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Symmetry
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Shape Dynamics
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Shape Dynamics

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (15 August 2022) | Viewed by 8584

Special Issue Editor

Dr. Flavio Mercati

E-Mail Website
Guest Editor
Federico II University of Naples, Italy
Interests: general relativity; shape dynamics; quantum gravity; noncommutative geometry

Special Issue Information

Dear Colleagues,

Shape dynamics is the modern incarnation of ‘relationalism’: in essence, the idea that all geometric, kinematical, and dynamical structures in physics should be determined by the internal state of the universe, rather than by something external to it. This leads to the need to systematically remove all ‘absolute’ structures from physics, such as, for example, special coordinate systems (diffeomorphism symmetry), asymptotic boundary conditions, absolute scales (conformal symmetry), and special time parameters (reparametrization invariance).

More specifically, shape dynamics is a formulation of general relativity in terms of a dynamical three-dimensional conformal field theory, which is time-reparametrization-invariant. This theory was born in the 1970s, from some important results in mathematical relativity with relationalist ideas developed by J. Barbour and B. Bertotti. The former led to the proof, due to A. Lichnerowicz, Y. Choquet-Bruhat and J. York, of the well-posedness of the initial value problem of general relativity, which indicates that the physical degrees of freedom of general relativity can be written as 3D conformal geometries (on certain hypersurfaces of simultaneity of ‘constant mean extrinsic curvature’). Combining this with relationalist ideas led to a proposal to describe gravity (and all other known field theories) as a theory describing the relative change of conformally-invariant degrees of freedom.

Revisiting well-known solutions of general relativity in shape-dynamical terms has shed some new light on classical gravity, in particular on the nature of singularities, on the problem of the arrow of time, and black holes and gravitational collapse.

This Special Issue will cover current research in shape dynamics and closely-related areas (e.g., N-body toy models), with a particular focus on the consequences of the distinctive symmetries of the theory: scale/conformal invariance and time reparametrization invariance.

Dr. Flavio Mercati
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • shape dynamics
  • general relativity
  • conformal invariance
  • gravitational singularities
  • big bang
  • black holes
  • arrow of time
  • relationalism

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Published Papers (4 papers)

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22 pages, 3491 KiB  
Article
Dendrographic Hologram Theory: Predictability of Relational Dynamics of the Event Universe and the Emergence of Time Arrow
by Oded Shor, Felix Benninger and Andrei Khrennikov
Symmetry 2022, 14(6), 1089; https://doi.org/10.3390/sym14061089 - 25 May 2022
Cited by 3 | Viewed by 2158
Abstract
Recently we started the development of Dendrographic Hologram Theory (DH-theory). It is based on the novel mathematical representation of the relational event universe (in the spirit of Smolin et al.). Elementary events are represented by branches of dendrograms, finite trees that are generated [...] Read more.
Recently we started the development of Dendrographic Hologram Theory (DH-theory). It is based on the novel mathematical representation of the relational event universe (in the spirit of Smolin et al.). Elementary events are represented by branches of dendrograms, finite trees that are generated from data with clustering algorithms. In this context, we studied the dynamics of the event universe generated by the appearance of a new event. Generally, each new event can generate the complete reconstruction of the whole dendrogramic universe. However, we found (via numerical simulation) unexpected stability in this universe. Its events are coupled via the hierarchic relational structure, which is relatively stable even with respect to the random generation of new events. We also observed the regularity patterns in the location of new events on dendrograms. In the course of evolution, the dendrogram’s complexity increases and determines the arrow of time in the event universe. We used the complexity measure from particle shape dynamics, which was shown to increase in both directions away from a Janus point and thus determine the arrow of time in symmetrical manner away from a Janus point. The particle shape dynamics theory is a relational theory with close ideological resemblance to DH-theory, as both rely on Mach’s principle and Leibniz’s relationalism and principles. By using the complexity measure on dendrograms and its p-adic string representation, we demonstrate the emergence of a time arrow from the p-adic zero-dimensional field, where space and time are absent. Full article
(This article belongs to the Special Issue Shape Dynamics)
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11 pages, 259 KiB  
Article
Shape Dynamics of the TT¯ Deformation
by Vasudev Shyam
Symmetry 2021, 13(12), 2242; https://doi.org/10.3390/sym13122242 - 24 Nov 2021
Cited by 1 | Viewed by 1363
Abstract
I will show how the flow triggered by deforming two-dimensional conformal field theories on a torus by the TT¯ operator is identical to the evolution generated by the (radial) quantum Shape Hamiltonian in 2 + 1 dimensions. I will discuss how [...] Read more.
I will show how the flow triggered by deforming two-dimensional conformal field theories on a torus by the TT¯ operator is identical to the evolution generated by the (radial) quantum Shape Hamiltonian in 2 + 1 dimensions. I will discuss how the gauge invariances of the Shape Dynamics, i.e., volume-preserving conformal invariance and diffeomorphism invariance along slices of constant radius are realized as Ward identities of the deformed quantum field theory. I will also comment about the relationship between the reduction to shape space on the gravity side and the solvability of the irrelevant operator deformation of the conformal field theory Full article
(This article belongs to the Special Issue Shape Dynamics)
22 pages, 514 KiB  
Article
Total Collisions in the N-Body Shape Space
by Flavio Mercati and Paula Reichert
Symmetry 2021, 13(9), 1712; https://doi.org/10.3390/sym13091712 - 16 Sep 2021
Cited by 4 | Viewed by 2044
Abstract
We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. [...] Read more.
We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model. Full article
(This article belongs to the Special Issue Shape Dynamics)
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20 pages, 325 KiB  
Article
Scale Symmetry and Friction
by David Sloan
Symmetry 2021, 13(9), 1639; https://doi.org/10.3390/sym13091639 - 6 Sep 2021
Cited by 8 | Viewed by 2074
Abstract
Dynamical similarities are non-standard symmetries found in a wide range of physical systems that identify solutions related by a change of scale. In this paper, we will show through a series of examples how this symmetry extends to the space of couplings, as [...] Read more.
Dynamical similarities are non-standard symmetries found in a wide range of physical systems that identify solutions related by a change of scale. In this paper, we will show through a series of examples how this symmetry extends to the space of couplings, as measured through observations of a system. This can be exploited to focus on observations that can be used to distinguish between different theories and identify those which give rise to identical physical evolutions. These can be reduced into a description that makes no reference to scale. The resultant systems can be derived from Herglotz’s principle and generally exhibit friction. Here, we will demonstrate this through three example systems: the Kepler problem, the N-body system and Friedmann–Lemaître–Robertson–Walker cosmology. Full article
(This article belongs to the Special Issue Shape Dynamics)
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