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2.2 Geometric structures

The global geometry of vacuum spacetimes in 2+1 dimensions is described mathematically by the theory of geometric structures  [256Jump To The Next Citation Point62147Jump To The Next Citation Point] (for examples of geometric structures, see  [248] ; for a slightly more detailed version of this section, see  [72]). For simplicity, let us begin with the case of a vanishing cosmological constant. If the spacetime manifold M is topologically trivial, then by Equation (1View Equation) the vacuum field equations imply that (M, g) is simply a subset of ordinary Minkowski space (V 2,1,j) . If M is topologically nontrivial, it can still be covered by contractible coordinate patches Ui, each isometric to V 2,1, with the standard Minkowski metric j mn on each patch. The geometry is then encoded entirely in the transition functions gij on the intersections Ui /~\ Uj, which determine how these patches are glued together. Since the metrics in Ui and Uj are identical, these transition functions must be isometries of jmn, that is, elements of the Poincaré group ISO(2, 1) . Similarly, if /\ /= 0, a vacuum spacetime can be built by patching together pieces of de Sitter or anti-de Sitter space by appropriate isometries: SO(3, 1) for /\ > 0 and SO(2, 2) or SL(2,R) × SL(2,R)/Z2 for /\ < 0 .

Such a construction is an example of a geometric structure, in the flat case a Lorentzian or (ISO(2,1), V 2,1) structure. In general, a (G, X) manifold is one locally modeled on X, much as an ordinary n -dimensional manifold is modeled on Rn . More precisely, let G be a Lie group that acts analytically on some n -manifold X, the model space, and let M be another n -manifold. A (G, X) structure on M is then a set of coordinate patches Ui for M with “coordinates” fi : Ui --> X taking their values in X and with transition functions gij = fi o fj -1| Ui /~\ Uj in G .

View Image

Figure 1: The curve g is covered by coordinate patches Ui , with transition functions gi (- G . The composition g o ...o g 1 n is the holonomy of the curve.
A fundamental ingredient in the description of a (G, X) structure is its holonomy group, which can be viewed as a measure of the failure of a single coordinate patch to extend around a closed curve. Let M be a (G, X) manifold containing a closed path g . As illustrated in Figure  1View Image, we can cover g with coordinate charts
fi : Ui --> X, i = 1,...,n, (8)
with constant transition functions gi (- G between Ui and Ui+1, i.e.,
f | U /~\ U = g o f |U /~\ U , i i i+1 i i+1 i i+1 (9) fn|Un /~\ U1 = gn o f1| Un /~\ U1.
Let us now try to analytically continue the coordinate f1 from the patch U1 to the whole of g . We can begin with a coordinate transformation in U2 that replaces f2 by f2'= g1 o f2, thus extending f1 to U1 U U2 . Continuing this process along the curve, with fj'= g1 o ...o gj-1 o fj, we will eventually reach the final patch U n, which again overlaps U 1 . If the new coordinate function ' fn = g1 o ...o gn-1 o fn agrees with f1 on Un /~\ U1, we will have covered g with a single patch. Otherwise, the holonomy H(g) = g1 o ...o gn measures the obstruction to such a covering.

It may be shown that the holonomy of a curve g depends only on its homotopy class  [256Jump To The Next Citation Point] . In fact, the holonomy defines a homomorphism

H : p1(M ) --> G. (10)
H is not quite uniquely determined by the geometric structure, since we are free to act on the model space X by a fixed element h (- G, changing the transition functions gi without altering the (G, X) structure of M . Such a transformation has the effect of conjugating H by h, and it may be shown that H is unique up to such conjugation  [256] . The space of holonomies is thus the quotient
M = hom(p1(M ),G)/ ~, (11) r1 ~ r2 if r2 = h .r1 .h-1,h (- G.

Note that if we pass from M to its universal covering space M, we will no longer have noncontractible closed paths, and f1 will be extendible to all of M . The resulting map D : M --> X is called the developing map. At least in simple examples, D embodies the classical geometric picture of development as “unrolling” - for instance, the unwrapping of a cylinder into an infinite strip.

The holonomies of the geometric structure in (2+1)-dimensional gravity are examples of diffeomorphism-invariant observables, which, as we shall see below, are closely related to the Wilson loop observables in the Chern-Simons formulation. It is important to understand to what extent they are complete - that is, to what extent they determine the geometry. It is easy to see one thing that can go wrong: If we start with a flat three-manifold M and simply cut out a ball, we can obtain a new flat manifold without affecting the holonomy. This is a rather trivial change, though, and we would like to know whether it is the only problem.

For the case of a vanishing cosmological constant, Mess  [200Jump To The Next Citation Point] has investigated this question for spacetimes with topologies R × S . He shows that the holonomy group determines a unique “maximal” spacetime M - specifically, a domain of dependence of a spacelike surface S . Mess also demonstrates that the holonomy group H acts properly discontinuously on a region W < V 2,1 of Minkowski space, and that M can be obtained as the quotient space W/H . This quotient construction can be a powerful tool for obtaining a description of M in reasonably standard coordinates, for instance in a time-slicing by surfaces of constant mean curvature. Similar results hold for anti-de Sitter structures. Some instructive examples of the construction of spacetimes with /\ < 0 from holonomies are given in  [133] .

For de Sitter structures, on the other hand, the holonomies do not uniquely determine the geometry  [200Jump To The Next Citation Point] . An explicit example of the resulting ambiguity has been given by Ezawa  [117Jump To The Next Citation Point] for the case of a topology R × T 2 (see also Section 4.5 of  [81Jump To The Next Citation Point]). A similar ambiguity occurs for (2+ 1)-dimensional gravity with point particles, where, as Matschull has emphasized  [194Jump To The Next Citation Point], it may imply a physical difference between the metric and Chern-Simons formulations of (2+1)-dimensional gravity.

We close this section with a partial description of the space of solutions of the vacuum Einstein field equations on a manifold R × S, where S is a compact genus g two-manifold, that is, a surface with g “handles.” The fundamental group of such a spacetime, p (M ) - ~ p (S) 1 1, is generated by g pairs of closed curves (Ai,Bi), with the single relation

-1 -1 -1 - 1 - 1 -1 A1B1A1 B1 A2B2A2 B2 ...AgBgAg Bg = 1. (12)
By Equation (11View Equation), the space of holonomies is the space of homomorphisms from p1(S) to G (where G is ISO(2, 1) for /\ = 0, SO(3, 1) for /\ > 0, or SO(2, 2) for /\ < 0) modulo overall conjugation. For g > 1, this space of homomorphisms has dimension 12g - 12 : p1(S) has 2g generators and one relation, and the identification by conjugation leaves 2g - 2 choices of elements of a six-dimensional group G 1 1 For g = 0 , p1(S) is trivial, and there is only one geometric structure. The case of g = 1 will be discussed below in Section   2.7 . There are two subtleties that prevent the space (11View Equation) from being the exact moduli space of solutions of the vacuum field equations. First, as noted above, the holonomies do not always determine a unique geometric structure. In particular, for /\ > 0 one may need an additional discrete variable to specify the geometry. Second, not all homomorphisms from p1(S) to G give geometric structures that correspond to smooth manifolds. The space of homomorphisms (11View Equation) is not connected  [148], and, in general, only one connected component gives our desired geometry. Even once these caveats are taken into account, though, we still have a (12g - 12) -dimensional space of solutions that can, in principle, be described completely.
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