Spacetimes between Einstein and Kaluza-Klein

Extended general relativity is the extension of general relativity from the tensor algebra to its own universal covering algebra. The universal covering algebra involves all direct sums of tensors of different rank, elements called tensor multinomials. Focusing on a subalgebra consisting of direct sums of scalar, vector, and second rank tensor fields only, results in geometric objects as spacious as those in five-dimensional Kaluza-Klein spacetimes while remaining defined over only four space-time dimensions, hence may be thought of as being between the theories of Einstein and Kaluza and Klein. It is possible to deduce the equivalent of Christoffel symbols from the equivalent geodesic equation, and from the Christoffel symbols, the extended Ricci curvature. The extended Ricci curvature yields the Ricci tensor of general relativity, Maxwell's electromagnetism, and a Kaluza-Klein scalar field equation as components of a single second-rank tensor multinomial equation. The scalar field couples naturally to the square of the electric charge, leading to a new force that may be significant only in collapsed matter or in a cosmological setting. There are numerous possible new Lagrangians unique to this formalism, and rich new ways of introducing scalar fields and accelerated cosmological expansion. In one of many such solutions, a scalar field source that falls off with expanding volume forces a dark energy equation of state.