37 research outputs found
Variational principle for the Einstein-Vlasov equations
The Einstein-Vlasov equations govern Einstein spacetimes filled with matter which interacts only via gravitation. The matter, described by a distribution function on phase space, evolves under the collisionless Boltzmann equation, corresponding to the free geodesic motion of the particles, while the source of the gravitational field is given by the stress-energy tensor defined in terms of momenta of the distribution function. As no variational derivation of the Einstein-Vlasov system appears to exist in the literature, we here set out to fill this gap. In our approach we treat the matter as a generalized type of fluid, flowing in the tangent bundle instead of the spacetime. We present the actions for the Einstein-Vlasov system in both the Lagrangian and Eulerian pictures
Isometric embeddings of 2-spheres by embedding flow for applications in numerical relativity
We present a numerical method for solving Weyl's embedding problem which
consists of finding a global isometric embedding of a positively curved and
positive-definite spherical 2-metric into the Euclidean three space. The method
is based on a construction introduced by Weingarten and was used in Nirenberg's
proof of Weyl's conjecture. The target embedding results as the endpoint of an
embedding flow in R^3 beginning at the unit sphere's embedding. We employ
spectral methods to handle functions on the surface and to solve various
(non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity
range from quasi-local mass and momentum measures to coarse-graining in
inhomogeneous cosmological models.Comment: 18 pages, 14 figure
Mechanics of multidimensional isolated horizons
Recently a multidimensional generalization of Isolated Horizon framework has
been proposed by Lewandowski and Pawlowski (gr-qc/0410146). Therein the
geometric description was easily generalized to higher dimensions and the
structure of the constraints induced by the Einstein equations was analyzed. In
particular, the geometric version of the zeroth law of the black hole
thermodynamics was proved. In this work we show how the IH mechanics can be
formulated in a dimension--independent fashion and derive the first law of BH
thermodynamics for arbitrary dimensional IH. We also propose a definition of
energy for non--rotating horizons.Comment: 25 pages, 4 figures (eps), last sections revised, acknowledgements
and a section about the gauge invariance of introduced quantities added;
typos corrected, footnote 4 on page 9 adde
Quasi--local angular momentum of non--symmetric isolated and dynamical horizons from the conformal decomposition of the metric
A new definition of quasi--local angular momentum of non--axisymmetric
marginally outer trapped surfaces is proposed. It is based on conformal
decomposition of the two--dimensional metric and the action of the group of
conformal symmetries. The definition is completely general and agrees with the
standard one in axi--symmetric surfaces.Comment: Final version to appear in Classical and Quantum Gravity. One
reference adde
Evolution of a periodic eight-black-hole lattice in numerical relativity
The idea of black-hole lattices as models for the large-scale structure of
the universe has been under scrutiny for several decades, and some of the
properties of these systems have been elucidated recently in the context of the
problem of cosmological backreaction. The complete, three-dimensional and fully
relativistic evolution of these system has, however, never been tackled. We
explicitly construct the first of these solutions by numerically integrating
Einstein's equation in the case of an eight-black-hole lattice with the
topology of S3.Comment: 21 pages, 13 figures. Corrected and clarified discussio
A quasi-static approach to structure formation in black hole universes
JD and TC both acknowledge support from the STFC under grant STFC ST/N504257/1
The magnetic part of the Weyl tensor, and the expansion of discrete universes
42 pages, 27 figures42 pages, 27 figures42 pages, 27 figures42 pages, 27 figures42 pages, 27 figures42 pages, 27 figuresTC is supported by the STFC and DG by an AARMS postdoctoral fellowship
The Method of Images in Cosmology
31 pages, 18 figures31 pages, 18 figuresWe apply the method of images to the exact initial data for cosmological models that contain a number of regularly arranged discrete masses. This allows us to join cosmological regions together by throats, and to construct wormholes in the initial data. These wormholes allow for the removal of the asymptotically flat "flange" regions that would otherwise exist on the far side of black holes. The method of images also provides us with a way to investigate the definition of mass is cosmology, and the cosmological consequences of the gravitational interaction energies between massive objects. We find evidence that the interaction energies within clusters of massive objects do indeed appear to contribute to the total energy budget in the cosmological regions of the space-time
Persistent black holes in bouncing cosmologies
In this paper we explore the idea that black holes can persist in a universe
that collapses to a big crunch and then bounces into a new phase of expansion.
We use a scalar field to model the matter content of such a universe {near the
time} of the bounce, and look for solutions that represent a network of black
holes within a dynamical cosmology. We find exact solutions to Einstein's
constraint equations that provide the geometry of space at the minimum of
expansion and that can be used as initial data for the evolution of
hyperspherical cosmologies. These solutions illustrate that there exist models
in which multiple distinct black holes can persist through a bounce, and allow
for concrete computations of quantities such as the black hole filling factor.
We then consider solutions in flat cosmologies, as well as in
higher-dimensional spaces (with up to nine spatial dimensions). We derive
conditions for the black holes to remain distinct (i.e. avoid merging) and
hence persist into the new expansion phase. Some potentially interesting
consequences of these models are also discussed.Comment: 37 pages, 16 figure
Covariant coarse-graining of inhomogeneous dust flow in General Relativity
A new definition of coarse-grained quantities describing the dust flow in
General Relativity is proposed. It assigns the coarse--grained expansion, shear
and vorticity to finite-size comoving domains of fluid in a covariant,
coordinate-independent manner. The coarse--grained quantities are all
quasi-local functionals, depending only on the geometry of the boundary of the
considered domain. They can be thought of as relativistic generalizations of
simple volume averages of local quantities in a flat space. The procedure is
based on the isometric embedding theorem for S^2 surfaces and thus requires the
boundary of the domain in question to have spherical topology and positive
scalar curvature. We prove that in the limit of infinitesimally small volume
the proposed quantities reproduce the local expansion, shear and vorticity. In
case of irrotational flow we derive the time evolution for the coarse-grained
quantities and show that its structure is very similar to the evolution
equation for their local counterparts. Additional terms appearing in it may
serve as a measure of the backreacton of small-scale inhomogeneities of the
flow on the large-scale motion of the fluid inside the domain and therefore the
result may be interesting in the context of the cosmological backreaction
problem. We also consider the application of the proposed coarse-graining
procedure to a number of known exact solutions of Einstein equations with dust
and show that it yields reasonable results.Comment: 17 pages, 5 figures. Version accepted in Classical and Quantum
Gravity