Interacting vector fields in Relativity without Relativity

Barbour, Foster and \'{O} Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This paper completes the recovery of the presently known bosonic sector within the 3-space approach. We show, for a rather general ansatz, that 3-vector fields can interact among themselves only as Yang--Mills fields minimally coupled to gravity.Comment: Replaced with final version accepted by Classical and Quantum Gravity (14 pages, no figures

Scale-Invariant Gravity: Geometrodynamics

We present a scale-invariant theory, conformal gravity, which closely resembles the geometrodynamical formulation of general relativity (GR). While previous attempts to create scale-invariant theories of gravity have been based on Weyl's idea of a compensating field, our direct approach dispenses with this and is built by extension of the method of best matching w.r.t scaling developed in the parallel particle dynamics paper by one of the authors. In spatially-compact GR, there is an infinity of degrees of freedom that describe the shape of 3-space which interact with a single volume degree of freedom. In conformal gravity, the shape degrees of freedom remain, but the volume is no longer a dynamical variable. Further theories and formulations related to GR and conformal gravity are presented. Conformal gravity is successfully coupled to scalars and the gauge fields of nature. It should describe the solar system observations as well as GR does, but its cosmology and quantization will be completely different.Comment: 33 pages. Published version (has very minor style changes due to changes in companion paper

New interpretation of variational principles for gauge theories. I. Cyclic coordinate alternative to ADM split

I show how there is an ambiguity in how one treats auxiliary variables in gauge theories including general relativity cast as 3 + 1 geometrodynamics. Auxiliary variables may be treated pre-variationally as multiplier coordinates or as the velocities corresponding to cyclic coordinates. The latter treatment works through the physical meaninglessness of auxiliary variables' values applying also to the end points (or end spatial hypersurfaces) of the variation, so that these are free rather than fixed. [This is also known as variation with natural boundary conditions.] Further principles of dynamics workings such as Routhian reduction and the Dirac procedure are shown to have parallel counterparts for this new formalism. One advantage of the new scheme is that the corresponding actions are more manifestly relational. While the electric potential is usually regarded as a multiplier coordinate and Arnowitt, Deser and Misner have regarded the lapse and shift likewise, this paper's scheme considers new {\it flux}, {\it instant} and {\it grid} variables whose corresponding velocities are, respectively, the abovementioned previously used variables. This paper's way of thinking about gauge theory furthermore admits interesting generalizations, which shall be provided in a second paper.Comment: 11 page

Penta­aqua­(4,6-dihy­droxy­benzene-1,3-disulfonato-κO 1)zinc penta­hydrate

The ZnII atom in the title complex, [Zn(C6H4O8S2)(H2O)5]·5H2O, is coordinated by five water mol­ecules and an O atom of a 4,6-dihy­droxy­benzene-1,3-disulfonate dianion. The coord­ination geometry is distorted octa­hedral, with the Zn—Osulfonate bond relatively long compared to the Zn—Owater bonds. The coordinated and lattice water mol­ecules inter­act with each other and with the hy­droxy groups and sulfonate ligand through O—H⋯O hydrogen bonds, generating a tightly held three-dimensional network

Triangleland. I. Classical dynamics with exchange of relative angular momentum

In Euclidean relational particle mechanics, only relative times, relative angles and relative separations are meaningful. Barbour--Bertotti (1982) theory is of this form and can be viewed as a recovery of (a portion of) Newtonian mechanics from relational premises. This is of interest in the absolute versus relative motion debate and also shares a number of features with the geometrodynamical formulation of general relativity, making it suitable for some modelling of the problem of time in quantum gravity. I also study similarity relational particle mechanics (`dynamics of pure shape'), in which only relative times, relative angles and {\sl ratios of} relative separations are meaningful. This I consider firstly as it is simpler, particularly in 1 and 2 d, for which the configuration space geometry turns out to be well-known, e.g. S^2 for the `triangleland' (3-particle) case that I consider in detail. Secondly, the similarity model occurs as a sub-model within the Euclidean model: that admits a shape--scale split. For harmonic oscillator like potentials, similarity triangleland model turns out to have the same mathematics as a family of rigid rotor problems, while the Euclidean case turns out to have parallels with the Kepler--Coulomb problem in spherical and parabolic coordinates. Previous work on relational mechanics covered cases where the constituent subsystems do not exchange relative angular momentum, which is a simplifying (but in some ways undesirable) feature paralleling centrality in ordinary mechanics. In this paper I lift this restriction. In each case I reduce the relational problem to a standard one, thus obtain various exact, asymptotic and numerical solutions, and then recast these into the original mechanical variables for physical interpretation.Comment: Journal Reference added, minor updates to References and Figure

2-[((E)-2-{2-[(E)-2-Hy­droxy­benzyl­idene]hydrazinecarbon­yl}hydrazinyl­idene)meth­yl]phenol

The mol­ecule of the title compound, C15H14N4O3, is completed by the application of crystallographic twofold symmetry, with the carbonyl group lying on the rotation axis. The mol­ecule is close to planar: the greatest deviation of a torsion angle from 0° is 7.3 (2)° about the bond linking the phenol ring to the rest of the mol­ecule. An intra­molecular O—H⋯N(imine) hydrogen bond is formed in each half of the mol­ecule. The carbonyl O atom is anti with respect to the amine H atoms and this allows for the formation of N—H⋯O(hydrox­yl) hydrogen bonds in the crystal, which results in supra­molecular layers lying parallel to (100)

N′-[(2Z)-4-Oxo-4-phenyl­but-2-en-2-yl]pyridine-4-carbohydrazide

There are significant twists in the title compound, C16H15N3O2, as seen in the dihedral angle between the benzene and adjacent but-2-enal group [29.26 (4)°] and between the pyridine ring and amide group [24.79 (6)°]. A twist is also evident around the hydrazine bond [the C—N—N—C torsion angle is −138.25 (13)°]. The conformation about the ethene bond is Z. An intra­molecular N—H⋯O hydrogen bond involving the benzoyl O atom and leading to an S(6) motif is formed. Significant delocalization of π-electron density is found in this part of the mol­ecule. In the crystal, helical supra­molecular chains aligned along the b axis and mediated by N—H⋯O hydrogen bonds are formed

Chlorido{4-cyclo­hexyl-1-[1-(pyridin-2-yl-κN)ethyl­idene]thio­semicarbazidato-κ2 N 1,S}diphenyl­tin(IV)

The distorted octa­hedral geometry about the SnIV atom in the title compound, [Sn(C6H5)2(C14H19N4S)Cl], is defined by the N,N,S-tridentate Schiff base ligand, two mutually trans ipso-C atoms of the Sn-bound phenyl groups, and the Cl atom which is trans to the azo N atom. The two five-membered chelate rings and pyridyl ring are almost coplanar with the dihedral angle between the outer five-membered chelate and pyridine rings being 5.39 (8)°. Centrosymmetric dimers feature in the crystal packing mediated by N—H⋯S hydrogen bonds, leading to eight-membered {⋯HNCS}2 synthons. The dimeric aggregates are connected into a three-dimensional architecture by C—H⋯Cl and C—H⋯π inter­actions, as well as π–π inter­actions occurring between centrosymmetrically related pyridine rings [centroid–centroid distance = 3.6322 (13) Å]

Emergent Semiclassical Time in Quantum Gravity. Full Geometrodynamics and Minisuperspace Examples

I apply the preceding paper's semiclassical treatment to geometrodynamics. The analogy between the two papers is quite useful at the level of the quadratic constraints, while I document the differences between the two due to the underlying differences in their linear constraints. I provide a specific minisuperspace example for my emergent semiclassical time scheme and compare it with the hidden York time scheme. Overall, interesting connections are shown between Newtonian, Leibniz--Mach--Barbour, WKB and cosmic times, while the Euler and York hidden dilational times are argued to be somewhat different from these.Comment: References Update

Relational Particle Models. II. Use as toy models for quantum geometrodynamics

Relational particle models are employed as toy models for the study of the Problem of Time in quantum geometrodynamics. These models' analogue of the thin sandwich is resolved. It is argued that the relative configuration space and shape space of these models are close analogues from various perspectives of superspace and conformal superspace respectively. The geometry of these spaces and quantization thereupon is presented. A quantity that is frozen in the scale invariant relational particle model is demonstrated to be an internal time in a certain portion of the relational particle reformulation of Newtonian mechanics. The semiclassical approach for these models is studied as an emergent time resolution for these models, as are consistent records approaches.Comment: Replaced with published version. Minor changes only; 1 reference correcte