Matrix General Relativity: A New Look At Old Problems

We develop a novel approach to gravity that we call 'matrix general relativity' (MGR) or 'gravitational chromodynamics' (GCD or GQCD for the quantum version). Gravity is described in this approach not by one Riemannian metric (i.e. a symmetric two-tensor field) but by a multiplet of such fields, or by a matrix-valued symmetric two-tensor field that satisfies certain conditions. We define the matrix extensions of standard constructions of differential geometry including connections and curvatures, and finally, an invariant functional of the new field that reduces to the standard Einstein action functional in the commutative (diagonal) case. Our main idea is the analogy with Yang-Mills theory (QCD and the standard model). We call the new degrees of freedom of gravity associated with the matrix structure 'gravitational colour' or simply 'gravicolour' and introduce a new gauge symmetry associated with this degree of freedom. As in the standard model there are two possibilities. First of all, it is possible that at high energies (say at the Planckian scale) this symmetry is exact (symmetric phase), but at low energies it is badly broken, so that one tensor field remains massless (and gives general relativity) and the other ones become massive with masses of Planckian scale. The second possibility is that the additional degrees of freedom of the gravitational field are confined to the Planckian scale. What one sees at large distances are singlets (invariants) of the new gauge symmetry.