2,215 research outputs found
The paradox of soft singularity crossing and its resolution by distributional cosmological quantitities
A cosmological model of a flat Friedmann universe filled with a mixture of
anti-Chaplygin gas and dust-like matter exhibits a future soft singularity,
where the pressure of the anti-Chaplygin gas diverges (while its energy density
is finite). Despite infinite tidal forces the geodesics pass through the
singularity. Due to the dust component, the Hubble parameter has a non-zero
value at the encounter with the singularity, therefore the dust implies further
expansion. With continued expansion however, the energy density and the
pressure of the anti-Chaplygin gas would become ill-defined, hence from the
point of view of the anti-Chaplygin gas only a contraction is allowed.
Paradoxically, the universe in this cosmological model would have to expand and
contract simultaneously. This obviosly could not happen. We solve the paradox
by redefining the anti-Chaplygin gas in a distributional sense. Then a
contraction could follow the expansion phase at the singularity at the price of
a jump in the Hubble parameter. Although such an abrupt change is not common in
any cosmological evolution, we explicitly show that the set of Friedmann,
Raychaudhuri and continuity equations are all obeyed both at the singularity
and in its vicinity. We also prove that the Israel junction conditions are
obeyed through the singular spatial hypersurface. In particular we enounce and
prove a more general form of the Lanczos equation.Comment: 12 pages; to be published in Phys.Rev.
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
The arrow of time, black holes, and quantum mixing of large N Yang-Mills theories
Quantum gravity in an AdS spacetime is described by an SU(N) Yang-Mills
theory on a sphere, a bounded many-body system. We argue that in the high
temperature phase the theory is intrinsically non-perturbative in the large N
limit. At any nonzero value of the 't Hooft coupling , an
exponentially large (in N^2) number of free theory states of wide energy range
(of order N) mix under the interaction. As a result the planar perturbation
theory breaks down. We argue that an arrow of time emerges and the dual string
configuration should be interpreted as a stringy black hole.Comment: 50 pages 3 figures uses harvma
Emergent order in rheoscopic swirls
We discuss the reflection of light by a rheoscopic fluid (a suspension of
microscopic rod-like crystals) in a steady two-dimensional flow. This is
determined by an order parameter which is a non-oriented vector, obtained by
averaging solutions of a nonlinear equation containing the strain rate of the
fluid flow. Exact solutions of this equation are obtained from solutions of a
linear equation which are analogous to Bloch bands for a one-dimensional
Schrodinger equation with a periodic potential. On some contours of the stream
function, the order parameter approaches a limit, and on others it depends
increasingly sensitively upon position. However, in the long-time limit a local
average of the order parameter is a smooth function of position in both cases.
We analyse the topology of the order parameter and the structure of the generic
zeros of the order parameter field.Comment: 28 pages, 13 figure
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