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D. C. Salisbury

Publications and source records attributed to D. C. Salisbury.

15 recordsLinked to original sources
arXiv2010

Observables in classical canonical gravity: folklore demystified

We give an overview of some conceptual difficulties, sometimes called paradoxes, that have puzzled for years the physical interpetation of classical canonical gravity and, by extension, the canonical formulation of generally covariant theories. We identify these difficulties as stemming form some terminological misunderstandings as to what is meant by "gauge invariance", or what is understood classically by a "physical state". We make a thorough analysis of the issue and show that all purported paradoxes disappear when the right terminology is in place. Since this issue is connected with the search of observables - gauge invariant quantities - for these theories, we formally show that time evolving observables can be constructed for every observer. This construction relies on the fixation of the gauge freedom of diffeomorphism invariance by means of a scalar coordinatization. We stress the condition that the coordinatization must be made with scalars. As an example of our method for obtaining observables we discuss the case of the massive particle in AdS spacetime.

gr-qc↗
arXiv2009

Revisiting observables in generally covariant theories in the light of gauge fixing methods

We derive for generally covariant theories the generic dependency of observables on the original fields, corresponding to coordinate-dependent gauge fixings. This gauge choice is equivalent to a choice of intrinsically defined coordinates accomplished with the aid of spacetime scalar fields. With our approach we make full contact with, and give a new perspective to, the "evolving constants of motion" program. We are able to directly derive generic properties of observables, especially their dynamics and their Poisson algebra in terms of Dirac brackets, extending earlier results in the literature. We also give a new interpretation of the observables as limits of canonical maps.

gr-qc↗
arXiv2009

Gravitational observables, intrinsic coordinates, and canonical maps

It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However their relation to the symmetry group of diffeomorphism transformations has remained obscure. In a symmetry-inspired approach we construct invariants out of canonically induced active gauge transformations. These invariants may be intepreted as the full set of dynamical variables evaluated in the intrinsic coordinate system. The functional invariants can explicitly be written as a Taylor expansion in the coordinates of any observer, and the coefficients have a physical and geometrical interpretation. Surprisingly, all invariants can be obtained as limits of a family of canonical transformations. This permits a short (again geometric) proof that all invariants, including the lapse and shift, satisfy Poisson brackets that are equal to the invariants of their corresponding Dirac brackets.

gr-qc↗
arXiv2007

Rosenfeld, Bergmann, Dirac and the Invention of Constrained Hamiltonian Dynamics

In a paper appearing in Annalen der Physik in 1930 Leon Rosenfeld invented the first procedure for producing Hamiltonian constraints. He displayed and correctly distinguished the vanishing Hamiltonian generator of time evolution, and the vanishing generator of gauge transformations for general relativity with Dirac electron and electrodynamic field sources. Though he did not do so, had he chosen one of his tetrad fields to be normal to his spacetime foliation, he would have anticipated by almost thirty years the general relativisitic Hamiltonian first published by Paul Dirac.

physics.hist-ph↗
arXiv2005

The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity

Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic free particle and the resulting complete set time-dependent invariants for this exactly solvable model are displayed. In contrast with the free particle case, we show that gauge invariants that are simultaneously constants of motion cannot exist in general relativity. They vary with intrinsic time.

gr-qc↗
arXiv2003

Gauge Fixing and Observables in General Relativity

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.

gr-qc↗
arXiv2000

Gauge symmetries in Ashtekar's formulation of general relativity

It might seem that a choice of a time coordinate in Hamiltonian formulations of general relativity breaks the full four-dimensional diffeomorphism covariance of the theory. This is not the case. We construct explicitly the complete set of gauge generators for Ashtekar's formulation of canonical gravity. The requirement of projectability of the Legendre map from configuration-velocity space to phase space renders the symmetry group a gauge transformation group on configuration-velocity variables. Yet there is a sense in which the full four-dimensional diffeomorphism group survives. Symmetry generators serve as Hamiltonians on members of equivalence classes of solutions of Einstein's equations and are thus intimately related to the so-called "problem of time" in an eventual quantum theory of gravity.

gr-qc↗
arXiv1999

The gauge group in the real triad formulation of general relativity

We construct explicitly generators of projectable four-dimensional diffeomorphisms and triad rotation gauge symmetries in a model of vacuum gravity where the fundamental dynamical variables in a Palatini formulation are taken to be a lapse, shift, densitized triad, extrinsic curvature, and the time-like components of the Ricci rotation coefficient. Time-foliation-altering diffeomorphisms are not by themselves projectable under the Legendre transformations. They must be accompanied by a metric- and triad-dependent triad rotation. The phase space on which these generators act includes all of the gauge variables of the model.

gr-qc↗
arXiv1999

Gauge Transformations in Einstein-Yang-Mills Theories

We discuss the relation between spacetime diffeomorphisms and gauge transformations in theories of the Yang-Mills type coupled with Einstein's General Relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Legendre map. Although pure Yang-Mills gauge transformations are projectable by themselves, diffeomorphisms are not. Instead the projectable symmetry group arises from infinitesimal diffeomorphism-inducing transformations which must depend on the lapse function and shift vector of the spacetime metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer.

gr-qc↗
arXiv1999

Gauge group and reality conditions in Ashtekar's complex formulation of canonical gravity

We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar's complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac's method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.

gr-qc↗
arXiv1996

Gauge transformations in the Lagrangian and Hamiltonian formalisms of generally covariant theories

We study spacetime diffeomorphisms in Hamiltonian and Lagrangian formalisms of generally covariant systems. We show that the gauge group for such a system is characterized by having generators which are projectable under the Legendre map. The gauge group is found to be much larger than the original group of spacetime diffeomorphisms, since its generators must depend on the lapse function and shift vector of the spacetime metric in a given coordinate patch. Our results are generalizations of earlier results by Salisbury and Sundermeyer. They arise in a natural way from using the requirement of equivalence between Lagrangian and Hamiltonian formulations of the system, and they are new in that the symmetries are realized on the full set of phase space variables. The generators are displayed explicitly and are applied to the relativistic string and to general relativity.

gr-qc↗
arXiv1994

A Connection Approach to Numerical Relativity

We discuss a general formalism for numerically evolving initial data in general relativity in which the (complex) Ashtekar connection and the Newman-Penrose scalars are taken as the dynamical variables. In the generic case three gauge constraints and twelve reality conditions must be solved. The analysis is applied to a Petrov type \{1111\} planar spacetime where we find a spatially constant volume element to be an appropriate coordinate gauge choice.

gr-qc↗