17 research outputs found
Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration
We consider two strongly hyperbolic Hamiltonian formulations of general
relativity and their numerical integration with a free and a partially
constrained symplectic integrator. In those formulations we use hyperbolic
drivers for the shift and in one case also for the densitized lapse. A system
where the densitized lapse is an external field allows to enforce the momentum
constraints in a holonomically constrained Hamiltonian system and to turn the
Hamilton constraint function from a weak to a strong invariant.
These schemes are tested in a perturbed Minkowski and the Schwarzschild
space-time. In those examples we find advantages of the strongly hyperbolic
formulations over the ADM system presented in [arXiv:0807.0734]. Furthermore we
observe stabilizing effects of the partially constrained evolution in
Schwarzschild space-time as long as the momentum constraints are enforced.Comment: This version clarifies some points concerning the interpretation of
the result
Hamiltonian theory for the axial perturbations of a dynamical spherical background
We develop the Hamiltonian theory of axial perturbations around a general
time-dependent spherical background spacetime. Using the fact that the
linearized constraints are gauge generators, we isolate the physical and
unconstrained axial gravitational wave in a Hamiltonian pair of variables.
Then, switching to a more geometrical description of the system, we construct
the only scalar combination of them. We obtain the well-known Gerlach and
Sengupta scalar for axial perturbations, with no known equivalent for polar
perturbations. The strategy suggested and tested here will be applied to the
polar case in a separate article.Comment: 12 pages, accepted by Classical and Quantum Gravit
Gravitational collapse and naked singularities
Gravitational collapse is one of the most striking phenomena in gravitational
physics. The cosmic censorship conjecture has provided strong motivation for
researches in this field. In the absence of general proof for the censorship,
many examples have been proposed, in which naked singularity is the outcome of
gravitational collapse. Recent development has revealed that there are examples
of naked singularity formation in the collapse of physically reasonable matter
fields, although the stability of these examples is still uncertain. We propose
the concept of ``effective naked singularities'', which will be quite helpful
because general relativity has the limitation of its application for
high-energy end. The appearance of naked singularities is not detestable but
can open a window for new physics of strongly curved spacetimes.Comment: 12 pages, to appear in the Proceedings of the International
Conference on Gravitation and Cosmology (ICGC-2004), ed. by B.R. Iyer, V.
Kuriakose and C.V. Vishveshwara, published by Pramana, minor correction
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
From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
This article reviews some aspects in the current relationship between
mathematical and numerical General Relativity. Focus is placed on the
description of isolated systems, with a particular emphasis on recent
developments in the study of black holes. Ideas concerning asymptotic flatness,
the initial value problem, the constraint equations, evolution formalisms,
geometric inequalities and quasi-local black hole horizons are discussed on the
light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity.
Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24
November, 2006), part of the "General Relativity Trimester" at the Institut
Henri Poincare (Fall 2006). Comments and references added. Typos corrected.
Submitted to Classical and Quantum Gravit
Naked Singularity Formation In f(R) Gravity
We study the gravitational collapse of a star with barotropic equation of
state in the context of theories of gravity.
Utilizing the metric formalism, we rewrite the field equations as those of
Brans-Dicke theory with vanishing coupling parameter. By choosing the
functionality of Ricci scalar as , we
show that for an appropriate initial value of the energy density, if
and satisfy certain conditions, the resulting singularity would be naked,
violating the cosmic censorship conjecture. These conditions are the ratio of
the mass function to the area radius of the collapsing ball, negativity of the
effective pressure, and the time behavior of the Kretschmann scalar. Also, as
long as parameter obeys certain conditions, the satisfaction of the
weak energy condition is guaranteed by the collapsing configuration.Comment: 15 pages, 4 figures, to appear in GR
The Einstein-Vlasov System/Kinetic Theory
The main purpose of this article is to provide a guide to theorems on global
properties of solutions to the Einstein--Vlasov system. This system couples
Einstein's equations to a kinetic matter model. Kinetic theory has been an
important field of research during several decades in which the main focus has
been on non-relativistic and special relativistic physics, i.e., to model the
dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In
1990, Rendall and Rein initiated a mathematical study of the Einstein--Vlasov
system. Since then many theorems on global properties of solutions to this
system have been established.Comment: Published version http://www.livingreviews.org/lrr-2011-
Angular momentum near the black hole threshold in scalar field collapse
For the formation of a black hole in the gravitational collapse of a massless scalar field, we calculate a critical exponent that governs the black hole angular momentum for slightly non-spherical initial data near the black hole threshold. We calculate the scaling law by second-order perturbation theory. We then use the numerical results of a previous first-order perturbative analysis to obtain the numerical value ??0.76 for the angular momentum critical exponent. A quasi-periodic fine structure is superimposed on the overall power law
Self-similar spherically symmetric solutions of the massless Einstein-Vlasov system
We construct the general spherically symmetric and self-similar solution of the Einstein-Vlasov system (collisionless matter coupled to general relativity) with massless particles, under certain regularity conditions. Such solutions have a curvature singularity by construction, and their initial data on a Cauchy surface to the past of the singularity can be chosen to have compact support in momentum space. They can also be truncated at large radius so that they have compact support in space, while retaining self-similarity in a central region that includes the singularity. However, the Vlasov distribution function cannot be bounded. As a simpler illustration of our techniques and notation we also construct the general spherically symmetric and static solution, for both massive and massless particles.<br/
Charge scaling and universality in critical collapse
Consider any one-parameter family of initial data such that data with a parameter value p>p* form black holes, and data with p<p* do not. As p?p* from above ("critical collapse"), the black hole mass scales as M?(p-p*)?, where the critical exponent ? is the same for all such families of initial data. So far critical collapse has been investigated only for initial data with zero charge and zero angular momentum. Here, we allow for U(1) charge. In scalar electrodynamics coupled to gravity, with action R+|(?+iqA)?|2+F2, we consider initial data with spherical symmetry and a nonvanishing charge. From dimensional analysis and a previous calculation of Lyapunov exponents, we predict that in critical collapse the black hole mass scales as M?(p-p*)?, and the black hole charge as Q?(p-p*)?, with ?=0.374±0.001 (as for the real scalar field) and ?=0.883±0.007. We conjecture that, where there is no mass gap, this behavior generalizes to other charged matter models, with ?>~2?. We suggest the existence of universality classes with respect to parameters such as q