Gravity from Invariance Under Transformations of Minkowski Metric

The Minkowski manifold serves as the standard model of spacetime for relativistic theories of non-gravitating matter. It admits global coordinate systems and a Minkowski metric. In fact, it admits more than one Minkowski metric. In special relativity, one of these metrics is chosen as the non-dynamical physical metric. The theory is not invariant under a transformation that maps this metric to another Minkowski metric. We reformulate special relativity to make it invariant under such transformations by introducing a non-dynamical auxiliary tensor field. When we promote this field to be a dynamical gravitational field, it induces a dynamical physical metric. In order to determine the dynamics of this field, we propose an invariance principle, the principle of \\(\\eta\\)-Covariance, which posits that any theory set on a nonempty open subset of the Minkowski manifold must be independent of the choice of the Minkowski metric. We construct a three-parameter theory of gravity which obeys this principle of \\(\\eta\\)-Covariance. The action functional and the field equations of the theory are identical to the ones in New General Relativity, the theory of teleparallel gravity. In addition, the theory admits (i) the traceless canonical energy momentum tensor of the gravitational field, (ii) the divergenceless total canonical energy momentum tensor, (iii) the divergenceless total energy current vector and momentum current vectors and (iv) the conserved total energy and total momentum. All these quantities are independent of the choice of the Minkowski metric. The theory also admits the symmetric, divergenceless total energy momentum tensor.