Progress in Clifford Space Gravity

Clifford-space Gravity is revisited and new results are found. The Clifford space (C-space) generalized gravitational field equations are obtained from a variational principle and which is based on an extension of the Einstein-Hilbert-Cartan action. One of the main results of this work is that the C-space connection requires torsion in order to have consistency between the Clifford algebraic structure and the zero nonmetricity condition ${\nabla{\kappa g}^{MN} = 0}$ . A discussion on the field equations, the cosmological constant, dark energy and multi-metric theories of gravity follows. We continue by pointing out the relations of Clifford space gravity to Lanczos-Lovelock-Cartan (LLC) higher curvature gravity with torsion. We finalize by describing the Clifford bundle approach to C-space gravity and by pointing out that C-space gravity involves higher-spins beyond spin 2 and argue why one could view the LLC higher curvature actions, and other extended gravitational theories based on f(R), f(R _μν ), . . . actions, for polynomial-valued functions, as effective actions after integrating the C-space gravitational action with respect to all the poly-coordinates, except the vectorial ones x ^μ .