Gauged Gravity via Spectral Asymptotics of non-Laplace type Operators

Abstract

We construct invariant differential operators acting on sections of vector bundles of densities over a smooth manifold without using a Riemannian metric. The spectral invariants of such operators are invariant under both the diffeomorphisms and the gauge transformations and can be used to induce a new theory of gravitation. It can be viewed as a matrix generalization of Einstein general relativity that reproduces the standard Einstein theory in the weak deformation limit. Relations with various mathematical constructions such as Finsler geometry and Hodge-de Rham theory are discussed.

Matrix gravity and massive colored gravitons

“…However, in [7] all gravitons have unbroken U(N) symmetry and are all massless which is non-physical. More recently he obtained a unique gravity action based on the spectral expansion of non-Laplace type operator [8]. That formulation differs from the approach considered here as we take the mixing between the SL(2, C) space-time symmetry and the U(N) internal symmetry to be non-trivial, and where after symmetry breaking, all gravitons except one become massive.…”

Non-Laplace Type Operators on Manifolds With Boundary

“…For non-Laplace operators on manifolds without boundary even the invariant A 4 is not known, in general (for some partial results see [13,10,12] and the review [14]). For natural non-Laplace type differential operators on manifolds without boundary the coefficients A 0 and A 2 were computed in [13].…”

“…Seeley [42] showed that there are no logarithmic terms in the asymptotic expansion of the trace of the heat kernel, which are possible on general grounds, and that the heat invariants do depend on the boundary condition at the singular set; the neglect of that simple fact lead to some controversy on the coefficient A 2 in the past until this question was finally settled in [42,11]. [8,9,10,12], when instead of a single Riemannian metric there is a matrixvalued symmetric 2-tensor, which we call a "non-commutative metric". Matrix geometry is motivated by the relativistic interpretation of gauge theories and is intimately related to Finsler geometry (rather a collection of Finsler geometries) (see [8,9,10]).…”

“…Therefore, it is useless to try to use a Riemannian structure to cast the heat invariants in an invariant form. Rather, a non-Laplace type operator defines a collection of Finsler geometries (a matrix geometry in the terminology of [8,9,10,12]). Therefore, it is the matrix geometry that should be used to study the geometric structure of the spectral invariants of non-Laplace type operators.…”

Kinematics in matrix gravity

“…Only the total mass of a gravitating particle is observed. For more details and discussions see [2,3].…”