Sunday, August 20, 2006

Anyons, Weyl Curvature & Arrow of Time

1. The Weyl curvature tensor is zero in 2 + 1 spacetime.
2. Therefore, the gravity entropy is zero there. There is only a Ricci tensor.
3. In the World Hologram, the anyonic 2 + 1 spacetime boundary of 3 + 1 space-time is more fundamental.
4. The 2 + 1 space-time can only have zero gravity entropy if Penrose is right.
This suggests the the initial singularity is 2 + 1 space-time with zero-entropy in order to have the correct Arrow of Time.
5. Note that the dark energy and dark matter as zero point energy of negative and positive pressures respectively on larger and short scales respectively are virtual sources of zero entropy Ricci tensor.

This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphite or the quantum Hall effect. In space of three dimensions (or more), elementary particles have tightly constrained quantum numbers and, in particular, are restricted to being fermions or bosons. In two-dimensional systems, however, quasiparticles are observed whose quantum states range continuously between fermionic and bosonic, taking on any quantum value in between. Frank Wilczek coined the term "anyons" in 1982 to describe such particles.
Let's say we have two identical particles on a plane. If we interchange both particles so that each particle travels counterclockwise for half a cycle around the center of both particles, the wave function of the system changes by a factor of eiθ where θ is an angle which only depends upon the type of particle in question. If θ is zero, we have a boson and if θ is π we have a fermion. For any other value, we have an anyon. If we have two particles a and b, which may or may not be identical, then their mutual statistics is the change in the phase factor, which is picked up after particle b is rotated counterclockwise around particle a for one full cycle. The mutual statistics may be completely unrelated to the interchange angle between two identical particles.

Weyl curvature hypothesis
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This article concerns Roger Penrose's 1979 Weyl curvature hypothesis, which would justify spatial homogeneity and isotropy of the observable part of the Universe in the Big Bang model. A different article discusses Weyl's postulate, which is an assumption relating to separation of space and time in the Big Bang model.
The Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 [1] in an attempt to provide explanations for two of the most fundamental issues in physics. On the one hand one would like to account for a Universe which on its largest observational scales appears remarkably spatially homogeneous and isotropic in its physical properties (and so can be described by a simple Friedmann-Lemaître model), on the other hand there is the deep question on the origin of the second law of thermodynamics.
Penrose suggests that the resolution of both of these problems is rooted in a concept of the entropy content of gravitational fields. Near the initial cosmological singularity (the Big Bang), he proposes, the entropy content of the cosmological gravitational field was extremely low (compared to what it theoretically could have been), and started rising monotonically thereafter. This process manifested itself e.g. in the formation of structure through the clumping of matter to form galaxies and clusters of galaxies. Penrose associates the initial low entropy content of the Universe with the effective vanishing of the Weyl curvature tensor of the cosmological gravitational field near the Big Bang. From then on, he proposes, its dynamical influence gradually increased, thus being responsible for an overall increase in the amount of entropy in the Universe, and so inducing a cosmological arrow of time.
The Weyl curvature represents such gravitational effects as tidal fields and gravitational radiation. Mathematical treatments of Penrose's ideas on the Weyl curvature hypothesis have been given in the context of isotropic initial cosmological singularities e.g. in the articles [2] ,[3] ,[4] ,[5]. Penrose views the Weyl curvature hypothesis as a physically more credible alternative to cosmic inflation (a hypothetical phase of accelerated expansion in the early life of the Universe) in order to account for the presently observed almost spatial homogeneity and isotropy of our Universe [6].

Weyl tensor
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In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. In other words, it is a tensor that has the same symmetries as the Riemann curvature tensor with the extra condition that its Ricci curvature must vanish.
In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero.
The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valent tensor (by contracting with the metric). The (0,4) valent Weyl tensor is then

where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and hOk denotes the Kulkarni-Nomizu product of two symmetric (0,2) tensors:

The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.
The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if g′ = f g for some positive scalar function then the (1,3) valent Weyl tensor satisfies W′ = W. For this reason the Weyl tensor is also called the conformal tensor. It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. It turns out that in dimensions ≥ 4 this condition is sufficient as well. In dimension 3 the vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat.
The Weyl tensor is given in components by

where Rabcd is the Riemann tensor, Rab is the Ricci tensor, R is the Ricci scalar (the scalar curvature) and [] refers to the antisymmetric part.

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