Large quantum gravity effects and nonlocal variables

We reconsider here the model where large quantum gravity effects were first found, but now in its Null Surface Formulation (NSF). We find that although the set of coherent states for $Z$, the basic variable of NSF, is as restricted as it is the one for the metric, while some type of small deviations from these states may cause huge fluctuations on the metric, the corresponding fluctuations on $Z$ remain small.Comment: 4 pages, accepted in PR

Differential equations and conformal structures

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature $(3,2)$. The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group $G_2$. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.Comment: Some typos in formulae concerning (3,2)-signature conformal metrics of Section 5.3 were correcte

Third order ODEs and four-dimensional split signature Einstein metrics

We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables

Absorbing boundary conditions for simulation of gravitational waves with spectral methods in spherical coordinates

We present a new formulation of the multipolar expansion of an exact boundary condition for the wave equation, which is truncated at the quadrupolar order. Using an auxiliary function, that is the solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary condition is simply written as a perturbation of the usual Sommerfeld radiation boundary condition. It is very easily implemented using spectral methods in spherical coordinates. Numerical tests of the method show that very good accuracy can be achieved and that this boundary condition has the same efficiency for dipolar and quadrupolar waves as the usual Sommerfeld boundary condition for monopolar ones. This is of particular importance for the simulation of gravitational waves, which have dominant quadrupolar terms, in General Relativity.Comment: 14 pages, 4 figures. Strongly modified version, accepted for publication in Journal of Computational Physics (new title, new figures and removal of the description of multidomain spectral methods

Non-Life Insurance Pricing: Multi Agents Model

We use the maximum entropy principle for pricing the non-life insurance and recover the B\"{u}hlmann results for the economic premium principle. The concept of economic equilibrium is revised in this respect.Comment: 6 pages, revtex

On the Constant that Fixes the Area Spectrum in Canonical Quantum Gravity

The formula for the area eigenvalues that was obtained by many authors within the approach known as loop quantum gravity states that each edge of a spin network contributes an area proportional to sqrt{j(j+1)} times Planck length squared to any surface it transversely intersects. However, some confusion exists in the literature as to a value of the proportionality coefficient. The purpose of this rather technical note is to fix this coefficient. We present a calculation which shows that in a sector of quantum theory based on the connection A=Gamma-gamma*K, where Gamma is the spin connection compatible with the triad field, K is the extrinsic curvature and gamma is Immirzi parameter, the value of the multiplicative factor is 8*pi*gamma. In other words, each edge of a spin network contributes an area 8*pi*gamma*l_p^2*sqrt{j(j+1)} to any surface it transversely intersects.Comment: Revtex, 7 pages, no figure

Einstein's equations in Ashtekar's variables constitute a symmetric hyperbolic system

We show that the 3+1 vacuum Einstein field equations in Ashtekar's variables constitutes a first order symmetric hyperbolic system for arbitrary but fixed lapse and shift fields, by suitable adding to the system terms proportional to the constraint equations.Comment: 4 pages, revte

Quantum Geometry and Black Hole Entropy

A `black hole sector' of non-perturbative canonical quantum gravity is introduced. The quantum black hole degrees of freedom are shown to be described by a Chern-Simons field theory on the horizon. It is shown that the entropy of a large non-rotating black hole is proportional to its horizon area. The constant of proportionality depends upon the Immirzi parameter, which fixes the spectrum of the area operator in loop quantum gravity; an appropriate choice of this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the same choice of the Immirzi parameter, this result also holds for black holes carrying electric or dilatonic charge, which are not necessarily near extremal.Comment: Revtex, 8 pages, 1 figur

3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs

Quantum Aspects of Black Hole Entropy

This survey intends to cover recent approaches to black hole entropy which attempt to go beyond the standard semiclassical perspective. Quantum corrections to the semiclassical Bekenstein-Hawking area law for black hole entropy, obtained within the quantum geometry framework, are treated in some detail. Their ramification for the holographic entropy bound for bounded stationary spacetimes is discussed. Four dimensional supersymmetric extremal black holes in string-based N=2 supergravity are also discussed, albeit more briefly.Comment: 13 Pages Revtex with 3 eps figures; based on plenary talk given at the International Conference on Gravitation and Cosmology, Kharagpur, India, January, 2000 One reference adde