9,094 research outputs found
On the Weyl Curvature Hypothesis
The Weyl curvature hypothesis of Penrose attempts to explain the high
homogeneity and isotropy, and the very low entropy of the early universe, by
conjecturing the vanishing of the Weyl tensor at the Big-Bang singularity.
In previous papers it has been proposed an equivalent form of Einstein's
equation, which extends it and remains valid at an important class of
singularities (including in particular the Schwarzschild, FLRW, and isotropic
singularities). Here it is shown that if the Big-Bang singularity is from this
class, it also satisfies the Weyl curvature hypothesis.
As an application, we study a very general example of cosmological models,
which generalizes the FLRW model by dropping the isotropy and homogeneity
constraints. This model also generalizes isotropic singularities, and a class
of singularities occurring in Bianchi cosmologies. We show that the Big-Bang
singularity of this model is of the type under consideration, and satisfies
therefore the Weyl curvature hypothesis.Comment: 10 pages, slides at
http://www.sciencedirect.com/science/article/pii/S000349161300171
The Geometry of Warped Product Singularities
In this article the degenerate warped products of singular semi-Riemannian
manifolds are studied. They were used recently by the author to handle
singularities occurring in General Relativity, in black holes and at the
big-bang. One main result presented here is that a degenerate warped product of
semi-regular semi-Riemannian manifolds with the warping function satisfying a
certain condition is a semi-regular semi-Riemannian manifold. The connection
and the Riemann curvature of the warped product are expressed in terms of those
of the factor manifolds. Examples of singular semi-Riemannian manifolds which
are semi-regular are constructed as warped products. Applications include
cosmological models and black holes solutions with semi-regular singularities.
Such singularities are compatible with a certain reformulation of the Einstein
equation, which in addition holds at semi-regular singularities too.Comment: 14 page
Dichotomies for evolution equations in Banach spaces
The aim of this paper is to emphasize various concepts of dichotomies for
evolution equations in Banach spaces, due to the important role they play in
the approach of stable, instable and central manifolds. The asymptotic
properties of the solutions of the evolution equations are studied by means of
the asymptotic behaviors for skew-evolution semiflows.Comment: 22 page
Turing test, easy to pass; human mind, hard to understand
Under general assumptions, the Turing test can be easily passed by an appropriate algorithm. I show that for any test satisfying several general conditions, we can construct an algorithm that can pass that test, hence, any operational definition is easy to fulfill. I suggest a test complementary to Turing's test, which will measure our understanding of the human mind. The Turing test is required to fix the operational specifications of the algorithm under test; under this constrain, the additional test simply consists in measuring the length of the algorithm
On the wavefunction collapse
Wavefunction collapse is usually seen as a discontinuous violation of the
unitary evolution of a quantum system, caused by the observation. Moreover, the
collapse appears to be nonlocal in a sense which seems at odds with General
Relativity. In this article the possibility that the wavefunction evolves
continuously and hopefully unitarily during the measurement process is
analyzed. It is argued that such a solution has to be formulated using a time
symmetric replacement of the initial value problem in Quantum Mechanics. Major
difficulties in apparent conflict with unitary evolution are identified, but
eventually its possibility is not completely ruled out. This interpretation is
in a weakened sense both local and realistic, without contradicting Bell's
theorem. Moreover, if it is true, it makes Quantum Mechanics consistent with
General Relativity in the semiclassical framework.Comment: Available at: http://quanta.ws/ojs/index.php/quanta/article/view/4
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