Julian Barbour

Mach’s principle and the structure of dynamical theories

A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.

Shape Dynamics. An Introduction

Shape dynamics is a completely background-independent universal framework of dynamical theories from which all absolute elements have been eliminated. For particles, only the variables that describe the shapes of the instantaneous particle configurations are dynamical. In the case of Riemannian three-geometries, the only dynamical variables are the parts of the metric that determine angles. The local scale factor plays no role. This leads to a shape-dynamic theory of gravity in which the four-dimensional diffeomorphism invariance of general relativity is replaced by three-dimensional diffeomorphism invariance and three-dimensional conformal invariance. Despite this difference of symmetry groups, it is remarkable that the predictions of the two theories – shape dynamics and general relativity – agree on spacetime foliations by hypersurfaces of constant mean extrinsic curvature. However, the two theories are distinct, with shape dynamics having a much more restrictive set of solutions. There are indications that the symmetry group of shape dynamics makes it more amenable to quantization and thus to the creation of quantum gravity. This introduction presents in simple terms the arguments for shape dynamics, its implementation techniques, and a survey of existing results.

The physical gravitational degrees of freedom

When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving three-dimensional conformal Riemannian geometries obtained by imposing two general principles: (1) time is derived from change; (2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation, but also all the equations used in Yorks conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.

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 Weyls 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.

Relational Concepts of Space and Time

According to Leibniz, space is not something substantial but is merely the order of coexisting things; similarly, time is merely the successive order of things. Historically, this view was opposed by Newton, with his concepts of absolute space and absolute time. It was clear to Newton-but perhaps not so clear to Leibniz-that a distinction between his absolute concepts and Leibnizs relational concepts becomes meaningful only at the level of dynamical theories of motion. Indeed, Hermann Weyl [I949], who inclines more to the Newtonian point of view, concedes that at the level of kinematics the relationist standpoint is irrefutable. The argument therefore hinges on the ability of the rival camps to formulate dynamical laws of motion. Here, the epistemological cogency of the relationist camp at the level of kinematics has been seriously weakened by its inability hitherto to construct genuinely relational theories of motion. What I should like to do in this paper is to identify the philosophical principles which are fundamental to the relationist case and show how they lead one, on the one hand, to epistemological doubts about the conventional conceptual framework of Newtonian and Einsteinian dynamics and, on the other, how they can be used positively in the construction of an alternative relationist framework.

Interacting vector fields in relativity without relativity

Barbour, Foster and O Murchadha have recently developed a new framework, called here 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 three-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.

Scale anomaly as the origin of time

We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schrödinger equation. As with the Wheeler–DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schrödinger equation with a repulsive cosmological force in the ‘late-time’ physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts.

On the Origin of Structure in the Universe

The most obvious thing about the universe in which we find ourselves is its structure. Indeed, according to the basic philosophical viewpoint adopted by thinkers such as Berkeley, Leibniz and Mach, it would not really be possible to speak of the universe at all if it did not exhibit differentiating structure and qualities. Leibniz’s monadology equates existence with variety: monads are defined by and simultaneously distinguished from other monads by virtue of their attributes and nothing else. You cannot remove the attributes of a thing and leave some mysterious “thisness” (haeccity). Remove the attributes and nothing is left.

Arrows of time in unconfined systems

Entropy and the second law of thermodynamcs were discovered through study of the behaviour of gases in confined spaces. The related techniques developed in the kinetic theory of gases have failed to resolve the apparent conflict between the time-reversal symmetry of all known laws of nature and the existence of arrows of time that at all times and everywhere in the universe all point in the same direction. I will argue that the failure may be due to unconscious application to the universe of the conceptual framework developed for confined systems. If, as seems plausible, the universe is an unconfined system, new concepts are needed.

The Fundamental Problem of Dynamics

In a world in which all objects are in relative motion, there arises the problem of equilocality: the identification of points in space that have the same position at different times. Newton recognized this as the fundamental problem of dynamics and to solve it introduced absolute space. Inspired by Mach, Einstein created general relativity in the hope of eliminating this controversial concept, but his indirect approach left the issue unresolved. I will explain how the general method of best matching always leads to dynamical theories with an unambiguous notion of equilocality. Applied to the dynamics of Riemannian 3-geometry, it leads to a radical rederivation of general relativity in which relativity of local scale replaces replaces relativity of simultaneity as a foundational principle. Whereas in the standard spacetime picture there is no unique notion of simultaneity or history, if this alternative derivation leads to the physically correct picture both are fixed in the minutest detail. New approaches to several outstanding problems, including singularities and the origin of time’s arrows, are suggested.