  We develop a novel approach to gravity that we call `matrix general
relativity' (MGR) or `gravitational chromodynamics' (GCD or GQCD for 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 Standard Model). We call the new degrees of freedom of gravity
associated with the matrix structure `gravitational color' or simply
`gravicolor' 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 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 the masses of Planckian scale. Second possibilty is that
the additional degrees of freedom of gravitational field are confined within
the Planckian scale. What one sees at large distances are singlets (invariants)
of the new gauge symmetry.
