Conserved charges in (Lovelock) gravity in first order formalism

Conserved charges in (Lovelock) gravity in first order formalism

Elias Gravanis*

Akropoleos 1 Nicosia 2101, CYPRUS(Received 5 December 2009; published 6 April 2010)

We derive conserved charges as quasilocal Hamiltonians by covariant phase space methods for a class

of geometric Lagrangians that can be written in terms of the spin connection, the vielbein, and possibly

other tensorial form fields, allowing also for nonzero torsion. We then recalculate certain known results

and derive some new ones in three to six dimensions hopefully enlightening certain aspects of all of them.

The quasilocal energy is defined in terms of the metric and not its first derivatives, requiring ‘‘regulari-

zation’’ for convergence in most cases. Counterterms consistent with Dirichlet boundary conditions in first

order formalism are shown to be an efficient way to remove divergencies and derive the values of

conserved charges, the clear-cut application being metrics with anti–de Sitter (or de Sitter) asymptotics.

The emerging scheme is: all is required to remove the divergencies of a Lovelock gravity is a boundary

Lovelock gravity.

DOI: 10.1103/PhysRevD.81.084013 PACS numbers: 04.20.Fy, 04.50.�h

I. INTRODUCTION

Defining energy in a consistent and general manner isexceedingly difficult in general relativity. One good reasonis that unlike any other field theory it is hard to localizeenergy in gravity: its density should be reasonably con-structed out of first derivatives of the metric which vanishat any point for inertial observers. Formally, energy is thegenerator of a local symmetry thus it should vanish locally.Another reason of a similar nature, is that energy, byshaping spacetime itself, is not defined in some absolutemanner: it depends strongly on boundary conditions, whichintroduce the asymptotics i.e. the class of metrics oneintends to study. Implicitly or explicitly the conditionsdefine a reference background, for which energy is zero.

In general relativity, the Bondi-Sachs mass [1,2] definedin the null infinity and the ADM mass [3] defined in thespatial infinity, are accepted definitions of energy forasymptotically flat spacetimes. ADM mass is interpretedas the total energy of spacetime. Abbott-Deser mass [4]provides a definition for asymptotically de Sitter space-times, in the spirit of ADM mass. All these definitions arewritten as an integral over a codimensional two spatialsphere. This is an implication of nonlocalizability.

Quasilocalization is a most interesting alternative. Theword means, we consider the total energy content of finitespatial regions, of size (and shape) otherwise arbitrary. Ofall definitions, some of which are mentioned in Sec. II C,the Brown-York (BY) quasilocal energy [5] is especiallyappealing. By the generalization of [6,7], which originatedfrom the ‘‘AdS/CFT correspondence’’ [8,9], the definitionapplies to spacetimes with anti–de Sitter (AdS) or even deSitter asymptotics in a natural manner. Moreover, both theBrown-York definition and the famous correspondence arein the spirit of ‘‘holography’’ [10,11]: the gravitational

degrees of freedom are fundamentally boundary degreesof freedom.Even if one is not particularly interested in the localiza-

tion problem, the Brown-York quasilocal energy, as ameans to define energy in spacetime, is technically attrac-tive. It derives from the invariant action functional as agenerator in the sense of Hamilton-Jacobi theory [5,12].Such a definition is still applicable when one departs fromgeneral relativity.It has become customary in recent years to consider

curved spacetimes of a dimension higher than four. Oncesuch a choice is made the elementary conditions thatuniquely determine the Einstein equations in four dimen-sions are not adequate. In dimension five or higher thereexist rank two, symmetric and covariantly conserved ten-sors involving up to second derivatives of the metric tensor.These are the Lovelock tensors [13]. Generic linear com-binations of them define equally good field equations forthe metric tensor. Moreover, the field equations are secondorder differential equations, giving ‘‘Lovelock gravity’’ afeeling of familiar ground. The Cauchy problem appearsalmost—though not entirely—analogous to that in generalrelativity [14].The action functional for Lovelock gravity is very inter-

esting (starting from the Einstein-Hilbert action itself).Written in tensor notation, each Lovelock Lagrangian hasthe form of a topological density of some lower (even)dimension, only now its indices are summed over a highernumber of dimensions. This dimensional translation pro-vides local dynamics for the metric tensor: Einstein andgeneral Lovelock field equations.This is mysteriously neat. The mystery is partly dis-

solved if we, as Zumino did [15], get more geometric.For our needs, useful mathematical references are[16,17]. The metric tensor may be replaced by the vielbeinEa i.e. a basis in an abstract Lorentz vector bundle withmetric �ab and volume form �a...b. The spacetime metric*[email protected]

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1550-7998=2010=81(8)=084013(21) 084013-1 � 2010 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.81.084013

tensor reads g ¼ �abEa � Eb. A connection !a

b on the

Lorentz bundle may or may not be Levi-Civita, and let�a

b:¼ d!a

b þ!ac!

cb denote its curvature. (The exte-

rior product is understood.) The general LovelockLagrangian is constructed by the Lorentz invariant form�ð� . . .�E . . .EÞ � �ab...cde...f�

ab . . . �cdEe . . .Ef. We

integrate this density over a manifold without a boundary,to get an action. The difference between Einstein andLovelock gravity is that the latter contains terms withmore than one � factors. First order formalism meanstreating E and ! as independent variables.

Varying with respect to ! the curvature changes by�� ¼ Dð�!Þ, where D is the covariant derivative. Thecurvature satisfies D� ¼ 0, the Bianchi identity. Varyingthe action with respect to ! we get, after an integration byparts, terms involving DE. If there are no E factors in theLagrangian the variation vanishes identically: it is a topo-logical density. The torsion, DE, vanishes iff the connec-tion is Levi-Civita. Upon restricting ourselves to suchconnections the variation vanishes identically (in Einsteingravity we have no choice); then the Lovelock tensors andfield equations are obtained, purely algebraically, by vary-ing the action with respect to E.

Technically there is a bonus in the first order formula-tion. Once one is free to allow for torsion, one may con-sider a generalization of Lovelock gravity such thatconnection is not Levi-Civita, as well as consider otherLorentz invariant densities built by contraction with theinvariant tensors �a

½b . . .�dc�. These are related to the

Pontryagin invariants as opposed to Lovelock terms whichare related to the Euler number. These Lorentz invariantswill be our toy model of a geometric gravitationalLagrangian. Most of our examples though will be inLovelock gravity.

In Sec. II A the covariant phase space methods as ap-plied to a gravitational action are discussed. In Sec. II B wederive the Hamiltonian in this framework based on thePoincare invariance of the symplectic form, and a distinc-tion between cases where this can and cannot be done ismade, type (I) and (II) cases, respectively. In Sec. II C thequasilocal value of the charges is introduced. In Sec. II Ethe type (II) case Hamiltonian is derived. BetweenSecs. III, IV, and V various results in Einstein gravityincluding torsion are collected and discussed. Matterssuch as a possibility for attributing a nonzero mass toMinkowski space in certain gravity theories are analyzed.In Sec. VI an example of Lovelock gravity, the case of fivedimensions, is discussed in detail. In Sec. VII a briefexcursion in dimension six shows that the Dirichlet bound-ary forms are adequate to remove the divergencies andobtain results obtained by other methods. It is also under-stood that there is a general scheme: at least in the firstorder formalism, all that is required to remove the diver-gencies of a Lovelock gravity is a boundary Lovelockgravity. In Sec. VIII the quasilocal mass is used to inter-

polate between AdS and flat spacetime results. A briefdiscussion on boundary forms is in Appendix B.As a matter of terminology, one should distinguish the

‘‘canonical generator’’ in the sense of Hamiltonian theorywhich involves Poisson brackets, from that in the sense ofHamilton-Jacobi theory where generators are defined asderivatives of Hamilton’s principal function that is theaction functional on-shell. Of course, the two kinds areintimately related; the covariant phase space frameworkprovides the link. But there are differences. One obtains,respectively, the value of the total (over all space) value ofthe charge, and the actual quasilocal value over a finitespatial region. But we shall work everything out in aquasilocal setting: the boundary Lagrangian will be ofcrucial importance. Thus, everything is quasilocal andwhat is what will be understood from context. After all,in the covariant phase formulation one can easily see thatthe family of functions (with radius as parameter) whoselimit is the Hamiltonian generator are the Hamilton-Jacobigenerators, Sec. II C. Having said that, we shall not be verycareful about terminology for the rest of this paper.

II. THE CONSERVED CHARGE AS CANONICALGENERATOR

A. Action and invariant symplectic form

Let a regionR be an arbitrary open set spacetimeM. Let�A... be a collection of vector valued forms over M. (TheLorentz indices of the bulk fields are now denoted bycapital letters.) Let fields intrinsic to the boundary @R ofthe region R be denoted by �a...

k . We will use this same

symbol �k for bulk fields which agree with the intrinsic

ones when pulled back into R.The induced fields into the boundary are i�k�

a.... In every

smooth component of the boundary @R, and as long asthese components are ‘‘orthogonal’’ [12], the intrinsicfields can be chosen to coincide with the induced valuesof the bulk fields: �a...

k ¼ i�k�a.... We shall be content with

the orthogonal case.Consider a local functional of the fields

S ¼ZRLþ

Z@R

i�kBðCÞ; (1)

where L is a functional of the fields �, the Lagrangian ofthe theory. BðCÞ is a form in R that depends on � and �kand it is such that under given conditions C for the varia-tions �ðCÞ of the fields � and �k, the action functional Shas an extremum

�ðCÞS ¼ 0; (2)

when the Euler-Lagrange equations are satisfied by � inR.We want first to find and analyze a formula for the total

energy and angular momentum, derived as canonical gen-erators conjugate to the associated diffeomorphisms, for

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the theory of fields � given by the Lagrangian L. Ourvariables � will be the vielbein E and the Lorentz con-nection !, as well as other ‘‘matter’’ fields which we shallnot explicitly mention. Our toy model, and quite the ob-jective, will be LagrangianLwhich is a linear combinationof local Lorentz invariants of the curvature� and E. Therecan also be matter Lagrangian terms which will be leftunderstood.

As a warm-up, consider an action in particle mechanics:

S ¼ Rt2t1 dtLðq; _qÞ. Let us define the phase space of the

theory to mean the space of solutions of the classicalequations [18], denoted compactly as E ¼ 0. On the phasespace of the histories q infinitesimal variations �q aretangent directions on that space and can be thought of asvector fields. They are solutions to the linearized equationsof motion �E ¼ 0. It is quite convenient to define anexterior derivative � over the phase space. Its nilpotence,�2 ¼ 0, acting on the fields means that we choose a multi-parameter family of fields twice continuously differentia-ble with repect to these parameters, variation meansdifferentiation with respect to a parameter and the orderof differentiations is irrelevant. We have

� 2S ¼ ½�p�q�t2t1 �Z t2

t1

dt�q � �E; (3)

where of course p ¼ @L=@ _q. By the nilpotence of � wehave that �2S ¼ 0. So for on-shell variations we get

½�p�q�t2t1 ¼ 0, i.e. �p�q is unchanged through time.

The importance of the phase space 2-form � :¼ �p�qis that it gives the clue to formulate the Hamiltonian theoryin a relativistically covariant way [18]. Phase space hashere explicit coordinates x ¼ ðx1; x2Þ ¼ ðp; qÞ. The form�is closed �� ¼ 0 and nondegenerate, i.e. its components�12 ¼ ��21 ¼ 1 form an invertible matrix. Now if on aphase space we can find a closed and nondegenerate2-formwe say we have a symplectic structure. The crucial theoremof Darboux [19] says that we can always find coordinatessuch that locally a symplectic structure can be written inthe form �p�q. Therefore we have the essential structureto do canonical theory without any unnecessary choice ofcoordinates that breaks covariance. If the symplectic struc-ture does not depend on time, as above, then it is a Poincareinvariant. Thus, one succeeds in formulating a covariantcanonical theory. Hamilton’s equations can be written as_xi�ij ¼ �@jH. Phase space components need not neces-

sarily appear as wewill see below. The symplectic methodsin Hamiltonian and Lagrangian mechanics are discussed indepth in [19,20].

Let us turn now to the theory of fields � and our toymodel. We have

�S ¼Z@R

�! � @L@�

þ �BðCÞ þZR�� E: (4)

� is thought of as anticommuting with the exterior deriva-

tive d, �dþ d� ¼ 0, and the same with all p-forms with podd.Taking the second variation we have

0 ¼ �2S ¼Z@R

�

��! � @L

@�þ �BðCÞ

�þ

ZR�� E:

(5)

The variations �� are solutions to the linearized equationsof motion �E ¼ 0.For a field theory over space and time the boundary @R

is a closed hypersurface in spacetime. Also an instant oftime means a spacelike hypersurface. Let two hypersurfa-ces �1 and �2 and let R be the region between them. Theboundary of R is closing with a timelike hypersurface atspatial infinity. To bring that a little closer let two spacelikehypersurfaces ��

1 and ��2 whose boundaries S�1 and S�2 are

connected by a (mostly) timelike hypersurface T� suchthat as � ! 1 the boundaries S� approach spheres atspatial infinity.In order to obtain a Poincare invariant symplectic struc-

ture in field theory we need a ‘‘no-leaking’’ condition: Thepart of the boundary integral over T� in (5) goes to zero as� ! 1:

i�T�

��

��! � @L

@�þ �BðCÞ

��! 0; (6)

as � ! 1. To proceed from this point one should deter-mine and define carefully the asymptotic symmetries atspatial infinity, define the space of fields whose asymp-totics preserve these symmetries, and make sure that theyare such that the quantity above vanishes in the limit. Thisis already quite involved in Einstein gravity and wewill nottry to be rigorous; to obtain a result we have to proceedsomehow else. We shall assume that there are asymptoticconditions on the fields such that the no-leaking condition(6) holds and we will see we get from that.Condition (6) implies that locally in phase space

i�T�

��! � @L

@�þ �BðCÞ

�! i�T��B1; (7)

in the limit � ! 1, for some phase space function B1.We define a ‘‘regularized’’ boundary form:Breg

ðCÞ :¼ BðCÞ �B1. Then

i�T�

��! � @L

@�þ �Breg

ðCÞ

�! 0; (8)

as � ! 1. Upon replacing BðCÞ with this boundary form

from the level of the Lagrangian (1), we work with aboundary form such that the no-leaking conditions holdin the form (8).Before proceeding let us digress briefly for some en-

lightening comments. First of all let us return to Eq. (4). If

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we restrict ourselves to variations � ¼ �ðCÞ which respect

the conditions C in the given region, by the very definitionof the boundary form we have

i�k

��ðCÞ! � @L

@�þ �ðCÞBðCÞ

�¼ 0; (9)

and S attains an extremum for E ¼ 0 in the interior of theregion.

Dirichlet boundary conditions amount to consider var-iations such that the induced fields are held fixed. Thatmeans thatBðCÞ is defined within a class of boundary forms

which satisfy (9) up an arbitrary functional of the inducedfields. The idea is that one might be able to find within thatclass a boundary form, denoted by Breg

ðCÞ, so that the

stronger no-leaking condition (8).Other boundary conditions such that a boundary form

satisfying both (9) and (8) are also possible. They arediscussed in Sec. II D.

If �ðCÞ are replaced with general variations the right-

hand side of (9) does not vanish. Instead it defines the‘‘quasilocal’’ energy momentum and spin tensors�quasilocal:

�! � @L@�

þ �BregðCÞ ¼ �� quasilocal; (10)

where the pullback into the boundary is understood. Theterminology comes from the fact that for Dirichlet bound-ary conditions �quasilocal is indeed the quasilocal energy

tensors of Brown and York [5]. The modification to theregularized boundary form originated from the AdS/CFTcorrespondence [8] whose framework suggests that localboundary counterterms should cure divergencies [9]. It wasthen applied successfully to AdS mass calculations in [6]following also Ref. [21]. (It was also shown that by thesame formal methods one can obtain results for de Sitterspacetimes [7] though there are minus signs ambiguitiesdue to nature of de Sitter space [22,23].)

The point is that if the quasilocal tensors are finite thenindeed the right-hand side of (10) goes to zero at infinity, as�� vanishes in that limit for any definition of the functionspace of the fields. In other words, the no-leaking condition(8) is satisfied. In other words, the symplectic form isPoincare invariant.

The basic problem then is whether we know how toconstruct the form B

regðCÞ such that (8) holds. It will be quite

useful in stating certain simple observations in this work todistinguish between the following three cases.

Type (I). We know how to construct BregðCÞ. Then the

conserved charges quasilocal in nature. Everything de-pends on the finiteness of the quasilocal tensor. This guar-anties that (7) can be satisfied within a reasonable fieldspace i.e. the asymptotics of the field � will allow forvariations of the parameters of the solutions in ��.Stronger conditions on asymptotics would invalidate thewhole analysis. We are clearly very heuristic here.

Type (II). We do not know how to write down a simpleenough B

regðCÞ. This is especially the case of asymptotically

flat spacetimes.Of course, the ‘‘we know’’ and ‘‘we do not know’’ part

in the above rather sketchy definitions depends on assump-tions. For example, for the asymptotically flat case, fairlycomplicated boundary terms have been presented in workssuch as [24–27] which do produce convergent results.These are related to works such as [28] where anotherboundary term is constructed by a ‘‘light-cone reference;’’this term produces a quasilocal mass with nice properties inthe ‘‘small sphere limit.’’ We will not attempt to add some-thing new on these matters so we have nothing to offer inthose directions.We would like, among other things, to obtain values for

the conserved charges by simple prescriptions in a uniformway in the various dimensions and theories (of interest).The boundary forms B

regðCÞ shall be the simplest polynomials

of forms constructed out of the vielbein and connectionone-forms and their derivatives. This is adequate in manycases, as far as the values for the charges are concerned.This is particularly useful in Lovelock gravity where for-mulas can get a lot messier than in Einstein gravity. Theuse of differential forms notation throughout is also achoice made in that spirit.Type (H) (‘‘H’’ stands for ‘‘Hybrid’’). This is a special

case of (I) such that no ‘‘corrections’’ are required: aminimal boundary form consistent with the boundary con-ditions is adequate for convergence.The following definition will be completely clear after

the formulas have been derived.Definition II.1—The first two cases correspond to two

different derivations of the Hamiltonian generators. If aHamiltonian is calculated as type (I) it will be called aBrown-York Hamiltonian. If it is calculated as a type (II)it will be called a relative Hamiltonian generator. As thelatter is defined through weaker conditions a Brown-York generator can always be thought of as a relativegenerator; one may have only to fix a �-integration con-stant appropriately. The hybrid case type (H) is onesuch that a relative generator is also a Brown-York gen-erator. In what follows the derivations are not essentiallydifferent in spirit from the covariant phase space methodsof Wald and collaborators [29–32] though the style iscloser to that of Ref. [18]. The difference lies in thatwe work throughout with first order formalism whichallows us to obtain the results in a closed form for theLagrangians under consideration, and in that we take theexistence of a well-defined Poincare invariant symplecticform as the guiding principle of the derivations.

B. The Hamiltonian generator in type (I)

Suppose we know how to construct explicitly a bound-ary form BðCÞ such that (8) holds.


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Note II.2—From now on we drop the index, reg, for aboundary form satisfying (8). We have a candidate for asymplectic form:

� :¼ �Z�

��! � @L

@�þ �BðCÞ

�; (11)

i.e. one that is independent of the spacelike hypersurface�. Finiteness is also required but will be left as an assump-tion. It will be rather clear from the very conditionsimposed.

� is a phase space 2-form which is by constructionclosed under the exterior derivative: �� ¼ 0. � it is nota symplectic form because it is degenerate: it vanisheswhen �� are gauge transformations. We see below that� will naturally project into a subspace normal to thegauge transformation directions.

Let � be a vector field over the spacetime manifold M.Then L�� can be regarded as a tangent vector over the

phase space. Let a phase space function H� such that

iL�� ¼ ��H�: (12)

(i� is the inner product on forms in phase space with thephase space vector �.) H� is the Hamiltonian generator in

phase space, or simply the Hamiltonian, of the transforma-tions generated by � in spacetime. Relations (12) areHamilton’s equations.

Here we got an action and a symplectic form � but not aHamiltonian. The transformations generated by �, thediffeomorphisms, are a symmetry of the action and there-fore of �. The Hamiltonian can be constructed through theexistence of the symmetry as a momentum mapping [20]essentially using a Noether type method.

If � is invariant along a vector � in phase space thisformally means that L�� ¼ 0. By the closure of � thismeans that �ði��Þ ¼ 0. We used Cartan’s relation for theLie derivative in phase space:L� ¼ �i� þ i��. Thus, thephase space one-form i�� must be exact: i�� ¼ ��H,for some phase space function H. That this holds globallyis an assumption. Now if the Hamiltonian H is given, andthe � is nondegenerate, then the last relation defines aHamiltonian vector field �H. Hamilton’s equations holdalong its integral curves L�� ¼ �H.

Now � being diffeomorphism invariant means that itdoes change along the directions on the tangent of thephase space that correspond to diffeomorphism transfor-mations of the fields. That is LL�� ¼ 0. Through a

reasoning similar to the above, we arrive at (12) for somephase space function H� which we must construct, the

Hamiltonian generators of transformations over M gener-ated by �.

A debt left from earlier is that � does project on thespace of the solutions’ modulo gauge transformations, byhaving no components in those directions. This statementtranslates to: iL�� ¼ 0. This is the case when � has a

compact support in M, or better corresponds to propergauge transformations in the language of [33,34]. ThenH�, like all gauge symmetry generators, vanish identically

on-shell, as generators of local symmetries should do, and� projects. When � are nontrivial asymptotically, areimproper diffeomorphisms, the generators become non-trivial tensorial quantities of the asymptotic symmetrygroups. They should of course be related to the asymptoticvalues of the fields.Regge-Teitelboim [33] produce the generators starting

with the Hamiltonian as the usual Legendre transform ofthe Lagrangian, and postulating additional suitable bound-ary integrals which make all the required functional de-rivatives well defined. There is quite a bit of analogybetween that construction and covariant phase space meth-ods which work with action functional, only in the latterthe canonical variables are not explicitly picked. The finalon-shell must agree wherever the methods make sensethough the results may differ by phase space constants,i.e. parameters that depend on the theory (couplings) andnot the solution. This happens explicitly in the case oflocally flat spacetime in three dimensions, as we shall see.Finally, without dwelling further into the symplectic

form of the Hamiltonian theory we will find an H� starting

from (12) and (11). We have

� ¼Z��! � � @L

@�: (13)

We have

iL�� ¼Z�L�! � � @L

@�þ �! �L�

@L@�

: (14)

By �L� ¼ L�� the integrant can also be written as

� �

�L�! � @L

@�

�þL�

��! � @L

@�

�: (15)

Using �� ¼ �D�! and L�! ¼ Dði�!Þ þ i�� one can

easily show that the first bracket reads

d

�i�! � @L

@�

�þ i�L� i�� E: (16)

This is a quite useful fact. Then it is easy to show that thewhole of (15) reads

��d

�i�! � @L

@�

�þ di�

��! � @L

@�

�þ �� i�E þ i�� E:

(17)

�i� ¼ �i�� is used often. We have kept the terms involv-

ing E and �E in this formula for completeness. So we havethat iL�� equals

� �ZS�i�! � @L

@�þ

ZS�i�

��! � @L

@�

�; (18)

where the limit � ! 1 is understood.


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Now S� is a boundary of both�� and T�. We may orientthe manifold T� to be such that � is tangent to it at theboundary S�. Then the no-leaking condition (8), extendedto the boundary of T�, tells us that

i�S�i��! � @L

@�

�! �i�S�i��BðCÞ; (19)

as � ! 1. Thus we have

iL�� ¼ ��

�ZS�i�! � @L

@�þ i�BðCÞ

�: (20)

By (12) we have

H� ¼ZS1

�i�! � @L

@�þ i�BðCÞ

�þH0

�: (21)

Equation (12) defines the Hamiltonian up to closed zero-form that is a constant H0

�.

Note II.3—It is important to remember that this formulawas obtained by treating S� as boundary of T�. ThereforeBðCÞ is constructed as a boundary form for the timelike

component of the boundary.To recapitulate, as long as (2) and (8) make sense, they

are all we need to construct explicitly a Hamiltonian for �in a fully covariant, first order formulation. TheHamiltonian H� stems from the existence of a finite qua-

silocal tensor, thus that notion is underlying the derivationin an essential way but the tensor itself is not explicitlyused.

Note II.4—We have taken for granted that the onlyboundary the spacelike hypersurfaces � have is a surfaceat large radial distances. In case this is not true an analo-gous condition (8) and corresponding additional terms inH� must be included, a procedure not really well defined

near actual singularities. This is a crucial detail. In whatfollows we would like to have the spatial infinity as theonly boundary of�. For that to be so, we shall assume fromnow on that singular energy distributions implied by themetrics we write down are replaced by smooth ones suchthat no timelike singularities arise.

The value of the ‘‘angular’’ integral over S1 in (21)cannot be shifted by the addition of a local term on theboundary action terms. The phase space constant H0 can-not be shifted. Its presence makes the generator potentiallyconsistent with other constructions of the generators but itis a quantity without much meaning when we discussquasilocal generators. Thus we shall give a name to theangular integral. We will refer to it as the Brown-Yorkgenerator:

H ¼ HBY þH0: (22)

C. Quasilocal generators and conservation

We mentioned already in the introduction that a majorproblem in gravity is localizing energy. All our formal

choices might appear natural from certain points of view,but they are essentially motivated by proposals addressingthe localization problem. The equivalence principle orgeneral covariance say that any energy density mustvanish.Going to the boundary was the brilliant move of Brown

and York [5]. It fits nicely in ’t Hooft’s holography idea:gravitational degrees of freedom are fundamentally bound-ary degrees of freedom [10]. There is a host of quasilocaldefinitions, see e.g. Ref. [35] and the works discussedthere. The Misner-Sharp mass [36] is another definitionof mass of quasilocal type, applying in spherical symmetry,with nice properties [37]; for a Lovelock gravity it wasdefined recently in [38].In our derivation we encountered the quantities

H�ð�Þ ¼ZS�

�i�! � @L

@�þ i�BðCÞ

�þH0

�: (23)

What is important about them is that they converge to thecanonical generator H� for � ! 1, where r� is the radius

of the hypersurface T�. One may regardH�ð�Þ as the valueof the associated charge contained in the spatial sectionsbounded by T�.Though this is not precise, as we will discuss in

Sec. VIII, there is a reason for thinking something likethat.H�ð�Þ is essentially the quasilocal mass of Brown and

York, in first order formalism; their mass is defined as acanonical generator in the sense of Hamilton-Jacobi theory[5,12]. In order to understand the claim, let T� be a time-like hypersurface and � be a vector field tangential to it. �is not a Killing vector. By manipulations learned in theprevious section it is straightforward to show that

ZT�

d

�i�! � @L

@�þ i�BðCÞ

�

¼ZT�

�L�! � @L

@�þL�BðCÞ � i� ðLþ dBðCÞÞþ i�� E

�:

(24)

The last term vanishes on-shell, E ¼ 0. The middle termvanishes as � is tangent on T�. Now, note first that theremaining term is nothing but the boundary term on T� ofthe action functional for Lie derivative variations of thefields, see (4). Also, we see from (10) that this readsL�� quasilocal. From precisely this term the quasilocal densities

are read off from in the Hamilton-Jacobi analysis ofRefs. [5,12]. Now, the left-hand side of the equation isnothing but the difference of H� ð�Þ evaluated in the futureand past S�. In other words, we obtain explicitly that theintegrand involving the Brown-York quasilocal densities isa total derivative producingH� ð�Þ on the boundary sphere.Thus, as mentioned in the introduction, H� ð�Þ is essen-

tially the canonical generator in the sense of Hamilton-Jacobi theory.


084013-6

Conservation is easily obtained from the previous for-mula if we let � be a Killing vector: then the future and pastvalues of H� ð�Þ must agree. The time derivative of (23)

can also be calculated directly. Let a � be a timelike vectorfield tangential on T�. By @S� ¼ 0 the previous formulatells us that the rate of change _H�ð�Þ of the quasilocal

charge H�ð�Þ under the flow of � reads

_H�ð�Þ ¼ZS�

�i�

�L�! � @L

@�þL�BðCÞ

�

� i�i�ðLþ dBðCÞÞ þ i�ði�� EÞ�: (25)

For any Killing vector � we have that L�� ¼ 0 thus the

first term should vanish. (If � is an asymptotic Killingvector then conservation holds in the limit � ! 1.) If �is a timelike Killing vector the remaining term vanishes for� ¼ � by the identity iXiX ¼ 0. (In general, the innerproduct operator satisfies iXiY ¼ �iYiX for any vectorfields X and Y.) This a remnant of the ’’commutativity’’of the Hamiltonian with itself. Mass and angular momen-tum as given by (23) are indeed conserved on-shell.

D. Different boundary conditions

A question which arises is, To what extent does theHamiltonian depend on the conditions C? To answer thatone should look at the associated no-leaking condition,Eq. (8), which we may call FðCÞ. The phase space isdefined such that this condition holds. Let us denote byC1 and C2 two different boundary conditions. Then oneworks in two phase spaces where the conditions FðC1Þ andFðC2Þ hold. Two things may happen. One, the phase spaceshave a measure zero intersection. Then we may obtain twodifferent Hamiltonians. Second, the intersection of phasespaces is a nice subset of each one. Then we may restrictour variations in there. Subtracting the no-leaking condi-tions (8) we have that i�T�ð�BðC1Þ � �BðC2ÞÞ goes to zero

for � ! 1. The previous derivation shows that the twoHamiltonians (21) associated with C1 and C2 may differonly by a phase space constant. For example, a set ofcounterterms applying to asymptotically AdS spacetimeswere constructed and applied in Refs. [39–44] as boundaryterms respecting the conformal symmetry of the AdSboundary. Though those counterterms differ a lot fromthe simple boundary forms we are using the results areexpected to agree and they indeed do as we shall see inexamples. Additionally the same boundary forms applystraightforwardly with little formal modification to asymp-totically de Sitter spacetimes, along the lines of Ref. [7].From the point of view of Refs. [39–44] it is not clear whythis is so; from the point of view of the Dirichlet counter-terms it is not that surprising.

E. The Hamiltonian generator in type (II)

When spacetime is asymptotically flat, there is no lengthscale to facilitate regularization. It is then characteristicallydifficult to construct a boundary formBreg

ðCÞ that would leadto finite charges. On the other hand, in this case the ADMconstruction of charges [3], it is a consistent method forcalculating the energy and the angular momentum forasymptotically flat space. An extension of this is theAbbott-Deser mass [4] which calculates total energy forasymptotically de Sitter spacetimes.From the point of view of first order formulation these

are all subtraction methods. Energy and any charge ismeasured relatively to the background by studying andformulating things in terms of the asymptotic deviationsfrom the background.In such cases the form Breg should exist but one cannot

find a local expression of it, or does not care to find one. Itis not an accident that this applies to asymptotically flatspaces: the ‘‘holographic renormalization’’ does not workstraightforwardly in asymptotically flat spacetimes and thecorrections need to be nonlocal [45]. In specific cases onemay find local but not polynomial counterterms whichsuffice to remove divergencies but they do not apply toall asymptotically flat spaces [24–27]. For the general case,nonlocality suggests that the boundary form can be con-structed by involving another solution of the equations ofmotion, i.e. another point in phase space.Let � and �1 be two solutions of the equations of

motion. The indication ‘‘1’’ is simply a name to distin-guish the second solution; its meaning will be explainedbelow. Then subtracting conditions (7) applied on eachsolution we have�

�! � @L@�

þ �BðCÞ��

��! � @L

@�þ �BðCÞ

�1! �B;

(26)

pulled back into T�, � ! 1.B is a phase space function which is antisymmetric in

fields � and �1. B must be annihilated by �ðCÞ, that is,�B ¼ �Bði��; i��1Þ. This is an important fact: Choose afield �� to induce the same field with � into T�,

i�T�� ¼ i�T��; (27)

such that there is a well-defined limit �� ! �1 as � !1. Then by antisymmetry �B ¼ 0, i.e.

�! � @L@�

! ��BðCÞ þ��! � @L

@�þ �BðCÞ

�1

(28)

pulled back into T�, � ! 1. Condition (27) is not neces-sary but allows us solubility; without it we are simply backto the general (26).Let find now an explicit form for the1 part of the right-

hand side of (28) which we know is � exact. Let�1 be anysolution that can be continuously related to � by the


084013-7

parameters. We have ðiL��Þ1 ¼ ��H1;�. Then relation

(18) applied to�1 allows us to write the right-hand side of(28) as � exact, at least under the integral sign in theprocess of the derivation of H�. This is all one needs.

Applying (28) in the derivation at the step of (18) one findseasily

H� ¼ZS1

�i�! � @L

@�þ i�BðCÞ �

�i�! � @L

@�þ i�BðCÞ

�1

�þH1;�: (29)

The case� ¼ �1 tells us that the �-integration constant iszero. This is our Hamiltonian for the cases (II), e.g. whenspacetimes are asymptotically flat. It is derived under thecondition (7) in field space and condition (27) for thesecond field �1.

Unlike the constant H0� in (21), H1;� is not a constant

function in phase space. It is the value at of theHamiltonian at �1. Now �H1;� does not vanish unless

�1 is an equilibrium point of the phase space. This is trueif�1 is an actual ‘‘ground state’’ of the system. Of coursein any case the H� satisfies (12) by construction.

Condition (27) is crucial for applications of (29). Ingeneral it cannot be satisfied if the fields are thought ofas components with respect to the same coordinates. Theinduced fields will agree on T� as geometric objects up tonecessary diffeomorphisms. There is nothing unnatural inthat when using manifestly invariant quantities. A condi-tion of the form (27) was introduced in [46] as a means toguaranty finiteness of a ‘‘physical’’ action Sð�Þ � Sð�1Þ,dragging also the conserved quantities to finiteness. Here itwas introduced as a convenient step integrating a Poincareinvariant symplectic form in phase space.

We can now give a specific definition of the relative(on-shell values of the) generators, first mentioned in defi-nition II.1.

Definition II.5—Every case can be regarded as type(II). So we may bypass the arbitrariness of H1 or usethe arbitrariness of H0 in type ðIÞ to define a relativegenerator Hrel such that: Hrel ¼ 0 if a certain conditionholds. The condition usually expresses the fact that wemeasure the charges such that their vacuum values areset to zero.

III. SPHERICAL AND TOPOLOGICAL METRICS

We start by studying some characteristic examples inthree, four, and five spacetime dimensions in Einsteingravity. Some interesting things arise when the angularmanifolds are not spheres. Einstein gravity will be de-scribed by the Lagrangian

L ¼ 1

2c1�ð�E . . .E|fflffl{zfflffl}

D�2

Þ ¼ 1

2c1�ð�ED�2Þ; (30)

where ðD� 2Þ!c1 ¼ ð8�GÞ�1, where G we define as

Newton’s constant in the given dimensions, following themost standard conventions.The standard form of the metrics we consider is

ds2 ¼ �g2dt2 þ dr2

g2þ r2d�2; (31)

and d�2 is the metric of a compact i.e. closed, constantcurvature codimensional 2 manifold. Let a frame ~Ei overthis manifold, i.e. d�2 ¼ �ik

~Ei � ~Ek and ~!ij a connection.

Its curvature will be ~�ij ¼ k ~Ei ~Ej, k ¼ �1, 0. Of coursek ¼ 1 refers to the spherically symmetric case. But inter-esting facts are related to a comparison of that case to theflat (k ¼ 0) manifold. The black holes associated with k aredifferent than one have been termed as topological (seebelow).Let the frame

E0 ¼ gdt; E1 ¼ g�1dr; Ei ¼ r ~Ei: (32)

Then of course ds2 ¼ �ABEA � EB. As long as torsion is

zero, and this is always the case for Einstein gravity invacuum, we have that the nonzero components of theconnection are

!01 ¼ gg0dt; !i

1 ¼ g ~Ei; !ij ¼ ~!i

j: (33)

The first one we would like to analyze is the moststandard of all: a four dimensional Schwarzschild metricwith zero cosmological constant. Then

g2 ¼ 1� 2Gm

r: (34)

We would like to analyze how m arises as the total energyof the system.Energy, or mass, will be calculated as the on-shell value

of the Hamiltonian for the Killing vector � ¼ @=@t. Theproblem, due to asymptotic flatness, is case (II) as onemay check for oneself. We use (29).For Einstein gravity and Dirichlet boundary conditions

the boundary form isBðCÞ ¼ � 12 c1�ðEEÞ, as discussed in

Appendix B. The form ¼ !�!k is constructed from

the induced connection !k ¼ i�!ab of a constant r ¼ r�hypersurface T�, where a, b ¼ 0, i. (The induced andintrinsic fields’ equalities, for the connection as well asEk ¼ i�E, are consistent for zero intrinsic torsion.)

Thus, we have 01 ¼ gg0dt and i1 ¼ g ~Ei. The first partof the integral in (29) readsZ

S2c1 �

�1

2�ði�!EEÞ � 1

2i��ðEEÞ

�¼ � 1

Gg2r; (35)

evaluated at r ¼ r� ! 1 as � ! 1. This term of (29)alone diverges. Before proceeding we digress for a coupleof comments.One should note that there are no derivatives in the

result. The divergence is due to this fact. This means thatthe no-leaking condition is not respected, thus, (35) is not a


084013-8

final result. Nonetheless, we may note that in fact all ourfinal formulas will not contain derivatives of the metricfunction.

Ref. [47] seems to have a general setting similar to ours.Various ingredients, e.g. first order formalism, boundaryforms, covariant phase space, are there. The results thoughdisagree, the Hamiltonian is convergent by construction.The difference lies in the definition of the phase space. [47]imposes conditions on the fields associated to asymptoticflatness. Without them certain ambiguities of the asymp-totic Poincare group, i.e of the energy-momentum defini-tions that arise in four dimensions. Doing so, one againimplicitly works with Minkowski as a reference spacetimefrom the very beginning, trying to make sense of theformulas under that condition. Moreover, the asymptoticconditions on ! are imposed using torsion freeness. Thisbrings the analysis closer to the second order formalism. Inall, the results of Ref. [47] are expected to be very differentfrom ours. The ambiguities themselves are a matter ofgeometry first of all, and any set of requirements on thefields which remove them could be applied when a defini-tion of energy has been adopted. We will not go into theseissues here.

We introduce fields �� such that the condition (27)holds: Let

ds2� ¼ �g2�dt2� þ dr02�

g2�þ r02�d�2; (36)

where g� ¼ g�ðr0�Þ is another solution, say for the parame-term�. The change of coordinates is given by r ¼ r�, r

0� ¼

r�, and gðr�Þt ¼ g�ðr�Þt� for some fixed radius r�. Thus,(27) holds.

What, in particular, (27) means is that

� ¼ @

@t¼ g

g�

@

@t�: (37)

Thus, the integral in the Hamiltonian (29) equals

� 1

Gðg2r� gg�rÞ; (38)

everything evaluated at r� which goes to infinity. This isconvergent and we have H� ¼ m�m1 þH1;�. In par-

ticular we may use Minkowski spacetime fieldsm1 ¼ 0 asthe �1 phase space point. As mentioned in the previoussection, for the asymptotically nontrivial Killing � thecorresponding canonical generators are tensorial quantitiesof the asymptotic symmetries. In particular as long as thesesymmetries indeed are Lorentz transformations thenH1;� ¼ 0 that is H� ¼ m, indeed.

Analyzing how the result arises one observes somethinginteresting. The finite part of (35) is 2m. The divergent partis due to the angular manifold (2-sphere) constant curva-

ture: ~�ij ¼ 1 � ~Ei ~Ej. Consider now the more general casegiven by (36), i.e. k ¼ �1, 0. Then the field equations areequivalent to ðrg2Þ0 ¼ k, that is

g2 ¼ k� r0r; (39)

where r0 is an integration constant. These solutions are thezero cosmological constant analogues of the AdS topologi-cal black holes [48–50].When k ¼ 0 the 2 manifold is flat. It can be a torus or a

Klein bottle. Its volume vol0 is arbitrary. In that case (35)converges. The problem is a type (H) case. The Brown-York mass reads

k ¼ 0: mBY ¼ vol04�

r0G: (40)

Let us recalculate the mass as a relative mass applying thetype (II) Hamiltonian (29) which here reads (38). Let g�be such that r0ð�Þ ! 0 for � ! 1. Clearly the result isagain (40). Thus, we indeed deal with a relative masswhich can also be regarded as a Brown-York mass, inaccordance with the definition II.1 of the type (H) case.This phenomenon persists even in the quite more com-

plicated case of Einstein-Gauss-Bonnet gravity as we willsee. This is due to the fact that g� ! 0 for the specificsolution. There is a finite discontinuity going from the AdSmetrics of length scale l to their l ¼ 1 analogues whenthey are (H) metrics, as we shall see.Let us turn to study anti–de Sitter vacuum metrics in

three, four, and five dimensions, in the presence also ofmass. An anti–de Sitter vacuum means that the equationsof motion read �ðf�þ l�2EEgED�3�EÞ ¼ 0, that is thebulk Lagrangian is

L ¼ 1

2c1�ð�ED�2Þ þ 1

2c1

D� 2

l2D�ðEDÞ: (41)

Again ðD� 2Þ!c1 ¼ ð8�GÞ�1.The three dimensional pure anti–de Sitter spacetime is

of the form (36), explicitly,

ds2 ¼ ��1þ r2

l2

�dt2 þ

�1þ r2

l2

��1dr2 þ r2d2: (42)

The vielbein is E0 ¼ gdt, E1 ¼ g�1dr, E2 ¼ rd, and thenonzero components of the connection read !0

1 ¼ gg0dt,!2

1 ¼ gd.We first calculateZ

S1c1 �

�1

2�ði�!EÞ � 1

2i��ðEÞ

�¼ �2�c1g

2; (43)

for � ¼ @=@t. This expression diverges

� 2�c1r2

l2� 2�c1: (44)

The problem is a type (I) case. The previous result wascalculated with the minimal Dirichlet boundary form, dis-cussed in Appendix B. Let a boundary form as

B ðCÞ ¼ �12c1�ðEÞ þ 1

2c1b0�ðEkEkÞ1: (45)

Note III.1—�ð. . .Þ1 shall mean from now on a contraction


084013-9

with the volume form �a...b1 on the intrinsic Lorentzbundle; it is identified with the induced ‘‘bulk’’ volumeform, where E1 in practice is always the radial covector. Atheoretical problem is that �a...b1 is not invariant undervariations, potentially contributing terms inexistent in ourderivations. This is not so. Let � a unit vector, �AB�

A�B ¼1, normal on the boundary. That is �a...b1 ¼ �a...bA�

A. ��A

is not zero but the unit length of � implies that ��A isnormal to �A, i.e. tangential. As a result it drops out thevariations of the intrinsic boundary forms as it contributesone tangential index too many.

The result (43) is ‘‘corrected’’ by �2�c1b0gr whichasymptotically reads

� 2�c1b0r2

l� �c1b0l: (46)

Canceling the divergence fixes b0 ¼ �l�1. So finally wehave

mBY ¼ �c1� ¼ � 1

8G: (47)

Expression (47) is a well known result.For the Poincare patch of the AdS3,

� r2

l2dt2 þ l2

r2dr2 þ r2d2; (48)

one similarly finds the correct answer mBY ¼ 0. Thingswork nicely. The three dimensional case will be discussedand commented on in detail in the next section.

A four dimensional anti–de Sitter Schwarzschild solu-tion is given in the k ¼ 1 case of (36) with

g2 ¼ r2

l2þ k� r0

r; (49)

where l is the AdS radius and r0 an integration constant.We will consider the general k. This is a case (I) and weapply (21) for a (nonminimal) boundary form:

B ðCÞ ¼ 12c1f��ðEEÞ þ b0�ðE3

kÞ1 þ b1�ð�kEkÞ1g: (50)

The constants are to be determined by finiteness. TheHamiltonian is given by the result (35) plus b corrections:

� 16�c1g2rþ 12�c1b0gr

2 þ 4�c1b1kg; (51)

times a factor volk=ð4�Þ where volk is the volume of the 2manifold. The expression above is evaluated at a constantr ¼ r� which goes to infinity. The divergencies are can-celed for b0 ¼ 4=ð3lÞ and b1 ¼ 2l. b1 is useless and notdefined for k ¼ 0. The Brown-York Hamiltonian turns outto be

mBY ¼ volk4�

r02G

: (52)

This holds for any k and regularization is required in allcases.

For the spherical case k ¼ 1 this is the well-knownSchwarzschild mass we found in the asymptotically flatcase, formally l ! 1. No discontinuity arises between thecase for finite l and l ! 1. Presumably, one may conclude,no energy can be associated when the vacuum itself arisesin four dimensions.Now for the flat angular 2 manifold, i.e. k ¼ 0, things

are different. The result (40) in the asymptotically flat casel ¼ 1 studied above is twice the result we find now forfinite l.Remark III.2—In a case (H), where no regularization is

needed for finiteness, the mass is twice the mass one findswhen the metric is looked at in the presence of an intrinsiclength, as it happens in the presence of the cosmologicalconstant. (It will turn out that the remark holds as long asthe Einstein term is present in the Lagrangian.)Let us see how things work out in the one dimension

higher than the usual macroscopic dimension of spacetime.The five-dimensional anti–de Sitter Schwarzschild solu-tion is a case where, as in three dimensions, it turns out thatthe energy of vacuum is not zero. The difference withrespect to three dimensions is that vacuum energy is notconvergent for l ! 1.Let the metric be given by the five-dimensional analogue

of (36) with

g2 ¼ r2

l2þ k� r20

r2; (53)

i.e. the angular 3 manifold has constant curvature k. TheHamiltonian will be given byZ

S3

�1

2c1�ði�!E3Þ þBðCÞ

�; (54)

for a boundary form

B ðCÞ ¼ 12c1f��ðE3Þ þ b0�ðE4

kÞ1 þ b1�ð�kE2kÞ1g: (55)

The result is

� 36�2c1g2r2 þ 24�2c1b0gr

3 þ 12�2c1b1kgr; (56)

times volk=ð2�2Þ. Divergencies cancel for

b0 ¼ 3

2l; b1 ¼ 3l

2; (57)

where again b1 is useless and not defined for k ¼ 0, and thefinite Brown-York mass is

18�2c1

�k2l2

4þ r20

�volk2�2

: (58)

For the usual definition 3!c1 ¼ ð8�GÞ�1 this agrees withthe known results in the spherically symmetric case [6,51].The couplings b0 and b1 satisfy the relation

� l2db0dl

¼ db1dl

: (59)

We will see that when Einstein gravity is supplemented by


084013-10

the Gauss-Bonnet term, the dependence of b0 and b1 on lchanges but relation (59) holds as it is.

Clearly this result is hopeless in the limit l ! 1. Forcomparison purposes, consider the pure Schwarzschildmetric g2 ¼ 1� r20r

�2. This is a case (II). We apply

(29) and following steps explained at the beginning ofthis section we find that the relative mass is given by thelimit of

� 36�2c1ðg� g�Þgr2 þmr0¼0; (60)

where we imply that the fields �1 are the Minkowskisolution r0 ¼ 0. We have

m ¼ 18�2c1r20 þmr0¼0: (61)

By the Lorentz invariance the pure Minkowski spacetimeenergy should vanish, mr0¼0 ¼ 0.

In all, formula (58) can be thought of as describing aSchwarzschild mass given by the previous result, and avacuum energy given by

mBY ¼ � 3�

32GK; (62)

where we wrote c1 in terms of the five-dimensionalNewton’s constant and K ¼ �l�2 is the constant curvatureof spacetime.

We will see the result (62) appearing again when weconsider Lovelock gravity in dimension five. It will only becorrected by a term depending only on the gravitationalcouplings, analogous to the three dimensional vacuumenergy.

The vacuum energy (62) is divergent for infinite l. Onthe other hand the case l ¼ 1, that is asymptotically flat, isnot one such that the relative mass is a Brown-York mass,i.e. is not type (H) and remark III.2 need not apply.

IV. POINT PARTICLE IN THREE DIMENSIONS

The asymptotically flat (local) three dimensional spheri-cal metric has been bypassed up to this point. It belongs tothe hybrid case, type (H), and we shall apply remark II.2.We shall discuss it at some length.

Einstein gravity in three dimensions is always an intri-guing problem. It is because it feels very elementary insome sense and to an unimaginable extent soluble. In threedimensions there is no curvature outside matter, i.e. nogravity in a vacuum and hence no propagation of gravitythrough gravitational waves. In other words the effect ofgravitational sources is not local, rather, they affect space-time in a global manner.

One implication of this is that singularities are inte-grable. On the other hand all the discussions ofHamiltonians that so emphasize spatial infinity is in thespirit that the gravitational effects are diminishing there.This is not the case in three dimensions, the deficit angledoes not cease to exist for large distances. Nevertheless

things seem to work and it is interesting to apply theformulas anyway.Interest in Einstein gravity in three spacetime dimen-

sions was essentially initiated by [52].Einstein gravity in three dimensions without the cosmo-

logical constant can be described by the Lagrangian

L ¼ 12c1�ð�EÞ; (63)

c1 ¼ ð8�GÞ�1. The field equations say that torsion TA ¼ 0and curvature �AB ¼ 0.The metric

ds2 ¼ �ðdtþ adÞ2 þ dr2

�2þ r2d2; (64)

is a spherically symmetric solution. 0< r <1 and 0<< 2�. � and a are constants.From the vielbein

E0 ¼ dtþ ad; E1 ¼ dr

j�j ; E2 ¼ rd; (65)

TA ¼ 0 gives the connection

!21 ¼ j�jd: (66)

Thus, � ¼ 0 and both field equations are satisfied.Space can be given in conformally flat coordinates. Let

r ¼ �1� . Then

ds2 ¼ �ðdtþ adÞ2 þ ��2 ðd�2 þ �2d2Þ; (67)

where �2 ¼ ð1� Þ2. The interesting thing here is theorigin; r ¼ 0 is mapped to the origin � ¼ 0 if < 1 andto � ¼ 1 if > 1. Also the metric is invariant under thetransformation � ¼ 1=�0 and ¼ 2� 0. The other coor-dinate system is not transformed: r0 ¼ r and j�0j ¼ j�j.So in some sense � ¼ 0 is dual to � ¼ 1. There is a

difference: if > 1, � ¼ 0 lies at infinite proper distance,R�0

0 d�� ¼ 1, from any point �0. Instead � ¼ 1 lies

at finite proper distance from an arbitrary point,R1�0d�� <1.

Let us calculate first the mass. The Hamiltonian for theKilling vector is � ¼ @=@t. It is amusing to do that in bothkinds of coordinates.As discussed, one can always treat a case as type (II)

and calculate a charge as a relative charge applying (29).We will do that for the moment. We chose �1 to be themetric with �2 ! 1. The crucial condition (27) is satisfiedtrivially here. T� is the hypersurface at some fixed radiusr� in both radial coordinates wewrite� and�� (they neednot be the same). Using (65) and (66) the first line of (29)equals

ZS1c1 �

�1

2�ði�!EÞ � 1

2i��ðEÞ

�¼ � 1

4Gj�j; (68)

thus


084013-11

m ¼ 1

4Gð1� j�jÞ þm�¼1: (69)

The sphere at infinity is a unit circle S1. ¼ !�!k is

calculated for the constant radius hypersurface r ¼ r�,where r� ! 1 for � ! 1. This is the surface T�. It is!k ¼ 0. Note that is the second fundamental form only

because by our derivation we have taken T� to be such that� is tangent to it.

Consider now the conformally flat space coordinatescase. The spatial vielbein now is

E1 ¼ �� d�; E2 ¼ �1� d: (70)

The connection is !21 ¼ �ð1� Þd.

� ¼ �. A hypersurface at constant � viewed as an outerboundary will have a normal vector in the direction of E1.When � ¼ þ it means that we take E1 in the direction ofgrowing � i.e. towards infinity. When � ¼ �, E1 is in thedirection of decreasing � i.e. towards the origin of thatcoordinate.

The formula for the Hamiltonian is obtained by treatingT� as an outer boundary. When � ¼ þ we integratearound the origin of � and when � ¼ � we integratearound the infinity of �.

When < 1 the infinity of � is at an infinite properdistance from any point. We take �1 to be the metric for ! 0. Formula (29) gives

m ¼ 1

4G þm ¼0: (71)

When > 1 the infinity of � lies at a finite distance froman arbitrary point �0 while the origin is at an infinite properdistance from it: Infinity is at � ¼ 0.

We may compactify the space, add a point at infinity,and have then two particles, one at an origin and one atinfinity. Without compactification, the ‘‘place’’ where aparticle can be is at � ¼ 1 in these coordinates.

Integrating around � ¼ 1means taking � ¼ �1. > 1so we cannot take�1 to the metric for ! 0. Wewill takethe ‘‘dual’’ choice, ! 2. Applying formula (29) we findnow

m ¼ 1

4Gð2� Þ þm ¼2: (72)

The two formulas we obtained are the two cases oneobtains from (69), from the two solutions the relations ofthe parameters �2 ¼ ð1� Þ2, and for the respective ‘‘lo-cations’’ one would expect form the radial transformation.This is of course due to general covariance.

For ¼ 1 both � ¼ 0 and � ¼ 1 lie at an infiniteproper distance from any �0. It is the case of the spacewith the topology of the cylinder. The candidate ‘‘places’’for a particle are the infinities of a real line. These cannotbe considered as locations of particles.

In [53] it was argued that in the case � 1 one cannotapply uniformly the asymptotic conditions one applies also

for < 1, because a (finite) Hamiltonian does not exist.One may think of that as a mathematical explanation in thecanonical framework of the arguments given above.Presumably, one may note that this discussion is hardly

needed if we use the Schwarzschild coordinates (64). Thecylindrical case corresponds to � ¼ 0 which is alreadyunacceptable in that form of the metric.Let us now calculate the Hamiltonian generators using

(21), that is, as type (I). Let us use the minimal boundaryform BðCÞ ¼ � 1

2 c1�ðEÞ. The result is essentially given

by (68). We have

mBY ¼ � 1

4Gj�j: (73)

The result is perfectly finite, without regularization, that iswe deal with a type (H) case.Spacetime (64) is ‘‘spinning.’’ Its angular momentum

equals minus the Hamiltonian H� for the Killing vector

@=@. We have

JBY ¼ �ZS1c1 �

�1

2�ði�!EÞ � 1

2i��ðEÞ

�¼ 1

4Gj�ja:

(74)

The term involving originating from the boundary doesnot actually contribute, as the Killing vector is tangent ontothe sphere. In terms of the quasi local mass (73) it is simplyexpressed as

JBY ¼ �mBYa: (75)

There is a number of things that can be said here. Recalldefinition II.5 for the relative generators. Let us explicitlydefine a relative mass mrel by the condition: mrel ¼ 0 forj�j ¼ 1, i.e. for Minkowski spacetime. This fixes the freeconstant H0

�. We have

mrel ¼ 1

4Gð1� j�jÞ: (76)

This is the value calculated in [52] and in any calculationthat integrates the matter energy tensor. Similarly one maydefine Jrel by the condition: Jrel ¼ 0 if a ¼ 0. We have

Jrel ¼�1

4G�mrel

�a: (77)

Surprisingly relations (75) and (77) are apparently un-known in the literature. In Ref. [52] only the case mrel ¼0 is calculated, obtaining a ¼ �4GJ. The general multi-particle massive and spinning metric was obtained in [54],but the integration constants were not carefully related tocanonical generators.Consider now the quasilocal mass mBY for a flat space-

time. For j�j ¼ 1 and a ¼ 0 we have:

mBY ¼ � 1

4G: (78)


084013-12

In accordance with remark III.2 this is indeed twice thevacuum energy (47) for AdS spacetime.

Reference [55], reasoning essentially along the lines offormula (73), suggests that the vacuum energy ofMinkowski spacetime in three dimensions is nonzero andequal to (78). On the other hand Ref. [53], using Regge-Teitelboim methods, finds the result on the right-hand sideof (76), in agreement with [52]. That is, the Minkowskispace mass is found to be simply zero. The results are notinconsistent: they differ by a phase space function which isan arbitrariness of the definition anyway, relation (22). Itdoes raise, though, issues but it is rather a matter of con-sistent application of definitions. If both kinds of defini-tions apply they should give equivalent physical results; ifone of them is more relevant, for example, in the generalsetting of AdS/CFT ideas, the Brown-York definition is therelevant one and the results should be taken seriously,whatever they are. For example, Minkowski spacetimeshould apparently be treated according to (78).

V. FIVE-DIMENSIONAL CHERN-SIMONS POINTPARTICLE

We wish now to turn to the nearest analogue of the pointparticle in three dimensional gravity. If analogously tothree dimensions we consider a Chern-Simons theory ofthe five-dimensional Lorentz group we deal with a gravitythat is described by a pure Gauss-Bonnet term. This is notexactly true unless we are in a pure vacuum, but theanalogies between three and five dimensions we willfind, should be traced to the Chern-Simons equivalenceof these gravities at the level of the Lagrangian.

Presumably, working in a first order formalism, we areable to construct Hamiltonians even in the presence oftorsion. The following are an example of this fact.

Consider the Lagrangian

L ¼ 12c2�ð��EÞ; (79)

where c2 is a coupling constant with dimensions of mass.In [56] it was observed that the gravity described by thisLagrangian admits a vacuum solution which is the obviousgeneralization of (64) for a ¼ 0, i.e. non spinning, and its(relative) mass was calculated.

We consider the obvious generalization of the full spin-ning metric (64):

� ðdtþ aÞ2 þ dr2

�2þ r2d�2; (80)

where a ¼ ai ~Ei for some Kerr constants ai, and d�2 ¼�ij

~Ei ~Ej is the metric of the unit 3-sphere, expressed con-

veniently in terms of a frame ~Ei. The frame satisfies therelation d ~Ei ¼ �ijk ~E

j ~Ek, where we may identify �ijk ��01ijk.

Now if by habit we insist that torsion is zero, then (80)does not solve the vacuum fields equations.

In Einstein gravity, when a form like dtþ a is intro-duced in the metric one assumes da ¼ 0, precisely to avoidintroducing torsion. We will impose

da ¼ T0: (81)

Under this condition the zero torsion connection of themetric with ai ¼ 0, that is !i1 ¼ � ~Ei and !ij ¼ �ijk ~E

k, is

also a connection for (80), but with nontrivial torsion

T0 ¼ ai�ijk ~Ej ~Ek: (82)

One should note two things. First, unlike the threedimensional Einstein case the metric (80) is not flat. Thecurvature reads:�ij ¼ ð1� �2Þ ~Ei ~Ej ¼ ð1� �2Þr�2EiEj.In fact there is a curvature singularity at r ¼ 0. There isalso a torsion singularity there, T0

ij ¼ ai�ijkr�2. The situ-

ation is somewhat analogous: �-function singularities incurvature and torsion in three dimensions [57] are replacedby r�2 singularities in five.Second, the torsion T0 needs no source. The vacuum

field equations are satisfied everywhere outside the singu-larity r ¼ 0. Also, just like in the three dimensionalEinstein gravity, the singularity is integrable [56]. Putdifferently, it is a case of type (H).We may verify that indeed no regularization is needed.

The mass is calculated as the Hamiltonian (21) for theKilling vector � ¼ @=@t from the formulaZ

S3c2 �

��ði�!�EÞ � i��

�

��k þ 1

32�E

��: (83)

The BðCÞ form used is the well-known boundary term for

Dirichlet conditions constructed in Ref. [58] (see alsoAppendix B). One finds easily that the quasilocal mass isgiven by

mBY ¼ �8�2c2j�jð3� �2Þ: (84)

This presumably gives a Brown-York vacuum forMinkowski spacetime in this theory equal to

mBY ¼ �16�2c2: (85)

It is the five-dimensional analogue of the �2�c1 ¼�ð4GÞ�1 result of Ref. [55] for the three dimensions weencountered in the previous section.Defining a relative mass similarly to the three dimen-

sions by mrel ¼ 0 for � ¼ 1 we obtain here

mrel ¼ 8�2c2f2� j�jð3� �2Þg: (86)

This is the result found in [56] by integrating the matterenergy tensor. It was shown that it can be calculated veryelegantly using the Gauss-Bonnet theorem; the constantterm, which is minus the alleged Minkowski vacuum en-ergy, is related to the Euler number of a spatial 4 ball.It perhaps interesting to note that, for c2 > 0, the massm

in this example is bounded from below, with a minimum at�2 ¼ 1. This is to be contrasted with the definite bounded-ness from above (76) in three dimensions, which was


084013-13

especially discussed in Ref. [53]. In fact, the same happensin Chern-Simons theories in all dimensions which aremultiples of four modulo one and three, respectively.

Now let �i be Killing vectors dual to ~Ei. The quasilocalangular momentum is calculated as minus theHamiltonians, given by (83), for the vectors �i. We have

JiBY ¼ 8�2c2j�jð1� �2Þai: (87)

It amusing to note the relation

� 3Ji ¼ �@

@�mai; (88)

which also holds in three dimensions but with a factor �1instead of �3. Thus, the angular momentum is the scalingof the mass for the Chern-Simons particles. Relation (88)properly generalizes (75).

VI. BOULWARE-DESER-CAI METRICS IN 5d

We now turn to the full Lovelock gravity in five dimen-sions. We study the spherically symmetric metrics ofBoulware and Deser [59], discovered at the same time byJ. T. Wheeler [60], and the generalization by Cai [61].These are solutions to the so-called Einstein-Gauss-Bonnet gravity in five dimensions described by theLagrangian

L ¼ 1

2c2�ð��EÞ þ 1

2c1�ð�E4Þ � �

5!�ðE5Þ: (89)

The last term is formally equivalent to a matter energytensor TA

B ¼ ��AB. � is not an integration constant thus it

is not a phase space parameter; it operates as one moregravitational coupling. For the Boulware-Deser-Cai solu-tion is a vacuum solution for this Lagrangian everywhereoutside the origin r ¼ 0 (which is not included in space-time anyway). Torsion is assumed zero which solves theconnection field equations.

Abbott-Deser types of calculations were done in [62–64]for quadratic curvature gravities. The resulting formulascompute the mass relative to the asymptotic vacuum. Theyare rather involved and hard to generalize. (The massparameter in Boulware-Deser-Cai metrics given usuallyin the literature is calculated by these formulas.)

A basic feature of Lovelock gravity is that its solutionsare multivalued and its branches have different asymp-totics. Relative mass depends on the reference background,i.e. the asymptotic vacuum. Thus, one cannot associate avalue of energy with an entire solution.

Brown-York computations are in no better shape in thatrespect; they too depend on asymptotics: the counter-termsdepend on the branch. On the other hand the Brown-Yorkdefinition is more self-consistent: the branch of the solu-tion we work at is already chosen at the level of theLagrangian. Then a value of energy is obtained for thesolution.

Stability analysis based on energy considerations [62–64] is treacherous in Lovelock gravity. The theory regardsboth branches as available states. The Brown-York defini-tion emphasizes strongly that we work one branch at atime, there is no overview of the solution. The issue ofstability of vacua was touched upon in Ref. [65]. We shallshown in a separate work that instabilities are far moregeneric in this gravity that what is envisaged in [65]. Herewe shall point out that Brown-York masses make an un-expected appearance in semiclassical calculations.Quasilocal calculations in Lovelock gravity have beenpresented in various works, see e.g. [39–44,66–72].The metrics can be given the general form considered in

Sec. III:

ds2 ¼ �g2dt2 þ dr2

g2þ r2d�2: (90)

d�2 is the metric of a compact, constant curvature k 3manifold: denoting a frame on it by ~Ei, i.e. d�2 ¼ �ik

~Ei �~Ek its curvature will be ~�ij ¼ k ~Ei ~Ej, where k ¼ �1, 0.The metric function g2 is given by

g2 � k ¼ 3c12c2

r2�1þ s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2�

27c21þ C

r4

s �; (91)

with s2 ¼ 1. Thus there are two solutions obtained, onlyone of which can be asymptotically flat. This is one maycalled the Einstein branch. The other, s ¼ þ1, is the‘‘exotic’’ branch of the Boulware-Deser solution. Muchof the novelty arising by these metrics is ought to itsexistence.The metrics (91) appear in the literature more often in

terms of the parameters �2, �, and m:

3!c1 � ��2; � � c26c1

¼ �2c2;

C ¼ m8�2�

3�2

2�2

volk:

(92)

The first relation implies also that �2 ¼ 8�G, in terms ofthe five-dimensional Newton’s constant. volk is the volumethe 3 manifold of constant curvature k.Our basic formula (21) for the Hamiltonian generator

reads here:ZS3c1 �

�1

2�ði�!E3Þ � 1

2i��ðE3Þ

�

þ c2 ��ði�!�EÞ � i��

�

��k þ 1

32�E

��; (93)

where S3 is a large 3 manifold of constant curvature k. TheKilling vector � is @=@t. The result is

� 36�2c1g2r2 � 24�2c2g

2

�k� g2

3

�; (94)

times volk=ð2�2Þ, where volk is the volume of a unit 3


084013-14

manifold of constant curvature k. This quantity divergesfor large r.

It will be adequate to ‘‘correct’’ the minimal boundaryform with terms similar to those used in the five-dimensional Einstein gravity in Sec. III:

12 c1b0�ðE4

kÞ1 þ 12c1b1�ð�kE2

kÞ1: (95)

The ‘‘correction’’ to (94) is to

24�2c1b0gr3 þ 12�2c1b1kgr; (96)

again times volk=ð2�2Þ.Define the length l by

� l�2 � K ¼ � 3c12c2

�1þ s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2�

27c21

s �; (97)

where K is the spacetime constant curvature read off fromthe asymptotics of the metric g2 ¼ kþ r2l�2 þOðr4Þ.The curvature form is �AB ¼ KEAEB. This is of coursemeaningful as long as K of (97) is real.

Remark VI.1—Amost important fact about the structureof the counterterms is that they are determined solely by theasymptotic form. In other words: one needs only to fix theconstants b0 and b1 so that divergencies are removed whenthe metric function is exactly g2 ¼ kþ r2l�2.

One may consider for simplicity the AdS case K < 0,the caseK > 0 can be treated analogously at least formally.One finds that divergencies are removed for

b0 ¼ 3

2l

�1� 2c2

9c1l2

�; b1 ¼ 3l

2

�1þ 2c2

3c1l2

�: (98)

Then b0 and b1 remove the divergencies for theBoulware-Deser-Cai metric if l is given by (97). b1 isuseless and not defined for k ¼ 0. b0 and b1 make senseas long as l�1 � 0, i.e. for all cases except the asymptoti-cally flat spacetime s ¼ �1 and � ¼ 0. On the other hand,if k ¼ 0 the asymptotically flat case makes sense trivially.We discuss this interesting solution separately below. Thespecific counterterms have also appeared in [70] but not inthis form.

If we compare them to the couplings b0 and b1 we foundin five-dimensional Einstein gravity in Sec. III, the firstterms are similar but there are � corrections. Interestinglythe corrected couplings satisfy the exact same differentialrelation (59):

� l2db0dl

¼ db1dl

:

That is, the relation depends on the same asymptotics toAdS and (reasonably) the dimensionality but not on thetheory.

Finally the Brown-York mass reads

mBY ¼��9�2c2 þ 9

2�2c1l

2

�k2

volk2�2

þm: (99)

For the spherically case k ¼ 1, discussed below, this for-mula agrees with the result of [41] for five dimensionsderived by a set of counterterms applying to asymptoticallyAdS spacetimes. They were constructed in Refs. [39,40,42]as boundary terms respecting the conformal symmetry ofthe AdS boundary. (It was also obtained in [70] by counter-terms similar to ours, though only for the Einstein branchand not emphasized much.) This is in accordance with ourcomments in Sec. II D: If the quasi local mass exists, i.e.the large radius is convergent, most likely the result isindependent of the boundary conditions used.Let us write down the spherical case explicitly:

mBY ¼ �9�2c2 � 3�

32GKþm: (100)

We see the mass parameterm, whose setting to zero makesthe metric (91) a metric of constant curvature K. We alsosee a K dependent term which is exactly the same as whatwe obtained in Einstein gravity (62) in five dimensions. Wealso see a term which depends purely on the gravitationalconstant c2. This is analogous to the vacuum energy of AdSin three dimensions, �ð8GÞ�1, definitely a remnant of theChern-Simons nature of the Gauss-Bonnet term in fivedimensions discussed in the previous section.This result was obtained through regularization, so it is a

real type (I) case, not a type (H) one. This holds even if weset c1 ¼ 0, decoupling the Einstein term, which kills the Kdependent term above. Remark III.2 does not apply andone should note the difference between this first term in(99) and the Brown-York mass of Minkowski space (85) inpure Gauss-Bonnet gravity.The flat case k ¼ 0 is completely different. We noted

already writing down the counterterm coefficients (98),that in this case the limit l ! 1 makes sense trivially.(This of course happens only for Einstein branch, s ¼�1), i.e. the l ¼ 1 is a type (H) case. Remark III.2 mayapply.Indeed, formula (99) says that for l <1, where regu-

larization is still required, we have mBY ¼ m. But if we goto formula (94) for l ¼ 1, that � ¼ 0 and s ¼ �1, theresult is convergent and equal to

mBY ¼ 2m: (101)

This is in accordance with remark III.2 extended inLovelock gravity.Concluding, we would like to recalculate the mass for

this metric as a type (II) case, which we can always do,and obtain merely a relative value for this generator. Wewill define it in the usual way by: mrel ¼ 0 for m ¼ 0.We apply formula (29) under condition (27), using the

already calculated sphere integral, formula (94). One finds


084013-15

mrel ¼ �36�2c1fg2r2 � gg�r2g volk2�2

� 24�2c2

�g2�k� g2

3

�� gg�

�k� g2�

3

��volk2�2

;

(102)

where r ¼ r� ! 1 and m� ! 0 as � ! 1. From that wefind

mrel ¼ s2m ¼ m: (103)

The variable m is the standard quantity representing themass of the respective spacetime in this theory. We explic-itly saw here that this follows from a subtraction method ofcalculation. There are two things that should be remarkedupon.

First, the previous result holds except for k ¼ 0 and � ¼0 of the Einstein branch (s ¼ �1) of the Boulware-Deser-Cai metric, which is a type (H) metric as we alreadymentioned. In that case one finds

mrel ¼ 2m: (104)

When compared with (101), we see that they are equal.This is in accordance with the very definition of a type (H)metric: the relative mass can be regarded as a Brown-Yorkmass.

Second, there is a new thing brought up by consideringrelative values of generators in Lovelock gravity. All met-rics are at least double valued in this theory, thus so are thevacuum metrics. The relative mass in each branch is calcu-lated with respect to the constant curvature spacetime(vacuum) in that branch. As the vacua are very differentso is the meaning of parameter m as the mass in eachbranch. Put in different words, the masses in each caseare defined for different asymptotics and therefore theirvalues are not comparable. For � ¼ 0 this is all we have:the Einstein branch (s ¼ �1) is asymptotically flat.

When the cosmological constant is nonzero the Brown-York definition of the generators raises this fake degener-acy in an explicit though formal manner. The Brown-Yorkmasses of the two branches of a given Boulware-Desermetric differ by

mEinsteinBY �mexotic

BY ¼ �162�2 c21

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2�

27c21

s: (105)

Of course this subtraction is meaningful for comparisonpurposes when we subtract things of the same nature. Onthe other hand mBY is obtained by adding to the actioncounter-terms which depend on the scale l. l in turn,defined in (97), depends explicitly on � as well on thebranch sign s, distinguishing the branches already at thelevel of the Lagrangian. This is like comparing masses ofcompletely different gravitational theories. Therefore onecannot regard the Brown-York mass as a kind of absolute

mass and cannot regard the mBY in each branch as com-parable quantities. That is, (105) is also a formalsubtraction.On the other hand if we restrict ourselves to the sector of

solutions belonging to a given branch then the Brown-Yorkmass can be regarded as a kind of absolute definition in thefollowing sense: One calculates the mass a Hamiltoniangenerator of a given theory with couplings c1, c2 and �, andno reference background or subtractions are involved.Then, inspection of the formulas (100) and (97) showthat the Brown-York mass increases with � in theEinstein branch sector of solutions and decreases with �in the exotic branch sector. This is a reflection in thequasilocal calculation of the antigravity behavior of exoticbranch solutions: asymptotically the exotic branch metrichas an extra minus sign in the mass-dependent term.As a physical energy of some kind, Brown-York mass is

inherently related to quantum effects through the AdS/CFTcorrespondence [8,9,73]: mBY equals the vacuum energy(Casimir effect) of a conformal quantum field theory livingon the boundary manifold. This is a purely quantum effectcoming from the zero-point energies of the field oscillators.We shall not go into boundary dual field theories here butwe can point to another quantum place where the Brown-York mass makes an appearance.In the semiclassical description of false vacuum decay

(in the presence also of gravity) [74–76], one works inimaginary (Euclidean) time and considers configurationssuch that a bubble of real vacuum nucleates within a sea offalse vacuum. The quantity which gives the rate of nuclea-tion involves the on-shell Euclidean action. It is made ofpieces of the form: ðenergy densityÞ ðbubble volumeÞ,and ðsurface tensionÞ ðbubble areaÞ. For transitions be-tween vacua of some curvatures K one may verify that thebulk pieces of the on-shell action are proportional to

mBY � K2 � ðbubble volumeÞ; (106)

where mBY is explicitly given by (100) for m ¼ 0. Fromthe analogous formulas in Einstein gravity it is not veryclear that the Brown-York mass appeared there, or better aquantity equal to it, as the result is what one expects ondimensional grounds. In Lovelock gravity there are relativecoefficients that should match. We do not have a detailedexplanation for this phenomenon.

VII. BOULWARE-DESER METRICS IN 6d

Dimension five, being the minimum dimension for thequadratic Lovelock term to exist, exhibits certain peculiar-ities inexistent in dimension six or higher. This is shown,for example, in the qualitative differences of the black holesolutions between five and higher dimensions [61], or inthe novel topological black holes presented in [77] whichexist in dimension six and not in five due to a vanishingWeyl tensor. Five dimensions being minimal could betreacherous. The simple gauge theoretic construction of


084013-16

counterterms we advocate could fail in dimension higherthan five. This is not so. To our knowledge such a calcu-lation has been not been presented in the literature.

The Lagrangian of the theory is easily written down,according to what we have said already in the Introduction.It amounts to an insertion of one more factor of E in theLagrangian (89) (and divide � in the last term with 6!instead of 5!). The field equations are obtained by varyingthe action functional with respect to E. For the standardform of the metric (90) the field equations give

g� k ¼ c1r2

c2

�1þ s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2�

120c21þ C

r5

s �: (107)

C is an integration constant associated with the mass of thespacetime w.r.t. the asymptotic vacuum. Of course thereare two branches in the solution, s ¼ �. The relation of ourcouplings c1 and c2 to the usual couplings is: 4!c1 ¼ ��2

and � ¼ 2!�2c2, where �2 ¼ 8�G.

The Brown-York Hamiltonian (21), constructed out ofthe minimal Dirichlet boundary form, readsZ

S4c1 �

�1

2�ði�!E4Þ � 1

2i��ðE4Þ

�

þ c2 ��ði�!�E2Þ � i��

�

��k þ 1

32�E2

��; (108)

where S4 is a large 4 manifold of constant curvature k ¼�1, 0.

This diverges so we supplement the boundary formBðCÞwith the intrinsic counterterms

12 c1b0�ðE5

kÞ1 þ 12c1b1�ð�kE3

kÞ1 þ 12c1b2�ð�k�kEkÞ1:

(109)

One finds

� 256�2c1g2r3 � 256�2c2g

2

�k� g2

3

�r

þ 32�2c1b05gr4 þ 32�2c1b13kgr

2 þ 32�2c1b2k2g;

(110)

times 3volk=ð8�2Þ, where volk is the volume of the con-stant curvature k 4 manifold S4. This additional factor isequal to one when S4 is a sphere.

We may consider for simplicity asymptotically AdS (orflat) metrics. The asymptotic AdS length scale l and con-stant curvature K are defined by

� l�2 � K ¼ � c1c2

�1þ s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ c2�

120c21

s �: (111)

We have already mentioned that removing the divergen-cies for the asymptotic metric alone, that is when themetric is exactly given by g2 ¼ kþ r2l�2, suffices toremove the divergencies for all metrics (107). It is thenstraightforward to verify that the divergencies are removed

when

b0 ¼ 8

5l

�1� c2

3c1l2

�; b1 ¼ 4l

3

�1þ c2

c1l2

�;

b2 ¼ �l3�1� 3c2

c1l2

�:

(112)

b1 and b2 are useless and not defined for k ¼ 0.Now let l be given by (111). The Brown-York mass for

the spacetime with metric (107) in the respective branchreads

mBY ¼ Cc21c2

s2 ¼ Cc21c2

; (113)

times 3volk=ð8�2Þ. s ¼ � is the branch sign. This result isin agreement with quasilocal calculations in the literature[41,69,72]. We derived the result from scratch in a rela-tively easier manner.It is worth mentioning the following. The Brown-York

mass of pure AdS, or for that matter of pure de Sitterspacetime, given by C ¼ 0, is zero in six dimensions.The same we observed in four dimensions, and holds forall even dimensions [41,78]. As a result the Brown-York ofboth branches of the solution is given by a similar formula.From the above we reach the conclusion: If we have a

Lovelock gravity in the bulk, all is required to have is aLovelock gravity intrinsic to the boundary. Then all diver-gencies can be removed for a pure AdS background byappropriately fixing the couplings of the boundaryLovelock theory. Then the same boundary theory appliesalso to asymptotically AdS spacetimes. Formally, it shouldnot be very difficult to determine the b coefficients in thegeneral Lovelock gravity, or even to more general geomet-ric theories. A better understanding of the whole thingwould be much better and we shall leave such an analysisfor a separate work.

VIII. FLAT FROM ADS

The quasilocal charges H�ð�Þ, relation (23), were intro-

duced as candidates of the total content of a spacelikesection �� bounded by S� in the respective conservedcharge. Such a quantity should possess a number of prop-erties, discussed in the works cited in Sec. II C. Of them weprove none apart from the naturalness of their definition,i.e. them being essentially on-shell values of the Hamilton-Jacobi theory generators. On this basis at least, one maystudy them and realize that one can extract interestinginformation from them. For example they operate as inter-polations between the singular asymptotical flat case andthe nonzero cosmological constant metrics.As a first example consider an AdS metric in three

dimensions, discussed in Sec. III. The mass was calculatedthere as the large r limit of the quantity �2�c1g

2 �2�c1b0gr. After a little algebra one may write the resultin the form


084013-17

mBYðrÞ ¼ � 1

4G

�1þ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ðl=rÞ2p ��1: (114)

One first observes that this is an increasing function of r: itis larger the bigger the region is. Specifically goes from thevalue �ð4GÞ�1 obtained in the ‘‘small sphere’’ limit r !0, to the value �ð8GÞ�1 in the limit r ! 1. The lattervalue is of course the value of the Brown-York mass (47) ofAdS spacetime in dimension three.

On the other hand, the small sphere result �ð4GÞ�1 canbe recognized as the Minkowski spacetime Brown-Yorkmass (78) we found in Sec. IV. The reason is simple: Theresult depends on the lengths l and r only through thedimensionless ratio l=r. Therefore r ! 0 is equivalent tol ! 1. This is nothing but taking the asymptotically flatlimit before letting the size of the region go to infinity.Letting r become large is then trivial in this example andone obtains from the AdS space quasilocal charge (114) theflat space Brown-York mass.

Thus, there is actually continuity in the limit l ! 1.(This is also one more reason why we should apparentlytake a nonzero Brown-York defined mass for theMinkowski spacetime seriously.) The reason why it ispossible to obtain the flat space result for the AdS spaceone, is that the former is a type (H) case. One may verifythat everything said above applies also to the five-dimensional k ¼ 0 Boulware-Deser-Cai metric ofLovelock gravity.

We may now extend remark III.2 which concerns thetypes (H) cases:

Remark VIII.1—The quasilocal values of theHamiltonian generators in type (H) cases can be obtainedas the small sphere limit of the quasilocal generators of theassociated AdS metrics.

The need for the extension is that in the limit l ! 1 themass not always changes by a factor of 2. This can be seenby comparing the mass (100) of the Boulware-Deser metricand the mass (85) of the Minkowski spacetime in pureGauss-Bonnet gravity (i.e. no Einstein term is present)which is a type (H) case.

To make things completely explicit let us consider pureGauss-Bonnet gravity in the presence of a cosmologicalconstant. This is the theory (89) for c1 ¼ 0. Everything weneed can be obtained from the full Lovelock gravity resultsof the previous section, as neither the length l nor thecounter-term coefficients c1b0 and c1b1 are singular insetting c1 ¼ 0. In any case it is straightforward to verifythat the obtained counterterms work. One finally finds

mBYðrÞ ¼ 4�2c2 ��3r2

l2� 2

r4

l4

�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ l2

r2

s� 1

�

þ r2

l2� 4

�: (115)

In the limit r ! 1we obtain�9�2c2. This is the first termof the full formula (100) as expected. In the small sphere

limit r ! 0 one obtains �16�2c2 which indeed is the l ¼1 result (85).Now, if mBYðrÞ is the amount of total energy contained

in a sphere of radius r in these examples, why does it notvanish in the small sphere limit?First of all, one can always set a constant value of energy

to zero by suitably fixing the constant H0 in (21) and (23).Consider then the example of AdS3. If we fix H0 such thatmBYðrÞ vanishes in the small sphere limit then the Brown-York mass in spacetime is �ð4GÞ�1, instead of �ð8GÞ�1.But it is not that easy to change our minds and fiddle thelatter value; it can be rediscovered as the mass of a state inthe boundary conformal field theory which can be identi-fied with AdS3 [79]. It is then advisable to relax theinterpretation of the quasilocal mass as the mass ‘‘con-tained’’ in a spatial region. A better use of it has beenpointed out in this section.

IX. SUMMARYAND COMMENTS

Covariant phase space methods are used in the first orderformulation of Einstein and Lovelock gravity for the deri-vation of the Hamiltonian generators. Many known resultsand some new ones are derived and discussed in thiscontext. Relations between them and phenomena whicharise as we go from three to six dimensions are studied. Wefind that in the odd dimensions, if the higher possibleLovelock term is included, one should attribute a Brown-York energy in Minkowski spacetime. This is intimatelyrelated to the effect of the latter to create deficit angles’singularities in space. The relation of the flat spacetimeresults to those of the associated AdS spaces is discussed.In certain cases, where a minimal boundary Lagrangian isadequate for convergence, the flat space result derives fromthe small sphere limit of the AdS one. In higher dimen-sions, five and six, with Lovelock gravity turned on, thesimplicity of the first order formulation of the theory (indifferential forms notation) with Dirichlet boundary con-ditions and its agreement with other methods is empha-sized. This agreement is consistent with the analysis of thefirst sections of the paper which suggests that all conver-gent quasilocal definitions should agree up to a phase spaceconstant. The counterterm coefficients exhibit a structure,for example, in five dimensions those coefficients of thefull Lovelock gravity obey the same differential relationwith those of Einstein; something similar can be done indimension six. It becomes rather clear that a boundaryLovelock gravity can regularize the divergencies of aLovelock gravity in the bulk. Study of this general problemis left for future work.Well-defined Hamiltonian generators rest on the exis-

tence of a nondegenerate symplectic form, which we tac-itly assumed. This is not guaranteed in Lovelock gravity. Inan analogous particle system with Lagrangian Lðx; vÞ thesymplectic form would read dðdL=dvÞ ^ dx. This givestrouble if L has an inflection point. At such points accel-


084013-18

eration terms drop out the field equations. Neither such asystem nor Lovelock gravity have an a priori well-posedinitial value problem [14]. We may derive a Hamiltonian inthe space of solutions only away from those points [80].

An interesting way out of this problem, as well as out ofworst problems of causal evolution arising for piecewisesmooth solutions [81], was put forward in Ref. [82]. Thereis a class of Lovelock gravities, which are not too special,i.e. they are not of measure zero in coupling space, whoseactions are interpreted as a Weyl tube volume (see [82] andreferences therein). The idea is that as long as the Weyltube does not intersect itself, in which case a Lovelockaction does represent the tube volume, the associated grav-ity is safe. To the extent that this is correct, one couldconsistently do Hamiltonian theory in those theories.

ACKNOWLEDGMENTS

We thank S. Willison for helpful discussions. Part ofthis work was done during a pleasant visit to CECS (Centrode Estudios Cientıficos); the Center is thanked for itshospitality.

APPENDIX A: SECOND FUNDAMENTAL FORM

Consider a non-null hypersurface embedded in space-time with a unit vector field �A normal to it. � � � ¼ �1.We denote by i�k the pullback into the hypersurface. Let E

ak

be an intrinsic vielbein and !abk be an intrinsic connection.

Lower case Lorentz indices label tensor components nor-mal to � .

Impose a coincidence condition: i�k!ab ¼ !ab

k and

i�kEa ¼ Ea

k , i.e. the induced fields coincide with the intrin-

sic fields. This is in accordance with Sec. II A. This ispossible in each smooth component of the boundary.

Let us impose the conditions that the induced fields heldfixed:

i�k�!ab ¼ 0 and i�k�E

a ¼ 0: (A1)

Then also �Eak ¼ 0 and �!ab

k ¼ 0, by the coincidence

condition. In words: the induced fields are not varied underEuler-Lagrange variations; respecting the coincidence con-dition, the intrinsic fields are held fixed to their values. Thisis the ‘‘Dirichlet boundary conditions’’ in first orderformalism.

We now define

� :¼ i�kð!�!kÞ: (A2)

This is the second fundamental form of the embedding of

the hypersurface into spacetime. Clearly, the nonzero com-ponents of � have one index in the normal direction of thehypersurface.The fields !k and Ek can be regarded as a bulk field

which agrees with the intrinsic fields when pulled back intothe hypersurface. We often use the quantity � !�!k.Of course � ¼ i�k.

APPENDIX B: BOUNDARY FORMS

Consider for simplicity, Einstein gravity in three dimen-sions. The Lagrangian is L ¼ 1

2 c1�ð�EÞ.We have

��ð�EÞ ¼ d�ð�!EÞ þ �� E: (B1)

We have

i�k�ð�!EÞ ¼ i�k�ð�½!�!k�EÞ þ i�k�ð�!kEÞ¼ �fi�k�ð½!�!k�EÞg þ i�k�� Ebdy; (B2)

where Ebdy are the boundary equations of motion, essen-

tially the quasilocal energy-momentum and spin tensorsQquasilocal of Sec. II A. By (A1) this term vanishes.

Thus we have that under Dirichlet conditions

�ðCÞ�Z

R�ð�EÞ �

Z@R

�ð½!�!k�EÞ�¼

ZR�ðCÞ� � E:

(B3)

The symbol �ðCÞ, which we use in the main text, empha-

sizes that the variations are done under the specific bound-ary conditions. Therefore the action

S ¼ 1

2c1

ZR�ð�EÞ � 1

2c1

Z@R

�ðEÞ; (B4)

has an extremum on-shell under �ðCÞ. This is the Einstein-Hilbert action with a Gibbons-Hawking term. The bound-ary form BðCÞ used in the next is read off from it. The

generalization of this action in higher dimensions amountsto simply inserting an equal number of factors of E in thebulk and boundary form, as one may verify.In Einstein-Gauss-Bonnet gravity, which we mostly use

in this paper, the boundary form was written down inRef. [58], and for Lovelock gravity in Ref. [83]. This canbe constructed in all Lovelock gravities via a Chern-Weilprocedure which provides the transgression forms in theassociated topological problems, see e.g. [16]. In that spiritthey where constructed in [84]. A general method forconstruction of the boundary forms for general geometricLagrangians was presented in [85].


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Conserved charges in (Lovelock) gravity in first order formalism

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