Carlip S. Quantum Gravity in 2+1 Dimensions
Подождите немного. Документ загружается.
66 4 Geometric structures and Chern-Simons theory
It can be shown that the holonomy of a curve y depends only on its
homotopy class. In fact, the holonomy defines a group homomorphism
H :m(M
9
*)->G. (4.8)
Note that a coordinate change in the original patch U\
9
say
<j>\
—•
(f)\
o
h
9
will have the effect of conjugating all of the holonomies by some element
h £
G.
The homomorphism H is thus not quite unique, but is determined
only up to overall conjugation.
If we now pass from M to its universal covering space M, there are
no longer any noncontractible closed paths^By the argument above, a
coordinate
</>i
can thus be extended to all of M, giving a map D
:
M
—•
X.
This map is called the
developing
map of the geometric structure. For our
two-dimensional example, the holonomy group is precisely the Fuchsian
group F, while the developing map describes the tiling of the hyperbolic
plane by copies of the polygon P(L).
As a concrete example of this construction, consider the conical space-
time of a static point particle discussed in chapter
3.
At a fixed time, space
is described by a wedge W (figure 3.1), with edges identified by a rotation
R(P);
the whole spacetime is R x W, with the same identification at all
times.
The fundamental group of such a spacetime has a single generator
y, describing a loop around the apex of the cone. The corresponding
holonomy is H(y) = R(P), and the developing map is the map that 'un-
wraps' the cone into an infinite strip. Similarly, for a spinning particle,
the holonomy is a Poincare transformation that includes a rotation and a
time translation.
We
can now describe the three-dimensional cosmologies of the preceding
section in the same language. We start with the region [0,1] x P(L)
of figure 4.2, and identify sides [0,1] x E
t
and [0,1] x E[. Once again,
[0,1] xP(Z) may be viewed as a coordinate patch with transition functions
connecting the boundaries. Moreover, these transition functions may now
be treated as isometries of the full (2+l)-dimensional Minkowski metric,
thus describing a geometric structure.
But while these transition functions are isometries of the Minkowski
metric, they lie in a rather restricted group of such isometries. The
full group of isometries of the Minkowski metric is the Poincare group
ISO
(2,1).
The identifications of section 1, on the other hand, are isome-
tries of the two-dimensional metric (4.2), and thus lie in the subgroup
PSL(2,K) c ISO(2,1). The generalization we need is now clear: we
must allow transition functions that live in the full Poincare group. In
other words, we must look for Lorentzian
structures,
or (ISO
(2,1),
R
2+1
)
structures, on [0,1] x Z.
Cambridge Books Online © Cambridge University Press, 2009
43 The space of Lorentzian structures 67
4.3 The space of Lorentzian structures
To understand the classification of Lorentzian structures, we must first
deal with a technical problem arising from the fact that the space R x 2
is not compact. If M is a three-manifold with a flat Lorentzian metric,
we can always find a 'new' noncompact solution to the field equations
by simply cutting out a chunk of M. For instance, if we choose some
spacelike slice S in M and remove the entire future of S, we will be left
with a manifold with the same topology and the same geometric structure.
To avoid this rather trivial overcounting of solutions, we need some notion
of a 'maximal' spacetime. The appropriate concept turns out to be that of
a domain of dependence, essentially a spacetime that includes the entire
past and future of some spacelike slice (see appendix B for a more precise
definition).
Mess has shown that every maximal flat Lorentzian spacetime with
the topology [0,1] x Z is uniquely determined by its ISO (2,1) holonomy
group
[195].
Moreover, any homomorphism from 7Ci(2) to ISO(2,1) gives
an admissible holonomy group, provided only that its SO (2,1) projection
is Fuchsian. In other words, no new restrictions are needed on the
translations in the Poincare group; as long as the 'positions' generate a
Fuchsian subgroup of
SO
(2,1),
the 'momenta' are arbitrary.
As always, two holonomy groups give different geometric structures
unless they are conjugate. The solutions of the field equations are thus
parametrized by the space of group homomorphisms
2,1))/ ~, (4.9)
where
Pi~P2 if P2 = h'pi'h-\ he ISO
(2,1).
(4.10)
The subscript 0 in equation (4.9) means that we restrict ourselves to
homomorphisms whose SO(2,1) projections are Fuchsian; Goldman has
shown that this subspace is a connected topological component of the
space of all homomorphisms
[130].
One more subtlety must be taken into account if we want a complete
description of the space of solutions. Diffeomorphic spacetimes should
not be counted as distinct solutions, and while the space M is not affected
by diffeomorphisms that can be smoothly deformed to the identity, it is
acted on by 'large' diffeomorphisms. To understand this phenomenon,
note first that the large diffeomorphisms of [0,1] x 2 can be characterized
completely by their actions on Z. The group of large diffeomorphisms
of a surface 2, also known as the mapping class group, is discussed in
appendix A. It is generated by Dehn twists around closed curves in E, that
is,
diffeomorphisms obtained by cutting £ open along a curve y, twisting
Cambridge Books Online © Cambridge University Press, 2009
68
4
Geometric
structures and Chern-Simons theory
one
end by
2n
9
and
regluing
the
ends.
(See
figure
A.I for an
illustration
of
a Dehn twist
of a
torus.)
It is
fairly easy
to see
that such Dehn twists will
mix
up the
generators
of
the fundamental group rci(L);
in
fact,
it may be
shown that they generate
all
of^the outer automorphisms
of n\(L) [37].
This means that
if pi e Jt and p2 G M are two
homomorphisms
that differ only
by an
automorphism
of
TTI(S), they will give
the
same
flat metric
on [0,1] x S.
Equivalently,
the
geometry
of
spacetime should
depend only
on the
fundamental group TTI(Z)
as an
abstract group,
and
not
on any
particular choice
of
generators.
The
true space
of
flat spatially
closed (2+l)-dimensional spacetimes
is
thus parametrized
by
M = Jf/Out(ni(L)).
(4.11)
This characterization
of Jt
makes
it
simple
to
determine
the
number
of empty space solutions
to the
Einstein equations.
A
heuristic counting
argument proceeds
as
follows.
The
fundamental group
of a
genus
g sur-
face
2 has 2g
generators
and one
relation. Each generator
is
mapped into
the six-dimensional space
ISO
(2,1),
giving
a
total
of
6(2g
—
1) parameters.
Conjugate groups must still
be
identified, however,
so six of
these param-
eters
are
redundant.
We
thus expect there
to be a
(12g
—
12)-dimensional
space
of
solutions. This result confirms the ADM computations
of
chapter
2,
in
which
we
found that solutions were labeled
by
6g
—
6
positions
and
6g
—
6
momenta.
The
counting
is
slightly more complicated when
2 is a
sphere
or a
torus,
but it is not
hard
to
check that
the
number
of
solutions
again agrees with
the
ADM results.
We
can
establish
an
even closer connection with
the ADM
method
if
we note that
the
group
ISO (2,1)
is
geometrically
the
cotangent bundle
of
SO
(2,1);
that
is,
translations
in
ISO (2,1)
can be
viewed
as
cotangent
vectors
to
Lorentz transformations. Indeed,
if
11—>
A(t) is a
curve
in the
group manifold
ISO
(2,1),
a
cotangent vector
at A(0) is a
vector
of the
form
dt
t=o
(4.12)
Given
two
curves
A\(t) and
A2(i) through
a
common point
in the
group
manifold
of
SO
(2,1),
the
'product cotangent vector'
is
thus
t=0
(4.13)
giving precisely
the
right semidirect product structure
for
ISO
(2,1).
This
result implies,
in
turn, that
Jt is
itself
a
cotangent bundle,
Jt = T* Jf,
with
a
base space
Jf
=
JfI Out (7ti(S)), (4.14)
Cambridge Books Online © Cambridge University Press, 2009
4.4 Adding
a
cosmological constant
69
with
P1~P2
if
P2
=
h'
P
i'h-
1
, heS0(2,l). (4.15)
This characterization will
be
crucial when
we
consider
the
quantization
of
(2+l)-dimensional gravity
in the
connection representation.
Observe that
Jf is
once again
the
ordinary moduli space
of Z,
that
is,
the space
of
constant negative curvature metrics,
or
hyperbolic structures,
on
Z.
This again confirms
the
structure
we
found
for the
reduced phase
space
of
chapter
2, in
which
the
momenta
p
a
were cotangents
to the
moduli
m
a
.
4.4 Adding
a
cosmological constant
A similar construction exists when
the
cosmological constant
is
nonzero,
but some
new
subtleties appear.
Let us
first take
A to be
negative,
say
A
=
—I//
2
.
In
this case,
the
model space analogous
to
Minkowski space
is (2+l)-dimensional anti-de Sitter space (adS),
or,
strictly speaking,
the
universal covering space
adS.
This space
may be
represented concretely
by starting with
the
flat four-dimensional space
R
2
'
2
,
with coordinates
(XuX
2
,
Tu T
2
) and
metric
dS
2
= dX
{
2
+ dX
2
2
-
dTi
2
- dT
2
2
,
(4.16)
and restricting
to the
submanifold
X
2
+ X
2
2
- Ti
2
- T
2
2
= -t
1
.
(4.17)
We
can
write
the
coordinates
of R
2
'
2
in
matrix form,
and anti-de Sitter space
is
then determined
by the
condition
detX
= l,
(4.19)
i.e.,
X G
SL(2,R).
Jn
this representation,
it is
easy
to
check that
the
isometry group
of
adS
is
SX(2,
R)
x
SL(2,
R)/Z2, where
the two
factors
of
SL(2,
R)
act by
left
and
right multiplication,
X ->
R
+
XR~, with (i?
+
,
R~) ~
(-R
+
,-R~).
Anjinti-de Sitter structure
is
thus obtained
by
gluing together patches
of
adS
with transition functions
in
SL(2,R)
x
SL(2,R)/Z
2
.
As in the
Cambridge Books Online © Cambridge University Press, 2009
70 4
Geometric
structures and Chern-Simons theory
preceding section, the space of holonomies on a manifold [0,1] x Z with
Z spacelike is
),
SL(2,R) x SL(2,R)/Z
2
)/ ~, (4.20)
where
P\~Pi
if p2 = h'
Pl
-h-\ h€ SL(2,R) x SL(2,R)/Z
2
.
(4.21)
The subscript 0 in (4.20) again means that we must restrict ourselves to
homomorphisms whose projections on each of the SL(2,R) factors are
Fuchsian.
As in the case of a vanishing cosmological constant, it may be shown
that a holonomy p e M determines a unique maximal spacetime, and
that such spacetimes are parametrized by Jt modulo the action of the
mapping class group
[195].
In contrast to the Aj= 0 case, however, the
space (4.20) is not a cotangent bundle. Rather, Jt is, roughly speaking, a
product of two copies of the ordinary moduli space (4.15) of E, each with
its own symplectic structure.
For the case of a positive cosmological constant, the relevant model
space is de Sitter space, which can be obtained from R
1
'
3
, with coordinates
(T,X, Y,Z) and metric
dS
2
= -dT
2
+ dX
2
+ dY
2
+ dZ
2
, (4.22)
by restricting to the submanifold
-T
2
+X
2
+
Y
2
+Z
2
= ~. (4.23)
It is evident from this construction that (2+l)-dimensional de Sitter space
has isometry group
SO
(3,1).
As in the previous cases, solutions of the field
equations with A > 0 may be described by de Sitter geometric structures,
that is, by piecing together patches of de Sitter space with transition
functions in
SO
(3,1).
For the case of de Sitter structures, however, Mess has shown that
the holonomies do not determine the geometry
[195].
Rather, an infinite
discrete set of spacetimes have the same holonomy. We shall see a specific
example of this phenomenon at the end of the next section. This ambiguity
may indicate the presence of a new discrete quantum number in quantum
gravity. Alternatively, as Witten has argued, we may choose to take the
holonomies as the fundamental observables; in that case, spacetimes that
differ classically would be quantum mechanically indistinguishable
[290].
Cambridge Books Online © Cambridge University Press, 2009
4.5 Closed universes as quotient spaces 71
4.5 Closed universes as quotient spaces
In sections 1 and 2, we encountered two equivalent descriptions of a two-
dimensional surface, as a quotient space H
2
/F and as a geometric space
modeled on H
2
with holonomy group F. We succeeded in generalizing
the latter representation to three dimensions. It is natural to ask whether
the quotient space representation can be extended as well.
In general, it cannot. Consider, for example, the conical spacetime of a
point particle with an irrational deficit angle /?. Rotations by /? are not
periodic, and enough rotations will bring an initial point
XQ
arbitrarily
close to any other point x at the same radius. Consequently, the group
generated by the rotation R(fi) does not act properly discontinuously -
it does not 'separate points' - and the quotient space R
2
/(!?(/?)) is not
well-behaved.^
For cosmological solutions with the topology [0,1] x S, on the other
hand, the situation is more favorable. Mess has shown that if 2 is spacelike,
any such spacetime can be written as a quotient space iV/F, where F is
the holonomy group and (for genus greater than one) N is a region in
the interior of the light cone of Minkowski space
[195].
In principle, this
makes it possible to construct the spacetime [0, l]xl explicitly: one need
merely find a set of coordinates upon which the group F acts nicely and
form the quotient space by identifying appropriate coordinate values.
In practice, this task is already almost unmanageable in two dimensions.
Even for genus two, such coordinate identifications are extraordinarily
difficult to describe explicitly. For genus one, however - that is, for
spacetimes with the spatial topology of a torus - the quotient space
construction allows a complete, simple, and explicit description of all flat
Lorentzian metrics with spacelike hypersurfaces T
2
. We now turn to this
simple case.
We must first determine the possible holonomy groups of the spacetime
M = [0,1] x T
2
. The fundamental group of the torus, and thus of M, is the
abelian group Z©Z, with one generator for each of the two independent
circumferences of T
2
. The holonomy group must therefore be generated
by two commuting Poincare transformations, say (Ai,ai) and (A2,fl2)»
We begin by analyzing the
SO
(2,1) components Ai and
A2.
Any Lorentz
transformation in 2+1 dimensions fixes a vector n, and for Ai and A2 to
commute, they must fix the same vector. This vector may be spacelike,
For the cone, there is a way out: if we remove the line r = 0 from Minkowski space, and
then form the universal covering space (by allowing the polar angle
<\>
to range from —00
to 00), the conical spacetime can be expressed as the quotient of this covering space by
R(P).
Similar constructions have been found in other specific instances, but a systematic
generalization is not known.
Cambridge Books Online © Cambridge University Press, 2009
72 4 Geometric structures and Chern-Simons theory
null, or timelike, and the space of holonomies correspondingly splits into
three components.
(A similar splitting occurs for spaces of higher genus as well. For
g > 1, the components are genuinely topologically disconnected, and
only one component corresponds to a Lorentzian structure on [0,1] x
Z
[130].
For the torus, on the other hand, the components are not
completely disconnected, and the space of holonomies is a complicated,
non-Hausdorff space. For a careful description of this space, see reference
[181].)
If n is timelike or null, it may be shown that the toroidal slices T
2
are not spacelike: roughly speaking, a T
2
slice must contain n. In this
case,
the quotient construction fails. If n is timelike, for example, Ai
and A2 are conjugate to rotations, and the problem is essentially the
same as that of constructing a conical geometry as a quotient space.
Nonstandard quotient constructions may still be possible, as described in
the footnote on page 71; for the toroidal universe with timelike or null
n, this possibility is explored in references [108, 181]. For most 'physical'
applications, however, we are interested in solutions like those that arise
in the ADM description, for which the T
2
slices are spacelike.
For such solutions, two possibilities remain: either Ai and A2 fix a
spacelike vector, say (0,0,1), or Ai and A2 both vanish. Let us start with
the latter, simpler possibility. The holonomy group is now generated by a
pair of translations a\ and #2, which we take to be spacelike in order to
ensure that the T
2
slices are spacelike. We may then choose coordinates
such that a\ = (0,a,0) and ai = (0,aTi,ai2). A fundamental region for
the action of the holonomy group on a spatial slice of constant t is now
a parallelogram with vertices at (0,0), (a,0), (aTi,at2), and {a(x\ + I),ai2).
The spatial geometry on each slice is that of a torus with modulus
T
=
TI
+ii2 and area axi, and the resulting spacetime is precisely the static
torus geometry (3.83).
If, on the other hand, Ai and A2 stabilize a spacelike vector, they
must both be boosts. We can then use our remaining freedom of overall
conjugation to put the two generators of the holonomy group in the form
H(y 1) : (t, x, y) -» (t cosh k + x sinh k, x cosh k + t sinh
k
9
y + a)
H(72) '
(U
x, y)
—•
(t cosh
ju
+ x sinh
\x,
x cosh
fi
+ t sinh
//,
y + b).
(4.24)
This action simplifies if we define new coordinates
t = — cosh
w,
x = — sinh
w,
(4.25)
Cambridge Books Online © Cambridge University Press, 2009
4.5 Closed universes as quotient spaces
73
Fig. 4.4. A flat torus universe is foliated by surfaces of constant TrK. Each slice
is a parallelogram (with opposite edges identified), but the area and modulus vary
from slice to slice. Note that at the initial singularity, the parallelogram collapses
to a line.
in which the Minkowski metric becomes
(4.26)
The coordinate T has been chosen so that surfaces of constant T have
extrinsic curvature TrK =
—
T, in order to allow comparison with the
results of chapter 3.
In the new coordinates, the transformations (4.24) reduce to
:(T,u,y)
H(y
2
):(T
9
u
9
y)
(4.27)
On a constant T surface, a fundamental region for the action of the
holonomy group (H(yi)
9
H(y2)) is thus simply the torus (u
9
y) ~ (u + x,
y + a) ~ (u + \i
9
y + b). A typical three-manifold obtained by such identi-
fications is illustrated in figure 4.4.
To put this metric in a more standard form, we can transform to new
Cambridge Books Online © Cambridge University Press, 2009
74 4 Geometric structures and Chern-Simons theory
spatial coordinates (at fixed T)
' ( 2^ P\~
l
( ^
k
\
x = la +j2 ) \,
a
y
+
f2
u
)
y>
= h + #\ ' (Q^j
.
(4.28)
The spatial metric is then
da
2
= (a
2
H
j)(
dx
'
2
+
d
y'
2
)>
(
4
-
29
)
with a periodicity
A
2
A"
1
(4.30)
From the definition of the modulus T, da
2
is therefore the metric of a
torus with
(4.31)
The corresponding momentum conjugate to
T
is easily found to be
. (4.32)
In contrast to the case of purely translational holonomies, such a metric is
clearly dynamical: the shape of a toroidal cross-section changes with time.
Indeed, the solutions we have constructed are precisely the four-parameter
family of solutions (3.80) that we already found in chapter 3, with A = 0.
Let us next attempt to generalize this quotient construction to the case
of a nonvanishing cosmological constant. For A = —I//
2
< 0, maximal
spacetimes are again determined by their (anti-de Sitter) holonomies,
and most of the preceding discussion can be carried through with only
small changes. For the [0,1] x T
2
topology, for example, the holonomies
now fall into six sectors, since the isometry in each SX(2,R) factor can
fix a timelike, null, or spacelike vector. Spacelike T
2
slices occur in
the 'spacelike-spacelike' sector, in which the two commuting holonomies
Cambridge Books Online © Cambridge University Press, 2009
4.5 Closed universes as quotient spaces 75
(Rt,Rf) € SL(2,R) x SL(2,R) and (R^,^) e SL(2,R) x SL(2,R) may be
conjugated to the form
4.
/V«
±/2
o \
*-(„
e
-^>
(433)
These holonomies act on the coordinates X\, Xi, T\, and Ti of equation
(4.16),
and the geometry of the resulting quotient space can be translated
back into the form (3.80) of chapter 3 through the coordinate transforma-
tion
= Fe~
t/2
si
= Fe~
t/2
cosh(>lx + /ay)
(^) (4.34)
with
^ = \{rt
+
rn
H =
\ixt
+
n).
(4.35)
A straightforward computation then shows that the isometries generated
by the holonomies (4.33) reduce to the identifications x ~ x + 1 and
y ~ y + 1.
A slice of this spacetime at fixed York time has a two-geometry that
can be easily computed from equation (3.81) of chapter 3. The results are
most elegantly expressed in terms of a new time coordinate 1, defined by
the condition
T =
-^cotj,
(4.36)
or in terms of the coordinate t of equation (3.68),
One finds that
Cambridge Books Online © Cambridge University Press, 2009