General relativity as an extended canonical gauge theory (original) (raw)

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.

The generic model of General Relativity

Journal of Physics: Conference Series, 2009

We develop a generic spacetime model in General Relativity which can be used to build any gravitational model within General Relativity. The generic model uses two types of assumptions: (a) Geometric assumptions additional to the inherent geometric identities of the Riemannian geometry of spacetime and (b) Assumptions defining a class of observers by means of their 4-velocity u a which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3 decomposition results in two major results. The 1+3 decomposition of u a;b defines the kinematic variables of the model (expansion, rotation, shear and 4-acceleration) and defines the kinematics of the gravitational model. The 1+3 decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and define the dynamics of he model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the 4-velocity of the observers ua there exists a second universal vector field na in spacetime so that one has a so called double congruence (ua, na) which can be used to define the 1+1+2 decomposition of tensor fields. The 1+1+2 decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables. After presenting and discussing the results in their full generality we show how they are applied in practice by considering in a step by step approach the case of a string fluid in Bianchi I spacetime for the comoving observers.

On canonical transformations between equivalent Hamiltonian formulations of General Relativity

2008

Two Hamiltonian formulations of General Relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to the constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for constrained systems.

Quadratic curvature theories formulated as covariant canonical gauge theories of gravity

Physical Review D

The covariant canonical gauge theory of gravity is generalized by including at the Lagrangian level all possible quadratic curvature invariants. In this approach, the covariant Hamiltonian principle and the canonical transformation framework are applied to derive a Palatini type gauge theory of gravity. The metric g μν , the affine connection γ λ μν and their respective conjugate momenta, k μνσ and q η αξβ tensors, are the independent field components describing the gravity. The metric is the basic dynamical field, and the connection is the gauge field. The torsion-free and metricity-compatible version of the spacetime Hamiltonian is built from all possible invariants of the q η αξβ tensor components up to second order.

New constraints for canonical general relativity

Nuclear Physics B, 1995

Ashtekar's canonical theory of classical complex Euclidean GR (no Lorentzian reality conditions) is found to be invariant under the full algebra of infinitesimal 4-diffeomorphisms, but non-invariant under some finite proper 4-diffeos when the densitized dreibein,Ẽ a i , is degenerate. The breakdown of 4-diffeo invariance appears to be due to the inability of the Ashtekar Hamiltonian to generate births and deaths of E flux loops (leaving open the possibility that a new 'causality condition' forbidding the birth of flux loops might justify the non-invariance of the theory). A fully 4-diffeo invariant canonical theory in Ashtekar's variables, derived from Plebanski's action, is found to have constraints that are stronger than Ashtekar's for rankẼ < 2. The corresponding Hamiltonian generates births and deaths ofẼ flux loops. It is argued that this implies a finite amplitude for births and deaths of loops in the physical states of quantum GR in the loop representation, thus modifying this (partly defined) theory substantially. Some of the new constraints are second class, leading to difficulties in quantization in the connection representation. This problem might be overcome in a very nice way by transforming to the classical loop variables, or the 'Faraday line' variables of Newman and Rovelli, and then solving the offending constraints. Note that, though motivated by quantum considerations, the present paper is classical in substance.

On a Generalized Theory of Relativity

2010

The General Theory of Relativity (GTR) is essentially a theory of gravitation. It is built on the Principle of Relativity. It is bonafide knowledge, known even to Einstein the founder, that the GTR violates the very principle upon which it is founded i.e., it violates the Principle of Relativity; because a central equation i.e., the geodesic law which emerges from the GTR, is well known to be in conflict with the Principle of Relativity because the geodesic law, must in complete violation of the Principle of Relativity, be formulated in special (or privileged) coordinate systems i.e., Gaussian coordinate systems. The Principle of Relativity clearly and strictly forbids the existence/use of special (or privileged) coordinate systems in the same way the Special Theory of Relativity forbids the existence of privileged and or special reference systems. In the pursuit of a more Generalized Theory of Relativity i.e., an all-encampusing unified field theory to include the Electromagnetic, Weak & the Strong force, Einstein and many other researchers, have successfully failed to resolve this problem. In this reading, we propose a solution to this dilemma faced by Einstein and many other researchers i.e., the dilemma of obtaining a more Generalized Theory of Relativity. Our solution brings together the Gravitational, Electromagnetic, Weak & the Strong force under a single roof via an extension of Riemann geometry to a new hybrid geometry that we have coined the Riemann-Hilbert Space (RHS). This geometry is a fusion of Riemann geometry and the Hilbert space. Unlike Riemann geometry, the RHS preserves both the length and the angle of a vector under parallel transport because the affine connection of this new geometry, is a tensor. This tensorial affine leads us to a geodesic law that truly upholds the Principle of Relativity. It is seen that the unified field equations derived herein are seen to reduce to the well known Maxwell-Procca equation, the non-Abelian nuclear force field equations, the Lorentz equation of motion for charged particles and the Dirac equation.

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.

On canonical transformations of gravitational variables in extended phase space

Gravitation and Cosmology, 2011

Last years a certain attention was attracted to the statement that Hamiltonian formulations of General Relativity, in which different parametrizations of gravitational variables were used, may not be related by a canonical transformation. The example was given by the Hamiltonian formulation of Dirac and that of Arnowitt-Deser-Misner. It might witness for non-equivalence of these formulations and the original (Lagrangian) formulation of General Relativity. The problem is believed to be of importance since many authors make use of various representations of gravitational field as a starting point in searching a way to reconcile the theory of gravity with quantum principles. It can be shown that the mentioned above conclusion about non-equivalence of different Hamiltonian formulations is based on the consideration of canonical transformations in phase space of physical degrees of freedom only, while the transformations also involve gauge degrees of freedom. We shall give a clear proof that Hamiltonian formulations corresponding to different parametrizations of gravitational variables are related by canonical transformations in extended phase space embracing gauge degrees of freedom on an equal footing with physical ones. It will be demonstrated for the full gravitational theory in a wide enough class of parametrizations and gauge conditions.

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.