# Research Papers

Virtually all my research, which is still continuing actively, has been based on the conviction that time, motion and size are all relative. I call this the relational approach. My main collaborators for the earlier work were Bruno Bertotti, Niall Ó Murchadha, Edward Anderson, Brendan Foster, and Bryan Kelleher. I am currently working closely with Henrique Gomes, Sean Gryb, Tim Koslowski and Flavio Mercati. I list below the more significant papers that have resulted from this research, including papers that consider the implications of the relativity of time and motion for the quantum theory of the universe. It is worth emphasizing that Einstein set out to create a theory in which time and motion are relative in a brilliant but indirect way and that this has led to considerable confusion. In contrast, the approach of the papers below is direct. I believe their main value to be, first, the demonstration that the direct route to relativity of time and motion leads to the same theory as Einstein’s indirect route, second, that the inclusion of relativity of size leads to Shape Dynamics (Ideas), and third that this work identifies the dynamical features of gravity likely to be important in the creation of quantum gravity. For this, the last four papers (before the two on maximal variety) are the most relevant and relate to Shape Dynamics.

Finally, there are two papers on maximal variety, Leibniz’s idea that the universe in which we live is more varied than any other possible universe. This speculative idea was developed by Lee Smolin and myself. It requires much development, but is a good illustration of key features of Leibniz’s thinking that could be relevant to quantum gravity.

## Papers on the Relativity of Time and Motion

### Relative-distance Machian theories. Nature 249, 328 (1974). (PDF)

### Gravity and inertia in a Machian framework. (With Bruno Bertotti.) Nuovo Cimento 38B, 1 (1977).

The basic idea behind these two papers is that only relative separations occur in the action principle of the universe. It later transpired that the same idea had been developed by several authors, most notably by Schrödinger in 1925. The main value of such theories is that they show how Mach’s Principle can be implemented, but they lead to anisotropic inertial masses, which are ruled out experimentally. For a discussion of work along these lines, which was initiated by Mach himself, see the conference proceedings Mach’s Principle (books).

### Relational concepts of space and time. British Journal for the Philosophy of Science 33, 251 (1982). (Journal)

Explains precisely what one should expect of a Machian dynamics. It
develops an idea due to Poincaré.

### Mach’s principle and the structure of dynamical theories. (With Bruno Bertotti.) Proceedings of the Royal Society (London) A 382, 295 (1982). (PDF)

In this paper, Bertotti and I succeeded in finding a way to implement Mach’s Principle that does not lead to anisotropic masses. I now call this method best matching. It is universal, closely related to the gauge principle of modern high-energy physics, and leads to a direct dynamical and relational derivation of general relativity. All the subsequent papers on the relativity of time and motion listed below are based on this principle.

### Leibnizian time, Machian dynamics, and quantum gravity. In: Quantum Concepts in Space and Time, eds. R. Penrose and C. J. Isham, Oxford University Press, Oxford (1986). (PDF)

Explores the implications for quantum gravity of the Machian structure of general relativity found in the previous paper. I argue that the quantum mechanics of the universe will be very different from the existing form of quantum mechanics, which is valid in the framework provided by the universe.

### The part played by Mach’s Principle in the genesis of relativistic cosmology. In: Modern Cosmology in Retrospect, eds. B Bertotti et al, Cambridge University Press, Cambridge (1990). (PDF)

### The development of Machian themes in the twentieth century. In: The Arguments of Time, ed. J Butterfield, Oxford University Press, Oxford (1999).

The two above papers give an account of Einstein’s attempts to implement Mach’s idea about the origin of inertia. My main conclusion is that Einstein created much confusion by never clearly identifying precisely how he should implement Mach’s proposal but that despite this succeeded to a large degree. The Machian motivation behind the creation of general relativity was very great.

### Time and complex numbers in canonical quantum gravity. Physical Review D 47, 12 (1993). Journal (PDF, 7.5MB)

Complex numbers play a very significant role in quantum mechanics because the fundamental time-dependent Schrödinger equation is complex. However the putative equation of quantum gravity is real, and it is not clear how complex numbers should enter the theory. The above paper focuses on this issue, which has still not attracted the attention that it warrants. I gave a seminar on this at the Perimeter Institute.

### The emergence of time and its arrow from timelessness. In: Physical Origins of Time Asymmetry, eds. J. Halliwell et al, Cambridge University Press, Cambridge (1994). (PDF)

### The timelessness of quantum gravity I, II. Classical and Quantum Gravity 11, 2853, 2875 (1994). (Journal, Journal)

These papers show that the notion of an independently existing time is redundant if one is considering the dynamics of the universe. Change does not occur in time. Rather dynamics relates all changes in the universe to each other. I believe that the implications of this fact for the quantum theory of the universe (quantum gravity) are profound. The quantum universe is likely to be static. Ironically, I believe that in this timeless scenario it will be easier to explain the arrow of time, which I trace to the marked asymmetry of the configuration space of the universe. I conjecture that this concentrates the wave function of the universe on configurations that contain what we interpret as mutually consistent records of a past. My The End of Time (books) gives a popular account of these ideas.

### Dynamics of pure shape, relativity and the problem of time. In: Decoherence and Entropy in Complex Systems (Proceedings of the Conference DICE, Piombino 2002, ed. H.-T Elze), Springer Lecture Notes in Physics 2003. (arXiv:gr-qc/0309089)

In 1999 I began a very fruitful collaboration with Niall O'Murchadha that greatly extended my earlier work with Bruno Bertotti on time and Mach's principle. This paper gives a useful introductory overview of the results that we obtained in collaboration with Brendan Foster, Edward Anderson, and Bryan Kelleher. Details can be found in the five following papers.

### Relativity without relativity. (With Brendan Z Foster and Niall Ó Murchadha.) Classical and Quantum Gravity 19, 3217 (2002). (arXiv:gr-qc/0012089)

I think this may prove to be my most important research paper; it greatly extends my work with Bertotti and demonstrates the power of the notion of best matching. The key insight is due to Niall O'Murchadha, who realized that it is very difficult to create a consistent theory that meets the relational requirements; consistency becomes a powerful tool for the finding of theories. The paper considers the construction of a relational theory that describes the dynamical evolution of three-dimensional Riemannian geometry and shows that the simplest nontrivial consistent theory of this kind is general relativity. If in addition one attempts to allow other fields to interact with the dynamical Riemannian geometry, then the simplest realization of such interaction enforces the emergence of a universal light cone, gauge theory, and the equivalence principle. This result needs to be put into its historical perspective. Newton introduced absolute space and time in order to formulate dynamics, but Leibniz and Mach argued that position and time are relative, so that dynamics must be relational. This paper completely vindicates the relational standpoint and shows that its simplest implementation leads directly and inexorably to all the fundamental principles of modern physics except for quantum theory.

### Interacting vector fields in relativity without relativity. (With Edward Anderson.) Classical and Quantum Gravity 19 3249 (2002). (arXiv:gr-qc/0201092)

This paper, largely the work of Edward Anderson, gives the details of
the emergence of gauge theory in the framework of the previous paper
“Relativity without relativity”.

### Scale-invariant gravity: particle dynamics. Classical and Quantum Gravity 20 1543 (2003). (arXiv:gr-qc/0211021)

In a fully relational theory, not only time and position should be relative but also size – if all distances in the universe were suddenly doubled, nothing observable would change. This paper extends the principle of relational best matching to the relativity of size for the case of particle dynamics. It presents a dynamics of pure shape and demonstrates that one can exactly recover Newtonian inertia, gravity, and electrostatics for subsystems of the universe together with one additional force that acts significantly only over scales of the order of the complete universe and is closely analogous to Einstein’s cosmological constant. This paper, together with the two following papers, highlights a most strange feature of general relativity and the Big Bang cosmology: in these theories, overall size is absolute, in contrast to everything else. This is the feature of general relativity that allows the ‘expansion of the universe’. In the standard model, the universe is doing two things simultaneously: it is expanding and changing its shape (it is becoming more inhomogeneous). If the universe were perfectly relational, it could only change its shape. Attractive as this idea is, it appears to be in strong conflict with the evidence from cosmology. For me, this is a great mystery and a stimulus to further research.

### Scale-invariant gravity: geometrodynamics. (With Edward Anderson, Brendan Z Foster, and Niall Ó Murchadha.) Classical and Quantum Gravity 20 1571 (2003). (arXiv:gr-qc/0211022)

In this paper the principle of best matching is extended to create a theory of relational dynamical Riemannian three-geometry in which position, time, and size are completely relative. The resulting theory has many attractive features and is essentially identical to general relativity as regards processes that take place below intergalactic scales. It therefore passes all the same stringent observational tests as general relativity except those on cosmological scales, for which it fails badly. It cannot be a realistic theory of the universe. For me this is the mystery noted in the discussion of the previous paper: why does the universe fail to be perfectly relational? The significance of this question is emphasized by the following paper.

### The physical gravitational degrees of freedom. (With Edward Anderson, Brendan Z Foster, Bryan Kelleher, and Niall Ó Murchadha.) Classical and Quantum Gravity 22 1795 (2005). (arXiv:gr-qc/0407104)

This paper shows the precise sense in which general relativity, treated as a dynamical theory, just fails to be fully scale invariant. A Riemmanian three-geometry is characterized by three degrees of freedom at each space point P. Two of them determine angles at P (the conformal part of the geometry) while the third determines the scale at P. This is analogous to the way in which two angles determine the shape of a triangle, while scale determines its size. It is intuitively clear that shape is more fundamental than size. The above paper shows that general relativity can be represented as a theory in which the two local degrees of freedom at each space point interact with each other and with one single extra global degree of freedom, essentially the rate of change of the volume of the universe. It is very odd that the local scales play no role but the rate of change of the global scale (the volume of the universe) does. This is what allows the universe to expand.

### Constraints and gauge transformations: Dirac's theorem is not always valid (with Brendan Foster) (arXiv:0808.1223)

This paper is related to a widely held belief that there is no genuine dynamical evolution in classical general relativity. The belief relies on a famous theorem proved by Dirac in his Lectures on Quantum Mechanics (1964), which is used to interpret the so-called Hamiltonian constraint in canonical quantum gravity. Our paper shows that in fact Dirac's theorem is not universally valid and thus calls into question the orthodox belief.

### The definition of Mach's Principle Found. Phys. 40 1263-1284 (2010) (arXiv:1007.3368)

There is much confusion surrounding the definition of Mach's principle. In this paper, which includes historical and conceptual material, I provide what I believe is the only sensible definition.

### Conformal superspace: the configuration space of general relativity (with Niall O'Murchadha) (arXiv:1009.3559)

I regard this as one of my most important papers. It shows that general relativity can be characterized as a theory of the dynamics of local shapes of space. This perfectly matches the definition of Mach's principle as given in the above paper.

### Einstein gravity as a 3D conformally invariant theory (by Henrique Gomes, Sean Gryb, and Tim Koslowski) (arXiv:1010.2481)

This paper and the following, both by my collaborators, complement the one above very usefully, making effective use of Dirac's theory of constrained dynamical systems.

### The link between general relativity and shape dynamics (by Henrique Gomes and Tim Koslowski) (arXiv:1101.5974)

### A Gravitational Origin of the Arrows of Time (by Julian Barbour, Tim Koslowski, Flavio Mercati) (arXiv:1310.5167 and PRL)

The only widely accepted explanation for the various arrows of time that everywhere and at all epochs point in the same direction is the `past hypothesis': the Universe had a very special low-entropy initial state. We present the first evidence for an alternative conjecture: the arrows exist in all solutions of the gravitational law that governs the Universe and arise because the space of its true degrees of freedom (shape space) is asymmetric. We prove our conjecture for arrows of complexity and information in the Newtonian N-body problem. Except for a set of measure zero, all of its solutions for non-negative energy divide at a uniquely defined point into two halves. In each a well-defined measure of complexity fluctuates but grows irreversibly between rising bounds from that point. Structures that store dynamical information are created as the complexity grows. Recognition of the division is a key novelty of our approach. Each solution can be viewed as having a single past and two distinct futures emerging from it. Any internal observer must be in one half of the solution and will only be aware of one past and one future. The `paradox' of a time-symmetric law that leads to observationally irreversible behaviour is fully resolved. General Relativity shares enough architectonic structure with the N-body problem for us to prove the existence of analogous complexity arrows in the vacuum Bianchi IX model. In the absence of non-trivial solutions with matter we cannot prove that arrows of dynamical information will arise in GR, though they have in our Universe. Finally, we indicate how the other arrows of time could arise.

### Entropy and the Typicality of Universes (by Julian Barbour, Tim Koslowski, Flavio Mercati) (arXiv:1507.06498)

The universal validity of the second law of thermodynamics is widely attributed to a finely tuned initial condition of the universe. This creates a problem: why is the universe atypical? We suggest that the problem is an artefact created by inappropriate transfer of the traditional concept of entropy to the whole universe. Use of what we call the relational N-body problem as a model indicates the need to employ two distinct entropy-type concepts to describe the universe. One, which we call entaxy, is novel. It is scale-invariant and decreases as the observable universe evolves. The other is the algebraic sum of the dimensionful entropies of branch systems (isolated subsystems of the universe). This conventional additive entropy increases. In our model, the decrease of entaxy is fundamental and makes possible the emergence of branch systems and their increasing entropy. We have previously shown that all solutions of our model divide into two halves at a unique 'Janus point' of maximum disorder. This constitutes a common past for two futures each with its own gravitational arrow of time. We now show that these arrows are expressed through the formation of branch systems within which conventional entropy increases. On either side of the Janus point, this increase is in the same direction in every branch system. We also show that it is only possible to specify unbiased solution-determining data at the Janus point. Special properties of these `mid-point data' make it possible to develop a rational theory of the typicality of universes whose governing law, as in our model, dictates the presence of a Janus point in every solution. If our self-gravitating universe is governed by such a law, then the second law of thermodynamics is a necessary direct consequence of it and does not need any special initial condition.

## Papers on Maximal Variety

### Extremal variety as the foundation of a cosmological quantum theory. (With Lee Smolin.) Unpublished. (arXiv: hep-th/9203041; no figures!)

### The deep and suggestive principles of Leibnizian philosophy. The Harvard Review of Philosophy 11, Spring (2003). (PDF)

Leibniz was mocked for claiming that we live in the best of all possible worlds. (Voltaire’s Candide asks: “If this is the best, what are the others like?”) In fact, Leibnizian philosophy is really concerned with variety, and in his Monadology Leibniz postulated that the universe is created “with as much variety as possible, but with the greatest order possible”. Strangely, no one seems to have attempted to express this idea in concrete mathematical form before Smolin and I found a realization in the form of various models for which an intrinsic variety can be defined and maximized. Unlike Shannon information, the information content of such models is intrinsic and can be 'read off' directly from them. We initially hoped these models would cast light on the mysteries of quantum mechanics, but, despite several intriguing properties of these models, any direct link to quantum mechanics is clearly still a long way off.