60 research outputs found
Holographic Special Relativity
We reinterpret special relativity, or more precisely its de Sitter
deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime
geometry. An inertial observer, usually described by a geodesic in spacetime,
becomes instead a choice of ways to reverse the conformal compactification of a
Euclidean vector space up to scale. The observer's "current time," usually
given by a point along the geodesic, corresponds to the choice of scale in the
decompactification. We also show how arbitrary conformal 3-geometries give rise
to "observer space geometries," as defined in recent work, from which spacetime
can be reconstructed under certain integrability conditions. We conjecture a
relationship between this kind of "holographic relativity" and the "shape
dynamics" proposal of Barbour and collaborators, in which conformal space takes
the place of spacetime in general relativity. We also briefly survey related
pictures of observer space, including the AdS analog and a representation
related to twistor theory.Comment: 17 pages, 5 illustration
Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
Einstein gravity in both 3 and 4 dimensions, as well as some interesting
generalizations, can be written as gauge theories in which the connection is a
Cartan connection for geometry modeled on a symmetric space. The relevant
models in 3 dimensions include Einstein gravity in Chern-Simons form, as well
as a new formulation of topologically massive gravity, with arbitrary
cosmological constant, as a single constrained Chern-Simons action. In 4
dimensions the main model of interest is MacDowell-Mansouri gravity,
generalized to include the Immirzi parameter in a natural way. I formulate
these theories in Cartan geometric language, emphasizing also the role played
by the symmetric space structure of the model. I also explain how, from the
perspective of these Cartan-geometric formulations, both the topological mass
in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the
Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the
language of Cartan geometry provides a guiding principle for elegantly
reformulating any 'gauge theory of geometry'.Comment: Article prepared for special journal issue dedicated to Elie Carta
Geometrodynamics and Lorentz symmetry
We study the dynamics of gauge theory and general relativity using fields of
local observers, thus maintaining local Lorentz symmetry despite a space/time
splitting of fields. We start with Yang--Mills theory, where observer fields
are defined as normalized future-timelike vector fields. We then define
observers without a fixed geometry, and find these play two related roles in
general relativity: splitting fields into spatial and temporal parts, and
"breaking" gauge symmetry, effectively reducing the spacetime SO(n,1)
connection to an observer-dependent spatial SO(n) connection. In both gauge
theory and gravity, the observer field reduces the action to canonical form,
without using gauge fixing. In the 4d gravity case, the result is a manifestly
Lorentz covariant counterpart of the Ashtekar-Barbero formulation. We also
explain how this leads geometrically to a picture of general relativity in
terms of "observer space" rather than spacetime---a setting where both
spacetime symmetry and the dynamical description are simultaneously available.Comment: 11 pages. Submission for the proceedings of "3Quantum: Algebra,
Geometry, Information", Tallinn, July 201
Linking Covariant and Canonical General Relativity via Local Observers
Hamiltonian gravity, relying on arbitrary choices of "space," can obscure
spacetime symmetries. We present an alternative, manifestly spacetime covariant
formulation that nonetheless distinguishes between "spatial" and "temporal"
variables. The key is viewing dynamical fields from the perspective of a field
of observers -- a unit timelike vector field that also transforms under local
Lorentz transformations. On one hand, all fields are spacetime fields,
covariant under spacetime symmeties. On the other, when the observer field is
normal to a spatial foliation, the fields automatically fall into Hamiltonian
form, recovering the Ashtekar formulation. We argue this provides a bridge
between Ashtekar variables and covariant phase space methods. We also outline a
framework where the 'space of observers' is fundamental, and spacetime geometry
itself may be observer-dependent.Comment: 8 pages; Essay written for the 2012 Gravity Research Foundation
Awards for Essays on Gravitatio
Exotic Statistics for Strings in 4d BF Theory
After a review of exotic statistics for point particles in 3d BF theory, and
especially 3d quantum gravity, we show that string-like defects in 4d BF theory
obey exotic statistics governed by the 'loop braid group'. This group has a set
of generators that switch two strings just as one would normally switch point
particles, but also a set of generators that switch two strings by passing one
through the other. The first set generates a copy of the symmetric group, while
the second generates a copy of the braid group. Thanks to recent work of
Xiao-Song Lin, we can give a presentation of the whole loop braid group, which
turns out to be isomorphic to the 'braid permutation group' of Fenn, Rimanyi
and Rourke. In the context 4d BF theory this group naturally acts on the moduli
space of flat G-bundles on the complement of a collection of unlinked unknotted
circles in R^3. When G is unimodular, this gives a unitary representation of
the loop braid group. We also discuss 'quandle field theory', in which the
gauge group G is replaced by a quandle.Comment: 41 pages, many figures. New version has minor corrections and
clarifications, and some added reference
The geometric role of symmetry breaking in gravity
In gravity, breaking symmetry from a group G to a group H plays the role of
describing geometry in relation to the geometry the homogeneous space G/H. The
deep reason for this is Cartan's "method of equivalence," giving, in
particular, an exact correspondence between metrics and Cartan connections. I
argue that broken symmetry is thus implicit in any gravity theory, for purely
geometric reasons. As an application, I explain how this kind of thinking gives
a new approach to Hamiltonian gravity in which an observer field spontaneously
breaks Lorentz symmetry and gives a Cartan connection on space.Comment: 4 pages. Contribution written for proceedings of the conference
"Loops 11" (Madrid, May 2011
2-Group Representations for Spin Foams
Just as 3d state sum models, including 3d quantum gravity, can be built using
categories of group representations, "2-categories of 2-group representations"
may provide interesting state sum models for 4d quantum topology, if not
quantum gravity. Here we focus on the "Euclidean 2-group", built from the
rotation group SO(4) and its action on the group of translations of 4d
Euclidean space. We explain its infinite-dimensional unitary representations,
and construct a model based on the resulting representation 2-category. This
model, with clear geometric content and explicit "metric data" on triangulation
edges, shows up naturally in an attempt to write the amplitudes of ordinary
quantum field theory in a background independent way.Comment: 8 pages; to appear in proceedings of the XXV Max Born Symposium: "The
Planck Scale", Wroclaw, Polan
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