Approaches to Quantum Gravity

62 J. Stachelinvestigate (3 + 1) and (2 + 2) decompositions of the first order form of the fieldequations and of the compatibility conditions between metric and affine connection(see Sections 4.3 and earlier in 4.6), and in first order formulations of initialvalue problems. If the n-connection is metric, then “normal” has the additionalmeaning of “orthogonal” (see discussion above). The (t, t) surface affine connectionis (uniquely) compatible with the surface metric; the (t, n)(n, t) curvatures areequivalent; and the (n, n) torsion reduces to an infinitesimal rotation. On a hypersurface(p = n − 1), the torsion vanishes, and the (t, n) and (n, t) curvatures areequivalent to the second fundamental form of the hypersurface.The Ashtekar connection combines the (t, t) and (n, t) curvatures into a singlethree-connection. Extension of the Ashtekar variables, or some generalization ofthem, to null hypersurfaces is currently under investigation. 37 In the two-plus-twodecomposition, there is a pair of second fundamental forms and the (n, n) rotationis non-vanishing. For a formulation of the two-plus-two initial value problemwhen the metric and connection are treated as independent before imposition of thefield equations, see [25]. Whether some analogue of the Ashtekar variables can beusefully introduced in this case remains to be studied.4.7 Background space-time symmetry groupsThe isometries of a four-dimensional pseudo-Riemannian manifold are characterizedby two integers: the dimension m ≤ 10 of its isometry group (i.e. its groupof automorphisms or motions) and the dimension o ≤ min(4, m) of this group’shighest-dimensional orbits (see, e.g., [36; 15]). There are two extreme cases.The maximal symmetry group: (m = 10, o = 4). Minkowski S-T is the uniqueRicci-flat S-T in this group. Its isometry group is the Poincaré or inhomogeneousLorentz group, acting transitively on the entire S-T manifold. Special-relativisticfield theories involving field equations that are invariant under this symmetrygroup are the most important example of background-dependent theories (seeIntroduction). At the other extreme isThe class of generic metrics: (m = 0, o = 0). These S-Ts have no non-trivialisometries. The class of all solutions to a set of covariant field equations (seeSection 4.5.2) will include a subclass – by far the largest – of generic metrics. 3837 For a review of some results of a generalization based on null hypersurfaces, see [24]. D’Inverno and coworkershave researched null Ashtekar variables.38 This global, active diffeomorphism group should not be confused with the groupoid of passive, local coordinatetransformations. Nor must the trivial freedom to carry out active diffeomorphisms acting on all structureson the manifold, including whatever fixed background metric field (such as the Minkowski metric) may bepresent, be confused with the existence of a subgroup of such diffeomeorphisms that constitutes the isometrygroup of this background metric.