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 $S_{g,n}$ of genus $g$ with $n$ punctures and the associated surface $S_{g,n}\setminus D$ with a disc $D$ 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 $\pi_1(S_{g,n}\setminus D)$ 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 $S_{g,n}$ 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 $R\times S_g$, where $S_g$ is an oriented two-surface of genus $g>1$, 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 $\Theta_\Lambda$ 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 $\Theta_\Lambda$-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 $\Theta_\Lambda$.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 $R\times S_g$, where $S_g$ is an orientable two-surface of genus $g>1$. 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 $\lambda$. 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 $\theta$ satisfying $\theta^2=0$.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⁻¹ 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 (¹³⁷Cs) 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 ¹³⁷Cs-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 ¹³⁷Cs 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 ¹³⁷Cs 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