510 research outputs found
On Singularity Formation of a Nonlinear Nonlocal System
We investigate the singularity formation of a nonlinear nonlocal system. This
nonlocal system is a simplified one-dimensional system of the 3D model that was
recently proposed by Hou and Lei in [13] for axisymmetric 3D incompressible
Navier-Stokes equations with swirl. The main difference between the 3D model of
Hou and Lei and the reformulated 3D Navier-Stokes equations is that the
convection term is neglected in the 3D model. In the nonlocal system we
consider in this paper, we replace the Riesz operator in the 3D model by the
Hilbert transform. One of the main results of this paper is that we prove
rigorously the finite time singularity formation of the nonlocal system for a
large class of smooth initial data with finite energy. We also prove the global
regularity for a class of smooth initial data. Numerical results will be
presented to demonstrate the asymptotically self-similar blow-up of the
solution. The blowup rate of the self-similar singularity of the nonlocal
system is similar to that of the 3D model.Comment: 28 pages, 9 figure
Metric connections in projective differential geometry
We search for Riemannian metrics whose Levi-Civita connection belongs to a
given projective class. Following Sinjukov and Mikes, we show that such metrics
correspond precisely to suitably positive solutions of a certain projectively
invariant finite-type linear system of partial differential equations.
Prolonging this system, we may reformulate these equations as defining
covariant constant sections of a certain vector bundle with connection. This
vector bundle and its connection are derived from the Cartan connection of the
underlying projective structure.Comment: 10 page
General solutions of the Wess-Zumino consistency condition for the Weyl anomalies
The general solutions of the Wess-Zumino consistency condition for the
conformal (or Weyl, or trace) anomalies are derived. The solutions are
obtained, in arbitrary dimensions, by explicitly computing the cohomology of
the corresponding Becchi-Rouet-Stora-Tyutin differential in the space of
integrated local functions at ghost number unity. This provides a purely
algebraic, regularization-independent classification of the Weyl anomalies in
arbitrary dimensions. The so-called type-A anomaly is shown to satisfy a
non-trivial descent of equations, similarly to the non-Abelian chiral anomaly
in Yang-Mills theory.Comment: 9 pages. RevTeX fil
Classification of Generalized Symmetries for the Vacuum Einstein Equations
A generalized symmetry of a system of differential equations is an
infinitesimal transformation depending locally upon the fields and their
derivatives which carries solutions to solutions. We classify all generalized
symmetries of the vacuum Einstein equations in four spacetime dimensions. To
begin, we analyze symmetries that can be built from the metric, curvature, and
covariant derivatives of the curvature to any order; these are called natural
symmetries and are globally defined on any spacetime manifold. We next classify
first-order generalized symmetries, that is, symmetries that depend on the
metric and its first derivatives. Finally, using results from the
classification of natural symmetries, we reduce the classification of all
higher-order generalized symmetries to the first-order case. In each case we
find that the generalized symmetries are infinitesimal generalized
diffeomorphisms and constant metric scalings. There are no non-trivial
conservation laws associated with these symmetries. A novel feature of our
analysis is the use of a fundamental set of spinorial coordinates on the
infinite jet space of Ricci-flat metrics, which are derived from Penrose's
``exact set of fields'' for the vacuum equations.Comment: 57 pages, plain Te
A scalar invariant and the local geometry of a class of static spacetimes
The scalar invariant, I, constructed from the "square" of the first covariant
derivative of the curvature tensor is used to probe the local geometry of
static spacetimes which are also Einstein spaces. We obtain an explicit form of
this invariant, exploiting the local warp-product structure of a 4-dimensional
static spacetime, , where is
the Riemannian hypersurface orthogonal to a timelike Killing vector field with
norm given by a positive function, on . For a static
spacetime which is an Einstein space, it is shown that the locally measurable
scalar, I, contains a term which vanishes if and only if is
conformally flat; also, the vanishing of this term implies (a)
is locally foliated by level surfaces of , , which are totally
umbilic spaces of constant curvature, and (b) is locally a
warp-product space. Futhermore, if is conformally flat it
follows that every non-trivial static solution of the vacuum Einstein equation
with a cosmological constant, is either Nariai-type or Kottler-type - the
classes of spacetimes relevant to quantum aspects of gravity.Comment: LaTeX, 13 pages, JHEP3.cls; The paper is completely rewritten with a
new title and introduction as well as additional results and reference
Projective dynamics and first integrals
We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure
The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions
We derive explicit formulae for a set of constraints for the Einstein
equations on a null hypersurface, in arbitrary dimensions. We solve these
constraints and show that they provide necessary and sufficient conditions so
that a spacetime solution of the Cauchy problem on a characteristic cone for
the hyperbolic system of the reduced Einstein equations in wave-map gauge also
satisfies the full Einstein equations. We prove a geometric uniqueness theorem
for this Cauchy problem in the vacuum case.Comment: 83 pages, 1 figur
Projective dynamics and classical gravitation
Given a real vector space V of finite dimension, together with a particular
homogeneous field of bivectors that we call a "field of projective forces", we
define a law of dynamics such that the position of the particle is a "ray" i.e.
a half-line drawn from the origin of V. The impulsion is a bivector whose
support is a 2-plane containing the ray. Throwing the particle with a given
initial impulsion defines a projective trajectory. It is a curve in the space
of rays S(V), together with an impulsion attached to each ray. In the simplest
example where the force is identically zero, the curve is a straight line and
the impulsion a constant bivector. A striking feature of projective dynamics
appears: the trajectories are not parameterized.
Among the projective force fields corresponding to a central force, the one
defining the Kepler problem is simpler than those corresponding to other
homogeneities. Here the thrown ray describes a quadratic cone whose section by
a hyperplane corresponds to a Keplerian conic. An original point of view on the
hidden symmetries of the Kepler problem emerges, and clarifies some remarks due
to Halphen and Appell. We also get the unexpected conclusion that there exists
a notion of divergence-free field of projective forces if and only if dim V=4.
No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure
A minimal set of invariants as a systematic approach to higher order gravity models: Physical and Cosmological Constraints
We compare higher order gravity models to observational constraints from
magnitude-redshift supernova data, distance to the last scattering surface of
the CMB, and Baryon Acoustic Oscillations. We follow a recently proposed
systematic approach to higher order gravity models based on minimal sets of
curvature invariants, and select models that pass some physical acceptability
conditions (free of ghost instabilities, real and positive propagation speeds,
and free of separatrices). Models that satisfy these physical and observational
constraints are found in this analysis and do provide fits to the data that are
very close to those of the LCDM concordance model. However, we find that the
limitation of the models considered here comes from the presence of
superluminal mode propagations for the constrained parameter space of the
models.Comment: 12 pages, 6 figure
- âŠ