5,284 research outputs found
Mathematics of random growing interfaces
We establish a thermodynamic limit and Gaussian fluctuations for the height
and surface width of the random interface formed by the deposition of particles
on surfaces. The results hold for the standard ballistic deposition model as
well as the surface relaxation model in the off-lattice setting. The results
are proved with the aid of general limit theorems for stabilizing functionals
of marked Poisson point processes.Comment: 12 page
Degree Correlations in Random Geometric Graphs
Spatially embedded networks are important in several disciplines. The
prototypical spatial net- work we assume is the Random Geometric Graph of which
many properties are known. Here we present new results for the two-point degree
correlation function in terms of the clustering coefficient of the graphs for
two-dimensional space in particular, with extensions to arbitrary finite
dimension
Weak Cosmic Censorship: As Strong as Ever
Spacetime singularities that arise in gravitational collapse are always
hidden inside of black holes. This is the essence of the weak cosmic censorship
conjecture. The hypothesis, put forward by Penrose 40 years ago, is still one
of the most important open questions in general relativity. In this Letter, we
reanalyze extreme situations which have been considered as counterexamples to
the weak cosmic censorship conjecture. In particular, we consider the
absorption of scalar particles with large angular momentum by a black hole.
Ignoring back reaction effects may lead one to conclude that the incident wave
may overspin the black hole, thereby exposing its inner singularity to distant
observers. However, we show that when back reaction effects are properly taken
into account, the stability of the black-hole event horizon is irrefutable. We
therefore conclude that cosmic censorship is actually respected in this type of
gedanken experiments.Comment: 4 page
Twistors, special relativity, conformal symmetry and minimal coupling - a review
An approach to special relativistic dynamics using the language of spinors
and twistors is presented. Exploiting the natural conformally invariant
symplectic structure of the twistor space, a model is constructed which
describes a relativistic massive, spinning and charged particle, minimally
coupled to an external electro-magnetic field. On the two-twistor phase space
the relativistic Hamiltonian dynamics is generated by a Poincare scalar
function obtained from the classical limit (appropriately defined by us) of the
second order, to an external electro-magnetic field minimally coupled, Dirac
operator. In the so defined relativistic classical limit there are no Grassman
variables. Besides, the arising equation that describes dynamics of the
relativistic spin differs significantly from the so called Thomas Bergman
Michel Telegdi equation.Comment: 39 pages, no figures, few erronous statements (not affecting anything
else in the papper) on page 23 delete
Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling
In the paper we investigate the evolution of the relativistic particle
(massive and massless) with spin defined by Hamiltonian containing the terms
with momentum-spin-orbit coupling. We integrate the corresponding Hamiltonian
equations in quadratures and express their solutions in terms of elliptic
functions.Comment: 18 page
Strict inequalities of critical values in continuum percolation
We consider the supercritical finite-range random connection model where the
points of a homogeneous planar Poisson process are connected with
probability for a given . Performing percolation on the resulting
graph, we show that the critical probabilities for site and bond percolation
satisfy the strict inequality . We also show
that reducing the connection function strictly increases the critical
Poisson intensity. Finally, we deduce that performing a spreading
transformation on (thereby allowing connections over greater distances but
with lower probabilities, leaving average degrees unchanged) {\em strictly}
reduces the critical Poisson intensity. This is of practical relevance,
indicating that in many real networks it is in principle possible to exploit
the presence of spread-out, long range connections, to achieve connectivity at
a strictly lower density value.Comment: 38 pages, 8 figure
Spinor calculus on 5-dimensional spacetimes
Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to
the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's
spinor calculus as the spin covariant derivative, the curvature spinors or the
definition of the spin coefficients on a spin frame can be carried over to the
spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and
differential properties of the curvature spinors are studied in detail and as
an application we extend the well-known 4-dimensional Newman-Penrose formalism
to a 5-dimensional spacetime.Comment: Convention mismatch and minor typos fixed. To appear in Journal of
Mathematical Physic
Positive cosmological constant in loop quantum cosmology
The k=0 Friedmann Lemaitre Robertson Walker model with a positive
cosmological constant and a massless scalar field is analyzed in detail. If one
uses the scalar field as relational time, new features arise already in the
Hamiltonian framework of classical general relativity: In a finite interval of
relational time, the universe expands out to infinite proper time and zero
matter density. In the deparameterized quantum theory, the true Hamiltonian now
fails to be essentially self-adjoint both in the Wheeler DeWitt (WDW) approach
and in LQC. Irrespective of the choice of the self-adjoint extension, the big
bang singularity persists in the WDW theory while it is resolved and replaced
by a big bounce in loop quantum cosmology (LQC). Furthermore, the quantum
evolution is surprisingly insensitive to the choice of the self-adjoint
extension. This may be a special case of an yet to be discovered general
property of a certain class of symmetric operators that fail to be essentially
self-adjoint.Comment: 36 pages, 6 figures, RevTex
Approximate twistors and positive mass
In this paper the problem of comparing initial data to a reference solution
for the vacuum Einstein field equations is considered. This is not done in a
coordinate sense, but through quantification of the deviation from a specific
symmetry. In a recent paper [T. B\"ackdahl, J.A. Valiente Kroon, Phys. Rev.
Lett. 104, 231102 (2010)] this problem was studied with the Kerr solution as a
reference solution. This analysis was based on valence 2 Killing spinors. In
order to better understand this construction, in the present article we analyse
the analogous construction for valence 1 spinors solving the twistor equation.
This yields an invariant that measures how much the initial data deviates from
Minkowski data. Furthermore, we prove that this invariant vanishes if and only
of the mass vanishes. Hence, we get a proof of the positivity of mass.Comment: 18 pages, corrected typos, updated reference
The Measure Problem in Cosmology
The Hamiltonian structure of general relativity provides a natural canonical
measure on the space of all classical universes, i.e., the multiverse. We
review this construction and show how one can visualize the measure in terms of
a "magnetic flux" of solutions through phase space. Previous studies identified
a divergence in the measure, which we observe to be due to the dilatation
invariance of flat FRW universes. We show that the divergence is removed if we
identify universes which are so flat they cannot be observationally
distinguished. The resulting measure is independent of time and of the choice
of coordinates on the space of fields. We further show that, for some
quantities of interest, the measure is very insensitive to the details of how
the identification is made. One such quantity is the probability of inflation
in simple scalar field models. We find that, according to our implementation of
the canonical measure, the probability for N e-folds of inflation in
single-field, slow-roll models is suppressed by of order exp(-3N) and we
discuss the implications of this result.Comment: 22 pages, 6 figures. Revised version with clarifying remarks on
meaning of adopted measure, extra references and minor typographical
correction
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