156 research outputs found
Geometry and observables in (2+1)-gravity
We review the geometrical properties of vacuum spacetimes in (2+1)-gravity
with vanishing cosmological constant. We explain how these spacetimes are
characterised as quotients of their universal cover by holonomies. We explain
how this description can be used to clarify the geometrical interpretation of
the fundamental physical variables of the theory, holonomies and Wilson loops.
In particular, we discuss the role of Wilson loop observables as the generators
of the two fundamental transformations that change the geometry of
(2+1)-spacetimes, grafting and earthquake. We explain how these variables can
be determined from realistic measurements by an observer in the spacetime.Comment: Talk given at 2nd School and Workshop on Quantum Gravity and Quantum
Geometry (Corfu, September 13-20 2009); 10 pages, 13 eps figure
Dual generators of the fundamental group and the moduli space of flat connections
We define the dual of a set of generators of the fundamental group of an
oriented two-surface of genus with punctures and the
associated surface with a disc removed. This dual is
another set of generators related to the original generators via an involution
and has the properties of a dual graph. In particular, it provides an algebraic
prescription for determining the intersection points of a curve representing a
general element of the fundamental group with the
representatives of the generators and the order in which these intersection
points occur on the generators.We apply this dual to the moduli space of flat
connections on and show that when expressed in terms both, the
holonomies along a set of generators and their duals, the Poisson structure on
the moduli space takes a particularly simple form. Using this description of
the Poisson structure, we derive explicit expressions for the Poisson brackets
of general Wilson loop observables associated to closed, embedded curves on the
surface and determine the associated flows on phase space. We demonstrate that
the observables constructed from the pairing in the Chern-Simons action
generate of infinitesimal Dehn twists and show that the mapping class group
acts by Poisson isomorphisms.Comment: 54 pages, 13 .eps figure
Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
We relate the geometrical and the Chern-Simons description of
(2+1)-dimensional gravity for spacetimes of topology , where
is an oriented two-surface of genus , for Lorentzian signature and general
cosmological constant and the Euclidean case with negative cosmological
constant. We show how the variables parametrising the phase space in the
Chern-Simons formalism are obtained from the geometrical description and how
the geometrical construction of (2+1)-spacetimes via grafting along closed,
simple geodesics gives rise to transformations on the phase space. We
demonstrate that these transformations are generated via the Poisson bracket by
one of the two canonical Wilson loop observables associated to the geodesic,
while the other acts as the Hamiltonian for infinitesimal Dehn twists. For
spacetimes with Lorentzian signature, we discuss the role of the cosmological
constant as a deformation parameter in the geometrical and the Chern-Simons
formulation of the theory. In particular, we show that the Lie algebras of the
Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz
algebra over a commutative ring, characterised by a formal parameter
whose square is minus the cosmological constant. In this
framework, the Wilson loop observables that generate grafting and Dehn twists
are obtained as the real and the -component of a Wilson loop
observable with values in the ring, and the grafting transformations can be
viewed as infinitesimal Dehn twists with the parameter .Comment: 50 pages, 6 eps figure
Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
We propose a description of open universes in the Chern-Simons formulation of
(2+1)-dimensional gravity where spatial infinity is implemented as a puncture.
At this puncture, additional variables are introduced which lie in the
cotangent bundle of the Poincar\'e group, and coupled minimally to the
Chern-Simons gauge field. We apply this description of spatial infinity to open
universes of general genus and with an arbitrary number of massive spinning
particles. Using results of [9] we give a finite dimensional description of the
phase space and determine its symplectic structure. In the special case of a
genus zero universe with spinless particles, we compare our result to the
symplectic structure computed by Matschull in the metric formulation of
(2+1)-dimensional gravity. We comment on the quantisation of the phase space
and derive a quantisation condition for the total mass and spin of an open
universe.Comment: 44 pages, 3 eps figure
Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
We relate the geometrical construction of (2+1)-spacetimes via grafting to
phase space and Poisson structure in the Chern-Simons formulation of
(2+1)-dimensional gravity with vanishing cosmological constant on manifolds of
topology , where is an orientable two-surface of genus
. We show how grafting along simple closed geodesics \lambda is
implemented in the Chern-Simons formalism and derive explicit expressions for
its action on the holonomies of general closed curves on S_g. We prove that
this action is generated via the Poisson bracket by a gauge invariant
observable associated to the holonomy of . We deduce a symmetry
relation between the Poisson brackets of observables associated to the Lorentz
and translational components of the holonomies of general closed curves on S_g
and discuss its physical interpretation. Finally, we relate the action of
grafting on the phase space to the action of Dehn twists and show that grafting
can be viewed as a Dehn twist with a formal parameter satisfying
.Comment: 43 pages, 10 .eps figures; minor modifications: 2 figures added,
explanations added, typos correcte
A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
We define and discuss classical and quantum gravity in 2+1 dimensions in the
Galilean limit. Although there are no Newtonian forces between massive objects
in (2+1)-dimensional gravity, the Galilean limit is not trivial. Depending on
the topology of spacetime there are typically finitely many topological degrees
of freedom as well as topological interactions of Aharonov-Bohm type between
massive objects. In order to capture these topological aspects we consider a
two-fold central extension of the Galilei group whose Lie algebra possesses an
invariant and non-degenerate inner product. Using this inner product we define
Galilean gravity as a Chern-Simons theory of the doubly-extended Galilei group.
The particular extension of the Galilei group we consider is the classical
double of a much studied group, the extended homogeneous Galilei group, which
is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure
of the doubly extended Galilei group, and quantise the Chern-Simons theory
using a Hamiltonian approach. Many aspects of the quantum theory are determined
by the quantum double of the extended homogenous Galilei group, or Galilei
double for short. We study the representation theory of the Galilei double,
explain how associated braid group representations account for the topological
interactions in the theory, and briefly comment on an associated
non-commutative Galilean spacetime.Comment: 38 pages, 1 figure, references update
Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
We define a theory of Galilean gravity in 2+1 dimensions with cosmological
constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke
group, extending our previous study of classical and quantum gravity in 2+1
dimensions in the Galilean limit. We exhibit an r-matrix which is compatible
with our Chern-Simons action (in a sense to be defined) and show that the
associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the
classical double of the extended Heisenberg algebra. We deduce that, in the
quantisation of the theory according to the combinatorial quantisation
programme, much of the quantum theory is determined by the quantum double of
the extended q-deformed Heisenberg algebra.Comment: 22 page
Soil loss by wind (SoLoWind): a new GIS-based model to identify risk areas
The focus of wind erosion studies in Germany is located in the Northern and Eastern parts of the country, where wind erosion is a major soil threat and environmental concern. One of the most susceptible regions not only within Germany, but also within Europe (1, 2) is Western Saxony even though no high resolution erosion risk map exists for that region yet. A new wind erosion model for modeling soil loss by wind called SoLoWind was developed and tested for Western Saxony (3). SoLoWind extends the existing DIN model (DIN standard 19706) applied by the public authorities in Germany to a multidirectional model with new causal factors. The new factors are combined by fuzzy logic with the original DIN factors into four modules. The “Natural Wind Erosions Susceptibility” (SUS) module determines the regional soil erodibility with respect to soil texture, soil organic content, soil moisture and wind speeds. A “Soil Cover” (COV) module distinguishes between bare soil and covered soil in satellite images. Furthermore, the modules “Mean Field Length” (MFL) and “Mean Protection Zones” (MPZ) are parameters for the wind erosions avalanching effect and sheltering of windbreaks. Both modules are weighted according to the frequency of wind directions. The application showed that about one-third of all arable land in Western Saxony have either high (26.9%) or very high soil erosion risk (3.6%) by wind. As such, wind erosion is a serious land degradation threat for the region as it is in the adjacent federal states. According to the modeled off-site effects of wind erosion, a potential danger of reduced visibility by windblown dust to sections of the highway A72 could clearly be identified which calls for immediate protection measures. The transparency, adaptability, and user-friendliness of the model suggest that SoLoWind might serve as a planning tool for soil conservation strategies not merely in Western Saxony, but also in other regions
Soil erosion in an avalanche release site (Valle d'Aosta: Italy): towards a winter factor for RUSLE in the Alps
Soil erosion in Alpine areas is mainly related to extreme topographic and
weather conditions. Although different methods of assessing soil erosion
exist, the knowledge of erosive forces of the snow cover needs more
investigation in order to allow soil erosion modeling in areas where the snow
lays on the ground for several months. This study aims to assess whether the
RUSLE (Revised Universal Soil Loss Equation) empirical prediction model,
which gives an estimation of water erosion in t ha yr<sup>−1</sup> obtained from
a combination of five factors (rainfall erosivity, soil erodibility,
topography, soil cover, protection practices) can be applied to mountain
areas by introducing a winter factor (<i>W</i>), which should account for the soil
erosion occurring in winter time by the snow cover. The <i>W</i> factor is
calculated from the ratio of Ceasium-137 (<sup>137</sup>Cs) to RUSLE erosion rates.
Ceasium-137 is another possible way of assessing soil erosion rates in the
field. In contrast to RUSLE, it not only provides water-induced erosion but
integrates all erosion agents involved. Thus, we hypothesize that in mountain
areas the difference between the two approaches is related to the soil
erosion by snow. In this study we compared <sup>137</sup>Cs-based measurement of
soil redistribution and soil loss estimated with RUSLE in a mountain slope
affected by avalanches, in order to assess the relative importance of winter
erosion processes such as snow gliding and full-depth avalanches. Three
subareas were considered: DS, avalanche defense structures, RA, release area,
and TA, track area, characterized by different prevalent winter processes.
The RUSLE estimates and the <sup>137</sup>Cs redistribution gave significantly
different results. The resulting ranges of <i>W</i> evidenced relevant differences
in the role of winter erosion in the considered subareas, and the application
of an avalanche simulation model corroborated these findings. Thus, the
higher rates obtained with the <sup>137</sup>Cs method confirmed the relevant role
of winter soil erosion. Despite the limited sample size (11 points), the
inclusion of a <i>W</i> factor in RUSLE seems promising for the improvement of
soil erosion estimates in Alpine environments affected by snow movements
Cosmological measurements, time and observables in (2+1)-dimensional gravity
We investigate the relation between measurements and the physical observables
for vacuum spacetimes with compact spatial surfaces in (2+1)-gravity with
vanishing cosmological constant. By considering an observer who emits lightrays
that return to him at a later time, we obtain explicit expressions for several
measurable quantities as functions on the physical phase space of the theory:
the eigentime elapsed between the emission of a lightray and its return to the
observer, the angles between the directions into which the light has to be
emitted to return to the observer and the relative frequencies of the lightrays
at their emission and return. This provides a framework in which conceptual
questions about time, observables and measurements can be addressed. We analyse
the properties of these measurements and their geometrical interpretation and
show how they allow an observer to determine the values of the Wilson loop
observables that parametrise the physical phase space of (2+1)-gravity. We
discuss the role of time in the theory and demonstrate that the specification
of an observer with respect to the spacetime's geometry amounts to a gauge
fixing procedure yielding Dirac observables.Comment: 38 pages, 11 eps figures, typos corrected, references update
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