Gravity, gauge theories and geometric algebra (original) (raw)
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A Gauge-theoretical Treatment of the Gravitational Field: Classical
2008
In the geometrodynamical setting of general relativity one is concerned mainly with Riemannian metrics over a manifold M . We show that for the space M := Riem(M), we have a natural principal fiber bundle (PFB) structure Diff(M) →֒ M π → M/Diff(M), first hinted at in [1]. This construction makes the gravitational field amenable to exactly the same gauge-theoretic treatment given in [2], where it is used to separate rotational and vibrational degrees of freedom of n-particle systems, both classically and quantum mechanically. Furthermore, we show how the gauge connection in this PFB setting can be seen as a realization of Mach’s Principle of Relative Motion, in accordance with Barbour’s et al work on timeless gravitational theories [3] using best-matching. We show Barbour’s reconstruction of GR is obtained by requiring the connection to be the one induced by the deWitt metric in M. As a simple application of the gauge theory, we put the ADM lagrangian in a Kaluza-Klein context, in wh...
Proceedings of 7th International Conference on Mathematical Methods in Physics — PoS(ICMP 2012), 2013
Pure gauge theories for de Sitter, anti de Sitter and orthogonal groups, in four-dimensional Euclidean spacetime, are studied. It is shown that, if the theory is asymptotically free and a dynamical mass is generated, then an effective geometry may be induced and a gravity theory emerges.
Geometric scalar theory of gravity
Journal of Cosmology and Astroparticle Physics, 2013
We present a geometric scalar theory of gravity. Our proposal will be described using the "background field method" introduced by Gupta, Feynman, Deser and others as a field theory formulation of general relativity. We analyze previous criticisms against scalar gravity and show how the present proposal avoids these difficulties. This concerns not only the theoretical complaints but also those related to observations. In particular, we show that the widespread belief of the conjecture that the source of scalar gravity must be the trace of the energy-momentum tensor-which is one of the main difficulties to couple gravity with electromagnetic phenomenon in previous models-does not apply to our geometric scalar theory. Some consequences of the new scalar theory are explored.
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.
Cartan gravity, matter fields, and the gauge principle
Annals of Physics, 2013
Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang-Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: 1) a 'contact vector' V A which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being 'rolled' on top of it, and 2) a gauge connection A AB µ , here taken to be valued in the Lie algebra of SO(2, 3) or SO(1, 4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan's geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. Furthermore, the energy-momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy-momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang-Mills fields. This suggests a way to unify a U (1) gauge field with gravity into a SO(1, 5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1, 3) × U (1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.
Gauge Freedom and Relativity: A Unified Treatment of Electromagnetism, Gravity and the Dirac Field
The geometric properties of General Relativity are reconsidered as a particular nonlin- ear interaction of fields on a flat background where the perceived geometry and coordi- nates are “physical” entities that are interpolated by a patchwork of observable bodies with a nonintuitive relationship to the underlying fields. This more general notion of gauge in physics opens an important door to put all fields on a similar standing but requires a careful reconsideration of tensors in physics and the conventional wisdom surrounding them. The meaning of the flat background and the induced conserved quantities are discussed and contrasted with the “observable” positive definite energy and probability density in terms of the induced physical coordinates. In this context, the Dirac matrices are promoted to dynamic proto-gravity fields and the keeper of “phys- ical metric” information. Independent sister fields to the wavefunctions are utilized in a bilinear rather than a quadratic lagrangian in these fields. This construction greatly enlarges the gauge group so that now proving causal evolution, relative to the physical metric, for the gauge invariant functions of the fields requires both the stress-energy conservation and probability current conservation laws. Through a Higgs-like coupling term the proto-gravity fields generate a well defined physical metric structure and gives the usual distinguishing of gravity from electromagnetism at low energies relative to the Higgs-like coupling. The flat background induces a full set of conservation laws but results in the need to distinguish these quantities from those observed by recording devices and observers constructed from the fields.
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.
Canonical transformation path to gauge theories of gravity
Physical Review D, 2017
In this paper, the generic part of the gauge theory of gravity is derived, based merely on the action principle and on the general principle of relativity. We apply the canonical transformation framework to formulate geometrodynamics as a gauge theory. The starting point of our paper is constituted by the general De Donder-Weyl Hamiltonian of a system of scalar and vector fields, which is supposed to be form-invariant under (global) Lorentz transformations. Following the reasoning of gauge theories, the corresponding locally form-invariant system is worked out by means of canonical transformations. The canonical transformation approach ensures by construction that the form of the action functional is maintained. We thus encounter amended Hamiltonian systems which are form-invariant under arbitrary spacetime transformations. This amended system complies with the general principle of relativity and describes both, the dynamics of the given physical system's fields and their coupling to those quantities which describe the dynamics of the spacetime geometry. In this way, it is unambiguously determined how spin-0 and spin-1 fields couple to the dynamics of spacetime. A term that describes the dynamics of the "free" gauge fields must finally be added to the amended Hamiltonian, as common to all gauge theories, to allow for a dynamic spacetime geometry. The choice of this "dynamics" Hamiltonian is outside of the scope of gauge theory as presented in this paper. It accounts for the remaining indefiniteness of any gauge theory of gravity and must be chosen "by hand" on the basis of physical reasoning. The final Hamiltonian of the gauge theory of gravity is shown to be at least quadratic in the conjugate momenta of the gauge fields-this is beyond the Einstein-Hilbert theory of general relativity.
General relativity is a gauge type theory
Letters in Mathematical Physics, 1981
It is shown that the Einstein-Maxwell theory ofinteracting electromagnetism and gravitation, can be derived from a first-order Lagrangian, depending on the electromagnetic field and on the curvature of a symmetric affine connection r on the space-time M. The variation is taken with respect to the electromagnetic potential (a connection on a D(1) principal fiber bundle on M) and the 'gravitational potential' r (a connection on the GL(4, R) principal fiber bundle of frames on M). The metric tensor g does not appear in the Lagrangian, but it arises as a momentum canonically conjugated to r.The Lagrangians of this type are calculated also for the Proca field, for a charge d scalar field interacting with electromagnetism and gravitation, and for a few other interesting physical theories.