metric, namely Rij proportional to gij , using this last
scaling we obtain
Rij =(n1)gij , DiDjV=V gij .(2.3)
This implies
Di(|DV |2
gV2) = 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
gV2=|~
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(rn/2), Dkhij =O(rn/2),
V=O(r),|Dkhij |2
gVL1,(RR)VL1.(3.6)
There exists δ > 0such that if
khkL+kDhkL< δ
and if |dV |gV, then we have
mZMhRR+n2
8n|Dh|2
giV dµg
1
2ZM(|ˇ
ψ|2
gˇ
ψiDiφ)V2hkiˇ
ψi+φˇ
ψkDkVg.(3.7)
A sharper bound can be found in (3.15) below.
It follows from (2.5) that |dV |gVholds 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
VRR=D+Q − hkiˇ
ψi+1
2φˇ
ψkDkV , (3.8)
where
D:= DiV gmj giℓ Dmhjℓ Dhj m+Dih(gmjgki gij gkm)hjmDkVi
+1
2DihVgkℓ gjk Djgiℓ gik Djgjℓ i
|{z }
()
+1
2Dih3hiℓhk
+gik|h|2
gDkVi
+1
2Dihkiφ+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
g1
4|Dˆ
h|2
g
+1
2ˆ
hiℓˆ
hjmRℓmij +n+ 2
2nφˆ
hij Rij +n24
8n2λφ2
+1
2|ˇ
ψ|2
gˇ
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
=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=hn+ 2
8n|D φ|2
g1
4|Dˆ
h|2
g+1
2|ˆ
h|2
gn24
8nφ2
+O|h|3
g+O(|h|g|Dh|2
g)iV
+1
2|ˇ
ψ|2
gˇ
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 dµg1
nZh(|Dˆ
h|2
g− |Dˆ
h|2
g− |div ˆ
hˆ
hdV |2
g)V
+Dj(ˆ
hikDiVˆ
hjk )ig.(3.13)
8 Hamed Barzegar et al.
with Ddefined in (A.1). This leads to
ZQgZhn+ 2
8n|D φ|2
g+n2
4n|Dˆ
h|2
g+n24
8nφ2
+1
2n|Dˆ
h|2
g+|div ˆ
hˆ
hdV |2
giV
+1
2nDj(ˆ
hikDiVˆ
hjk )+1
2|ˇ
ψ|2
gˇ
ψiDiφ)V
+O|h|3
gV+O(|h|g|Dh|2
gV) + O|h|2
g|Dh|g|DV |gg.(3.14)
Hence
mZMhRR+n+ 2
8n|D φ|2
g+n2
4n|Dˆ
h|2
g+n24
8nφ2
+1
2n|Dˆ
h|2
g+|div ˆ
hˆ
hdV |2
giV dµg
ZM 1
2|ˇ
ψ|2
gˇ
ψiDiφ)Vhkiˇ
ψi+1
2φˇ
ψkDkV
+O|h|3
gV+O(|h|g|Dh|2
gV) + O|h|2
g|Dh|g|DV |g!g.(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
DkDk+δΓk,(4.2)
On the total mass of asymptotically hyperbolic manifolds 9
where δΓkis a (1,2) tensor equal to δΓ··k:= Γ··kΓ··k. For example, applying
Dkon a vector component viwe get
Dkvi=Dkvi+δΓijk vj,(4.3)
where
δΓijk =1
2giℓ Djgkℓ +Dkgℓj Dgjk
=1
2giℓ Djhkℓ +Dkhℓj Dhjk .(4.4)
The Riemann tensors of the metrics gki and gki are related to each other via
the following equation
Rkimj =Rkimj +DmδΓkij DjδΓkim +δΓkmℓδΓij δΓkjδΓim .(4.5)
Contracting the first and third indices, one obtains
Rij =Rij +DkδΓkij DjδΓkik +δΓkkℓδΓij δΓkj δΓik .(4.6)
Inserting
δΓkik =1
2gkℓ Dihkℓ +Dkhiℓ Dhki =1
2gkℓDihkℓ (4.7)
into (4.6), we obtain
Rij =Rij +1
2"Dkgkℓ Dihjℓ +Djhℓi Dhj i
+gkℓ DkDihjℓ +DkDjhℓi DkDhj iDjgkDihkℓ
gkℓDjDihk+1
2gkpgℓqDhpk Dihj q +Djhiq Dqhij
1
2gkpgℓq Djhℓp +Dhpj Dphj Dihkq +Dkhiq Dqhki#.(4.8)
10 Hamed Barzegar et al.
And the Ricci scalar reads
R=gij Rij
=Rijgij +1
2gij "Dkgkℓ 2Dihjℓ Dhj i
+2gkℓ DkDihjℓ DkDhj i
DjgkℓDihkℓ +1
2gkpgℓq Dhpk 2Dihjq Dqhij
1
2gkpgℓq Djhℓp +Dhpj DphjℓDihkq +Dkhiq Dqhki#.
(4.9)
Using
1
2gij gkpgℓq Djhℓp +Dhpj DphjℓDihkq +Dkhiq Dqhki
=1
2gij gkpgℓqDphj 2Dqhki Dkhiq,(4.10)
this can be rewritten as
R=Rijgij +gij gkℓ DkDihjℓ DkDhj i
+1
2gij hDkgkℓ 2Dihjℓ Dhj i
DjgkℓDihkℓ +1
2gkpgℓqhDhpk 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 Dhj i
Dkgij gkℓDihjℓ Dhji
+1
2gij hDkgkℓ 2Dihjℓ Dhj iDjgkDihkℓ
+1
2gkpgℓq Dhpk 2Dihjq Dqhij Dphjℓ 2Dqhki Dkhiqi.
(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 Dhj iDkgijgkℓ Dihjℓ Dhj i
+1
2gij hgkpgℓqDkhpq 2Dihj Dhji+gkpgℓq Djhpq Dihkℓ
+1
2gkpgℓq Dhpk 2Dihjq Dqhij Dphjℓ 2Dqhki Dkhiqi
=Rijgij +Dkgij gkℓ Dihjℓ Dhj i
+gkℓgip gjqDkhpq +gij gkpgℓq DkhpqDihjℓ Dhji
+1
2gij gkpgℓqhDkhpq 2DihjDhji+DjhpqDihkℓ
+1
2Dhpk 2Dihjq Dqhij Dphj2Dqhki Dkhiqi.(4.14)
After some simplifications one gets
R=Rijgij +Dkgij gkℓ Dihjℓ Dhji+Q , (4.15)
where
Q:= 1
4gij gkpgℓq2DphjDqhki
|{z }
=:Q1
DhkpDqhij Dihpq Djhkℓ .(4.16)
12 Hamed Barzegar et al.
We note that
gij =gij hij +χij ,(4.17)
where
hi=gikhk, hij =gikgj hkℓ ,(4.18)
and
χij := gikgj gmn 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=RR+Rijhij DkgkhjiDgij ˇ
ψkQ
|{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
2DkgklDlφR
nφ
=Rij ˆ
hij R
n(φφ)Dkgkℓhj iDgij +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 Dhj i+Q
=VRij gij +Q+σ , (4.22)
where
σ:= VDkgij gkℓ DihjDhji
=DkV gij gkℓ Dihj Dhjigij gkℓ Dihjℓ Dhj iDkV
|{z }
=:
.(4.23)
On the total mass of asymptotically hyperbolic manifolds 13
Then
=Digij gkℓhjℓDkV+hj Di(gij gkDkV)
+Dgij gkℓhjiDkVhj iDgij gkDkV
=D(gij gkℓ gℓj gki)hjiDkV+hjℓ(gij gkgℓj 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 Dkh
jDhki+O|h|g|Dh|2
gV
=1
2VgkℓDigjkDjgiℓ +O|h|g|Dh|2
gV
=1
2VnDihgkℓ gjkDjgiℓ gik Djgjℓ i+gkℓDigik Djgj
gkℓgik Rmi gmℓ Rmij 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 Djgj
χijRij +hihjmRmij +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
gχijRij +hihjmRmij 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
Rijk=R
n(n1) δi
kgjℓ δi
gjk =δi
kgjℓ δi
gjk ,(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
gφ2+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)DkV1
2gkℓ gjkDjgiℓ gik Djgjℓ DiV
=hjℓ ψjgk+gijDigkDigℓj gki gℓj ψkDkV
+1
2hkℓgj kDjgiℓ hkℓgik ψDiV
=1
2hhkjψj+ 3hiDigkℓ +gkiDi|h|2
g2φψk+O|h|2
g|Dh|giDkV
=1
2h2hkiψi2φψk
|{z }
=:Ak,taken care of in (4.34)
3Dihiℓhk
+gkiDi|h|2
g
| {z }
=:Pk,taken care of in (4.31)
+O|h|2
g|Dh|giDkV , (4.29)
where we used
3hiDigkℓ =3hiDihkℓ +O|h|2
g|Dh|g
=3Dihiℓhk
+ 3Dihiℓhk
+O|h|2
g|Dh|g
=3Dihiℓhk
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
+gik|h|2
gDkVi
+1
23hiℓhk
gik|h|2
gVRik λgik +O|h|2
g|Dh|g|DV |
=1
2Dih3hiℓhk
+gik|h|2
gDkVi+1
2n3hiℓhk
Rik
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
+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
2hkigiℓDφ+φgkℓDφDkV
=hkiˇ
ψi+φˇ
ψkDkV
+1
2hkℓDφ+1
2gkℓDφ2+O|h|2
g|Dh|gDkV
=hkiˇ
ψi+φˇ
ψkDkV+1
2(Dhkℓφ+1
2gkℓφ2DkV
+ (ψkˇ
ψk)φDkV
|{z }
=1
4[D(gkℓφ2DkV)gkℓ φ2DDkV+O(|h|2
g|Dh|g)|DV |g]
+ˇ
ψkφDkV
hkℓφ+1
2gkℓφ2DkDV+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
2Dhkℓφ+1
4gkℓφ2DkV
+1
2n
4+ 1λφ21
42hkRkℓφV
+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
2Dhkℓφ+1
4gkℓφ2DkV
1
2hn
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 gkgℓj gki)DiDkV
=VRhij DiDjV+λgij V+Rij χij
+hjℓ gij gkgℓjgki + ˇχijklDiDkV
=VR+χij + ˆχij Rij λˆχij gij
=VhRhihℓj Rij +hijRijφ+R|h|2
g
λφ2+ (n2)|h|2
g+O|h|3
gi,(4.36)
where
ˇχiℓjk := 2gj[ih]kgk[hi]j+hj[ih]k+gj[iχ]k
+gk[χi]jhj[iχ]khk[χi]j+χj[iχ]k
= 2 gj[ih]kgk[hi]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 gjhik gji hℓk gkℓhij +gkihℓj +O|h|2
ggik
=φ2+ (n2)|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 (n1 + λ)ˆχij gij
=VR+Vφ2− |h|2
g+O|h|3
g.(4.40)
Summarizing we obtain, quite generally,
VRR=D+Q+G,(4.41)
where
D:= DiV gmj giℓ DmhjDhjm+ (gmj gki gij gkm)hj mDkV
+1
2DihVgkℓ gjk Djgiℓ gik Djgjℓ i
+1
2Dih3hiℓhk
+gik|h|2
gDkVi
+1
2Dihkiφ+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 DhkpDqhij +Dihpq Djhkℓ
+1
2|ψ|2
gχijRij +hihjmRmij
+3
2hiℓhk
Rik 1
2R+λ(3 n)|h|2
g
+1
2n
4+ 1λφ21
42hkRkℓφ
hihℓj Rij +hij Rijφ+R|h|2
g
λφ2+ (n2)|h|2
g+O|h|3
g+O|h|g|Dh|2
g)V
+O|h|2
g|Dh|g|DV |g
=n1
4||2
g1
4|Dh|2
g+1
2|ψ|2
g+1
2hiℓhj mRℓmij
+1
2hij Rijφ+1
2Rλ(n1)|h|2
g
+1
8(n4) λRφ2+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=h1
4gij gkpgℓq DhkpDqhij +Dihpq Djhkℓ
+1
2|ψ|2
gφ2+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
=h1
4||2
g1
4|Dh|2
g+1
2|ψ|2
gn
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
g1
4|Dˆ
h|2
g+1
2ˆ
hiℓˆ
hjm Rℓmij +n+ 2
2nφˆ
hij Rij
+n24
8n2λφ2+1
2|ˇ
ψ|2
gˇ
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=hn+ 2
8n|D φ|2
g1
4|Dˆ
h|2
g+1
2|ˆ
h|2
gn24
8nφ2
+O|h|3
g+O(|h|g|Dh|2
g)iV
+1
2|ˇ
ψ|2
gˇ
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:= V1ˆ
hij DjV , (A.1)
and note that L= div. For any symmetric tensor ˇ
hwe have (cf., e.g., [9,
Section 3])
(DD+L L )ˇ
h= (DD+ Ric Riem)ˇ
h , (A.2)
where
[(Ric Riem)ˆ
h]ij =1
2(Rikˆ
hkj+Rjk ˆ
hki2Rikjℓ 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 ig
=1
n+ 1 Zh(|Dˆ
h|2
g− |Dˆ
h|2
g− |div ˆ
hˆ
hdV |2
g)V
+Dj(ˆ
hikDiVˆ
hjk ) + |ˆ
hdV |2
gVig.(A.4)
In a coordinate system in which the (suitably-normalised) anti-de Sitter
metric greads
g=(r2+ 1)dt2+dr2
r2+ 1 +r2d2(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 dµg1
nZh(|Dˆ
h|2
g− |Dˆ
h|2
g− |div ˆ
hˆ
hdV |2
g)V
+Dj(ˆ
hikDiVˆ
hjk )ig.(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 dµg1
nZ|Dˆ
h|2
gg.(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
ZRikjˆ
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
hikλ|ˆ
h|2
gVig.(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
ZRikjˆ
hkℓˆ
hij λ|ˆ
h|2
gV dµg
=Zh|Dˆ
h|2
g− |Dˆ
h|2
g− | ˆ
ψn+ 2
2nD φ|2
g
+2( ˆ
ψkn+ 2
2nDkφ)(ˆ
hdV )kV+Dj(ˆ
hikDiVˆ
hjk )ig.(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=−|γ1D φ +γˆ
hdV |2
g+|γ1D φ|2
g+|γˆ
hdV |2
g(A.12)
leads to the following version of (A.11):
ZRikjˆ
hkℓˆ
hij λ|ˆ
h|2
gn+ 2
2n|γˆ
hdV |2
gV dµg
=Zh |Dˆ
h|2
g− |Dˆ
h|2
g− | ˆ
ψkn+ 2
2nD φ|2
g+ 2 ˆ
ψk(ˆ
hdV )k
n+ 2
2n|γ1+γˆ
hdV |2
g+n+ 2
2n|γ1D φ|2
gV
+Dj(ˆ
hikDiVˆ
hjk )ig.(A.13)
Suppose that there exist constants c0 and ε > 0 such that for all φand
ˆ
hwe have
1
2ˆ
hiℓˆ
hjmRℓmij +n+ 2
2nφˆ
hij Rij +n24
8n2λφ2
cRikjˆ
hkℓˆ
hij λ|ˆ
h|2
gn+ 2
2n|γˆ
hdV |2
gε|ˆ
h|2
g.(A.14)
22 Hamed Barzegar et al.
Integrating (3.10) over the manifold in the gauge ˇ
ψ0, using (A.13)-(A.14)
and the decay conditions on hwe obtain
ZQgZnhn+ 2
8n|D φ|2
g1
4|Dˆ
h|2
gε|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
2nγ2n+ 2
2n||2
gVog.(A.15)
The right-hand side will be strictly negative, as desired, for all sufficiently
small khkLand kDhkL, provided that V > 0, that V1|DV |gis bounded,
and that
0< c < 1
4, cγ2n+ 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
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