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2.4 The ADM approach

We next turn to a more traditional approach to classical general relativity, the conventional metric formalism in the space/time splitting of Arnowitt, Deser, and Misner  [25] . As Moncrief  [206Jump To The Next Citation Point] and Hosoya and Nakao  [159Jump To The Next Citation Point] have shown, this metric formalism can also be used to give a full description of the solutions of the vacuum field equations, at least for spacetimes with the topology R × S .
View Image

Figure 2: The ADM decomposition is based on the Lorentzian version of the Pythagoras theorem.
We start with the ADM decomposition of the spacetime metric gmn,
2 2 2 i i j j ds = N dt - gij(dx + N dt)(dx + N dt), (28)
as illustrated in Figure  2View Image . The action then takes the usual form 2 2 In this section I use standard ADM notation: g ij and R refer to the induced metric and scalar curvature of a time slice and Kij is the extrinsic curvature of such a slice, while the full spacetime metric and curvature are denoted by (3)g mn and (3)R .
integral V~ ----- integral integral ( ) Igrav = d3x - (3)g ((3)R - 2/\) = dt d2x pijgij- N iHi - N H , (29) S
with canonical momentum pij = V~ g-(Kij - gijK) and the momentum and Hamiltonian constraints
j -1-- ik jl ij kl V~ -- Hi = -2 \~/ jp i, H = V~ g-gijgkl(p p - p p )- g(R - 2/\). (30)
To solve the constraints, we can choose the York time-slicing  [284], in which the mean (extrinsic) curvature is used as a time coordinate, ij V~ -- -K = gijp / g = T . Andersson et al. have shown that this is a good global coordinate choice for classical solutions of the vacuum field equations  [20Jump To The Next Citation Point] . We next select a useful parametrization of the spatial metric and momentum. Up to a diffeomorphism, any two-metric on S can be written in the form  [1Jump To The Next Citation Point121]
gij = e2cgij(ma), (31)
where gij(ma) are a finite-dimensional family of metrics of constant curvature k (k = 1 for the two-sphere, 0 for the torus, and - 1 for spaces of genus g > 1). These standard metrics are labeled by a set of moduli ma that parametrize the Riemann moduli space of S . As in Section  2.2, such constant curvature metrics can be described in terms of a geometric structure - for genus g > 1 an 2 (H ,PSL(2, R)) structure - with moduli parametrizing the homomorphisms (10View Equation). We can count these just as in Section  2.2 ; now, since PSL(2, R) is three-dimensional, we find that a constant negative curvature surface of genus g > 1 is described by 6g - 6 parameters.

The corresponding decomposition of the conjugate momentum is described in  [206Jump To The Next Citation Point] : Up to a diffeomorphism, the trace-free part of pij can be written as a holomorphic quadratic differential pij, that is, a transverse traceless tensor with respect to the covariant derivative compatible with g ij . The space of such quadratic differentials parametrizes the cotangent space of the moduli space  [1], and the reduced phase space becomes, essentially, the cotangent bundle of the moduli space.

With the decomposition of  [206Jump To The Next Citation Point], the momentum constraints Hi = 0 become trivial, while the Hamiltonian constraint becomes an elliptic differential equation that determines the scale factor c in Equation (31View Equation) as a function of gij and ij p,

1- 2 2c 1-[ -1 ik a jl a ] - 2c k- Dc - 4 (T - 4/\)e + 2 g gij(ma)gkl(ma)p (p )p (p ) e - 2 = 0, (32)
where a p are the momenta conjugate to the moduli,
integral a 2 ij--@-- p = d x p @ma gij. (33) S
The theory of elliptic equations ensures that Equation (32View Equation) determines a unique scale factor c . The action (29View Equation) then simplifies to a “reduced phase space” action, involving only the physical degrees of freedom,
integral ( ) Igrav = dT padma--- H(m, p,T ) , (34) dT
with a time-dependent Hamiltonian
integral 2 V~ -- 2c(m,p,T ) H = d x g e . (35) ST
The classical Poisson brackets can be read off directly from Equation (34View Equation):
{ma, pb}= dba, {ma, mb}= {pa,pb}= 0. (36)

Three-dimensional gravity again reduces to a finite-dimensional system, albeit one with a complicated time-dependent Hamiltonian. The physical phase space is parametrized by (ma, pb), which may be viewed as coordinates for the cotangent bundle of the moduli space of S . For a surface of genus g > 1, this gives us 12g - 12 degrees of freedom, matching the results of Section  2.2 .

If /\ = 0, this correspondence can be made more explicit: For G = ISO(2, 1) and M -~ R × S, the space (11View Equation) of geometric structures is itself a cotangent bundle, whose base space is the space of hyperbolic structures on S . This follows from the fact that the group ISO(2, 1) is the cotangent bundle of SO(2, 1) . Concretely, in the first-order formalism of Section  2.3, the curvature equation (16View Equation) with /\ = 0 implies that w is a flat SO(2, 1) connection; and if w(s) is a curve in the space of such flat connections, the tangent vector e = dw(s)/ds satisfies the torsion equation (17View Equation). For /\ /= 0, I know of no such direct correspondence, and the general relationship between the ADM and first-order solutions seems less transparent.

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