On a class of invariant coframe operators with application to gravity

TL;DR

This paper characterizes a class of invariant coframe operators on 4D manifolds, constructs related field equations, and identifies those compatible with classical tests of gravity, including Einstein's equations.

Contribution

It introduces a broad class of invariant second-order operators and derives covariant field equations, highlighting their relation to Einstein's gravity and experimental indistinguishability.

Findings

Identifies operators invariant under coordinate and SO(1,3) transformations.

Constructs field equations derivable from action principles.

Finds equations consistent with classical tests of gravity.

Abstract

Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend on the coframe variables. The paper exhibits the class of operators that are invariant under a general change of coordinates, and, also, invariant under the global SO(1,3)-transformation of the coframe. A general class of field equations is constructed. We display two subclasses in it. The subclass of field equations that are derivable from action principles by free variations and the subclass of field equations for which spherical-symmetric solutions, Minkowskian at infinity exist. Then, for the spherical-symmetric solutions, the resulting metric is computed. Invoking the Geodesic Postulate, we find all the equations that are experimentally (by the 3 classical tests) indistinguishable from Einstein field equations. This family includes, of course, also Einstein equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool employed in the paper is an invariant formulation reminiscent of Cartan's structural equations. The article sheds light on the possibilities and limitations of the coframe gravity. It may also serve as a general procedure to derive covariant field equations.