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

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Secondly, with this choice of temporal gauge, the

reduced action given by [35] simplifies to

reduced



TTij

@

ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g



x ½38

from which one can read off an effective reduced

Hamiltonian density,

reduced

ð; ;p

Þ¼ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

½39

and an effective reduced Hamiltonian,

reduced

ð; ; p

Þ¼

ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

¼ðÞ

d

¼ðÞ

volðM; gÞ½40

where vol(M, g) =

d

is the volume of the

Riemannian manifold (M, g). Thus in terms of the

physical variables (g

, 

), the reduced Hamiltonian

reduced

at ‘‘time’’  is simply the volume of the CMC

slice with mean curvature  rescaled by the factor

()

. With this reduced Hamiltonian density, the

reduced action [38] takes the canonical form

reduced

TTij

@

H

reduced



x ½41

As the third consideration for the given choice of the

time function, we note that rescaling the physical

volume vol(M, g) by the factor ()

yields a dimen-

sionless quantity. Indeed, as we have seen, the spatial

physical volume has the dimension ‘

and the constant

mean curvature  has the dimension ‘

1

,sothatthe

reduced Hamiltonian ()

vol(M, g) is dimensionless.

The main advantage of having a dimensionless

reduced Hamiltonian is that only such a reduced

Hamiltonian can have a topological significance,

and indeed, the infimum of H

reduced

is closely related

to a dimensionless topological invariant of M (see

the remarks after [48]).

In terms of the conformal variables (, p

), the

reduced Hamiltonian is found from [21] and [40] to

be given for n  3by

reduced

ð; ; p

Þ¼ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

¼ðÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

detð’

4=ðn2Þ

Þ

¼ðÞ

ð’

4=ðn2Þ

n=2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det 

¼ðÞ

’

2n=ðn2Þ

d



½42

where d



is the Riemannian measure on M

determined by  (locally, d



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det 

x)and

’ = ’(, , p

) is the conformal factor which,

through the solution of the Hamiltonian constraint

[17], is expressed as a function of the ‘‘time’’ t and

the independent conformal (or canonical) variables

(, p

In the special case n = 2, M = S

, p  2, a simple

formula for H

reduced

can be derived. In terms of the

conformal variables (, p

), we find from [40],

[10], [14], [17], [21], [22], and [32] that

reduced

ð;;p



ðÞ

d

¼2





ðdetgÞ

1







RðgÞ



d

¼2



ðdetðe

2’

ÞÞ

1

p



d

ðe

2’

Þ

2



RðgÞd

¼2



ðe

2’

2

ðdetÞ

1

p



2’

d



8ðS

¼2



2’

ðdet Þ

1

p



d



þ16ðp 1Þ½43

where ’=’(, ,p

), (S

)=2(1 p) is the Euler

characteristic of the genus p surface S

, and where

we have used the Gauss–Bonnet theorem



RðgÞd

¼ 4ð S

Þ¼8ð1  pÞ½44

Since

reduced

ð; ; p

¼ 2



2’

ðdet Þ

1

ðp

 p

Þd



þ 16ðp  1Þ16ðp  1Þ½45

the infimum of H

reduced

is attained precisely when

= 0 and this infimum coincides with the topo-

logical invariant 8(S

) = 16(p  1), which char-

acterizes the surface S

(see also [51] below). As we

shall see shortly, an analogous result holds for

n  3.

A straightforward but lengthy calculation, which

is valid in arbitrary dimensions, shows that the

reduced Hamiltonian is strictly monotonically

decreasing in the direction of cosmological expan-

sion except for a family of continuously self-similar

spacetimes for which this Hamiltonian is constant

(Fischer and Moncrief 2002b). The latter solutions

exist if and only if M admits a Riemannian metric

 2M

1

which is an Einstein metric, that is, for

which the Ricci tensor satisfies Ric() = (1=n).

Using the mean curvature as a convenient time

coordinate, that is, temporarily taking t = , the

616 Hamiltonian Reduction of Einstein’s Equations

corresponding self-similar vacuum spacetime metrics

then have the line element

¼





d

ðn  1Þ





½46

In the case that n = 3, the Einstein metric  is

actually hyperbolic with constant sectional curvature

K() = 1=6 and Ricci curvature Ric() = (1=3).

Although the conformal variables (, p

) = (,0) are

static in this model, the physical variables (g, )are

not. In this case, the resulting spacetimes (which

depend on the underlying topology of M) have

expanding closed hyperbolic spacelike hypersurfaces

where the physical volume vol(M, g) ‘‘starts’’ at zero

at the big bang and expands to infinity in the forward

time direction, as befits a universe endlessly expand-

ing from the big bang. Such a universe is depicted in

Figure 1, where the genus-2 surface is used to

represent a generic closed hyperbolic 3-manifold.

The Bianchi and Thurston types of this model are

discussed in the next section.

The line element [46] is locally isometric to the

vacuum Friedmann–Lemaıˆtre–Robertson–Walker

(FLRW) k = 1 spacetime, which is well known to

be flat. Although these spatially compactified mod-

els are technically not classical FLRW spacetimes

since the expanding compact hypersurfaces are not

homogeneous (and thus not isotropic), they are

Lorentz-covered by the FLRW k = 1 spacetime

and thus are locally isometric to this classical

spacetime.

The same result leading to [46] holds even if

matter sources are allowed, provided they satisfy a

suitable energy condition, in which case the corre-

sponding reduced Hamiltonian will only be station-

ary in the vacuum limit and then only when the

metric is of the above type; otherwise it mono-

tonically decays. This result even has a quasilocal

generalization expressible in terms of the corre-

sponding quasilocal reduced Hamiltonian defined

for an arbitrary domain D



within the CMC slice

 = constant by restricting H

reduced

in [42] to the

domain D



, so that for n  3,



ð; ; p

Þ¼ðÞ



d

¼ðÞ



’

2n=ðn2Þ

d



½47

If D



is determined from its specification on some

initial slice  = 

, by letting the domain flow along

the normal trajectories of the CMC foliation, one

can then verify that H



is monotonically decreasing

except for the vacuum solutions of self-similar type

described above, in which case H



is constant. This

result is independent of the initial domain chosen.

We remark that one cannot use the quasilocal

Hamiltonian to get equations of motion (even

quasilocally) since the full true Hamiltonian is

nonlocal and so one gets contributions from the

whole manifold.

Since the reduced Hamiltonian H

reduced

as well as

its quasilocal variant H



is monotonically decreasing

for generic solutions of Einstein’s equations, it is

natural to ask what its infimum is and whether this

infimum is ever attained, at least asymptotically, by

solutions of the field equations. The infimum of the

reduced Hamiltonian for n  3andforaspatial

manifold M of negative Yamabe type can be character-

ized in terms of a certain topological invariant of M

called the sigma constant (M)ofM. For manifolds of

negative Yamabe type, this quantity can be defined in

terms of the infimum of the volume of all metrics which

range over the space of conformal metrics M

1

.The

precise definition leads to the formula

ðMÞ¼ inf

2M

1

volðM;Þ



2=n

½48

Interestingly, this equation defines the topological

invariant (M) by a purely geometrical equation

involving the volume functional restricted to M

1

We also remark that [48] is a dimensionless

equation, the left-hand side being dimensionless

since it is a topological invariant of M and the

right-hand side being dimensionless since the con-

formal metric and its volume are dimensionless (see

the remarks after [34]).

Although the -constant can be defined for all

Yamabe types, [48] holds only for manifolds of

negative Yamabe type. From this equation, one can

conclude that for such manifolds

ðMÞ0 ½ 49

Figure 1 Expansion of the physical universe in the Bianchi V,

Thurston type H

, spatially compactified FLRW flat spacetime

cosmology.

Hamiltonian Reduction of Einstein’s Equations 617

One can relate the foregoing to the reduced

Hamiltonian by showing that the infimum of

reduced

defined for arbitrary <0 as a functional

on the reduced phase space



T¼T



1



½50

is given by

inf

ð;p

Þ2T



reduced

ð;;p

8ðS

Þ¼16ðp 1Þ



n 1

ððMÞÞ



n=2

if n ¼2

and p 2

if n 3

½51

where for n 3, M is of negative Yamabe type and

thus (M) 0 (see [49]).

One proves this result by first showing that within

an arbitrary fiber of the cotangent bundle



1

), one minimizes H

reduced

by setting the

fiber variable p

to zero. In this case, the solution for

the conformal factor ’ reduces to a spatial constant

which is a function of  alone (see [33]), and thus the

formula for H

reduced

given in [42] reduces to

reduced

ð; ;0Þ¼

n  1



n=2

volðM;Þ½52

The infimum over all conformal metrics  2M

1

this latter functional yields the -constant as outlined

above. If matter sources obeying a suitable energy

condition are allowed, the argument goes through in

much the same way with the additional implication

that the infimum is achieved only for a vacuum

solution so that in fact the matter must be ‘‘turned off.’’

Thus, as a consequence of the above analysis,

one has

reduced

ð;; p

 H

reduced

ð;; 0Þ¼

n  1



n=2

volðM;Þ



n  1



n=2

inf



1

vol M;

ðÞ

n  1

ððMÞÞ



n=2

½53

where the last equality follows by inverting [48] to give

inf

2M

1

volðM;Þ¼ððMÞÞ

n=2

½54

Moreover, if  2M

1

actually achieves the

-constant, that is, if vol(M, ) = ((M))

n=2

(and

not just asymptotically approaches it as a curve or

sequence), then  must be an Einstein metric with

RicðÞ¼

 ½55

If, additionally, n = 3, then  must be hyperbolic

(with constant sectional curvature K() = 1=6).

Although Thurston’s conjectures do not refer to

the -constant, Anderson (1997) has been able to

reformulate and somewhat refine the Thurston

geometrization conjectures for 3-manifolds of arbi-

trary Yamabe type in terms of conjectured proper-

ties of the -constant. Additionally, if Perelman’s

results are technically complete, they would provide

a proof of Anderson’s conjectures as well as those of

Thurston’s (see Anderson (2003)).

The conjectured behavior for a sequence of

conformal metrics {

}, 

1

, i = 1, 2, ..., which

seeks to minimize the volume of a stand-alone

K(p, 1) 3-manifold M of negative Yamabe type can

be described as follows:

1. If M is hyperbolizable, then (M) < 0 is attained

by a hyperbolic metric 

1

, unique up to

diffeomorphism, and the sequence of conformal

metrics {

} converges to this metric in a suitable

function space topology.

2. If M is a pure graph manifold, then (M) = 0and

the sequence {

} of conformal metrics ‘‘volume

collapses’’ M with bounded curvature. Typically

this occurs through collapse of circular or toroidal

fibers in the associated circle or 2-torus bundle

structure (see examples 3, 4, and 5 in the section

‘‘Topological Background’’ and see also the penul-

timate section). The six manifolds of flat type are

not included here as they are of zero Yamabe type.

3. If M is a generic K(p, 1)-manifold (not of type 1

or 2 above), then M can be decomposed along

incompressible tori into its final finite-volume-

type hyperbolizable and (possibly empty set of)

graph-manifold pieces. In this case, (M)< 0 and

the sequence {

} of conformal metrics collapses

the graph-manifold components and converges to

finite-volume complete hyperbolic metrics on the

hyperbolizable components (normalized to have

R() = 1) yielding a -constant that is entirely

determined by the volumes of these final hyper-

bolic components (see the final section).

We shall return to this conjectured characteriza-

tion of sequences of conformal metrics in the next

two sections.

Reduction of Bianchi Models

and Conformal Volume Collapse

For manifolds of negative Yamabe type, the strict

monotonic decay of H

reduced

in the direction of

cosmological expansion along nonconstant integral

curves of the reduced Einstein equations suggests

618 Hamiltonian Reduction of Einstein’s Equations

that the reduced Hamiltonian is seeking to achieve

its infimum inf H

reduced

= ((n=(n  1))((M)))

n=2

But does this ever happen? Does the reduced

Einstein flow of the conformal geometry asymptoti-

cally approach inf H

reduced

in the limit of infinite

cosmological expansion?

To answer this question, one can consider for n = 3

known locally homogeneous vacuum solutions of

Einstein’s equations which spatially compactify to

manifolds of negative Yamabe type. Applying the

theory of Hamiltonian reduction to these classical

models, one can show that the reduced Hamiltonian

behaves as expected under the reduced Einstein flow

defined by these models. Since these models existed

long before this theory, it is somewhat satisfying to see

that they can be interpreted in terms of Hamiltonian

reduction and how, with this interpretation, new

properties of these classical solutions can be found.

Since H

reduced

is a strictly monotonically decreasing

function along nonconstant integral curves of the

reduced Einstein flow, it is expected that under certain

conditions, the reduced Hamiltonian is monotonically

seeking to decay to its infimum. Thus, it is of interest to

look at Hamiltonian reduction under the consequence

of the following two assumptions:

1. The reduced Einstein field equations give rise

to the existence of a positive semiglobal non-

constant solution ( (t), p

(t)) defined for all

t 2 (0, 1) (or equivalently, for all  2 (1, 0);

2. The reduced Hamiltonian strictly monotonically

decays to its infimum along nonconstant integral

curves,

reduced

ððtÞ;ðtÞ; p

ðtÞÞ

! inf H

reduced

as t !1 ½56

From [40] and [51], in terms of the physical

variables (g, ) (or (g, k)), [56] can be written

equivalently as



volðM;gÞ¼ðtr

kÞ

volðM;gÞ!

ððMÞÞ



3=2

as t! 1 ½ 57

As a consequence of these assumptions, it follows

from [53] that the conformal volume vol(M,) must

also decay to its infimum [54] (although not

necessarily monotonically),

volðM;ðtÞÞ! inf

2M

1

volðM;Þ

¼ððMÞÞ

3=2

as t ! 1 ½ 58

Now suppose that (M) = 0. A large class of

manifolds for which this is true are the graph

manifolds (and thus also the Seifert manifolds) of

negative Yamabe type since (M)  0 for graph

manifolds in general and since (M)  0 for mani-

folds of negative Yamabe type. In this case the curve

(t) 2M

1

of conformal metrics must necessarily

(conformally) volume collapse M in the direction of

cosmological expansion,

volðM;ðtÞÞ!ððMÞÞ

3=2

¼ 0ast ! 1 ½ 59

Consequently, the curve of conformal metrics (t)

must undergo some form of degeneration as its

volume collapses. The details of this metric degen-

eration are of importance and are discussed below.

Not all locally homogeneous vacuum Bianchi

models admit spatially compact quotients. Fortu-

nately, the general theory of which Bianchi models

admit spatially compact quotients has been worked

out in detail by Tanimoto, Koike, and Hosoya (see

Tanimoto et al. (1997) and the references therein).

These Bianchi models together with their corre-

sponding Thurston classification and typical exam-

ples of their closed quotient manifolds are listed in

Table 1, where ‘‘K–S’’ indicates ‘‘Kantowski–Sachs,’’

‘‘P,’’ ‘‘Z,’’ and ‘‘N’’ denote manifolds of Yamabe

type positive, zero, and negative, respectively (see

the section ‘‘Topological Background’’), ‘‘Seifert’’

means Seifert fibered, ‘‘Hyper’’ means hyperboliz-

able, ‘‘?’’ indicates ‘‘unknown, but conjectured to be

so,’’ and ‘‘manifold collapse’’ denotes the type of

collapse that the conformal manifold (M, (t)) goes

through as the conformal volume vol(M, (t))

collapses. We also remark that all of the manifolds

Table 1 Bianchi, Thurston, and Yamabe type of a connected closed oriented irreducible 3-manifold

Bianchi type Thurston type Typical examples Yamabe type -constant Manifold structure Manifold collapse

K–S S

 RS

 S

P >0 Seifert

IX S

Nontrivial S

-bundles over S

P >0 Seifert

I R

Z 0 Seifert

II Nil Nontrivial S

-bundles over T

N 0 Seifert Total

III H

 R S

 S

, p  2 N 0 Seifert Pancake

VIII

SL(2, R) Nontrivial S

-bundles over S

N 0 Seifert Pancake

Sol Nontrivial T

-bundles over S

N 0 Graph Barrel

V, VII

Closed hyperbolizable manifolds N < 0 ? Hyper None

Hamiltonian Reduction of Einstein’s Equations 619

listed in the ‘‘Typical examples’’ column are irredu-

cible with the exception of S

 S

, which is prime

but not irreducible. Also, in this column, p  2.

In this table, the eight Thurston types are grouped

into three sets according to their Yamabe type. The

first set of such Bianchi models are those that

spatially compactify to yield 3-manifolds of positive

Yamabe type which allow metrics with positive

constant scalar curvature, for example, Bianchi IX

models defined over spherical space forms. The

second set (consisting of one type) yields manifolds

of zero Yamabe type which allow zero scalar

curvature metrics but not constant positive scalar

curvature metrics, for example, Bianchi I models

defined over T

or one of the other five manifolds of

flat type finitely covered by T

. The third set (the

last five entries in Table 1 and the set of most

interest in this article) yields manifolds of negative

Yamabe type which do not allow metrics with zero

scalar curvature.

These latter models are the five Bianchi models of

types II, III, VIII, VI

, and V (and in part VII

which in turn correspond in Thurston’s classification

to manifolds of type Nil, H

 R,

SL(2, R ), Sol, and

, respectively.

In the first three cases, the models of Bianchi type

II, III, and VIII compactify to a nontrivial S

-bundle

over T

or to a trivial or nontrivial S

-bundle over

, p  2, respectively. Each of these spaces is Seifert

fibered. In the fourth case, the model of Bianchi type

compactifies to a nontrivial T

-bundle over S

which is an irreducible graph manifold. Since each

of these manifolds is also of negative Yamabe type,

in each of these four cases, as discussed in the

beginning of this section, (M) = 0. In the fifth case,

we consider vacuum Bianchi V metrics as well

as a special case of Bianchi type VII

which

compactify to an arbitrary closed oriented hyperbo-

lizable manifold M.

For these latter five Bianchi models that spatially

compactify to manifolds of negative Yamabe type,

one can consider the classical solutions from

the point of view of Hamiltonian reduction. The

starting point for this point of view is to use

explicitly known vacuum metrics for the

simplest ‘‘standard’’ metric forms, given, for exam-

ple, in Wainwright and Ellis (1997). One need not

consider all such possible spatially compact quoti-

ents, even though that would appear to be quite

feasible, but one need only consider some represen-

tative examples for each of the Bianchi types listed.

It can be shown by explicit calculation, using the

known solutions, that in the four nonhyperbolizable

cases where (M) = 0, each of the classical Bianchi

solutions gives rise to the existence of a positive

semiglobal nonconstant solution to the reduced Einstein

field equations and that along this solution, the reduced

Hamiltonian asymptotically approaches 0 under the

reduced Einstein flow, thereby confirming the expecta-

tion that the reduced Hamiltonian asymptotically

approaches its infimum ((3=2)((M)))

3=2

= 0. Thus

in these cases the reduced Einstein flow conformally

volume-collapses the 3-manifold.

The explicit calculations also show the details of

this collapse. In the second and third models of

Bianchi type III, Thurston type H

 R,and

Bianchi type VIII, Thurston type

SL(2, R), respec-

tively, the conformal metric degenerates along

embedded circular fibers and this metric degenera-

tion causes M to collapse to its base manifold S

p  2. Since the collapse is along one-dimensional

fibers and since the two-dimensional base mani-

fold S

does not collapse, we refer to this type of

collapse as pancake collapse (see Figure 2).

In the fourth model of Bianchi type VI

, Thurston

type Sol, the conformal metric degenerates along

embedded T

-fibers and this metric degeneration

causes M to collapse to its base manifold S

. Since

the collapse is along two-dimensional fibers and

since the one-dimensional base manifold S

does not

collapse, we refer to this type of collapse as barrel

collapse (see Figure 3).

In the first model of Bianchi type II, Thurston type

Nil, as in the second and third models, the

conformal metric degenerates along embedded cir-

cular fibers. Additionally, not only do the circular

fibers collapse but simultaneously the flat quotient

2-torus base manifold T

’ M=S

of M modulo its

circular fibers also collapses. Thus the metric

degeneration collapses M to a point, exhibiting a

t = 0

= ∞

Figure 2 Bianchi III, Thurston type H

 R, M = S

 S

pancake collapses to S

, p = 2. The conformal geometry starts

with an infinite S

-fiber at the big bang (t = 0

) and pancake

collapses with bounded curvature to S

at infinite cosmological

expansion (t !1).

620 Hamiltonian Reduction of Einstein’s Equations

case of total collapse. Thus these model universes

provide examples of nonflat almost-flat manifolds

that exhibit total collapse with bounded curvature.

Since the conformal geometries of these model

universes collapse to a point, they aptly deserve

their name Nil (see Figure 4).

Remarkably, in each of these four cases of

collapse, the collapse occurs with bounded curva-

ture, precisely as occurs in the totally different

setting of the Cheeger–Gromov theory of collapsing

Riemannian manifolds, recognized many years ago

to be of importance in the understanding of the

behavior of sequences of metrics with uniform

curvature bound (see Gromov (1999) for references

and Anderson (2004) for other applications of

Cheeger–Gromov theory to general relativity).

What is somewhat remarkable is that the above

cosmological models were constructed completely

independently of that setting and thus provide

naturally occurring cosmological models whose

closed spatial hypersurfaces undergo conformal

volume collapse and metric degeneration exactly as

occurs in the theory of collapsing Riemannian

manifolds.

Of course, this volume collapse and metric

degeneration only occur as described in the con-

formal variables. The physical variables behave

differently. Indeed, in contrast to the conformal

volume which collapses to zero in the first four cases

and is constant in the hyperbolizable case (see

below), the volume of the physical metric in all

five cases goes to infinity since the flow is

temporally oriented in the direction of infinite

cosmological expansion.

In the fifth case where M is hyperbolizable, (M)

is conjectured to be negative and to be determined

by the hyperbolic volume, (M) = (vol(M, 

))

2=3

of the hyperbolic conformal metric 

normalized so

that R(

) = 1. In this case, 

together with

= 0 is a fixed point for the reduced Einstein

flow so that trivially the conformal volume does

not collapse. Moreover, if (M) is determined by

the volume of 

, then the constant reduced

Hamiltonian also trivially achieves its infimum

reduced

(, 

,0)= ((3=2) ((M)))

3=2

= (3=2)

3=2

(vol

(M, 

)), again confirming the expectation for the

behavior of H

reduced

on these Bianchi models.

Note that for this static case, the physical

variables behave as described after [46] and as

shown in Figure 1. Also note that in contrast to

Figures 2–4 where the conformal geometry is

depicted, Figure 1 depicts the physical geometry.

Overall, in all five cases, subject in the hyper-

bolizable case to a hyperbolic metric realizing the

-constant, the reduced Hamiltonian asymptotically

approaches its -constant infimum along the flow

lines of the reduced Einstein system. In doing so, the

volumes of the conformal metrics either go to zero

(in the first four cases) or to the hyperbolic volume

(in the hyperbolic case). In all five cases, the

curvature of the conformal metrics is uniformly

bounded.

t = 0

t = ∞

Figure 3 Bianchi VI

, Thurston type Sol, nontrivial T

-bundle

over S

, barrel collapses to S

. The conformal geometry evolves

from a base manifold S

at the big bang (t = 0

). Instanta-

neously after the big bang, flat T

-fibers bloom out of the

collapsed S

state. The conformal metric then expands to a

maximum volume and then barrel collapses with bounded

curvature back to the base manifold S

at infinite physical

cosmological expansion ( t !1). The two facial 2-tori are flat

and are glued together by an orientation-reversing toral

automorphism so as to give a nontrivial T

-bundle over S

The gray-scale density grading along the tube also indicates the

nontriviality of the bundle.

t = 0

t = ∞

Figure 4 Bianchi II, Thurston type Nil, nontrivial S

-bundle

over T

, totally collapses to a point. The conformal geometry

evolves from a point at the big bang (t = 0

). Instantaneously

after the big bang, the full 3-manifold blooms out from that point.

The conformal geometry then evolves to a metric of maximum

volume and then totally collapses with bounded curvature back

to a point at infinite physical cosmological expansion (t !1).

The two 2-tori, represented here by doughnuts, are flat and

are glued together by an orientation-reversing toral automorph-

ism so as to give a nontrivial S

-bundle over T

Hamiltonian Reduction of Einstein’s Equations 621

Because the reduced Einstein field equations behave

as expected for the Bianchi models that we have

considered with spatially compactified manifolds

being either Seifert fibered, graph, or hyperbolizable,

it seems plausible that for a more complicated starting

manifold M, the reduced Einstein flow may induce a

decomposition of M into geometric pieces. Indeed,

Anderson’s conjectures (Anderson 1997) predict how

a sequence of geometries with bounded curvature

approaching (M) degenerate. Assuming these con-

jectures, the asymptotic behavior of large classes of

Einstein spacetimes may perhaps be characterized

rather explicitly in terms of the geometrization

program of 3-manifolds (see the next section).

Conversely, it is conceivable that the damped

hyperbolic system of equations defined by the reduced

Einstein flow (with its strictly monotonically decreas-

ing reduced Hamiltonian on nonconstant curves)

could be used to try to establish some form of the

geometrization conjectures for 3-manifolds, much like

the parabolic system of equations defined by Ricci

flow is currently being used. If such a program were to

be successful, it would amount to a spectacular

consequence of Einstein’s equations, implying as it

does that geometrization may actually occur in nature.

Possible Cosmological Applications

of the Reduced Hamiltonian

Astronomical observations strongly support the view

that in a sufficiently coarse-grained sense, the universe

is homogeneous and isotropic. Furthermore, it is

expanding at such a rate, relative to its observable

energy density, that it will continue to expand forever.

The simplest cosmological model consistent with these

properties and which has a vacuum limit is the k = 1

FLRW model. Spatially compactified variants of this

model are still locally homogeneous and isotropic even

though they are no longer globally so (see the

discussion after [46]). Evidence for one or another of

the infinitely many compactifications possible could be

sought in patterns of fluctuations of the cosmic

microwave background radiation and the detection

of such patterns could be strong evidence for a

spatially closed universe.

However, is one really justified in extrapolating

local observations of that portion of the universe

visible to astronomers to a conclusion about its

global topology? Could it be instead that there is a

dynamical reason, provided by Einstein’s equations,

for the observed fact that the universe seems to be

locally homogeneous and isotropic and in such a

state as to continue expanding forever?

Suppose for the sake of argument that the

universe has a more complicated topology, such as

that of one of the generic K(p, 1)-manifolds which

does not admit a locally homogeneous and isotropic

metric even though its hyperbolizable components

would each individually do so. A plausible scenario

suggested by the results in this article is that under

the Einstein evolution, the reduced Hamiltonian given

by [40] consisting of the rescaled spatial volume

becomes asymptotically dominated in the future

direction of cosmological expansion by the contribu-

tion of the hyperbolizable components. On each of

these components, the limiting conformal metric

approaches local homogeneity and isotropy with the

relative contribution of the graph-manifold constitu-

ents, if any are present, collapsing asymptotically to a

negligible fraction of the whole. The idea is that if

structure formation develops sufficiently late in the

evolution of such a universe, then it should occur, with

overwhelmingly high probability, in those regions

which dominate the conformal volume and admit an

asymptotically locally homogeneous and isotropic

metric of constant negative curvature, locally indis-

tinguishable from a k = 1 FLRW model.

One can speculate still further and imagine what

happens if the spatial topology is not of prime type

but rather consists of a connected sum of several

K(p, 1)’s together perhaps with nontrivial spherical

manifolds S

=G and handles S

 S

. Here it seems

conceivable, especially in view of the expected

tendency of spherical manifolds to ‘‘recollapse,’’ that

the evolving universe would develop pinch-off singu-

larities along the essential 2-spheres that separate the

individual prime factors. Such singularities might

occur in finite time between connected sums of

spherical recollapsing factors or in infinite time

between connected sums of K(p, 1)-factors. Similar

patterns of singularity formation are seen to occur in

Ricci flow and must be treated in the resolution of

the 3-manifold geometrization program.

Of course there is no proof of such behavior for the

full (3 þ 1)-dimensional Einstein gravity but for the

model problem of Einstein’s theory in (2 þ 1) dimen-

sions, something close to a proof of the analogous

conjecture is already at hand. In the vacuum case,

which can be described rather explicitly, one can

construct the generic solution for a higher genus

surface topology by cutting open the corresponding

k = 1 FLRW model and gluing in the so-called

Kazner wedges. These wedges play the role of the

graph-manifold constituents of a generic K(p, 1)-

manifold in three dimensions and evolve anisotropi-

cally. However, it is known rigorously in this case that

the rescaled spatial area H

reduced

(, , p

) =

()

Area(

, g) is asymptotically exhausted by the

FLRW components with the contribution from the flat

Kazner anisotropic pieces shrinking to zero in this

622 Hamiltonian Reduction of Einstein’s Equations

limit. If certain types of matter sources are included,

for example, those analogous to terms which result

from Kaluza–Klein reduction of vacuum gravity in

(3 þ 1)-dimensions, then a similar result can be proved

at least for sufficiently small but fully nonlinear

perturbations away from the vacuum backgrounds

(see Choquet-Bruhat (2004)).

In fully general (3 þ 1)-dimensional gravity, there

are few known topologically general results beyond

those mentioned earlier and the problem is compli-

cated by the presence of gravitational waves (which

are absent in (2 þ 1) dimensions) and the fact that

on such more general manifolds, there are no known

‘‘background’’ solutions to perturb about. However,

for the special case of (future) vacuum evolution on

a pure closed hyperbolizable manifold, one can show

that if the initial data is sufficiently close to that of an

FLRW model, then the fully nonlinear gravitational

perturbations eventually die out leaving a locally

homogeneous and isotropic model in the asymptotic

limit (see Andersson and Moncrief (2004)). It seems

likely that this result can be generalized to allow for

the inclusion of various types of matter sources as in

the (2 þ 1)-dimensional case.

Acknowledgments

This research was supported in part by NSF grant

PHY-0354391 to Yale University. We would also

like to thank the Institut des Hautes E

tudes

Scientifiques, the Max-Planck-Institut fu¨ r Gravita-

tionsphysik, Albert-Einstein-Institut, and The Erwin

Schro¨ dinger International Institute for Mathematical

Physics for their hospitality and support for several

periods during which this research was carried out.

See also: Computational Methods in General Relativity:

The Theory; Einstein Equations: Exact Solutions;

Einstein Equations: Initial Value Formulation; Einstein

Manifolds; General Relativity: Overview; Geometric

Analysis and General Relativity.

Further Reading

Anderson M (1997) Scalar curvature and geometrization con-

jectures for 3-manifolds. In: Grove K and Petersen P (eds.)

Comparison Geometry. Mathematical Science Research Insti-

tute Publication Cambridge: Cambridge University Press.

Anderson M (2003) Remarks on Perelman’s papers, http://

www.math.sunysb.edu/anderson/papers.html.

Anderson M (2004) Cheeger–Gromov theory and applications to

general relativity. In: Chrus

ciel PT and Friedrich H (eds.) The

Einstein Equations and the Large Scale Behavior of Gravita-

tional Fields (Cargese 2002), pp. 347–377. Basel: Birkha¨ user.

Andersson L and Moncrief V (2003) Elliptic–hyperbolic systems

and the Einstein equations. Annales Henri Poincare´ 4: 1–34.

Andersson L and Moncrief V (2004) Future complete vacuum

spacetimes. In: Chrus

ciel PT and Friedrich H (eds.) The

Einstein Equations and the Large Scale Behavior of Gravita-

tional Fields, pp. 299–330. Basel: Birkha¨ user.

Arnowitt R, Deser S, and Misner C (1962) The dynamics of

general relativity. In: Witten L (ed.) Gravitation: An Introduc-

tion to Current Research. New York: Wiley.

Cao H, Chow B, Chu S, and Yau ST (eds.) (2003) Collected

Papers on Ricci Flow, Series in Geometry and Topology, vol.

37. Somerville, MA: International Press.

Carlip S (1998) Quantum Gravity in 2 þ 1 Dimensions. Cambridge:

Cambridge University Press.

Choquet-Bruhat Y (2004) Future complete U(1) symmetric Einstei-

nian spacetimes, the unpolarized case. In: Chrus

ciel PT and

Friedrich H (eds.) The Einstein Equations and the Large Scale

Behavior of Gravitational Fields, pp. 251–298. Basel: Birkha¨user.

Choquet-Bruhat Y and York J (1980) The Cauchy problem.

In: Held A (ed.) General Relativity and Gravitation,vol.1,

New York: Plenum.

Fischer A (1970) The theory of superspace. In: Carmeli M, Fickler S,

and Witten L (eds.) Relativity. New York: Plenum.

Fischer A and Marsden J (1972) The Einstein equations of

evolution – a geometrical approach. Journal of Mathematical

Physics 28: 1–38.

Fischer A and Marsden J (1975) Deformations of the scalar

curvature. Duke Mathematical Journal 42: 519–547.

Fischer A and Moncrief V (2000) The reduced Hamiltonian of

general relativity and the -constant of conformal geometry.

In: Cotsakis S and Gibbons GW (eds.) Proceedings of the 2nd

Samos Meeting on Cosmology, Geometry and Relativity,

Mathematical and Quantum Aspects of Relativity and

Cosmology, Samos, August 31–September 4, 1998, Lecture

Notes in Physics, vol. 537, pp. 70–101. Berlin: Springer.

Fischer A and Moncrief V (2002a) Conformal volume collapse of

3-manifolds and the reduced Einstein flow. In: Holmes P,

Newton PK, and Weinstein A (eds.) Geometry, Mechanics,

and Dynamics: Volume in Honor of the 60th Birthday of

J. E. Marsden, pp. 463–522. New York: Springer.

Fischer A and Moncrief V (2002b) Hamiltonian reduction and

perturbations of continuously self-similar (n þ 1)-dimensional

Einstein vacuum spacetimes. Classical Quantum Gravity 19:

5557–5589.

Fischer A and Tromba A (1984) On a purely ‘‘Riemannian’’ proof of

the structure and dimension of the unramified moduli space of a

compact Riemann surface. Mathematische Annalen 267: 311–345.

Gromov M (1999) Metric Structures for Riemannian and Non-

Riemannian Spaces. Boston: Birkha¨user.

Hempel J (1976) 3-Manifolds, Annals of Mathematical Studies,

Number 86. Princeton: Princeton University Press.

Isenberg J (1995) Constant mean curvature solutions of the

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Quantum Gravity 12: 2249–2274.

Jaco W (1980) Lectures on three-manifold topology. In Con-

ference Board of the Mathematical Sciences, number 43,

Regional Conference Series in Mathematics. Providence, RI:

American Mathematical Society.

Lichnerowicz A (1955) The´ories Relativistes de la Gravitation

et de Electromagnetisme. Paris: Masson.

Tanimoto M, Koike T, and Hosoya A (1997) Hamiltonian

structures for compact homogeneous universes. Journal of

Mathematical Physics 38: 6560–6577.

Thurston W (1997) Three-Dimensional Geometry and Topology

vol. 1. Princeton: Princeton University Press.

Wainwright J and Ellis GFR (eds.) (1997) Dynamical Systems in

Cosmology. Cambridge: Cambridge University Press.

Hamiltonian Reduction of Einstein’s Equations 623

Hamiltonian Systems: Obstructions to Integrability

M Irigoyen, Universite

P.-M. Curie, Paris VI, Paris,

France

Introduction

In the study of differential systems, and particularly

of Hamiltonian differential equations, a fundamen-

tal problem is the question of their integrability.

Because there are different definitions of this notion,

a system which is integrable according to one

definition can be nonintegrable according to another

one. The notion of integrability is connected to the

existence of a sufficiently large number of first

integrals, which are linked to conservati on laws. For

a real analytic Hamiltonian system with n degrees of

freedom, the ‘‘complete integrability’’ means the

existence of n first integrals, which are functionally

independent, and ‘‘in involution,’’ in the entire phase

space. These integrals can be functions of class C

(r finite), C

, or analytic.

For the classical pro blems of Hamiltonian

mechanics which are integrable, their first integrals

can be continued into the complex domain of the

variables, as one-valued holomorphic, or mero-

morphic, functions of complex time. This fact leads

to the concept of ‘‘complex integrability’’ of a

system. Note that a real Hamiltonian system which

is integrable may be nonintegrable in the complex

domain, if the real first integrals cannot be con-

tinued as one-valued holomorphic functions of the

complex time.

Generally, the branchin g of solutions of a system,

as functions of complex time, is an obstruction to

the existence of one-valued first integrals. To study

this problem, one can, following Poincare´, expand

the solutions in convergent series of a small

parameter: this is the base of ‘‘perturbation meth-

ods,’’ and the main fact is that a small perturbation

of an integrable Hamiltonian system generally

destroys its integrability. Another metho d of proving

nonintegrability consists of studying the linearized

equations along a particular solution. This last

direction has been exploited recently, in particular,

through methods based on algebraic results inspired

by differential Galois theory.

Hamiltonian Systems and Mechanics

Let us consider a conservative holonomic real

dynamical system with n degrees of freedom: the

positions of this system are points of an

n-dimensional real manifold N (the state space or

configuration space) with local coordinates

, x

, ..., x

. If the velocities are denoted by

= dx

=dt, we consider the Lagrangian function L

associated to this system:

Lðx

; ...; x

;

; ...;

¼ Tðx

; ...; x

;

; ...;

ÞþVðx

; ...; x

where

x = (

, ...,

) is a tangent vector to the

manifold N at the point x = (x

, ..., x

). The kinetic

energy T(x,

x) is a positive-definite quadratic form in

, ...,

,andV(x) is the potential energy, whose

gradient determines the forces acting on the system.

The motions (x

(t), x

(t), ..., x

(t)) of the system

on the manifold N are the extremals of the action

integral:

L(x

, ..., x

, ...,

)dt (‘‘principle

of stationary action of Hamilton’’) and t hey a re

the solutions of the Euler–Lagrange system, which

consists in n differential equations of second order

for the coordinates x

, x

,..., x

(Whittaker 1904):





¼ 0; 1  i  n

This system can be written in the Hamiltonian

form: the Lagrangian L is a function defined on the

tangent bundle TN of the state space N,withlocal

coordinates x

, ..., x

, ...,

(i.e., an element of

TN consists in a point x of N, joint with a tangent

vector to N at x). Now, we consider the cotangent

bundle T



N: an element of T



N consists in a point x

of N joint with a cotangent vector to N at x,thatis,a

linear form defined in the tangent space to N at x.In

local coordinates, the components of this linear form

are y

, ..., y

, defined by: y

=@L=@

; y

,..., y

are

called the generalized momenta, or impulsions. x

and y

are called conjugate canonical variables.

The mapping from TN to T



N thus defined is the

Legendre transformation (Abraham and Marsden

1967). Through it, the Euler–Lagrange equations

become a system of 2n differential equations of first

order:

;

¼

; 1  i  n

where H(x

, ..., x

, y

, ..., y

) = T(x

, ..., x

, ..., y

)  V(x

, ..., x

H is the Hamiltonian function of this system. The

solutions of these differential equations are curves

on the 2n-dimensional manifold T



N, whose projec-

tions in the n-dimensional state manifold N coincide

with the solutions of the Lagrangian system. T



N is

624 Hamiltonian Systems: Obstructions to Integrability

called the phase sp ace of the system. The second

members of the differential system define a vector

field in the phase space.

Let M = T



N. On this 2n-dimensional manifold,

consider the standard symplectic form =

i = 1

^ dx

.Iff and g are C

-functions on M,

we define their Poisso n bracket {f , g} in local

coordinates by

f ; g

i¼1





It defines the space C

(M) as a Lie algebra over R.

Then, if H 2 C

(M) is the Hamiltonian fun ction

associated to a system, the corre sponding Hamilto-

nian equations can be written as the following 2n

‘‘canonical equations’’ (Arnol’d 1976):

¼ x

; H

;

¼ y

; H

¼

;

1  i  n ½1

AfunctionF 2 C

(M) is a (first) integral of eqns [1]

if it is constant along any solution of [1], that is, if it

verifies: {F , H} = 0. Thus, a first integral is a quantity

which is preserved along a solution (‘ ‘conservation

law’’). In particular, H itself is a first integral of the eqns

[1]. It represents the ‘‘total energy’ ’ of the system.

1. The simplest example of Hamiltonian system is

the harmoni c oscillator defined by the one degree of

freedom Hamiltonian:

Hðx; yÞ¼

It possesses the energy integral H. Thus, the trajectories

in the phase space R

(phase plane) are given by

þ y

= 2h, which are concentric circles if the

constant energy verifies h  0. The phase space R

foliated by these circles. The system is said to be

‘‘integrable.’’ Obviously, it is also possible to construct

Hamiltonian systems with n degrees of freedom

(n  1), by coupling n harmonic oscillators, with a

Hamiltonian defined by

Hðx

; ...; x

; y

; ...; y

Þ¼

i¼1

with n constant coefficients a

 0.

2. Another example of Hamiltonian system with one

degree of freedom is the simple mathematical pendu-

lum. The state coordinate is the angle  of the

pendulum with the vertical axis, defined modulo 2.

The phase space is: M = S

 R (x = (mod 2) 2

, y 2 R),that is, a cylinder. The Hamiltonian function

is: H(x, y) = (1=2) y

 cos x; H is a first integral of the

differential equations, the system is integrable and the

trajectories on the cylinder S

 R are defined by

(1=2)y

 cos x = h. According to the constant value

of h on each phase curve, the solutions are periodic

oscillations of the pendulum (if h  h

), periodic

solutions of rotation where the angle varies mono-

tonically with time (if h  h

), two equilibria (one

stable, one unstable) and solutions which ‘‘begin’’

when t !1at the unstable equilibrium and ‘‘finish’’

when t !þ1 at the same point (if h = h

): the

corresponding phase curves are called ‘‘separatrices.’’

3. The system of He´non– Heiles (He´non and Heiles

1964) is a system with two degrees of freedo m. The

phase space is R

 R

and the Hamiltonian is

defined by

Hðx

; x

; y

ðy

þ y

Þþ

ðx

þ x

Þþx

 x

where  is a real constant. This system is ‘‘integrable’’

for some isolated values of the parameter  (Ziglin

1983) and ‘‘nonintegrable’’ otherwise. Of course, it is

necessary to define the integrability of a Hamiltonian

system, although according to Poincare´: ‘‘A system of

differential equations is only more or less integrable.’’

Integrability of Hamiltonian Systems

Generally, if a differential system is of order p,itis

necessarytoknowp first integrals to integrate it. But if

the system is Hamiltonian of order 2n, only n first

integrals are sufficient to integrate it ‘‘by quadratures,’’

that is, by ‘‘algebraic’’ operations such as integrations

and inverting of functions. The reason is that the

existence of one first integral allows us to reduce the

order of the system by two: a system of order 2n with

one first integral can be reduced to order 2n –2.

Theorem of Liouville (see Arnol’d (1976)). Sup-

pose that F

, F

, ..., F

2 C

(M) are n first integrals

of the Hamiltonian system [1] which are ‘‘in

involution,’’ that is, such that:{F

, F

} = 0, 8i, j, and

suppose that they are functionally independent, that

is, the n differentials,dF

, are linearly independent at

each point of the level set M

defined by

¼ x

; ...; x

; y

; ...; y

ðÞ2M : F

ðx

; ...; y

; ...Þ

¼ f

; i ¼ 1; 2; ...; n

Then

(i) the set M

is a manifold which is invariant

along the solutions of the system [1];

(ii) if M

is compact and connected, it is diffeo-

morphic to an n-dimensional torus

¼ S

 S

S

¼ð’

; ...;’

Þ : ’

2 R=2Z

;

Hamiltonian Systems: Obstructions to Integrability 625