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2.8 Dynamics

For the torus universe of the preceding Section  2.7, the dynamics can be read off from the metric (40View Equation). The area of a slice of constant t is essentially the Hamiltonian (44View Equation); it increases from 0 at t = 0 to a maximum at t = pl/4, and then shrinks to zero at t = pl/2 . At the “big bang” and “big crunch” the modulus (42View Equation) is purely real, t2 = 0 . This means that even apart from the “crunch” in volume, the geometry is singular: A real value of t represents a torus that has collapsed to a line. For /\ > 0, the final big crunch disappears, and the torus universe expands forever from an initial big bang. The initial spatial geometry is again degenerate.

It is not hard to check that as time increases, the modulus (42View Equation) moves along a semicircle in the upper half of the complex plane, with a center on the real axis. Such a curve is a geodesic in the natural Weil-Petersson (or Poincaré) metric on the torus moduli space  [159135] . Because of the invariance under the mapping class group (50View Equation), however, the true physical motion in the moduli space of the torus - the space of physical configurations with the large diffeomorphisms modded out - is much more complicated; there are arbitrarily long geodesics, and the flow is, in fact, ergodic  [93] .

For spacetimes R × S with S being a surface of genus g > 1, no explicit metrics analogous to Equation (40View Equation) are known, except for the special case of solutions with constant moduli. The problem is in part that no simple form such as Equation (41View Equation) for the “standard” constant curvature metrics exists, and in part that the ADM Hamiltonian becomes a complicated, nonlocal function of the moduli. For the case of an asymptotically flat genus g space, some interesting progress has been made by Krasnov  [172] ; I do not know whether these methods can be extended to the spatially closed case.

One can write down the holonomies of the geometric structure for a higher genus surface, of course - though even there, it is nontrivial to ensure that they represent spacetimes with spacelike genus g slices - but to a physicist, these holonomies in themselves give fairly little insight into the dynamics. In principle, the ADM and Chern-Simons approaches might be viewed as complimentary: As Moncrief has pointed out, one could evaluate the holonomies in terms of ADM variables in a nice time-slicing, set these equal to constants, and thereby solve the ADM equations of motion  [207Jump To The Next Citation Point] . In practice, though, this approach seems intractable except for the genus one case. For /\ = 0, it may be possible to extract a useful physical picture from the geometrical results of  [55Jump To The Next Citation Point], which relate holonomies to the structure of the initial singularity and the asymptotic future geometry, but the implications have not yet been explored in any depth.

A number of qualitative statements nevertheless remain possible. The singular behavior of the torus universe carries over to higher genus: Spacetimes with /\ < 0 expand from a big bang and recollapse in a big crunch, while those with /\ > 0 expand forever  [200Jump To The Next Citation Point20] . Moreover, the degeneration of the spatial geometry at the initial singularity carries over to the higher genus case  [200Jump To The Next Citation Point55Jump To The Next Citation Point] . By introducing a global “cosmological time” and exploiting recent results in two- and three-dimensional geometry, Benedetti and Guadagnini have shown that when /\ = 0, a set of parameters describing the initial singularity and a second set describing the geometry in the asymptotic future together completely determine the spacetime  [55] . It seems likely that these two sets are canonically conjugate, and a better understanding of the symplectic structure could be useful for quantum gravity.


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