Gravity, gauge theories and geometric algebra

A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the position and orientation of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic’ relations between physical fields. For a wide range of phenomena, including all present experimental tests, the theory reproduces the predictions of general relativity. Differences do emerge, however, through the first-order nature of the equations and the global properties of the gauge fields, and through the relationship with quantum theory. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action which is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology, and to the formation and properties of black holes. Insistence on finding global solutions, together with the first-order nature of the equations, leads to a new understanding of the role played by time reversal. This alters the physical picture of the properties of a horizon around a black hole. The existence of global solutions enables one to discuss the properties of field lines inside the horizon due to a point charge held outside it. The Dirac equation is studied in a black hole background and provides a quick (though ultimately unsound) derivation of the Hawking temperature. Some applications to cosmology are also discussed, and a study of the Dirac equation in a cosmological background reveals that the only models consistent with homogeneity are spatially flat. It is emphasised throughout that the description of gravity in terms of gauge fields, rather than spacetime geometry, leads to many simple and powerful physical insights. The language of `geometric algebra’ best expresses the physical and mathematical content of the theory and is employed throughout. Methods for translating the equations into other languages (tensor and spinor calculus) are given in appendices.

A. N. Lasenby, C. J. L. Doran and S. F. Gull, Gravity, gauge theories and geometric algebra, Phil. Trans. R. Soc. Lond. A356, 487-582 (1998)