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Topic: Weyl curvature

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	Weyl curvature - Wikipedia, the free encyclopedia
		In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor.
		In dimensions 2 and 3 the Weyl curvature tensor vanishes identically.
		In dimension 3 the vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat.
	en.wikipedia.org /wiki/Weyl_tensor (304 words)

	Curvature of Riemannian manifolds - Wikipedia, the free encyclopedia
		the curvature tensor measures noncommutativity of the covariant derivative.
		It is the Gauss curvature of the σ-section at p; here σ-section is a locally-defined piece of surface which has the plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of σ under the exponential map at p.
		For a manifold of constant curvature, the Weyl tensor is zero.
	en.wikipedia.org /wiki/Curvature_of_Riemannian_manifolds (924 words)

	Curvature tensor
		In two dimensions, the curvature tensor is determined by the scalar curvature - which is the full trace of the curvature tensor.
		In three dimensions, the curvature tensor is specified by the Ricci curvature - which is a partial trace of the curvature tensor.
		For dimension n>3, the curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor.
	www.guajara.com /wiki/en/wikipedia/c/cu/curvature_tensor.html (728 words)

	Encyclopedia article: Weyl curvature (Site not responding. Last check: 2007-11-05)
		In differential geometry (additional info and facts about differential geometry), the Weyl curvature tensor is the traceless component of the Riemann curvature tensor (additional info and facts about Riemann curvature tensor).
		In other words, it a tensor (Any of several muscles that cause an attached structure to become tense or firm) that has the same symmetries as the Riemann curvature tensor (additional info and facts about Riemann curvature tensor) with the extra condition that its Ricci curvature (additional info and facts about Ricci curvature) must vanish.
		The Weyl tensor has the special property that it is invariant under conformal (additional info and facts about conformal) changes to the metric (A system of related measures that facilitates the quantification of some particular characteristic).
	www.absoluteastronomy.com /encyclopedia/w/we/weyl_curvature.htm (380 words)

	Curvature of Riemannian manifolds bei eLexi - das Onlinelexikon (Site not responding. Last check: 2007-11-05)
		Curvature of Riemannian manifolds bei eLexi - das Onlinelexikon
		It is the Gauss curvature of the -section at p; here -section is a locally-defined piece of surface which has the plane as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of under the exponential map at p.
		The curvature is given by an antisymmetric matrix of 2-forms, or equivalently a 2-form with values in, the Lie algebra of the orthogonal group (which is the structure group of the tangent bundle of a Riemannian manifold).
	www.elexi.de /en/c/cu/curvature_of_riemannian_manifolds.html (981 words)

	4.1 Weyl’s theory (Site not responding. Last check: 2007-11-05)
		Weyl’s fundamental idea for generalising Riemannian geometry was to note that, unlike for the comparison of vectors at different points of the manifold, for the comparison of scalars the existence of a connection is not required.
		In his “addendum concerning the newest theory of Weyl”, he came as far as to show that Weyl’s connection is gauge invariant, and to point to the identification of the electromagnetic 4-potential.
		For Weyl, knowledge of the charge and mass of each particle, and of the extension of their “world-channels” were insufficient to determine the field uniquely.
	www.emis.de /journals/LRG/Articles/lrr-2004-2/articlesu9.html (4019 words)

	Space and Time
		In the early universe the Weyl curvature was probably zero, but in a dying universe the large number of fl holes, Penrose argues, will give rise to a high Weyl curvature.
		The Weyl tensor is that part of the curvature of space-time that is not locally determined by the matter through the Einstein equations.
		Stephen argued that there must be small quantum fluctuations in the initial state and thus pointed out that the hypothesis that the initial Weyl curvature is zero at the initial singularity is classical, and there is certainly some flexibility as to the precise statement of the hypothesis.
	www.fortunecity.com /emachines/e11/86/space.html (4186 words)

	PlanetMath: Ricci tensor
		It is also convenient to regard the Ricci tensor as a symmetric bilinear form.
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		It is the conformally invariant, trace-free part of the curvature tensor.
	planetmath.org /encyclopedia/RicciTensor.html (336 words)

	How does matter couple to space-time so that space-time becomes curved?
		Furthermore, the curvature of space-time at each event is completely described by a multilinear operator (a generalization of a linear operator) called the Riemann curvature tensor, which has 20 algebraically independent components at each event.
		If it were not for the Weyl tensor, this would mean that matter here could not have a gravitational influence on distant matter separated by a void (a vacuum free of mass-energy).
		The Weyl tensor turns out to be analogous in many ways to the electromagnetic field tensor, which you can think of as an antisymmetric four by four matrix (6 algebraically independent components at each event).
	www.physlink.com /Education/AskExperts/ae98.cfm (689 words)

	Citebase - Double forms, curvature structures and the $(p,q)$-curvatures
		In particular, for p=0, the (0,q)-curvatures coincide with the H. Weyl curvature invariants, for p=1 the (1,q)-curvatures are the curvatures of generalized Einstein tensors and for q=1 the (p,1)-curvatures coincide with the p-curvatures.
		Also, we prove that for an Einstein manifold of dimension n≥ 4 the second H. Weyl curvature invariant is nonegative, and that it is nonpositive for a conformally flat manifold with zero scalar curvature.
		Recall that the (2k)-H. Weyl curvature invariant is a polynomial of degree 2k with respect to the Riemannian curvature.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0404081 (752 words)

	[No title]
		We establish some connections between negative curvature of the Weyl structure and the hyperbolicity of W-flows, generalizing in dimension 2 the classical result of Anosov on Riemannian geodesic flows.
		In the case of the Weyl structure with a local potential we can formulate a counterpart of Proposition 5.5 because by \thetag{6.4} the form $\Cal J$ is not decreased at the collisions.
		The stability of an orbit depends on its length $l$ and the signed curvatures at the points of reflections $k_0,k_1$ ($k$ is negative for a strictly concave boundary).
	www.ma.utexas.edu /mp_arc/papers/00-51 (3569 words)

	5D Chapter 1 (Site not responding. Last check: 2007-11-05)
		Weyl sought to alter the geometry of the continuum while maintaining the number of dimensions at four.
		Like Weyl's basic concepts, Kaluza's ideas have led to many modifications and extensions, but unlike Weyl's theory the five-dimensional structure built by Kaluza has never been proven wrong and still stands as an independent theory as it was originally conceived.
		The tensor describing the four-dimensional behavior corresponded to the curvature tensor in General Relativity, although a torsion-tensor was introduced representing the displacement along the normal.
	members.aol.com /yggdras/paraphysics/5dchap1.htm (12861 words)

	Research in Applied Mathematics
		The prototype is metric symmetry, but the study extends to homotheties, affine collineations, conformal and projective symmetries and Ricci, matter and curvature collineations.
		On a different topic, further work is planned on the remarkable (almost one to one) correspondence between the metric and the sectional curvature function on space-time established some years ago by Hall and the consequential idea of using the latter function as an alternative field variable for general relativity.
		Weyl connections) and has also been applied to the sectional curvature function.
	www.maths.abdn.ac.uk /maths/department/test-research/public-prospectus/xapplied/xapplied.html (892 words)

	Re: Does the Electromagnetic field have a Gravitational field?
		John can speak for himself, of course, but I am pretty sure I know what he had in mind: Weyl curvature can propagate from distant regions, so it can arrive from "outside", whereas Ricci curvature at some event is always due entirely to the -immediate presence- at that even of some nongravitational mass-energy.
		In this model, the interior of the star comprises a conformally flat region of spacetime, so all the curvature there is due to the Ricci tensor (or equivalently, to the Einstein tensor).
		That -is- a tensorial sum, but don't forget that the EFE is nonlinear, and in particular, the coupling of Weyl curvature to energy/momentum is nonlinear.
	www.lns.cornell.edu /spr/2005-01/msg0066560.html (742 words)

	CERN Courier - Superluminal phenomena shed - IOP Publishing - article
		This depends explicitly on the curvature, in violation of the SEP. The photon velocity is changed and light no longer follows the shortest possible path.
		Indeed we now know that for propagation in vacuum space-times (solutions of Einstein's field equations in regions with no matter present, such as the neighbourhood of the event horizon of fl holes), there is a general theorem showing that if one photon polarization has a conventional subluminal velocity, the other polarization is necessarily superluminal.
		In fact, gravity affects the photon velocity in two distinct ways: the first through the energy momentum of the gravitating matter; and the second through the component of the curvature of space-time that is not determined locally by matter, the so-called Weyl curvature.
	cerncourier.com /main/article/42/3/13 (1899 words)

	Ephilosopher :: Philosophy of Religion Forum :: God and the Philosophy of Physics (Site not responding. Last check: 2007-11-05)
		In fact, one of the most prevailing current hypotheses in physics, the Weyl Curvature Hypothesis, claims that at a minimum, the initial conditions of the universe could not have occurred by chance.
		The Weyl Curvature Hypothesis is not a debatable premise.
		The whole thrust of the Weyl hypothesis is that the notion of God in its most basic form is actually natural or consistent with the physical laws.
	www.ephilosopher.com /phpBB_14-action-viewtopic-topic-3124.html (3998 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		space with a nonvanishing scalar curvature to be a recurrent Weyl space is given.
		A number of theorems concerning the hypersurfaces of a concircularly recurrent Weyl space are given.
		R = 0 then the concircular curvature tensor becomes identical with the curvature tensor.
	www.science.az /acm/Web/Hakan.htm (643 words)

	Re:Does the Electromagnetic field have a Gravitational field?
		If that were true, long range gravitational interactions would be impossible, because the immediate presence of matter here and now (which according to the EFE creates Ricci curvature here and now) could not curve up a surrounding vacuum region, because the Weyl curvature would not "couple" to Ricci curvature.
		Not sure I understood the question, but roughly speaking, if you suddenly vary a distribution of charges, which is concentrated in some compact region, in such a way that the distribution of EM field energy varies aspherically and rapidly, then you can expect to create outgoing gravitational radiation, which will accompany the outgoing EM radiation.
		As we have seen, in gtr, Ricci curvature -does- couple to Weyl curvature, and almost anything you do to generate EM waves is likely to also produce (much, much, muuuuch weaker) accompanying gravitational waves.
	www.lns.cornell.edu /spr/2005-01/msg0066310.html (1008 words)

	Time Travel: (Site not responding. Last check: 2007-11-05)
		Penrose's proposal is that the Weyl tensor should vanish at one end of time but not the other.
		Penrose's theory is not CPT invariant and means that if the Weyl tensor had been exactly zero in the early universe, it would have been exactly homogeneous and isotropic and would have remained so for all time.
		Hawking states that the Weyl tensor can and will be small but can never reach zero because this would be a direct violation of the uncertainty principle and instead there would have been small fluctuations that later grew into galaxies and bodies.
	www.cakes.mcmail.com /StarTrek/time.htm (2110 words)

	2.3 Asymptotically flat space-times
		The equation is not conformally invariant since the conformal rescaling of a vacuum metric generates Ricci curvature in the unphysical space-time by Equation (111
		coincides with the Weyl tensor, which is the source of tidal forces acting on test particles moving in space-time.
		This equation for the rescaled Weyl tensor is an important sub-structure of the Einstein equation because it is conformally invariant, in contrast to the Einstein equation itself.
	relativity.livingreviews.org /Articles/lrr-2004-1/articlesu3.html (1560 words)

	ricci.weyl (Site not responding. Last check: 2007-11-05)
		The Weyl tensor tells the REST of the story about what happens to the ball.
		When we are in truly empty space, there's no Ricci curvature, so actually our ball of coffee grounds doesn't change volume.
		But there can be Weyl curvature due to gravitational waves, tidal forces, and the like.
	math.ucr.edu /home/baez/gr/ricci.weyl.html (560 words)

	Dr. Harish Seshadry
		A class of nonpositively curved Kahler manifolds biholomorphic to the unit ball in C^n (with Kaushal Verma), to appear in C. Acad.
		Weyl curvature and the Euler characteristic in dimension four, to appear in Differential Geometry and its Applications.
		Positive scalar curvature and minimal hypersurfaces, Proceedings of the American Mathematical Society, 133 (2005), 1497-1504.
	math.iisc.ernet.in /~harish/publi.htm (126 words)

	4a
		Weyl relativity is one proposal for replacing the Einstein-Hilbert action with an action that is invariant under multiplication of the metric by a positive constant: under the variation above,
		There is an effective algorithm for re-expressing the ambient results in terms of tractors which then expand easily into formulae in terms of the underlying Riemannian curvature and its covariant derivatives.
		operator and the square of the Weyl curvature is here viewed as a multiplication operator.
	www.aimath.org /WWN/confstruct/articles/html/4a (2504 words)

	oz11
		From what has been said about the Weyl tensor, some of it's terms relate to boundary conditions.
		I am a mite concerned that the Weyl does include static curvature from distant bits.
		Indeed, Penrose makes a big deal out of his quasi-plausible `Weyl curvature hypothesis', which says that as we approach the big bang, the Weyl curvature must go to zero, for reasons similar to what you say, and so the Weyl curvature serves as a kind of `arrow of time', increasing as time passes.
	math.ucr.edu /home/baez/gr/oz11.html (826 words)

	Lecture Notes on Finsler Geometry
		Surprisingly, the curvatures of a spray can be defined by very simple formulas.
		We call B the Berwald curvature and R the Riemann curvature.
		D and W are called the Douglas curvature and the Berwald-Weyl curvature, respectively.
	www.math.iupui.edu /~zshen/Research/papers/lecture/lecture.html (501 words)

	A Swift and Simple Refutation of the Kalam Cosmological Argument?
		If we remove the initial cosmological singularity, we render the Weyl Curvature Hypothesis irrelevant and "we should be back where we were in our attempts to understand the origin of the second law."{39} Could the special initial geometry have arisen sheerly by chance in the absence of a cosmic singularity?
		{37}Weyl curvature is the curvature of space-time which is not due to the presence of matter and is described by the Weyl tensor.
		Space-time curvature due to matter is described by the Einstein tensor.
	www.leaderu.com /offices/billcraig/docs/kalam-oppy.html (6075 words)

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