647 research outputs found
Time-Translation Invariance of Scattering Maps and Blue-Shift Instabilities on Kerr Black Hole Spacetimes
In this paper, we provide an elementary, unified treatment of two distinct
blue-shift instabilities for the scalar wave equation on a fixed Kerr black
hole background: the celebrated blue-shift at the Cauchy horizon (familiar from
the strong cosmic censorship conjecture) and the time-reversed red-shift at the
event horizon (relevant in classical scattering theory).
Our first theorem concerns the latter and constructs solutions to the wave
equation on Kerr spacetimes such that the radiation field along the future
event horizon vanishes and the radiation field along future null infinity
decays at an arbitrarily fast polynomial rate, yet, the local energy of the
solution is infinite near any point on the future event horizon. Our second
theorem constructs solutions to the wave equation on rotating Kerr spacetimes
such that the radiation field along the past event horizon (extended into the
black hole) vanishes and the radiation field along past null infinity decays at
an arbitrarily fast polynomial rate, yet, the local energy of the solution is
infinite near any point on the Cauchy horizon.
The results make essential use of the scattering theory developed in [M.
Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for the
wave equation on Kerr black hole exteriors, preprint (2014) available at
\url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translation
invariance of the scattering map and the non-triviality of the transmission
map.Comment: 26 pages, 12 figure
Sensitivity of wardrop equilibria
We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by Δ or removes an edge carrying only an Δ-fraction of flow. We study how the equilibrium responds to such an Δ-change.
Our first surprising finding is that, even for linear latency functions, for every Δ>â0, there are networks in which an Δ-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most Δ.
Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an Δ-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1â+âΔ) p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Both bounds are shown to be tight.
Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded
Inextendibility of expanding cosmological models with symmetry
A new criterion for inextendibility of expanding cosmological models with
symmetry is presented. It is applied to derive a number of new results and to
simplify the proofs of existing ones. In particular it shows that the solutions
of the Einstein-Vlasov system with symmetry, including the vacuum
solutions, are inextendible in the future. The technique introduced adds a
qualitatively new element to the available tool-kit for studying strong cosmic
censorship.Comment: 7 page
Weak solutions to problems involving inviscid fluids
We consider an abstract functional-differential equation derived from the
pressure-less Euler system with variable coefficients that includes several
systems of partial differential equations arising in the fluid mechanics. Using
the method of convex integration we show the existence of infinitely many weak
solutions for prescribed initial data and kinetic energy
Stability and Instability of Extreme Reissner-Nordstr\"om Black Hole Spacetimes for Linear Scalar Perturbations I
We study the problem of stability and instability of extreme
Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we
consider solutions to the linear wave equation on a suitable globally
hyperbolic subset of such a spacetime, arising from regular initial data
prescribed on a Cauchy hypersurface crossing the future event horizon. We
obtain boundedness, decay and non-decay results. Our estimates hold up to and
including the horizon. The fundamental new aspect of this problem is the
degeneracy of the redshift on the event horizon. Several new analytical
features of degenerate horizons are also presented.Comment: 37 pages, 11 figures; published version of results contained in the
first part of arXiv:1006.0283, various new results adde
SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension
We prove that if is the entropy
solution to a strictly hyperbolic system of conservation laws with
genuinely nonlinear characteristic fields then up to a
countable set of times the function is in
, i.e. its distributional derivative is a measure with no
Cantorian part.
The proof is based on the decomposition of into waves belonging to
the characteristic families and the balance
of the continuous/jump part of the measures in regions bounded by
characteristics. To this aim, a new interaction measure \mu_{i,\jump} is
introduced, controlling the creation of atoms in the measure .
The main argument of the proof is that for all where the Cantorian part
of is not 0, either the Glimm functional has a downward jump, or there is
a cancellation of waves or the measure is positive
Gravitational instability of an extreme Kerr black hole
Aretakis has proved the existence of an instability of a massless scalar
field at the horizon of an extreme Kerr or Reissner-Nordstrom black hole: for
generic initial data, a transverse derivative of the scalar field at the
horizon does not decay, and higher transverse derivatives blow up. We show that
a similar instability occurs for linearized gravitational, and electromagnetic,
perturbations of an extreme Kerr black hole. We show also that the massless
scalar field instability occurs for extreme black hole solutions of a large
class of theories in various spacetime dimensions.Comment: 13 pages. v2: minor clarifications. v3: minor changes, published
versio
Boost invariant marginally trapped surfaces in Minkowski 4-space
The extremal and partly marginally trapped surfaces in Minkowski 4-space,
which are invariant under the group of boost isometries, are classified.
Moreover, it is shown that there do not exist extremal surfaces of this kind
with constant Gaussian curvature. A procedure is given in order to construct a
partly marginally trapped surface by gluing two marginally trapped surfaces
which are invariant under the group of boost isometries. As an application, a
proper star-surface is constructed.Comment: 13 pages, comment added in section
Hyperbolic Balance Laws with a Non Local Source
This paper is devoted to hyperbolic systems of balance laws with non local
source terms. The existence, uniqueness and Lipschitz dependence proved here
comprise previous results in the literature and can be applied to physical
models, such as Euler system for a radiating gas and Rosenau regularization of
the Chapman-Enskog expansion.Comment: 26 page
A note on spherically symmetric naked singularities in general dimension
We discuss generalizations of the recent theorem by Dafermos (hep-th/0403033)
forbidding a certain class of naked singularities in the spherical collapse of
a scalar field. Employing techniques similar to the ones Dafermos used, we
consider extending the theorem (1) to higher dimensions, (2) by including more
general matter represented by a stress-energy tensor satisfying certain
assumptions, and (3) by replacing the spherical geometry by a toroidal or
higher genus (locally hyperbolic) one. We show that the extension to higher
dimensions and a more general topology is straightforward; on the other hand,
replacing the scalar field by a more general matter content forces us to shrink
the class of naked singularities we are able to exclude. We then show that the
most common matter theories (scalar field interacting with a non-abelian gauge
field and a perfect fluid satisfying certain conditions) obey the assumptions
of our weaker theorem, and we end by commenting on the applicability of our
results to the five-dimensional AdS scenarii considered recently in the
literature.Comment: 16 pages, no figures, typos fixe
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