Nonzero energy of 2� 1 Minkowski space

Donald Marolf* and Leonardo Patino†

Physics Department, UCSB, Santa Barbara, California 93106, USA(Received 3 May 2006; published 10 July 2006)

We compute the energy of 2� 1 Minkowski space from a covariant action principle. Using Ashtekarand Varadarajan’s characterization of 2� 1 asymptotic flatness, we first show that the 2� 1 Einstein-Hilbert action with Gibbons-Hawking boundary term is both finite on shell (apart from past and futureboundary terms) and stationary about solutions under arbitrary smooth asymptotically flat variations ofthe metric. Thus, this action provides a valid variational principle and no further boundary terms arerequired. We then obtain the gravitational Hamiltonian by direct computation from this action. The resultagrees with the Hamiltonian of Ashtekar and Varadarajan up to an overall additive constant. This constantis such that 2� 1 Minkowski space is assigned the energy EMink2�1

� � 14G , while the upper bound on the

energy becomes E � 0. Any variational principle with a boundary term built only from the extrinsic andintrinsic curvatures of the boundary is shown to lead to the same result. Interestingly, our result is not the�! 0 limit of the corresponding energy EAdS2�1

� � 18G of 2� 1 anti-de Sitter (AdS) space.

DOI: 10.1103/PhysRevD.74.024009 PACS numbers: 04.60.Kz, 04.20.Fy

I. INTRODUCTION

In covariant approaches to quantum mechanics, classicalmechanics is recovered through a semiclassical approxi-mation in which the path integral is dominated by sta-tionary points of the action. Here it is critical that theaction be stationary under all variations tangent to thespace of paths over which the integral is performed. Thusone must consider all variations which preserve any bound-ary conditions and not just, for example, variations ofcompact support. In particular, requiring the action to bestationary should yield precisely the classical equations ofmotion, with all boundary terms in the associated compu-tation vanishing on any allowed variation.

We are concerned here with gravitational systems. Themost familiar action for gravity is the Einstein-Hilbertaction with Gibbons-Hawking boundary term,

SEH�GH �1

16�G

ZM

��������gp

R�1

8�G

Z@M

��������hp

K;

(1.1)

where hij is the induced metric on a timelike boundary@M and Kij is the extrinsic curvature of this boundary,with trace K � Kijhij. However, for spacetimes with non-compact spatial slices the action (1.1) is generally notstationary under the full class of allowed variations aboutsolutions. In the asymptotically anti-de Sitter (AdS) con-text, it was shown in [1] that adding the AdS countertermsof [2–10] results in a fully stationary action. In addition,[11] showed that a fully stationary action for asymptoti-cally flat spacetimes of dimension d � 3� 1 can be de-fined by adding an appropriate additional boundary term to(1.1). In the asymptotically flat cases most closely related

to what we study below, this additional boundary termreduces to the term proposed in [12] (see also [13]).1

Here we address the case of 2� 1 asymptotically flatdimensions. In contrast to the higher dimensional case, wefind that this Gibbons-Hawking term suffices and that nofurther boundary terms are required to make the actionfully stationary. In particular, following the characteriza-tion of 2� 1 asymptotic flatness given in [19] (reviewed inSec. II below), we show in Sec. III that (1.1) is both(i) finite on the space of solutions (apart from past andfuture boundary terms) and (ii) stationary on solutionsunder any variation which preserves this definition ofasymptotic flatness.

Thus, the action (1.1) provides a valid variational prin-ciple. As a result, one may calculate a gravitationalHamiltonian as the Legendre transform of the correspond-ing Lagrangian. The calculation is essentially that of [20],but without a background subtraction term. We performthis calculation in Sec. IV. The result agrees with theHamiltonian of Ashtekar and Varadarajan [19] up to anoverall additive constant. This constant is such that ourdefinition assigns 2� 1 Minkowski space the nonzeroenergy EMink � �

14�G , while the upper bound on the

energy becomes E � 0. From the Hamiltonian form ofthe action it is straightforward to show (Sec. III B) thatthe same result also follows from the boundary stress

*Electronic address: marolf@physics.ucsb.edu†Electronic address: leo_p@physics.ucsb.edu

1Fully stationary variational principles were developed some-what earlier in other formalisms. For example, the Regge-Teitelboim construction [14] defines a fully stationaryHamiltonian H, which immediately leads to a fully stationaryvariational principle of the form S �

Rdt��

R� ~�ab _gab� �H.

This method was applied to asymptotically AdS spaces in[15,16]. Similarly, in 3� 1 dimensions [17,18] provided a fullystationary variational principle in the Palatini formalism andfurther showed that the 3� 1 decomposition of that action ledto the analogue of the Arnowitt-Deser-Misner (ADM)Hamiltonians.

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tensor associated with (1.1). We close with a brief discus-sion in Sec. V.

II. ASYMPTOTIC FLATNESS FOR d � 2� 1

We now review the definition of 2� 1 asymptotic flat-ness given in [19]. Recall that the solution

ds2 � �dt2 � r�8 GM�dr2 � r2d�2� for r > 0 (2.1)

represents a point particle of mass M at the origin [21].Here,G is the three dimensional Newton’s constant and thecoordinates take values as follows: t 2 ��1;�1�, r 2�0;1�, and � 2 �0; 2��; i.e., the coordinates range over theusual values associated with familiar cylindricalcoordinates.

Curvature does not propagate in 2� 1 dimensions, sofor any solution the Riemann tensor vanishes whereverT�� � 0. In particular, the metric (2.1) is flat everywhereexcept at the origin where the particle is located. To seethis, we can perform the coordinate transformation

� r�

�; �� �� with � 1–4 GM; (2.2)

to cast the metric in the form

ds2 � �dt2 � d�2 � �2d ��2; (2.3)

from which the flatness of the metric is apparent.Given that �� 2 �0; 2���, one also sees that there is a

deficit angle which, despite the local flatness of the solutionaway from the origin, makes this spacetime not globallyequivalent to Minkowski space. As a result, the generaldefinition of 2� 1 asymptotic flatness is chosen to allowmetrics which asymptotically approach such canonicalspacetimes. In particular, Ashtekar and Varadarajan [19]consider a 2� 1 spacetime to be asymptotically flat if theline element admits an expansion of the form

ds2 � ��1�O�1=r��dt2 � r����1�O�1=r��dr2

� r2�1�O�1=r��d�2 �O�r�1��=2�dtd�; (2.4)

for large positive r if � 2 �0; 2�. We use this characteriza-tion of asymptotic flatness below.

III. FINITENESS AND STATIONARITY OF THEACTION ON THE SPACE OF SOLUTIONS

In this section, we show that the action (1.1) is both(i) finite apart from past and future boundary terms and(ii) fully stationary on the space of asymptotically flatsolutions, under any variation preserving asymptotic flat-ness. We then compare the situation in 2� 1 dimensionswith that of higher dimensions, in which more complicatedboundary terms are required. We will see that the higherdimensional boundary terms do not extend to the 2� 1case in an obvious way.

Let us begin by stating precisely the prescription forcomputing the boundary term associated with the timelike

part of the boundary @M, which we call @M1. Thenotation @M1 is shorthand for some one-parameter familyof boundaries of regions M� �M. We take the regionsM� to be an increasing family, so that they satisfy M� �M�0 whenever �<�0, and such that they cover M, i.e.[�M� �M. Any such family represents a way of in-troducing a cutoff for the spacetime M and then removingit as �! 1. Here we take � � r�O�r0�. This is theanalogue of what was termed a ‘‘cylindrical spatial cutoff’’in [11]. The definition of � has been chosen so that theinduced line element on any � � constant surface is of theform

ds2@M � hijdxidxj

� ��1�O�1=r��dt2 � �r2�� �O�r�1����d�2

�O�r�1��=2�dtd�; (3.1)

with inverse metric

hij �htt ht�

h�t h��

� �

��1�O�1=r� O�r�3��=2�

O�r�3��=2� r��2 �O�r��3�

" #: (3.2)

To obtain a finite action, we also require some cutoff onthe integral over time. Here we simply restrict the coor-dinate t of (2.4) to some finite range t 2 �T�; T�. Thiscorresponds to what was called a ‘‘cylindrical temporalcutoff’’ in [11].

One might also attempt to consider an analogue of thehyperbolic temporal cutoff from [11]. However, this is lessnatural in the 2� 1 context due to the lack of asymptoticLorentz invariance [19,22]. Furthermore, hyperbolic cut-offs are less natural in the context of the Hamiltonianmethods we wish to apply. For these reasons we consideronly the cylindrical cutoff below.

A. The action is finite and stationary

To fully define the space of asymptotically flat geome-tries, we need to specify the asymptotic behavior of thestress-energy tensor. Following [19], we take the compo-nents of T�� in Cartesian coordinates to be of orderO�r��3�, so that in polar coordinates we have

T�� �Ttt Ttr Tt�Trt Trr Tr�T�t T�r T��

264375

�O�r��3� O�r��3� O�r��2�

O�r��3� O�r��3� O�r��2�

O�r��2� O�r��2� O�r��1�

264

375: (3.3)

Using (2.4) to take the trace we see that R� g��T�� �O�r��3�. From (2.4) we also find

��������gp

� r1�� �O�r���,so that the integrand in the bulk part of (1.1) is of order r�2.It follows that the Einstein-Hilbert term is finite when

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evaluated on any smooth asymptotically flat 2� 1solution.

We now consider the boundary term on the timelikeboundary @M1:

1

8�G

Z@M1

��������hp

K: (3.4)

By direct calculation for metrics with the asymptotic be-havior (2.4) one finds

K�� �Ktt Kt�K�t K��

� ��

O�r�=2�2� O�r�2��

O�r�2� 12 �2� ��r

1��=2

" #;

(3.5)

so that to leading order in r one has

K � 12�2� ��r

�1��=2 �O�r�2��=2�: (3.6)

Using the fact that��������hp

� r1��=2 �O�r1=2��=2�, we seethat the Gibbons-Hawking integrand is given by

��������hp

K � 1� �=2�O�r�1=2�: (3.7)

It follows that, apart from past and future boundary terms,the action (1.1) is finite on asymptotically flat 2� 1solutions.

We now turn to stationarity of the action. We use the factthat under a general variation one has

SHE�GH �1

16�G

Z@M

��������hp

�Kij � Khij�hij: (3.8)

Recall that hij vanishes on the past and future boundaries,while near @M1 we may assemble the results (3.1), (3.2),and (3.5) to find

��������hp

�Kij � Khij�hij ���������hp

�Ktt � Khtt�htt �O�1=r�;

(3.9)

which vanishes in the limit r! 1. Thus, as claimed inSec. I, the action SEH�GH is stationary on 2� 1 asymptoti-cally flat solutions under arbitrary asymptotically flatvariations.

B. Comparison with higher dimensions

We showed above that, for asymptotically flat space-times in 2� 1 dimensions, the Gibbons-Hawking termprovides a sufficient boundary term to promote theEinstein-Hilbert action to a satisfactory variational princi-ple. However, additional terms are needed in higher di-mensions [11]. Here we briefly discuss the extrapolation to2� 1 dimensions of various additional boundary terms(called ‘‘counterterms’’) which have been proposed forhigher dimensional cases. We find that all such extrapola-tions either vanish identically or are ill-defined.

The simplest asymptotically flat counterterm to discussis the one proposed in [12], proportional to

R ��������hp ������

Rp

,where R is the Ricci scalar of the boundary metric.

Because a 1� 1 cylinder is flat, this term simply vanishesfor 2� 1 asymptotically flat geometries. Similarly, onemay consider the counterterm proposed in [13], which isproportional to R3=2��������������������

R2�RijRij

p , where Rij is the Ricci tensor

induced on @M1. When this ratio is well defined, it againvanishes in the limit in which the cutoff is removed.

A more complicated (but more general) counterterm wasintroduced in [11]. This term is given by

�1

8�G

Z@M

K; (3.10)

with K defined implicitly by the equation

R ij � KijK � KilKlj: (3.11)

However, this equation degenerates in 2� 1 dimensions.This is easiest to see by considering the stronger relation

R ikjl � KijKkl � KikKjl; (3.12)

which reduces to (3.11) when contracted with hkl. Note thatthe right-hand side of (3.12) has all of the symmetries ofthe Riemann tensor. As a result, in 1� 1 dimensions it hasonly a single independent component which in fact is equalto det�K�. Thus, Eq. (3.11) can determine at most det�K�and not the desired trace of Kij. We see that the counter-term (3.10) is not well defined for 1� 1 dimensionalboundaries.

Finally, one may ask if we may use as a counterterm anadditional Gibbons-Hawking term

�1

8�G

Z@M

KRef ; (3.13)

evaluated on some reference background (see e.g.,[20,23,24]). This typically requires embedding the bound-ary (@M, h) in the background and using this embeddingto define the reference extrinsic curvature KRef . As de-scribed in [11], such an approach fails in higher dimensionsbecause a generic boundary cannot be isometrically em-bedded in a given reference background. Roughly speak-ing, codimension one embeddings into a fixed manifold arespecified by a single relation between the coordinates ofthe target manifold, while more than one function of theembedded manifold is required to specify a generic (fullygauge fixed) metric. However, in 1� 1 dimensions wemay always choose, e.g., conformal gauge, in which anarbitrary geometry is specified by a single free function.Thus, this counting argument does not rule out the use ofbackground subtraction for 2� 1 dimensional bulkspacetimes.

Nonetheless, it remains far from clear that backgroundsubtraction can succeed in the 2� 1 context. The problemis that, at least locally, the intrinsic geometry of @M doesnot determine the extrinsic curvature �Kref�ij. Indeed, weargued above that �Kref�ij cannot be determined from(3.12). Of course, an embedding would also satisfy the

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remaining Gauss-Codazzi equations

r�R�jkl � DkKrefjl �DlKref

jk; (3.14)

where Dk is the covariant derivative compatible with hijand r� is the unit vector normal to @M1. However, sincethese are differential equations and our @M1 is a manifoldwith boundary, we still expect constants of integration inany solution to (3.14). Without some prescription for thedependence of such integration constants on the cutoff �,the �! 1 limit of the boundary term (3.13) is ill defined.Thus we see that any proposed counterterm from higherdimensional asymptotically flat space which is well de-fined in 2� 1 dimensions vanishes identically.

IV. THE GRAVITATIONAL ENERGY

We have seen that the action (1.1) is finite up to past andfuture boundary terms, and that an arbitrary asymptoticallyflat variation of (1.1) vanishes on the space of asymptoti-cally flat solutions. We now compute the gravitationalHamiltonian as the Legendre transform of the associatedLagrangian. We proceed by performing a 2� 1 decom-position of the action following [20], which in turn follows[25–28] in addressing the bulk terms. Finally, we quicklyshow that the same energy is obtained by considering theboundary stress tensor associated with the action (1.1).

A. The 2� 1 decomposition and the gravitationalHamiltonian

Let us begin by foliating the spacetime with a family ofspacelike surfaces �t labeled by the coordinate t. Weintroduce a vector t� such that t�r�t � 1. In terms ofthe unit normal vector to the surfaces �t we can decom-pose t� into the usual lapse function N and shift vector N�

through t� � Nn� � N�. We choose �t orthogonal to the1� 1 part of the asymptotic boundary2 @M1. We assumethere are no inner boundaries, so the total boundary con-sists of @M1 (tangent to n�) together with initial and finalsurfaces for which n� is a unit normal. Below, we includein (1.1) the Gibbons-Hawking terms on the initial and finalsurfaces as well as that on @M1.

To perform the 2� 1 decomposition of the action, weexpress the three dimensional scalar curvature in terms ofthe induced scalar curvature R and the extrinsic curvatureKtij of �t. As a first step, we write

R � 2�G�� � R���n�n�: (4.1)

From the usual initial value constraint, the first term is

2G��n�n� �R� Kt

���Kt��� � �Kt�2: (4.2)

For the second term we use the identity

R��n�n� � Kt 2 � Kt

��Kt�� �r��n

�r�n��

� r��n�r�n��: (4.3)

When integrated over spacetime, the two total derivativesin (4.3) yield boundary terms. The first is proportional ton�, so it contributes only on the initial and final surfaces.On these surfaces, it completely cancels the Gibbons-Hawking term. The second term is orthogonal to n� so itwill contribute only on @M1. Adding the Gibbons-Hawking term to the boundary terms (4.3) and expressingthe result in terms of the unit vector r� normal to @M1

one finds

1

8�G

Z@M1

��������hp

�r�r� � r�n

�r�n��

�1

8�G

Z@M1

��������hp

�g�� � n�n��r�r�; (4.4)

where g�� is the metric corresponding to the line element(2.4). Recognizing q�� � g�� � n�n� as the projectoronto the surface �t, we see that the integrand in (4.4) isthe trace of the extrinsic curvature of the curve Ct � �t \@M, in the surface �t. We will call this trace k.

Thus, the action (1.1) takes the form

S �ZNdt

�1

16�G

Z�t

���qp�R� �Kt�2 � Kt

���Kt����

�1

8�G

ZCt

������qCp

k�; (4.5)

where q is the determinant of the metric induced on �t andqC is the determinant of the metric induced on Ct.

Recalling that the extrinsic curvature Kt�� of �t is given

by

Kt�� �

1

2N� _q�� � 2D��N��; (4.6)

where D is the covariant derivative on �t compatible withq��, and the momentum ~��� conjugate to q�� is, as usual,

~��� �L _q��

�

���qp

16�G�Kt�� � Ktq���: (4.7)

Defining an undensitized momentum ��� � 16�G��qpN ~���, we

can write the action as

S �ZNdt

�1

16�G

Z�t

���qp���� _q�� �R� �2

� ������ � 2N�D����� �1

8�G

ZCt

������qCp

k�

�1

8�G

ZdtZCt

������qCp

N�r����: (4.8)

It is now easy to Legendre transform the Lagrangian toobtain

2In fact, we choose the spatial regulator � such that, forsufficiently large �, �t is orthogonal to each regulated boundary@M�.

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H � �1

16�G

Z�t

N���qp�R� �2 � ������

� 2N�D����� �1

8�G

ZCt�N

������qCp

k

�������qCp

N�r�����: (4.9)

However, the boundary term simplifies further when wemake explicit use of the asymptotic behavior (2.4). Thefirst term of the boundary integrand is, to leading order, thesame as we computed in (3.7). For the second term we have

�ij ��tt �t���t ���

� ��

O�r�1��� O�r�1���

O�r�1��� O�r1���

� �; (4.10)

while the vectors N� and r� satisfy

N� �O�r�1���;

rr � r�=2 �O�r�1���; and r� �O�r�1���:(4.11)

Thus the second term in the integral over Ct is of orderO�r�4�� and vanishes as r! 1.

As a result, we obtain

H � �1

16�G

Z�t

N���qp�R� �2 � ������

� 2N�D����� �

1

16�G

ZCt�2� ��: (4.12)

This coincides with the Hamiltonian obtained in [19] viaRegge-Teitelboim methods [14], except for the addition ofthe constant term � 1

16�G

RCt

2 � � 14G .

B. Energy from the boundary stress tensor

By using the Hamiltonain form of the action (4.8), it isstraightforward to compare the ADM expressions (4.12)for the energy with that defined via a boundary stresstensor. Recall [11,29] that while for asymptotically flatspacetimes one cannot vary the on shell action with respectto boundary conditions, a useful boundary stress tensor cannevertheless be defined by first considering the action S�

of each regulated spacetime M� defined in Sec. III. Onedefines

T����� � �2��������hp

S�

h��� �

2��������hp

S�

g��; (4.13)

and the boundary stress tensor energy is defined by

Ebst � lim�!1

ZCt

������qCp

n�n�T��: (4.14)

To compare with the Hamiltonian definition of energy(4.12), we simply express T�� in terms of variations of theaction with respect to the lapse N and shift N�, with thespatial metric q�� � g�� � n�n� held constant. To thisend, it is useful to compute certain partial derivatives.Using the relation dt � � 1

N n�dx� we find

g��N

��������q�

�N��n�n�� �

�2

Nn�n�: (4.15)

As a result, we have

Ebst �ZCt

������qCp

n�n�T�� �

ZCt

������qCp 2n�n��������

�hp

Sg��

� �ZCt

g��N

Sg��

� �ZCt

SN

��������q��

� �1

16�G

ZCt�2� ��: (4.16)

Since the bulk contribution to (4.12) vanishes on the con-straint surface, it is clear that the Hamiltonian and bound-ary counterterm definitions of energy agree.3

V. DISCUSSIONS

We have shown above that the Einstein-Hilbert actionwith Gibbons-Hawking term (1.1) provides a satisfactoryvariational principle for 2� 1 asymptotically flat space-times. In higher dimensions it is important to add morecomplicated boundary terms but, as discussed in Sec. III B,these do not extend naturally to 2� 1 dimensions.Defining the boundary by a cylindrical cutoff prescription,we have shown that on the space of the solutions the action(i) is finite (apart from the past and future boundary terms)and (ii) is stationary under any asymptotically flatvariation.

Because (1.1) is appropriately finite and defines a goodvariational principle, a Hamiltonian can be defined directlyvia the Legendre transform of the Lagrangian. The result ofthis Legendre transform is (4.12), which agrees with thestandard result [19] except for a shift of the zero of energy.Interestingly, this shift causes our Hamiltonian to takevalues in the range

H 2��

1

4G; 0�; (5.1)

and, in particular, assigns 2� 1 Minkowski space theenergy EMink2�1

� � 14G . Such a nonzero energy is possible

because the phase space of asymptotically flat gravity in2� 1 dimensions is not invariant under Lorentz transfor-mations [19,22].

As noted above, our shift sets the upper bound on theenergy of 2� 1 asymptotically flat spacetimes to zero.Physically, the upper bound arises because, at this valueof the energy, the deficit angle at infinity reaches 2� andthe asymptotic region ‘‘closes off.’’ Now, recall that anyspatially closed universe is naturally assigned zero energyas well. In particular, since there is no boundary M1, this

3Note that, assuming that the appropriate actions can bewritten in an appropriate canonical form, this argument providesa simple derivation of the main result of [30] and of certainresults from [31,32]

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assignment follows from a definition of the Hamiltonianfor such systems via a Legendre transform of theLagrangian as above; one finds that the Hamiltonian isconstrained to vanish. This continuity property of theenergy when interpolating between spatially compact andasymptotically flat spacetimes may merit furtherinvestigation.

To readers familiar with asymptotically AdS counter-terms (e.g., [2,3]), it may not seem surprising that (1.1)provides a valid variational principle. In 2� 1 dimensions,the only additional AdS counterterm required (beyond theterms in (1.1)) is proportional to 1

‘

R@M

��������hp

, where ‘ isthe AdS length scale. Thus, this additional AdS counter-term vanishes as ‘! 1. In contrast, in higher dimensionsAdS boundary conditions require boundary terms propor-tional to positive powers of ‘, and so do not readily admit aflat space limit.

Similarly, to AdS-familiar readers, it may not be surpris-ing that 2� 1 Minkowski space has a negative vacuumenergy. In 2� 1 dimensions, adding the appropriate cova-riant counterterms and using the boundary stress tensorprescription [3] yields an energy EAdS2�1

� �1=8G for2� 1 AdS space, and, in particular, one which is indepen-dent of the AdS length scale ‘. (In contrast, in higherdimensions the energy of AdS space vanishes as ‘! 1.)However, what is quite striking is that the energyEMink2�1

� �1=4G of 2� 1 Minkowski space is not thelimit as ‘! 1 of the AdS result. Instead, it differs by afactor of 2. One may hope that a deeper understanding ofthis result is related to a deeper understanding of the flatspace limit of AdS/conformal field theory (CFT).

One may ask how unique is our prescription for calcu-lating the energy. After all, we have obtained the result(4.12) from a particular choice of boundary terms in thecovariant action (1.1). Now, for asymptotically AdS spacesin 2� 1 dimensions, the boundary terms are entirely de-termined by the properties of locality and covariance [3].Thus the result EAdS2�1

� �1=8G does not suffer from the‘‘scheme dependence’’ that one sees in higher dimensions.We will now show that our results are similarly uniquegiven the requirement that one begin with a well-definedvariational principle built from a local covariant boundaryterm.

To this end, consider any choice of boundary term builtlocally from the extrinsic curvature Kij and the Riemanncurvature Rijk

l of @M1 and which leads to a valid varia-tional principle for asymptotically flat spacetimes. Anysuch boundary term will differ from the one in (1.1) bysome finite �Swhose variation vanishes within the class ofasymptotically flat spacetimes. In particular, it must beinvariant under (time-dependent) variations of the deficitangle at infinity,

�! �� ��t�: (5.2)

To see the effect of this requirement, let us evaluate �S on

the point particle spacetimes (2.1) using r � constant sur-faces to compute the boundary term. The Riemann tensorvanishes on such surfaces, and the only nonzero compo-nent of Kij is K�� �

12 �2� ��r

1��=2. The boundary metricitself can enter only through

��������hp

and h��, and so providesonly factors of r �1��=2�. If �S is to have a well-definedlimit at large r, the r-dependence of the metric can serveonly to cancel the r-dependence of K��. As a result, wehave �S � f�K��r

��1��=2���T� � T�� � f�2� ���T� �T��.

Now, consider a variation ��t� which is constant over atime interval �t constituting most of interval [T�, T�], butwhich vanishes at t � T to preserve the past and futureboundary conditions. Since the variation of �S under (5.2)contains a term proportional to df

d��t, our �S can be sta-tionary only if f � constant; i.e., if on such solutions f isindependent of Kij. But since the Riemann tensor on @Mvanishes, on such solutions we must have

�S � cZ@M1

��������hp

: (5.3)

Imposing either the requirement that �S be finite or that�S be stationary then sets c � 0, and thus �S � 0 on suchsolutions. But since �S is stationary under all asymptoti-cally flat variations, it is in fact constant on the space ofsolutions. Thus, �S � 0 on all solutions. As a result, itcannot affect the result (4.12) for the energy.

On dimensional grounds, a vacuum energy such as wehave found can arise only in 2� 1 dimensions, wherec2=G has dimensions of mass (without using any factorsof @). Indeed, calculations of the vacuum energy of d� 1Minkowski space from a covariant variational principle ford > 2 obtain EMinkowski � 0 [11,29]. Here comparison withAdS space is more subtle, as the AdS vacuum energydepends [3] on the particular choice of boundary term(i.e., the choice of ‘‘renormalization scheme’’) used todefine the covariant variational principle. Nevertheless, itis interesting to note that in the most common scheme thevacuum energy diverges in the limit �! 0. This againsuggests subtleties in the flat space limit of AdS/CFTwhich would be interesting to understand in detail.

ACKNOWLEDGMENTS

The authors wish to thank Jim Isenberg and GregGalloway for input on the two dimensional embeddingproblem, and Bernd Schroers for other useful discussions.The authors were supported in part by NSF GrantsNo. PHY0354978 and No. PHY99-07949, and in part byfunds from the University of California. L. P. also acknowl-edges support from UC Mexus-Conacyt. D. M. wishes tothank the Kavli Insititute of Theoretical Physics for itshospitality during this work.

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