Spacetime Geometry with Geometric Calculus (original) (raw)

Gauge Theory Gravity with Geometric Calculus

Foundations of Physics, 2005

A new gauge theory of gravity on flat spacetime has recently been developed by Lasenby, Doran, and Gull. Einstein's principles of equivalence and general relativity are replaced by gauge principles asserting, respectively, local rotation and global displacement gauge invariance. A new unitary formulation of Einstein's tensor illuminates long-standing problems with energy-momentum conservation in general relativity. Geometric calculus provides many simplifications and fresh insights in theoretical formulation and physical applications of the theory.

SPACETIME CALCULUS for GRAVITATION THEORY

A new gauge theory of gravitation on ∞at spacetime has recently been developed by Lasenby, Doran, and Gull in the language of Geometric Calculus. This paper provides a systematic account of the mathemati- cal formalism to facilitate applications and extensions of the theory. It includes formulations of difierential geometry, Lie derivatives and inte- grability theorems which are coordinate-free and gauge-covariant. Em- phasis is on use of the language to express physical and geometrical concepts.

Spacetime physics with geometric algebra

American Journal of Physics, 2003

This is an introduction to spacetime algebra (STA) as a unified mathematical language for physics. STA simplifies, extends and integrates the mathematical methods of classical, relativistic and quantum physics while elucidating geometric structure of the theory. For example, STA provides a single, matrixfree spinor method for rotational dynamics with applications from classical rigid body mechanics to relativistic quantum theory-thus significantly reducing the mathematical and conceptual barriers between classical and quantum mechanics. The entire physics curriculum can be unified and simplified by adopting STA as the standard mathematical language. This would enable early infusion of spacetime physics and give it the prominent place it deserves in the curriculum.

Geometric Algebra for Special Relativity and Manifold Geometry

This thesis is a study of geometric algebra and its applications to relativistic physics. Geometric algebra (or real Clifford algebra) serves as an efficient language for describing rotations in vector spaces of arbitrary metric signature, including Lorentzian spacetime. By adopting the rotor formalism of geometric algebra, we derive an explicit BCHD formula for composing Lorentz transformations in terms of their generators — much more easily than with traditional matrix representations. This is used to straightforwardly derive the composition law for Lorentz boosts and the concomitant Wigner angle. Later, we include a gentle introduction to differential geometry, noting how the Lie derivative and covariant derivative assume compact forms when expressed with geometric algebra. Curvature is studied as an obstruction to the integrability of the parallel transport equations, and we present a surface-ordered Stokes’ theorem relating the ‘enclosed curvature’ in a surface to the holonomy ...

Gravity, gauge theories and geometric algebra

Philosophical Transactions of The Royal Society A: Mathematical, Physical and Engineering Sciences, 1998

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 positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of

Fundamental geometric structures for the Dirac equation in General Relativity

Acta Applicandae Mathematicae, 1998

We present an axiomatic approach to Dirac's equation in General Relativity based on intrinsically covariant geometric structures. Structure groups and the related principal bundle formulation can be recovered by studying the automorphisms of the theory. Various aspects can be most neatly understood within this context, and a number of questions can be most properly addressed (specifically in view of the formulation of QFT on a curved background). In particular, we clarify the fact that the usual spinor structure can be weakened while retaining all essential physical aspects of the theory.

Reconsiderations on the formulation of general relativity based on Riemannian structures

General Relativity and Gravitation, 2011

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BFlike field theory structure of the Einstein-Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern-Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten's formulation of threedimensional gravity as a Chern-Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.

Geometry of Gravitation and Electromagnetism

Physical Review, 1959

An independent derivation is given of equations first derived by Rainich which show how, under certain circumstances, the combined theory of gravitation and electromagnetism of Einstein and Maxwell can be unified and described exclusively in terms of geometry. Some algebraic relations are presented between the Ricci tensor, the electromagnetic field tensor, and their principal null vectors. It is shown that in regions of space-time where the two invariants of the electromagnetic field both vanish, the unified theory cannot apply. Either such regions do not exist in nature or their description in terms of pure geometry has yet to be found. Advantage is taken of the correspondences between tensors and spinors to carry out most of the present calculations in spinor space.

Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity

Clifford Algebras, 2004

We review the applications of geometric algebra in electromagnetism, gravitation and multiparticle quantum systems. We discuss a gauge theory formulation of gravity and its implementation in geometric algebra, and apply this to the fermion bound state problem in a black hole background. We show that a discrete energy spectrum arises in an analogous way to the hydrogen atom. A geometric algebra approach to multiparticle quantum systems is given in terms of the multiparticle spacetime algebra. This is applied to quantum information processing, multiparticle wave equations and to conformal geometry. The application to conformal geometry highlight some surprising links between relativistic quantum theory, twistor theory and de Sitter spaces.