2,147 research outputs found
On the steady states of the spherically symmetric Einstein-Vlasov system
Using both numerical and analytical tools we study various features of
static, spherically symmetric solutions of the Einstein-Vlasov system. In
particular, we investigate the possible shapes of their mass-energy density and
find that they can be multi-peaked, we give numerical evidence and a partial
proof for the conjecture that the Buchdahl inequality , the quasi-local mass, holds for all such steady states--both
isotropic {\em and} anisotropic--, and we give numerical evidence and a partial
proof for the conjecture that for any given microscopic equation of state--both
isotropic {\em and} anisotropic--the resulting one-parameter family of static
solutions generates a spiral in the radius-mass diagram.Comment: 34 pages, 18 figures, LaTe
The formation of black holes in spherically symmetric gravitational collapse
We consider the spherically symmetric, asymptotically flat Einstein-Vlasov
system. We find explicit conditions on the initial data, with ADM mass M, such
that the resulting spacetime has the following properties: there is a family of
radially outgoing null geodesics where the area radius r along each geodesic is
bounded by 2M, the timelike lines are incomplete, and for r>2M
the metric converges asymptotically to the Schwarzschild metric with mass M.
The initial data that we construct guarantee the formation of a black hole in
the evolution. We also give examples of such initial data with the additional
property that the solutions exist for all and all Schwarzschild time,
i.e., we obtain global existence in Schwarzschild coordinates in situations
where the initial data are not small. Some of our results are also established
for the Einstein equations coupled to a general matter model characterized by
conditions on the matter quantities.Comment: 36 pages. A corollary on global existence in Schwarzschild
coordinates for data which are not small is added together with minor
modification
Global existence and asymptotic behaviour in the future for the Einstein-Vlasov system with positive cosmological constant
The behaviour of expanding cosmological models with collisionless matter and
a positive cosmological constant is analysed. It is shown that under the
assumption of plane or hyperbolic symmetry the area radius goes to infinity,
the spacetimes are future geodesically complete, and the expansion becomes
isotropic and exponential at late times. This proves a form of the cosmic no
hair theorem in this class of spacetimes
Existence of axially symmetric static solutions of the Einstein-Vlasov system
We prove the existence of static, asymptotically flat non-vacuum spacetimes
with axial symmetry where the matter is modeled as a collisionless gas. The
axially symmetric solutions of the resulting Einstein-Vlasov system are
obtained via the implicit function theorem by perturbing off a suitable
spherically symmetric steady state of the Vlasov-Poisson system.Comment: 32 page
Static cylindrically symmetric spacetimes
We prove existence of static solutions to the cylindrically symmetric
Einstein-Vlasov system, and we show that the matter cylinder has finite
extension. The same results are also proved for a quite general class of
equations of state for perfect fluids coupled to the Einstein equations,
extending the class of equations of state considered in \cite{BL}. We also
obtain this result for the Vlasov-Poisson system.Comment: Added acknowledgemen
Regularity results for the spherically symmetric Einstein-Vlasov system
The spherically symmetric Einstein-Vlasov system is considered in
Schwarzschild coordinates and in maximal-isotropic coordinates. An open problem
is the issue of global existence for initial data without size restrictions.
The main purpose of the present work is to propose a method of approach for
general initial data, which improves the regularity of the terms that need to
be estimated compared to previous methods. We prove that global existence holds
outside the centre in both these coordinate systems. In the Schwarzschild case
we improve the bound on the momentum support obtained in \cite{RRS} for compact
initial data. The improvement implies that we can admit non-compact data with
both ingoing and outgoing matter. This extends one of the results in
\cite{AR1}. In particular our method avoids the difficult task of treating the
pointwise matter terms. Furthermore, we show that singularities never form in
Schwarzschild time for ingoing matter as long as This removes an
additional assumption made in \cite{A1}. Our result in maximal-isotropic
coordinates is analogous to the result in \cite{R1}, but our method is
different and it improves the regularity of the terms that need to be estimated
for proving global existence in general.Comment: 25 pages. To appear in Ann. Henri Poincar\'
Critical collapse of collisionless matter - a numerical investigation
In recent years the threshold of black hole formation in spherically
symmetric gravitational collapse has been studied for a variety of matter
models. In this paper the corresponding issue is investigated for a matter
model significantly different from those considered so far in this context. We
study the transition from dispersion to black hole formation in the collapse of
collisionless matter when the initial data is scaled. This is done by means of
a numerical code similar to those commonly used in plasma physics. The result
is that for the initial data for which the solutions were computed, most of the
matter falls into the black hole whenever a black hole is formed. This results
in a discontinuity in the mass of the black hole at the onset of black hole
formation.Comment: 22 pages, LaTeX, 7 figures (ps-files, automatically included using
psfig
On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system
In a previous work \cite{An1} matter models such that the energy density
and the radial- and tangential pressures and
satisfy were considered in the context of
Buchdahl's inequality. It was proved that static shell solutions of the
spherically symmetric Einstein equations obey a Buchdahl type inequality
whenever the support of the shell, satisfies
Moreover, given a sequence of solutions such that then the
limit supremum of was shown to be bounded by
In this paper we show that the hypothesis
that can be realized for Vlasov matter, by constructing a
sequence of static shells of the spherically symmetric Einstein-Vlasov system
with this property. We also prove that for this sequence not only the limit
supremum of is bounded, but that the limit is
since for Vlasov matter.
Thus, static shells of Vlasov matter can have arbitrary close to
which is interesting in view of \cite{AR2}, where numerical evidence is
presented that 8/9 is an upper bound of of any static solution of the
spherically symmetric Einstein-Vlasov system.Comment: 20 pages, Late
Existence of maximal hypersurfaces in some spherically symmetric spacetimes
We prove that the maximal development of any spherically symmetric spacetime
with collisionless matter (obeying the Vlasov equation) or a massless scalar
field (obeying the massless wave equation) and possessing a constant mean
curvature Cauchy surface also contains a maximal Cauchy
surface. Combining this with previous results establishes that the spacetime
can be foliated by constant mean curvature Cauchy surfaces with the mean
curvature taking on all real values, thereby showing that these spacetimes
satisfy the closed-universe recollapse conjecture. A key element of the proof,
of interest in itself, is a bound for the volume of any Cauchy surface
in any spacetime satisfying the timelike convergence condition in terms of the
volume and mean curvature of a fixed Cauchy surface and the maximal
distance between and . In particular, this shows that any
globally hyperbolic spacetime having a finite lifetime and obeying the
timelike-convergence condition cannot attain an arbitrarily large spatial
volume.Comment: 8 pages, REVTeX 3.
Probabilistic study of the resistance of a simply-supported reinforced concrete slab according to Eurocode parametric fire
We present the application of a simple probabilistic methodology to determine the reliability of a structural element exposed to fire when designed following Eurocode 1-1-2 (EC1). Eurocodes are being used extensively within the European Union in the design of many buildings and structures. Here, the methodology is applied to a simply-supported, reinforced concrete slab 180 mm thick, with a standard load bearing fire resistance of 90 min. The slab is subjected to a fire in an office compartment of 420 m 2 floor area and 4 m height. Temperature time curves are produced using the EC1 parametric fire curve, which assumes uniform temperature and a uniform burning condition for the fire. Heat transfer calculations identify the plausible worst case scenarios in terms of maximum rebar temperature. We found that a ventilation-controlled fire with opening factor 0.02 m 1/2 results in a maximum rebar temperature of 448°C after 102 min of fire exposure. Sensitivity analyses to the main parameters in the EC1 fire curves and in the EC1 heat transfer calculations are performed using a one-at-a-time (OAT) method. The failure probability is then calculated for a series of input parameters using the Monte Carlo method. The results show that this slab has a 0.3% probability of failure when the compartment is designed with all layers of safety in place (detection and sprinkler systems, safe access route, and fire fighting devices are available). Unavailability of sprinkler systems results in a 1% probability of failure. When both sprinkler system and detection are not available in the building, the probability of failure is 8%. This novel study conducts for the first time a probabilistic calculation using the EC1 parametric curve, helping engineers to identify the most critical design fires and the probabilistic resistance assumed in EC1
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