Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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Theorem 4 (Jeffrey and Kirwan (1995), corrected

as in Jeffrey and Kirwan (1997)).

Let  2 H



(M) induce 

2 H



red

). Then we

have

red

ðÞe

red

¼ n

Res D

ðXÞ

F2F



ðXÞ½dX

½5

where n

is the order of the stabilizer in G of a

generic element of 

1

(0), and the constant C

defined by

ð1Þ

sþn

jWjvolðTÞ

½6

We have introduced s = dim G and l = dim T; here

= (s  l)=2 is the number of positive roots. Also,

F denotes the set of components of the fixed -point

set of T, and if F is one of these components, then

the meromorphic function h



on t  C is defined by



ðXÞ¼e

iðFÞðXÞ



ð XÞe

ðXÞ

½7

and the polynomial D: t !R is defined by D(X) =

>0

(X), where  runs over the positive roots of G.

The residue map Res is defined on (a subspace of)

the meromorphic differential forms on t  C: its

definition depends on some choices, but the sum of

the residues over all F 2F is independent of these

choices. When T = U(1), we define the residue on

meromorphic functions of the form e

iX

when

 6¼ 0 (for N 2 Z)by

Res

iX



¼ Res

X¼0

iX

; if >0

¼ 0; if <0

More generally, the residue is specified by certain

axioms (see Jeffrey and Kirwan (1995, proposition

8.11)), and may be defined as a sum of iterated

multivariable residues Res

= 

...Res

= 

for a

suitably chosen basis of t yielding coordinates

, ..., X

(see Jeffrey and Kirwan (1997)).

The Tolman–Weitsman Theorem

The Tolman and Weitsman (2002) theorem is as

follows:

Theorem 5 We have

KerðÞ¼

ðK



 K

Þ½8

Here, S is a generic circle subgroup of T and K



(resp. K

) denote the set of all equivariant cohomol-

ogy classes  whose restriction to F



(resp. F

) is

zero. Here,



¼fF 2F: 

ðFÞ > 0g

where 

is the component of the moment map in

the direction of the Lie algebra of S.

For more information, see Intersection Theo ry,

Moduli Spaces: An Introduction, and Equivariant

Cohomology and the Cartan Model.

Acknowledgments

The author acknowledges support from NSERC.

See also: Cotangent Bundle Reduction; Equivariant

Cohomology and the Cartan Model; Hamiltonian Fluid

Dynamics; Intersection Theory; Moduli Spaces:

An Introduction; Poisson Reduction; Stationary Phase

Approximation; Symmetry and Symplectic Reduction.

Further Reading

Atiyah MF (1982) Convexity and commuting Hamiltonians.

Bulletin of the London Mathematical Society 14: 1–15.

Berline N, Getzler E, and Vergne M (1992) Heat Kernels and

Dirac Operators, Grundlehren, vol. 298. Berlin–Heidelberg:

Springer.

Cannas da Silva A (2001) Lectures on Symplectic Geometry, Lecture

Notes in Mathematics, vol. 1764. Berlin: Springer.

Delzant T (1988) Hamiltoniens pe´riodiques et images convexes de

l’application moment. Bulletin de la Socie´te´ Mathe´matique de

la France 116: 315–339.

Guillemin V (1994) Moment Maps and Combinatorial Invariants

of Hamiltonian T

-Spaces, Progress in Mathematics, vol. 122.

Boston: Birkha¨user.

Guillemin V, Ginzburg V, and Karshon Y (2002) Moment Maps,

Cobordisms, and Hamiltonian Group Actions, Mathematical

Surveys and Monographs, vol. 98. Providence, RI: American

Mathematical Society.

Guillemin V and Kalkman J (1996) The Jeffrey–Kirwan localiza-

tion theorem and residue operations in equivariant cohomol-

ogy. Journal fu¨ r die Reine und Angewandte Mathematik 470:

123–142.

Guillemin V and Sternberg S (1982) Convexity properties of the

moment mapping I and II. Inventiones Mathematical 67:

491–513.

Guillemin V and Sternberg S (1984) Convexity properties of the

moment mapping I and II. Inventiones Mathematical 77:

533–546.

Guillemin V and Sternberg S (1990) Symplectic Techniques in

Physics. Cambridge: Cambridge University Press.

Henriques A and Metzler D (2004) Presentations of noneffective

orbifolds. Transactions of the American Mathematical Society

356: 2481–2499.

Jeffrey LC and Kirwan FC (1995) Localization for nonabelian

group actions. Topology 34: 291–327.

Jeffrey LC and Kirwan FC (1997) Localization and the quantiza-

tion conjecture. Topology 36: 647–693.

Jeffrey LC and Kirwan FC (1998) Intersection theory on moduli

spaces of holomorphic bundles of arbitrary rank on a

Riemann surface. Annals of Mathematics 148: 109–196.

Kalkman J (1995) Cohomology rings of symplectic quotients. Journal

fu¨r die Reine und Angewandte Mathematik 458: 37–52.

606 Hamiltonian Group Actions

Kirwan F (1984a) Cohomology of Quotients in Symplectic and

Algebraic Geometry. Princeton, NJ: Princeton University Press.

Kirwan F (1984b) Convexity properties of the moment mapping

III. Inventiones Mathematical 77: 547–552.

Kirwan F (1992) The cohomology rings of moduli spaces of

vector bundles over Riemann surfaces. Journal of the Amer-

ican Mathematical Society 5: 853–906.

McDuff D and Salamon D (1995) Introduction to Symplectic

Topology. Oxford: Oxford University Press.

Satake I (1957) The Gauss–Bonnet theorem for V-manifolds.

Journal of the Mathematical Society of Japan 9: 464–492.

Tolman S and Weitsman J (2003) The cohomology rings of

symplectic quotients. Communications in Analysis and Geo-

metry 11: 751–773.

Hamiltonian Reduction of Einstein’s Equations

A E Fischer, University of California, Santa Cruz,

CA, USA

V Moncrief, Yale University, New Haven, CT, USA

Introduction

In general relativity there are several levels within

the framework of symplectic reduction of Einstein ’s

equations at which one could attempt to define a

Hamiltonian for the gravitational dynamics of a

spatially closed universe. At the most basic unre-

duced level, this Hamiltonian is simply a linear

function of the Einstein constraints and thus

vanishes for any solution of the field equations. At

the other extreme, at the deepest fully reduced level,

one affects a transformation to a complete set of new

canonical variables, the so-called ‘‘observables,’’

which Poisson-commute with all of the constraints.

At this level, the relevant Hamiltonian vanishes

identically since each of the new canonical variables

is a constant of the motion.

There is, however, an intermediate level wherein,

after making a su itable choice of coordinate gauge

and imposing the constra int equations, one can

define a nonvanishing Hamiltonian that generates

the gauge-fixed and constrained evolution equations

and whose global infimum as a function on the

relevant reduced phase space has direct topological

significance. For the large class of manifolds on

which this Hamiltonian can be defined, it has the

attractive feature of globally monotonically decaying

in the direction of cosmological expansion and thus

evolves in such a way so as to seek and, in certain

cases at least, to asym ptotically attain its infimum

value in the limit of this expansion. This Hamilto-

nian provides in these cases a weak Lyapunov

function for the dynamics that can be used to

partially control its global behavior. Since under-

standing the global behavior of solutions to

Einstein’s equations and its dependenc e upon the

spatial topology is one of the central open problems

in classical general relativity, the mathematical

properties of this quantity are worth y of study.

Further information and details regarding the

authors’ work disc ussed in this article can be found

in Fischer and Moncrief (2000, 2002a, b) and in the

references therein.

Topological Background

Einstein’s field equations are nonvacuous and

compatible with the introdu ction of material sources

in (n þ 1) dimensions for all n  2, the case of most

physical interest being of course n = 3. For the field

equations to be deterministic in a classical sense,

that is, for the Cauchy problem to be well-posed, it

is essential that they be formulated on a manifold

that is globally hyperbolic and, in particular, has a

product topology M  R (roughly, space  time =

spacetime) where M is a smooth (C

) connected

manifold of dimension n and R is the real line. For

the case of spatially closed universes of interes t here,

M should be closed, that is, compact and without

boundary. To simplify the analysis further, we also

assume that M is oriented, that is, orientable and

an orientation has been chosen. Thus, unless stated

otherwise, throughout this article M will denote

a smooth closed connected oriented n-manifold,

n  2, and all maps will be smooth.

Let ‘‘’’ denote the diffeomorphic equivalence

relation between smooth manifolds. Let S

denote

the unit n-sphere in Euclidean (n þ 1) space

nþ1

, n  1. An n-manifold M is trivial if M  S

and nontrivial if M 6 S

The connected sum M#N of two closed connected

oriented n-manifolds M and N is constructed by

removing the interior of an embedded closed n-ball in

M and N, respectively, and then identifying the

resulting S

n1

-boundary components by an orienta-

tion-reversing diffeomorphism of the (n  1) spheres.

The resulting manifold is smooth, connected, closed,

and orientable, and is naturally oriented by the

orientations on M and N. Up to orientation-preserving

diffeomorphism, this construction is independent of

the choice of the embeddings of the n-balls and of the

choice of the orientation-reversing diffeomorphism

used to join the manifolds together.

Hamiltonian Reduction of Einstein’s Equations 607

Let M be a nontrivial closed connected oriented

n-manifold. Then M is prime if M  M

implies that either M

 S

or M

 S

(but not

both since we are assuming that M is nontrivial).

M is a composite if M can be written as a nontrivial

connected sum, that is, if M  M

where both

6 S

and M

6 S

Note that with this definition, S

itself is not

prime. This is analogous to the fact that for the

positive integers, the unit 1 is not prime.

Now let M be a connected n-manifold without

boundary (not necessarily compact or orientable) and

let p be a group. Then M is a K(p, 1)-manifold if M is

an Eilenberg–MacLane space, that is, if its first

homotopy group (or fundamental group) 

(M) = p

and if all of its higher homotopy groups are trivial, that

is, 

(M) = 0fori > 1 (equivalently, the universal

covering space

M of M is contractible). Since the

higher homotopy groups 

(M), i > 1, can be inter-

preted as the homotopy classes of continuous maps

! M, each such map must be homotopic to a

constant map. Thus a K(p, 1)-manifold is said to be

aspherical. Moreover, at the level of homotopy, all of

the information about the topology of M is contained

in 

(M) = p. Thus, in particular, if f is a map between

connected aspherical manifolds that induces an iso-

morphism on their fundamental groups, then f is a

homotopy equivalence. Consequently, any two con-

nected aspherical manifolds are homotopy equivalent

if and only if their fundamental groups are isomorphic.

It is useful to define a connected n-manifold M to

be hyperbolizable if there exists a complete Rieman-

nian metric g on M with constant negative sectional

curvature, K(g) = constant < 0. We introduce this

terminology to emphasize the underlying topology

of manifolds that can support hyperbolic metrics

rather than the geometry of such metrics. Similarly,

M is of flat type if M admits a complete flat

Riemannian metric g, K(g) = 0, and M is of spherical

type if M admits a complete Riemannian metric g on

M with constant positive sectional curvature,

K(g) = constant > 0. In this latter case, by the Bon-

net–Myers theorem, M is necessarily compact and if n

is odd, then by Synge’s theorem, M is necessarily

orientable. In fact, all such manifolds have been

classified. As an important example, we note that a

connected 3-manifold M is of spherical type if and only

if it is diffeomorphic to a spherical space form S

=G,

where G is a finite subgroup of SO(4) acting freely and

orthogonally, that is, isometrically, on S

Within the class of K(p, 1)-manifolds are all flat-

type and hyperbolizable n-manifolds, since any such

manifold is isometrically covered by R

in the flat case

and homothetically covered by H

in the hyperbolic

case, where H

is the standard single-sheeted spacelike

hyperboloid with constant sectional curvature K = 1

embedded in (n þ 1)-Minkowski space R

nþ1

We now return to our standard assumptions on M,

so that M is connected, closed, and oriented. For

n = 2, these assumptions restrict the possibilities to

, T

, and the orientable higher genus surfaces

= T

# # T

(p factors) consisting of the

connected sum of p copies of T

, p 2. However,

from the point of view of (2 þ 1) gravity, unless one

includes material sources or a cosmological constant,

the spherical case is vacuous in that there are no

vacuum solutions of the field equations on S

 R.

The torus case is nonvacuous but the solutions, the

so-called flat Kazner spacetimes, can all be found by

elementary means. Thus only the case of genus p  2

surfaces presents problems of interest.

For n = 3, although not essential for the program of

reduction, it is convenient to assume the elliptization

conjecture of 3-manifold topology. This conjecture

asserts that a closed connected 3-manifold M with

finite fundamental group 

(M) must be diffeo-

morphic to a spherical space form S

=G, where, in

such a quotient, G will always be a finite subgroup of

SO(4) acting freely and orthogonally on S

and thus G

is isomorphic to 

(M).

The simply connected case is the Poincare´con-

jecture. The full elliptization conjecture is equivalent

to the Poincare´ conjecture and a conjecture asserting

that the only free actions of finite groups on S

are equivalent to the standard orthogonal ones.

The elliptization conjecture is part of Thurston’s

geometrization program (Thurston 1997). For back-

ground information regarding 3-manifold topology,

see Hempel (1976) and Jaco (1980).

Under the assumption of the elliptization con-

jecture, the Kneser–Milnor prime decomposition

theorem asserts that if M is nontrivial, then up to

order, M is uniquely diffeomorphic to a finite

connected sum of the following form:

M 

# # S

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

k spherical factors



ðS

 S

Þ# # ð S

 S

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

l wormholes ðor handlesÞ



Kðp

; 1Þ# # Kðp

; 1Þ

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

m aspherical factors

½1

where k, l, and m are integers  0, k þ l þ m  1,

and if either k, l,orm is 0, terms of that type do not

appear. Moreover, if k  1, then each G

,1 i  k,

is a finite nontrivial (G

6¼ {I}) subgroup of SO(4)

608 Hamiltonian Reduction of Einstein’s Equations

acting freely and orthogonally on S

, and if m  1,

then each aspherical factor is a K(p

, 1)-manifold,

1  j  m, and thus is universally covered by a

contractible manifold.

We remark that although in general a contractible

3-manifold need not be R

, conjecturally the

universal covering manifold of a K(p, 1) 3-manifold

is diffeomorphic to R

In 3-manifold topology, a concept closely related

to that of a prime manifold is that of an irreducible

manifold. A closed 3-manifold M is irreducible if

every embedded 2-sphere in M is the boundary of an

embedded closed 3-ball.

An embedded 2-sphere that does not bound such a

3-ball is essential. Thus in the prime decomposition [1]

above, M is decomposed along essential 2-spheres. For

this reason, the prime decomposition is sometimes

referred to as the sphere decomposition.

With the exception of S

which is irreducible but

not prime (by definition of prime) and S

 S

which is prime but not irreducible, a closed oriented

3-manifold is prime if and only if it is irreducible.

We also remark that the Poincare´ conjecture, when

taken in the form that there do not exist any fake

3-cells, is equivalent to every K(p, 1) 3-manifold

being irreducible. Thus in this article, since we are

assuming the elliptization conjecture and hence the

Poincare´ conjecture, every K(p, 1) 3-manifold will

automatically be irreducible.

Examples of the kinds of K(p, 1)-factors that can

occur in the decomposition [1] are as follows (we will

explain the Seifert and graph designations below):

1. Non-Seifert manifolds. Closed oriented hyperboliz-

able manifolds diffeomorphic to H

=G, where G is a

discrete torsion-free (i.e., no nontrivial element has

finite order) co-compact subgroup of the Lie group

Isom

) of orientation-preserving isometries of

which is Lie-group isomorphic to the proper

orthochronous Lorentz group SO

(1, 3).

2. Seifert manifolds. T

and five other 3-manifolds of

flat type which are finitely covered by T

.Noting

that S

= T

, we remark that the product manifold

S

= S

T

= T

is included in this class.

3. Seifert manifolds. Product manifolds S

 S

, p  2.

4. Seifert manifolds . Nontrivial circle bundles over

, p 1.

5. Graph manifolds. Any 3-manifold which fibers

nontrivially over a circle with fiber S

, p  1. Any

such manifold is obtained by identifying the

boundary components of [0, 1]  S

with an

orientation-reversing diffeomorphism of S

Since the handle S

 S

and spherical manifolds

=G are well understood, under the assumption of

the elliptization conjecture the task of 3-manifold

topology now reduces to understanding the topology

of the (automatically irreducible) K(p, 1)-factors that

can occur in the prime decomposition [1]. Since

essential 2-spheres have already been used to

decompose M into its prime components, the idea

now is to use the next simplest 2-manifold, the

2-torus, to probe the irreducible K(p, 1)-factors.

Let i : T

! M be an embedding of T

into a

closed oriented 3-manifold M. Then the embedded

torus i(T

), identified with T

, is incompressible if

the induced mapping of fundamental groups



: 

) ! 

(M) is injective. Thus noncontracti-

ble loops in T

remain noncontractible when T

embedded in M, or, in other words, the ambient

manifold M does not fill in any homotopy hole that

exists in T

when standing alone.

A closed oriented 3-manifold M is a Seifert-

fibered space, or a Seifert manifold, if M admits a

foliation by circles. For example, if S

acts freely on

M, then M is the total space of an S

-bundle over a

surface M=S

and M is a Seifert-fibered space (see

examples 2, 3, and 4 above). More generally, if S

acts without fixed points (locally free), then M is a

Seifert-fibered space, and in either case the fibers of

M are the orbits of the S

-action.

All spherical 3-manifolds are Seifert fibered with

base S

. Also, the product manifold S

 S

is Seifert

fibered, as are all manifolds finitely covered by T

,and

thus all 3-manifolds of flat type are Seifert fibered.

The only nontrivial connected sum that is a Seifert-

fibered space is P

# P

. No hyperbolizable manifold

is Seifert fibered. Thus the remaining Seifert

manifolds are among the nonhyperbolizable nonflat

type K(p, 1)-manifolds (i.e., those for which M does

not admit either a hyperbolic or a flat Riemannian

metric).

A generalization of Seifert-fibered spaces are the

graph manifolds. A closed oriented 3-manifold M is a

graph manifold if there exists a finite collection {T

}of

disjoint embedded incompressible tori T

 M such

that each component M

of Mn[T

is a Seifert-fibered

space. Thus a graph manifold is a union of Seifert-

fibered spaces glued together by toral automorphisms

along toral boundary components. The collection of

tori may be empty so that, in particular, a Seifert-

fibered manifold is a graph manifold.

We remark that the manifolds described by example

5 above are graph manifolds. We also remark that

graph manifolds are closed under connected sums so

that a graph manifold may be a composite. This

contrasts with the situation for Seifert spaces which,

with the exception of P

# P

, are not composites.

Conjecturally, the most general K(p, 1)-manifold,

not included in the list above, consists of ‘‘gluing

together’’ across disjoint embedded incompressible

Hamiltonian Reduction of Einstein’s Equations 609

tori a finite collection of finite-volume-type hyperbo-

lizable manifolds, that is, noncompact manifolds that

admit a finite-volume complete hyperbolic metric,

together with a possibly empty finite collection of

irreducible graph manifolds with toral boundaries.

Thus, overall, in this picture, to decompose an

arbitrary closed oriented 3-manifold M into its

elementary constituents, one first cuts along essential

2-spheres to break M down into its prime factors, that

is, the nontrivial spherical S

=-factors, the wormhole

 S

)-factors, and the aspherical K(, 1)-factors, as

given by [1]. Then one cuts each nonelementary

K(p, 1)-factor along incompressible tori to separate

these factors into their final finite-volume-type hyper-

bolizable and irreducible graph manifold components.

The graph manifold components can then be further

broken down along incompressible tori into Seifert-

fibered pieces, finally yielding the toral decomposition

of Jaco, Shalen, and Johannson (see Anderson (1997),

Jaco (1980), and the end of the section ‘‘The Reduced

Hamiltonian’’ for further details).

The Thurston (1997) geometrization program,

which implies that every closed oriented 3-manifold

has the structure described by the above prime (or

spherical) and toroidal decomposition, has been the

subject of recent work by G Perelman (see Anderson

(2003) and the references therein) who has argued

that it can be proved by an enhancement of the Ricci

flow program of R Hamilton (see the collected

papers edited by Cao et al. (2003)). Without

entering into the technical issues surrounding the

completeness of Perelman’s proof, one can simply

limit one’s attention to 3-manifolds of the above type.

If geometrization is correct, then no 3-manifolds of

interest have been excluded.

Returning to the general case of n-manifolds, in the

program of Hamiltonian reduction of Einstein’s

equations, an important consideration is under what

topological conditions on M can the conformal classes

of M be uniquely represented by a given metric in each

class. To analyze that question, we introduce the

concept of the Yamabe type of a manifold.

Let M be a connected closed oriented n-manifold,

n  3. There is no topological obstruction to the

existence of Riemannian metrics with constant nega-

tive scalar curvature, so all such manifolds admit a

Riemannian metric g such that R(g) = 1. However,

there are topological obstructions for zero scalar

curvature and positive constant scalar curvature

metrics on M. To help categorize these topological

obstructions, we introduce the following terminology:

1. M is of positive Yamabe type if M admits a

Riemannian metric g

with scalar curvature

R(g

) = 1;

2. M is of zero Yamabe type if M admits a

Riemannian metric g

with R(g

) = 0, but no

Riemannian metric g with R(g) = 1; and

3. M is of negative Yamabe type if M admits no

Riemannian metric g with R(g) = 0.

The definition of Yamabe type partitions the class

of connected closed oriented n-manifolds, n  3,

into three classes that are mutually exclusive and

exhaustive. The following rather complete topologi-

cal information regarding 3-manifolds of negative

Yamabe type is known.

Let M be a connected closed oriented 3-manifold.

Assume that the Poincare´ conjecture is true. Then M

is of negative Yamabe type if and only if M satisfies

one of the following three mutually exclusive

conditions:

1. M is hyperbolizable (and thus is a K(p, 1)-

manifold; see example 1 of K(p, 1)-manifolds);

2. M is a nonhyperbolizable nonflat type K(p, 1)-

manifold (see examples 3, 4, and 5 of K(p, 1)-

manifolds);

3. M has a nontrivial connected sum decomposition

(i.e., M is a composite) in which at least one factor

is a K(p, 1)-manifold; that is, M  M

# K(p, 1),

where M

6 S

. In this case the K(p, 1)-factor may

be either of flat type or hyperbolizable.

We remark that (1) is the vast class of closed

oriented hyperbolizable 3-manifolds. We also

remark that the six closed orientable 3-manifolds

of flat type, although K(p, 1)-manifolds, are

excluded from (2) as they are not of negative

Yamabe type (they are of zero Yamabe type). Lastly

we remark that if M is of negative Yamabe type and

Seifert fibered, then M must be of type (2) (see

remarks on Seifert-fibered spaces above).

In any dimension n  3, a manifold M of negative

Yamabe type has the property that it admits no

Riemannian metric g having scalar curvature R(g)  0

everywhere on M, or, in other words, every Riemannian

metric on M has scalar curvature which is negative

somewhere. For such a manifold M, Yamabe’s theorem

asserts that each Riemannian metric g on M is uniquely

globally conformal to a metric  with scalar curvature

R() = 1 (see also [21]). Thus one can represent

the conformal classes of Riemannian metrics on M

in a suitable function space setting by an infinite-

dimensional submanifold

1

¼M

1

ðMÞ¼f 2MjRðÞ¼1g½2

of the space M= M(M) = Riem(M) of Riemannian

metrics on M (see Fischer and Marsden (1975) for

details). For this reason, we refer to metrics  in

1

as conformal metrics.

610 Hamiltonian Reduction of Einstein’s Equations

The quotient of M

1

by the natural action of

= D

(M) = Diff

(M), the connected component

of the identity of the diffeomorphism group

D= D(M) = Diff(M)ofM, defines an orbit space

(not necessarily a manifold) T = T (M),

T¼

1

½3

which, when M is of negative Yamabe type, we

define as the Teichmu¨ ller space of conformal

structures on M.

In two dimensions in the case of a higher genus

manifold S

, p  2, this construction leads precisely

to the conventional Teichmu¨ ller space, as discussed

by Fischer and Tromba (1984). In this case the

resulting Teichmu¨ ller space

¼TðS

Þ¼

1

ðS

 R

6p6

½4

is then a manifold diffeomorphic to R

6p6

, which

then plays the role of the natural reduced configu-

ration space for the Einstein equations in (2 þ 1)

dimensions. Moreover, these constructions can be

carried out globally using known global cross

sections for the D

) action on M

1

). These

global cross sections can then be used to provide an

explicit model for the Teichmu¨ ller space T

as a

finite-dimensional subspace of M

1

For n = 3, T = T (M) plays the analogous role for

the reduced field equations in (3 þ 1) dimensions.

Moreover, for many 3-manifolds it is possible to

show that T is itself an infinite-dimensional con-

tractible manifold, rather than something more

general such as an orbifold or a stratified union of

manifolds. For technical simplicity, we shall assume

throughout this article that T is a manifold. Our

results remain valid in the more general case but in

that case one must work on stratified spaces (see

Fischer (1970) for results on the structure of orbit

spaces when they are not manifolds).

For higher-dimensional manifolds there is no

analog of the Thurston geometrization program.

Indeed, it is known that the set of closed n-

manifolds for n  4 is so rich that no purely

algebraic classification is possible. Nevertheless, for

manifolds of negative Yamabe type, every Rieman-

nian metric g is still uniquely conformal to a metric

 2M

1

so that the orbit space T = M

1

still

represents the Teichmu¨ ller space of conformal

equivalence classes on M. However, in these

higher-dimensional cases, very little is known

about the structure of T .

The Field Equations

Relative to a global time coordinate t = x

and local

spatial coordinates (x

, ..., x

) on a connected

closed oriented n-manifold M, one can express the

line element of an arbitrary (n þ 1)-Lorentzian

metric with signature (þþ)(n positive signs)

in the form

ðnþ1Þ







¼N

þ g

ðdx

þ X

dtÞðdx

þ X

dtÞ½5

where

(nþ1)



denotes the components of the space-

time metric, 0  ,   n, where the Riemannian

metric g with components g

is the first fundamental

form induced on each t = constant hypersurface,

where the time-dependent positive function

N = N(x, t) > 0 is referred to as the lapse function,

and where the time-dependent spatial vector field

X = X(x, t) with components X

(nþ1)

, where

denotes the inverse of the spatial metric g

,is

referred to as the shift vector field.

Let ‘ denote the dimension length. In this article

we use the convention that the spatial coordinates

, ..., x

) are always dimensionless, but the time

coordinate t may have a dimension (see [19] and

[36]). Since the line element ds

[5] has dimension ‘

and the spatial coordinates are dimensionless, the

physical spatial metric coefficients g

also have

dimension ‘

. If the time coordinate t has a

dimension, then the dimension of the lapse function

N is such that the quantity Ndt has dimension ‘ and

the dimension of the shift vector field X is such that

the quantity Xdt is dimensionless.

We now briefly consider the canonical formula-

tion of Einstein’s equations. For more information

regarding this formulation, see Arnowitt, Deser, and

Misner (1962) (ADM) or Fischer and Marsden

(1972) for a global perspective. We remark that

the canonical formulation of gravity itself is local

and is valid for any spatial topology of M. However,

as we shall see, Hamiltonian reduction of gravity

along the lines described in this article requires the

topological restriction that M be of negative

Yamabe type.

The standard definition of the second fundamen-

tal form k, or extrinsic curvature, induced on a

t = constant hypersurface leads to the coordinate

formula

¼

 X

ijj

 X

jji



½6

where the vertical bar signifies covariant differentia-

tion with respect to the spatial metric g and spatial

indices are raised and lowered using this metric. The

Hamiltonian Reduction of Einstein’s Equations 611

natural momentum variable conjugate to g turns out

to be the 2-contravariant symmetric tensor density 

(that is,  is a relative tensor of weight 1) whose

components in a positively oriented local coordinate

chart (x

, ..., x

), that is, in a chart in the orienta-

tion atlas of M, are given by



¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

ðtr

kÞg



½7

where k

= g

is the contravariant form of k,

and where

 ¼ ðg; kÞ¼tr

k ¼ g

½8

is the trace of the second fundamental form, or the

mean (extrinsic) curvature. From the coordinate

formula [6] for the extrinsic curvature, we see that

the components k

have dimension ‘

1

‘

= ‘ and

thus the mean curvature  = tr

k = g

has the

dimension ‘

2

‘ = ‘

1

Let

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

denote the (global) scalar density and

d

denote the (global) Riemannian measure on

M determined by the Riemannian metric g (note

that here d is not the exterior derivative). Similarly,

let 

denote the volume element, a nonvanishing

n-form on M, determined by g and the orientation

on M. In a positively oriented local coordinate chart

) = (x

,...,x

)onM,(

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

(d

)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

dx

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

x, where

x = dx

dx

is the Lebesque measure in R

and (

)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

^dx

^^dx

.Weadopt

the convention of suppressing the coordinate-chart

designation (x

) so that one can, for example, write

with some ambiguity

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

= (

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

)

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detg

We let

volðM; gÞ¼



d

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

x ½9

denote the volume of the Riemannian manifold

(M, g), given by either the integral of the volume

n-form 

or the Riemannian measure d

over M,

which is given in the last integral in its coordinate

form using the suppressed coordinate-chart conven-

tion adopted above. As expected, the spatial

physical volume has dimension (‘

)

n=2

= ‘

We shall refer to the canonical variables (g

, 

)as

the physical variables, in contrast to the reduced or

conformal variables (

,(p

)

) to be introduced

later.

Note that the mean curvature  = tr

k is a scalar

function on M whereas tr

 is a scalar density on M.

Taking the trace of [7] expresses the mean curvature

in terms of the canonical variables (g, ),

 ¼ ðg;Þ¼tr

k ¼

ðn  1Þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

 ½10

Using [10], eqn [7] can be inverted to give k in terms

of g and ,

¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g





ðn  1Þ

ðtr

Þg



½11

and then combined with [6] to give the kinematical

equation

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g





ðn  1Þ

ðtr

Þg



þ X

ijj

þ X

jji

½12

In terms of the canonical variables (g, ), a

Hamiltonian form for the action for Einstein’s

vacuum field equations can be expressed as

ADM

ðg;Þ¼





 NHðg;Þ

 X

ðg;Þ



x ½13

where I = [t

, t

]  R is a closed interval and where

the Hamiltonian (scalar) density H(g, ) and the

momentum (1-form) density J(g, ) are given by

Hðg;Þ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

   

n  1

ðtr

Þ





ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

RðgÞ

½14

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g





n  1

ðg







ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

RðgÞ½15

ðg;Þ¼2ð

Þ

¼2g



½16

where    is the g-metric contraction of  with itself,

and where, as above, R(g) is the scalar curvature of

the spatial metric. We also note that each of the three

terms in the integrand of [13] are global scalar

densities and thus can be integrated over M without

any further involvement of the metric g.

Variation of I

ADM

with respect to the lapse

function and shift vector field yields the constraint

equations

Hðg;Þ¼0 ½17

ðg;Þ¼0 ½18

which comprise that subset of the empty space

(n þ 1)-Einstein field equations corresponding to the

normal–normal and normal–tangential projections

of the Einstein tensor relative to a t = constant initial

hypersurface. Variation of I

ADM

with respect to 

reproduces the kinematical equation [12], whereas

612 Hamiltonian Reduction of Einstein’s Equations

variation of I

ADM

with respect to g

generates the

complementary tangential–tangential projections of

Einstein’s equations.

There are no evolution or constraint equations for

either the lapse function N or the shift vector field X

and therefore these quantities must be fixed by either

externally imposed or implicitly defined gauge condi-

tions. A convenient choice, for which a local existence

and well-posedness theorem for the corresponding

field equations can be established in any dimension

n  2, is given indirectly by imposing constancy of the

mean curvature and a spatial harmonic gauge condi-

tion on each t = constant slice (see Andersson and

Moncrief (2003, 2004)). These constant mean curva-

ture spatial harmonic (CMCSH) gauge conditions are

given, respectively, by the equations

t ¼  ½19

ð

ðgÞ

gÞÞ ¼ 0 ½20

where from [10],  is a function of the canonical

variables (g, ) and where

g is some convenient fixed

spatial reference metric (or background metric) on

M. The latter condition corresponds to the require-

ment that the identity map between the Riemannian

manifolds (M, g) and (M,

g) be harmonic. Neither of

these conditions involves the lapse function or shift

vector field directly but their preservation in time

implemented by the demand that the time deriva-

tives of the given conditions be enforced leads

immediately to a linear elliptic system for (N, X

)

which determines these variables. The foregoing

formalism is easily extended to the nonvacuum

field equations in the presence of suitable material

sources whose field equations are amenable to a

constrained Hamiltonian treatment. To simplify the

analysis, such sources will be ignored in the present

discussion.

For the special case of Einstein gravity in (2 þ 1)

dimensions, there is an elegant, alternative, triad-

based formulation of the action functional as an

Isom(R

)-invariant gauge-theoretic Chern–Simons

action, where Isom(R

) denotes the full isometry

group, or the Poincare´ group (= the inhomogeneous

Lorentz group), of (2 þ 1)-Minkowski space R

. For

nondegenerate triads the resulting field equations for

this alternative formulation can easily be shown to

be equivalent to those of the conventional formalism

when the latter is re-expressed in terms of triads but

the new formulation allows for meaningful field

equations in the case of degenerate triads as well

and thus suggests a potentially interesting general-

ization of the theory (see Carlip (1998) for details).

In any dimension n  2, there is a well-known

technique, pioneered by Lichnerowicz (1955), for

solving the constraint equations on a constant mean

curvature (CMC) hypersurface (see Choquet-Bruhat

and York (1980) and Isenberg (1995)). Of major

importance for the treatment of Hamiltonian reduc-

tion is that if n = 2andM = S

, p  2, or if n  3

and M is of negative Yamabe type, then every

Riemannian metric g on M is uniquely globally

pointwise conformal to a metric  which satisfies

R() = 1 (see remark above [2]). Thus, from now

on, we assume this topological condition on M.In

this case, every Riemannian metric g on M can be

uniquely expressed as

g ¼

2’

 if n ¼ 2 and M ¼ S

; p  2

’

4=ðn2Þ



if n  3 and M is of

negative Yamabe type

½21

with the conformal metric  normalized so that

R() = 1 and with the specific form of the

coefficient conformal factor being chosen to simplify

calculations involving the curvature tensors. In

the case n  3, ’ is positive and thus the space of

all Riemannian metrics on M is parametrized by

1

and the space of scalar functions ’>0onM.

The function ’ is then determined by solving the

Hamiltonian constraint [17] (see also the remark

before [33]).

In the given CMC slicing and imposing the

vacuum field equations, since by the momentum

constraint  must have zero divergence (see [16]

and [18]), one finds that 

must be expressible in

the form



¼ 



ðtr

Þg

½22

where 

is transverse (i.e., divergence-free) and

traceless with respect to g. In the nonvacuum case, 

picks up an additional summand determined by the

sources in the modified momentum constraint [18].

Substitution of the foregoing decompositions of

, 

) into the Hamiltonian constraint leads to a

nonlinear elliptic equation for ’ which, under the

conditions assumed here, determines this function

uniquely, provided  6¼ 0. No solutions exist for

 = 0 (equivalently, tr

 = 0) since from [14], [17],

and [22], the Hamiltonian constraint would then

immediately imply that

RðgÞ¼

det g

ð  Þ¼

det g



 



0 ½23

everywhere on M, which is not possible for a

manifold M of negative Yamabe type. Instantaneous

vanishing of the mean curvature, the defining

property of a maximal hypersurface, would corre-

spond to a moment at which an expanding universe

Hamiltonian Reduction of Einstein’s Equations 613

ceases to expand or a collapsing universe ceases to

collapse. From [23], such behavior is topologically

excluded here by the requirement that M be of negative

Yamabe type (see also the discussion after [36]).

In the unreduced formalism of I

ADM

, the role of a

super-Hamiltonian is played by the functional

super

ðg;Þ¼

ðNHðg;ÞþX

ðg;ÞÞd

x ½24

which evidently vanishes whenever the constraints

are satisfied. To achieve a fully reduced formulation

wherein again the effective Hamiltonian would

vanish, one could endeavor to solve the associated

Hamilton–Jacobi equations

Hðg

;S=g

Þ¼0 ½25

ðg

;S= g

Þ¼0 ½26

for a real-valued functional S = S(g, 

) of the metric

g and a set of additional independent parameters 

A complete solution S(g

, 

) would be one for

which an arbitrary solution (g

, 

) of the con-

straints could be realized as (g

, S=g

) for a

suitable (unique) choice of the 

. A complementary

set of reduced canonical variables 

(the momenta

conjugate to the 

’s) could then be defined by



= S=

and one could in principle solve the

equations



S

g

½27



S



½28

for (

, 

) as functionals of the canonical variables

, 

). This procedure, if it could be carried out,

would ensure that these functionals (

(g, ),



(g, )) Poisson-commute with all of the con-

straints and hence are conserved for an arbitrary

slicing of spacetime. Conversely, if a suitable set of

gauge conditions such as the CMCSH conditions

were imposed, one could in principle solve for the

remaining independent canonical variables as func-

tionals of the (

, 

) and an internal variable, such

as the mean curvature , which plays the role of

time, and hence solve the field equations for (g

, 

)

in the chosen gauge.

This proposal is purely heuristic in (3 þ 1) and

higher dimensions in that there is no known

procedure for finding the needed complete solution

of the Hamilton–Jacobi equations in these cases.

However, by exploiting the Chern–Simons analogy

discussed earlier in this section, a complete solution

can be found in (2 þ 1) dimensions and the

corresponding complete set of ‘‘observables’’

(

, 

) identified. The latter are equivalent, up to

a diffeomorphism of the associated reduced phase

space, to a complete set of traces of holonomies of

the flat Isom(R

)-connections defined in this Chern–

Simons formulation (see Carlip (1998) for more

details).

The Reduced Hamiltonian

We continue with the assumption that M is a

connected closed oriented n-manifold, with either

n = 2 and M = S

, p  2, or n  3 and M of negative

Yamabe type. We now define the reduced phase

space as the set of conformal variables given by

reduced

¼fð; p

Þj 2M

1

and p

is a

2-contravariant symmetric tensor density

that is transverse and traceless

with respect to g½29

We remark that the fully reduced phase space is

given by P

reduced

, where D

is the group of

diffeomorphisms of M isotopic to the identity.

However, here, for clarity of exposition, we work

on P

reduced

rather than the fully reduced phase space.

Given a scalar function ’, with ’>0ifn  3, the

physical variables (g, 

) are related to the con-

formal variables (, p

)by

g;



2’

; e

2’



if n ¼ 2

’

4=ðn2Þ

; ’

4=ðn2Þ



if n  3

(

½30

We adopt the convention that raising and low-

ering of indices on either momentum variable 

will be with respect to its own conjugate metric,

either g or , respectively. With this convention, the

mixed forms of 

and p

are equal, since for

n  3,

ð

¼ g



TTil

¼ ’

4=ðn2Þ



’

4=ðn2Þ

TTil

¼ 

TTil

¼ðp

½31

(and similarly for the n = 2 case). Thus the squared

norms of p

and 

are equal,

 p

¼ 



TTij

TTkl

¼ g



TTij



TTkl

¼ 

 

½32

where in the first term the center dot is -metric

contraction and in the last term the center dot is

g-metric contraction.

The uniquely determined scalar factor ’ relating

the physical metric g to the conformal metric  is

obtained by solving the Hamiltonian constraint

614 Hamiltonian Reduction of Einstein’s Equations

equation [17]. In the special case that p

= 0 (or

equivalently, from [30], that 

= 0), ’ is constant

and is given in the n  3 case by

’ ¼

ðn  1Þ



ðn2Þ=4

½33

Thus in this case

 ¼ ’

4=ðn2Þ

g ¼

ðn  1Þ



g ½34

In particular, since  has the dimension ‘

1

(see the

remark after [8]) and the components g

have

the dimension ‘

, we see from this formula that the

conformal metric 

is dimensionless. Although ’ is

not constant in the general case when p

6¼ 0, its

dimension, as in [33], is still ‘

(n2)=2

and thus the

components 

are still dimensionless in the general

case. Since in the conventions used in this article, the

spatial coordinates are dimensionless, the volume

vol(M, ) of the Riemannian manifold (M, ), as well

as all curvature tensors of , are also dimensionless.

Having a dimensionless conformal metric  with a

dimensionless volume has its advantages over the

physical metric g with dimension ‘

inasmuch, as we

shall see below, an infimum of the volume of the

conformal metric is related to a dimensionless

topological invariant of M (see [48] and the remark

thereafter).

If one now uses the conformal variables given by

[30] and the decomposition [22] in the ADM action

given by [13], one finds the reduced action to be

reduced

TTij

@



2ðn  1Þ

@

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g



@tr





x ½35

In this expression one can discard the final time

derivative which contributes only a boundary

integral and so does not contribute to the equations

of motion. Moreover, the conformal metric 

constrained to lie in the intersection of M

1

and a

slice for the action of D

on M

1

. This space can be

regarded as a local chart for the reduced configura-

tion space T = M

1

, under the technical

assumption that T is a manifold. Thus, taken

together, the conformal variables (

, p

TTij

) can be

viewed as local canonical coordinates for the

cotangent bundle T



T of Teichmu¨ ller space T ,

where T



T now plays the role of the reduced

phase space.

For n = 2, these constructions can be carried out

globally for the Teichmu¨ ller space T

of an arbitrary

closed oriented surface S

, p  2 (see the remarks

after [4]). Using these global constructions, the

reduced phase space T



for the (2 þ 1)-reduced

Einstein equations can be modeled explicitly.

Having restricted the slices to be CMC, one need

only choose the relationship between the time

coordinate and the CMC  in order to fix a

corresponding reduced Hamiltonian. The most

natural choice of time coordinate from the present

point of view is to take

t ¼ tðÞ¼

nðÞ

n1

½36

Note that this choice of time coordinate, although

also denoted t, is no longer dimensionless but has

dimension ‘

n1

This choice of time coordinate is motivated by

three considerations. Firstly, we remark that since

 = 0 is excluded in the setting used in this article

(see [23] and the discussion after),  can range in

either the domain R



= (1,0) or R

= (0, 1). The

usual convention on the sign of k, as adopted here,

is that the sign of k is negative when the tips of the

normals on a spacelike hypersurface are further

apart than their bases, as for example in the

expansion of a model universe, in which case

 = tr

k < 0. Thus, with this convention,  in the

range R



corresponds to an expanding universe and

 in the range R

corresponds to a collapsing one in

the future direction of increasing t. Thus for

manifolds of negative Yamabe type that we consider

here, the expected maximal range of the CMC  is R



for which  !1 corresponds to a ‘‘crushing

singular’’ big bang of vanishing spatial volume and

 ! 0



corresponds to the limit of infinite volume

expansion. Then, with the time function given by [36],

the coordinate time t ranges in the interval R

vanishes at the big bang, and tends to positive infinity

in the limit of infinite cosmological expansion.

We remark that to prove that a solution deter-

mined by Cauchy data prescribed at some initial

coordinate time t

2 R

actually exhausts the range

is a difficult global existence problem that is not

dealt with here. Nevertheless, one of the main

motivations for this work is the hope that Hamil-

tonian reduction will lead to advances in the study

of the global existence question for Einstein’s

equations.

We also remark that with the choice of temporal

gauge function given by [36] and with  in its

natural range R



d

2ðn  1Þ

ðÞ

> 0 ½37

so that this temporal coordinate choice preserves the

time orientation of the flow for all n  2.

Hamiltonian Reduction of Einstein’s Equations 615