February 15, 2007
In Euclidean relational particle mechanics (ERPM) only relative times, relative angles and relative separations are meaningful, while in similarity relational particle mechanics (SRPM) only relative times, relative angles and ratios of relative separations are. These theories are clearly of interest in the absolute or relative motion debate. In this paper, ERPM and SRPM are provided in fully reduced form for 3 particles in 2D, i.e. the classical dynamics on triangleland in 2D with and without scale. Exact solutions to each of these are then found, and simple Newton--Coulomb-like and harmonic oscillator-like SRPM models are studied numerically. The mathematics one arrives at thus overlaps in many ways with that which arises in the absolutist approach. The ERPM gives standard mathematics, while the SRPM has standard small-relative-scale behaviour and an unexpected but in itself standard universal large-relative-scale behaviour. One way in which SRPM is unusual is that it is a model in which a symmetry principle underlies an unexpected departure from standard physical behaviour at sufficiently large relative scales (interpolation between the abovementioned two behaviours). ERPM and SRPM are also theoretically interesting at the quantum level, both on their own merit and as toy models for the development of various approaches to the problems of time and of observables in quantum general relativity.
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September 6, 2008
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 prob...
September 20, 2008
Relational particle models are of value in the absolute versus relative motion debate. They are also analogous to the dynamical formulation of general relativity, and as such are useful for investigating conceptual strategies proposed for resolving the problem of time in quantum general relativity. Moreover, to date there are few explicit examples of these at the quantum level. In this paper I exploit recent geometrical and classical dynamics work to provide such a study base...
May 14, 2010
In scaled relational particle mechanics, only relative times, relative angles and relative separations are meaningful. It arose in the study of the absolute versus relative motion debate. It has then turned out to be a useful toy model of classical and quantum general relativity, such as for investigating conceptual strategies for the problem of time. This paper studies the 3-particle 2-d scaled relational particle model, for which the configurations are scaled triangles. The...
February 19, 2012
Relational particle mechanics models bolster the relational side of the absolute versus relational motion debate, and are additionally toy models for the dynamical formulation of General Relativity and its Problem of Time. They cover two aspects that the more commonly studied minisuperspace General Relativity models do not: 1) by having a nontrivial notion of structure and thus of cosmological structure formation and of localized records. 2) They have linear as well as quadra...
September 11, 2010
I investigate useful shape quantities for the classical and quantum mechanics of the relational quadrilateral in 2-d. This is relational in the sense that only relative times, relative ratios of separations and relative angles are significant. Relational particle mechanics models such as this paper's have many analogies with the geometrodynamical formulation of general relativity. This renders them suitable as toy models for 1) studying Problem of Time in Quantum Gravity stra...
September 15, 2005
It is argued that substantial portions of both Newtonian particle mechanics and general relativity can be viewed as relational (rather than absolute) theories. I furthermore use the relational particle models as toy models to investigate the problem of time in closed-universe canonical quantum general relativity. I consider thus in particular the internal time, semiclassical and records tentative resolutions of the problem of time.
November 14, 2005
This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question -- universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities -- is capable of accommodating a wide range of class...
June 27, 2007
Relational particle dynamics include the dynamics of pure shape and cases in which absolute scale or absolute rotation are additionally meaningful. These are interesting as regards the absolute versus relative motion debate as well as discussion of conceptual issues connected with the problem of time in quantum gravity. In spatial dimension 1 and 2 the relative configuration spaces of shapes are n-spheres and complex projective spaces, from which knowledge I construct natural...
November 7, 2011
This article contains a local solution to the notorious Problem of Time in Quantum Gravity at the conceptual level and which is actually realizable for the relational triangle. The Problem of Time is that `time' in GR and `time' in ordinary quantum theory are mutually incompatible notions, which is problematic in trying to put these two theories together to form a theory of Quantum Gravity. Four frontiers to this resolution in full GR are identified, alongside three further d...
January 7, 2010
Relational particle mechanics models (RPM's) are useful models for the problem of time in quantum gravity and other foundational issues in quantum cosmology. Some concrete examples of scalefree RPM's have already been studied, but it is the case with scale that is needed for the semiclassical and dilational internal time approaches to the problem of time. In this paper, I show that the scaled RPM's configuration spaces are the cones over the scalefree RPM's configuration spac...