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ON THE RECOVERY OF GEOMETRODYNAMICS FROM TWO DIFFERENT SETS OF FIRST PRINCIPLES
, 2006
"... The conventional spacetime formulation of general relativity may be recast as a dynamics of spatial 3geometries (geometrodynamics). Furthermore, geometrodynamics can be derived from first principles. I investigate two distinct sets of these: (i) Hojman, Kuchaˇr and Teitelboim’s, which presuppose th ..."
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The conventional spacetime formulation of general relativity may be recast as a dynamics of spatial 3geometries (geometrodynamics). Furthermore, geometrodynamics can be derived from first principles. I investigate two distinct sets of these: (i) Hojman, Kuchaˇr and Teitelboim’s, which presuppose that the spatial 3geometries are embedded in spacetime. (ii) The 3space approach of Barbour, Foster, Ó Murchadha and Anderson in which the spatial 3geometries are presupposed but spacetime is not. I consider how the constituent postulates of the conventional approach to relativity emerge or are to be built into these formulations. I argue that the 3space approach is a viable description of classical physics (fundamental matter fields included), and one which affords considerable philosophical insight because of its ‘relationalist’ character. From these assumptions of less structure, it is also interesting that conventional relativity can be recovered (albeit as one of several options). However, contrary to speculation in the earlier 3space approach papers, I also argue that this approach is not selective over which sorts of fundamental matter physics it admits. In particular, it does not imply the equivalence principle.
Relational particle models: I. Reconciliation with standard classical and quantum theory Classical and Quantum Gravity
"... 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 ..."
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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 classical physics phenomena. Furthermore, as I develop in Paper II, this relational particle model is a useful toy model for canonical general relativity. I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I provide many examples of quantized relational particle models and interpret them carefully from a relational perspective. Thus previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for closed universe quantum cosmology, while these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.
Relational particle models: II. Use as toy models for quantum geometrodynamics Classical and Quantum Gravity
 Classical and Quantum Gravity
, 2006
"... 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 perspe ..."
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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. Attempts at quantizing general relativity (GR) significantly underly Wheeler’s manyroutes perspective [1, 2, 3, 4, 5], in which Einstein’s [6] traditional ‘spacetime ’ route to GR is viewed not as the route to GR but as the first route to GR, which is additionally reachable along a number of a priori unrelated routes. E.g. six routes to GR are listed in [3]. Of these, the second (Einstein–Hilbert action) route, and the third and fourth routes (the twoway
Classical Machian Resolution of the Spacetime Reconstruction Problem
"... Following from a question of Wheeler, why does the Hamiltonian constraint H of GR have the particular form it does?. A first answer, by Hojman, Kuchař and Teitelboim, is that using embeddability into spacetime as a principle gives the form of H. The present paper culminates a second Machian answer – ..."
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Following from a question of Wheeler, why does the Hamiltonian constraint H of GR have the particular form it does?. A first answer, by Hojman, Kuchař and Teitelboim, is that using embeddability into spacetime as a principle gives the form of H. The present paper culminates a second Machian answer – initially by Barbour, Foster and ó Murchadha – in which space but not spacetime are assumed. Thus this answer is additionally a classicallevel resolution of the spacetime reconstruction problem. In this approach, mere consistency imposed by the Dirac procedure whittles down a general ansatz to one of four alternatives: Lorentzian, Galilean, or Carrollian relativity, or constant mean curvature slicing. These arise together as the different ways to kill off a 4factor obstruction term. It is novel for such an alternative to arise from principles of dynamics considerations (in contrast with the historical form of the dichotomy between universal local Galilean or Lorentzian relativity). It is furthermore intriguing that it gives constant mean curvature slicing – familiar from York’s work on the initial value problem – as a further option on a similar footing. That is related to a number of recent alternative theories/formulations of GR known collectively as ‘shape dynamics’. The original work did not treat this with Poisson brackets and a proper systematic Diractype analysis; we rectify this in this paper. It is also the first demonstration of how this approach solves the classical spacetime reconstruction problem via ‘hypersurface tensor dual nationality ’ and what can be interpreted as embedding equations arising. 1
The problem of time and quantum cosmology in the relational particle mechanics arena
, 2012
"... Most readers’ principal interest in this article will be its reviews of the problem of time in quantum GR in Secs 11 and 16. Namely, 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 ..."
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Most readers’ principal interest in this article will be its reviews of the problem of time in quantum GR in Secs 11 and 16. Namely, 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. This article also establishes relational particle models (RPM’s) as useful models for many parts of the study of the problem of time and of quantum cosmology. It also unlocks RPM’s, expositing how to understand the configuration spaces, classical dynamics and quantum mechanics of these in simple concrete examples. It then uses these to further our understanding of the histories, records, semiclassical, näıve Schrödinger interpretation and hidden time strategies toward resolving the Problem of Time, alongside consideration of Halliwell’s combination of the first three. This article’s relational wholeuniverse models are comparable to minisuperspace in amount of resemblance to full GR, but with a number of different resemblances including various midisuperspacelike ones which subsequently render RPM’s appropriate as Problem of Time models. One point of view is that the best one can currently do as regards quantum gravity is to compare multiple such toy models, each of which allows for a different range of calculations that are too difficult to carry out for GR to be substantially completed. The current article is then the RPM counterpart of Ryan’s book on minisuperspace models or Carlip’s on 2 + 1 GR, as regards each of these arenas being rendered open to detailed study. Other results covered include suitable variational techniques
Leibniz–Mach foundations for GR and fundamental physics
 Horizons in world physics
, 2005
"... Consider the configuration space Q for some physical system, and a continuous group of transformations G whose action on the configurations is declared to be physically irrelevant. G is to be implemented indirectly by adjoining 1 auxiliary g per independent generator to Q, by writing the action for ..."
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Consider the configuration space Q for some physical system, and a continuous group of transformations G whose action on the configurations is declared to be physically irrelevant. G is to be implemented indirectly by adjoining 1 auxiliary g per independent generator to Q, by writing the action for the system in an arbitrary Gframe, and then passing to the quotient Q/G thanks to the constraints encoded by variation w.r.t the g’s. I show that this arbitrary Gframe principle supercedes (and indeed leads to a derivation of) the Barbour–Bertotti best matching principle. I furthermore consider absolute external time to be meaningless for the universe as a whole. For various choices of Q and G, these Leibniz–Mach considerations lead to Barbour–Bertotti’s proposed absolutestructurefree replacement of Newtonian Mechanics, to Gauge Theory, and to the 3space approach (TSA) formulation of General Relativity (GR). For the TSA formulation of GR with matter fields, I clarify how the Special Relativity postulates emerge, discuss whether the Principle of Equivalence emerges, and study which additional simplicity postulates are required. I further develop my explanation of how a full enough set of fundamental matter fields to describe nature can be accommodated in the TSA formulation, and further compare the TSA with the ‘split spacetime formulation ’ of Kuchaˇr. I explain the emergence of broken and unbroken Gauge Theories as a consequence of the Principle of Equivalence. Whereas for GR one would usually quotient out 3diffeomorphisms, I also consider as further examples of the arbitrary Gframe principle the further quotienting out of conformal transformations or volumepreserving conformal transformations. Depending on which choices are made, this leads to York’s initial value formulation (IVF) of GR, new alternative foundations for the GR IVF, or alternative theories of gravity which are built out of similar conformal mathematics to the GR IVF and yet admit no GRlike spacetime interpretation. 1
DOES RELATIONALISM ALONE CONTROL GEOMETRODYNAMICS WITH SOURCES?
, 711
"... This paper concerns relational first principles from which the Dirac procedure exhaustively picks out the geometrodynamics corresponding to general relativity as one of a handful of consistent theories. This was accompanied by a number of results and conjectures about matter theories and general fea ..."
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This paper concerns relational first principles from which the Dirac procedure exhaustively picks out the geometrodynamics corresponding to general relativity as one of a handful of consistent theories. This was accompanied by a number of results and conjectures about matter theories and general features of physics – such as gauge theory, the universal light cone principle of special relativity and the equivalence principle – being likewise picked out. I have previously shown that many of these matter results and conjectures are contingent on further unrelational simplicity assumptions. In this paper, I point out 1) that the exhaustive procedure in these cases with matter fields is slower than it was previously held to be. 2) While the example of equivalence principle violating matter theory that I previously showed how to accommodate on relational premises has a number of pathological features, in this paper I point out that there is another closely related equivalence principle violating theory that also follows from those premises and is less pathological. This example being known as an ‘Einstein–aether theory’, it also serves for 3) illustrating limitations on the conjectured emergence of the universal light cone special relativity principle.
TRIANGLELAND. I. CLASSICAL DYNAMICS WITH EXCHANGE OF RELATIVE ANGULAR MOMENTUM
, 809
"... 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 ..."
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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 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 wellknown, e.g. S 2 for the ‘triangleland ’ (3particle) case that I consider in detail. Secondly, the similarity model occurs as a submodel 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.
Problem of Time and Background Independence: the Individual Facets
"... I lay out the problem of time facets as arising piecemeal from a number of aspects of background independence. Almost all of these already have simpler classical counterparts. This approach can be viewed as a facet by facet completion of the observation that Barbourtype relationalism is a backgroun ..."
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I lay out the problem of time facets as arising piecemeal from a number of aspects of background independence. Almost all of these already have simpler classical counterparts. This approach can be viewed as a facet by facet completion of the observation that Barbourtype relationalism is a background independent precursor to 2 of the 9 facets. That completion proceeds in an order dictated by the additional layers of mathematical structure required to support each. Moreover, the ‘nonlinear nature ’ of the interactions between the Problem of Time facets renders a joint study of them mandatory. The current article is none the less a useful prequel via gaining a conceptual understanding of each facet, prior to embarking on rendering some combinations of facets consistent and what further obstructions arise in attempting such joint considerations. See [20, 21, 26] for up to date studies of this more complicated joint version. I also identify new facets (threading based), subfacets (of observables and of reconstructions) and further source of variety from how far down the levels of mathematical structure these facets extend. 1