Spacetime Geometry with Geometric Calculus

Abstract

Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides mathematical tools that streamline the formulation and simplify calculations. The formalism automatically includes spinors so the Dirac equation is incorporated in a geometrically natural way.

Appendix. Comparison of flat and curved space formulations

This appendix is for readers who wish compare the gauge theory formulation of GR for curved space given in this paper with the flat space formulation given in [11].

Table 1 Coordinate frames for flat and curved spacetime

The flat space and curved space theories differ primarily in their use of coordinates. Corresponding quantities are listed in Table 1. I have deliberately used the same symbol $\underline{h}$ for the fiducial tensor in curved space and for the gauge tensor in flat space to facilitate comparison. In surveying Table 1 it will be noticed that the fiducial tensor corresponds to the inverse of the gauge tensor. That trivial difference has been introduced for notational reasons, but it emphasizes that the two tensors map most naturally in opposite directions. The really significant difference is that the fiducial tensor is coordinate dependent whereas the gauge tensor is not. This comes about because $\{\gamma _\mu =\underline{h}^{- 1}(\partial _\mu x)\}$ is necessarily an orthonormal frame in the fiducial case, whereas in the gauge case, $\{e_\mu =\partial _\mu x\}$ is an arbitrary coordinate frame that is completely decoupled from the gauge tensor. In other words, the remapping of events in spacetime is completely decoupled from changes in coordinates in the gauge theory, whereas the curved space theory has no means to separate passive coordinate changes from shifts in physical configurations. This crucial fact is the reason why in Gauge Theory Gravity the Displacement Gauge Principle has clear physical consequences, whereas in the curved space theory Einstein’s General Relativity Principle does not.

Table 2 Comparison of coderivatives and connexions

Mathematical features of the coderivative for flat and curved spacetime are compared in Table  2. Note, in particular, that expressions for $D_\mu M$ have the same form in each case. However, they behave differently under rotation gauge transformations. Whereas the “curved version” simply changes its functional form, the “flat version” transforms according to

$$\begin{aligned} \bar{L}: D_\mu M\quad \rightarrow \quad \bar{L}(D_\mu M)=D'_\mu M'=\partial _\mu M'+\omega '_\mu \times M'\,, \end{aligned}$$

(124)

induced by the active rotation gauge transformation

$$\begin{aligned} \bar{L}: M\quad \rightarrow \quad M'= \bar{L}M\equiv LML\,. \end{aligned}$$

(125)

In other words, rotation gauge transformations are represented as passive in the curved version but active in the flat version. This difference translates to a difference in physical interpretation. In this paper we have interpreted passive rotations as expressing equivalence of physics with respect to different inertial reference frames. In the flat theory, however, covariance under active rotations expresses physical equivalence of different directions in spacetime. Thus, “passive equivalence” is an equivalence of observers, while “active equivalence” is an equivalence of states. This distinction generalizes to the interpretation of any relativity (symmetry group) principle: Active transformations relate equivalent physical states; passive transformations relate equivalent observers.

As Table 2 shows, the use of common tools of Geometric Calculus for both curved and flat space versions of GR has enabled us to define a coderivative with the same form on both versions, despite differences in the way that fields are attached to the base manifold. It follows that all computations with coderivatives have the same mathematical form in both versions; this includes the curvature tensor and all its properties as well as the whole panoply of GR. Accordingly, all such results in this paper and in [11] are identical, so further discussion is unnecessary. By the way, this fact can be regarded as a proof of equivalence of Einstein’s curved space GR with flat space Gauge Theory Gravity.