Higher Derivative Gravity and Torsion from the Geometry of C-spaces

TL;DR

This paper explores a novel geometric framework called Clifford space ($C$-space), extending spacetime to include higher-dimensional objects, revealing a connection between derivatives in $C$-space and physical concepts like curvature and torsion, with implications for unifying theories.

Contribution

It introduces the geometry of curved $C$-space with holographic coordinates and links derivatives with curvature and torsion, advancing the geometric foundation for $M$-theory.

Findings

Curvature in $C$-space contains higher orders of ordinary space curvature.

Dependence on $\sigma^{\mu u}$ indicates torsion presence.

Dependence on vector quantities implies non-vanishing curvature.

Abstract

We start from a new theory (discussed earlier) in which the arena for physics is not spacetime, but its straightforward extension-the so called Clifford space ($C$-space), a manifold of points, lines, areas, etc..; physical quantities are Clifford algebra valued objects, called polyvectors. This provides a natural framework for description of supersymmetry, since spinors are just left or right minimal ideals of Clifford algebra. The geometry of curved $C$-space is investigated. It is shown that the curvature in $C$-space contains higher orders of the curvature in the underlying ordinary space. A $C$-space is parametrized not only by 1-vector coordinates $x^\mu$ but also by the 2-vector coordinates $\sigma^{\mu \nu}$, 3-vector coordinates $\sigma^{\mu \nu \rho}$, etc., called also {\it holographic coordinates}, since they describe the holographic projections of 1-lines, 2-loops, 3-loops, etc., onto the coordinate planes. A remarkable relation between the "area" derivative $\p/ \p \sigma^{\mu \nu}$ and the curvature and torsion is found: if a scalar valued quantity depends on the coordinates $\sigma^{\mu \nu}$ this indicates the presence of torsion, and if a vector valued quantity depends so, this implies non vanishing curvature. We argue that such a deeper understanding of the $C$-space geometry is a prerequisite for a further development of this new theory which in our opinion will lead us towards a natural and elegant formulation of $M$-theory.