Go to previous page Go up Go to next page

5.8 Homogeneity: Dynamics

The Hamiltonian constraint can be constructed in the standard manner and its matrix elements can be computed explicitly thanks to the simple volume spectrum. There are holonomy operators for all three directions, and so in the triad representation the constraint equation becomes a partial difference equation for y m1,m2,m3 in three independent variables. Its (lengthy) form can be found in [48Jump To The Next Citation Point] for the Bianchi I model and in [62Jump To The Next Citation Point] for all other class A models.

Simpler cases arise in so-called locally rotationally symmetric (LRS) models, where a non-trivial isotropy subgroup is assumed. Here, only two independent parameters m and n remain, where only one, e.g., n can take both signs if discrete gauge freedom is fixed, and the vacuum difference equation is, e.g., for Bianchi I,

V~ -------- 2d |n + 2d|(ym+2d,n+2d - ym- 2d,n+2d) 1 V~ ------- V~ ------- + 2( V~ |n +-d|- |n - d|)((m + 4d)ym+4d,n- 2mym,n + (m - 4d)ym-4d,n) - 2d |n- 2d|(y - y ) m+2d,n-2d m-2d,n- 2d = 0 (51)
from the non-symmetric constraint and
2d( V~ |n-+-2d|+ V~ |n| )(y - y ) V ~ ------- V~ ------- m+2d,n+2d m-2d,n+2d +( |n + d|- |n - d|)((m + 2d)ym+2d,n- mym,n + (m - 2d)ym -2d,n) V~ -------- V~ --- - 2d( |n - 2d|+ |n |)(ym+2d,n-2d- ym -2d,n-2d) = 0 (52)
from the symmetric version (see also [14Jump To The Next Citation Point]). This leads to a reduction between fully anisotropic and isotropic models with only two independent variables, and provides a class of interesting systems to analyze effects of anisotropies.
  Go to previous page Go up Go to next page