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	PlanetMath: Ricci tensor
		In Riemannian geometry, the Ricci tensor represents the average value of the sectional curvature along a particular direction.
		scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor
		This is version 6 of Ricci tensor, born on 2005-02-16, modified 2006-09-07.
	planetmath.org /encyclopedia/WeylTensor.html (319 words)

	Scalar curvature - Wikipedia, the free encyclopedia
		In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold.
		It is defined as the trace of the Ricci curvature tensor with respect to the metric:
		The trace depends on the metric since the Ricci tensor is a (0,2)-valent tensor; one must first contract with the metric to obtain a (1,1)-valent tensor in order take the trace (see musical isomorphisms).
	en.wikipedia.org /wiki/Scalar_curvature (245 words)

	Demystifying Einstein’s Field Equations
		Ricci Tensor in the field equation defines the deviation of the n-dimensional volume of the space in a curved space-time from the flat Euclidean space.
		Ricci Scalar is the result of the contraction of Ricci tensor and Metric tensor.
		Since Ricci tensor and Metric tensor are rank 2 tensors, the contraction between the two of them results in a rank zero tensor which is actually a scalar.
	www.hitxp.com /phy/rel/gr/210906.htm (2858 words)

	A Very Large Scale Pattern Matching Problem in MathTensor
		If one index is raised to its contravariant position and summed with its corresponding lower index, the Riemann tensor is equal to the so-called Ricci tensor which is symmetric in its two indices.
		Because of the antisymmetries in the pairs of indices, when one of the pairs is summed, the result is zero.
		If the indices of the Riemann tensor are summed across pairs or on the indices on the Ricci tensor, we obtain the Riemann Scalar.
	smc.vnet.net /wri_talk.html (2219 words)

	3.3 Competing theories of gravity (Site not responding. Last check: 2007-10-24)
		is the Ricci scalar of the ``Einstein'' metric
		Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of unification, such as string theory.
		If the mass of the scalar field is sufficiently large that its range is microscopic, then, on solar-system scales, the scalar field is suppressed, and the theory is essentially equivalent to general relativity.
	www.maths.tcd.ie /EMIS/journals/LRG/Articles/Volume4/2001-4will/node9.html (1048 words)

	Application to the scalar-tensor theories. (Site not responding. Last check: 2007-10-24)
		is the scalar gravitational field, g is the determinant of the metric components
		This effect is proportional to the amplitude of the scalar perturbation.
		This formula shows that the contribution of the scalar wave to the scintillation cannot be zero, whatever be the direction of observation of the distant light source.
	www.obs-hp.fr /www/preprints/pp129/node5.html (545 words)

	Curved Space Geometry (Site not responding. Last check: 2007-10-24)
		of Ricci tensor by summing all the components of Riemann tensor that their second index is i the forth index is k and the first and third indices are the same.
		Ricci tensor itself is diagonally symmetric, so in 4D time-space it has 10 independent components out of its 16 components.
		Ricci tensor can describe everything about 3D curvature and all the 81 elements of the Riemann tensor can be calculated from the Ricci tensor.
	www.polarhome.com:763 /~rafimoor/english/GRE2.htm (3550 words)

	LORENE: Metric class Reference
		To be a real scalar it must be divided by e.g.
		The allocated representation depends on the type of the input tensor indices.
		Pointer to the radial vector normal to a spherical slicing and pointing toward spatial infinity.
	www.lorene.obspm.fr /Refguide/classMetric.html (188 words)

	Re: GR curvature scalar? (Site not responding. Last check: 2007-10-24)
		The simplest is explained on John Baez's webpages http://math.ucr.edu/home/baez/einstein/einstein.html Next, the Ricci tensor happens to be symmetric, so we might try to employ the "averaging" interpretation of the trace of a symmetric matrix (thinking of R(X,X), X some vector field, as, roughly speaking, a quadratic form varying smoothly from event to event, i.e.
		the Ricci scalar of the hyperslice (-not- the same thing at all as the Ricci scalar of the spacetime itself) can be interpreted as an average over a very small unit sphere.
		A more important way in which the Ricci tensor--- rather than its "trace reverse", the Einstein curvature tensor--- enters directly into gtr is via some of the most important equations in this subject: the "Raychaudhuri equation", and its close relatives, the "optical equations".
	www.lns.cornell.edu /spr/2004-05/msg0061146.html (1204 words)

	5.1 Motion of a scalar charge (Site not responding. Last check: 2007-10-24)
		of a point scalar charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and affect its motion.
		That the observed mass is not conserved is a remarkable property of the dynamics of a scalar charge in a curved spacetime.
		Physically, this corresponds to the fact that in a spacetime with a time-dependent metric, a scalar charge radiates monopole waves and the radiated energy comes at the expense of the particle’s inertial mass.
	univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-6/articlesu26.html (2037 words)

	Green's Functions for Gravitational Waves in FRW Spacetimes -- from Mathematica Information Center
		Hadamard's general solution to Cauchy's problem for second-order, linear partial differential equations is applied to the FRW gravitational wave equation.
		The retarded Green's function may be calculated for any FRW spacetime, with curved or flat spatial sections, for which the functional form of the Ricci scalar curvature R is known.
		It is also shown that for all FRW spacetimes in which the Ricci scalar curvatures does not vanish, R 0, the Green's function violates Huygens' principle; the Green's function has support inside the light cone due to the scattering of gravitational waves off the background curvature.
	library.wolfram.com /infocenter/Articles/2937 (168 words)

	General relativity mathematics : physically acceptable ?
		Just take the Ricci tensor...then I suppose it's obvious that all the invariant under change of basis of the tensor are invariant under coordinate transformation..which is obvious since a change of coordinate induces a local basis change of the tensor.
		If I remember well, the Ricci scalar is obtained from some contraction (don't ask me which one now), of the Riemann tensor.
		When the Ricci scalar is zero, there is still the Weyl part of the Riemann tensor.
	www.physicsforums.com /showthread.php?t=72879&page=2 (2684 words)

	Christina Sormani's Research Interests (Site not responding. Last check: 2007-10-24)
		In particular, Jeff Cheeger and Tobias Colding have developed a method which compares a given distance function to a harmonic function in order to show that a region of a manifold is Gromov-Hausdorff close to a nice comparison manifold [7].
		While the Lichnerowitz Formula does relate the scalar curvature of a manifold to a laplacian, it is not the laplacian of a function but that of a spinor [15].
		Such analysis has led to the definition of enlargeble compact manifolds which descibes the behavior of the fundamental group of a scalar positive manifold [14].
	www.math.nyu.edu /phd_students/sormanic/research/interests.html (597 words)

	Citebase - Dual Nature of the Ricci Scalar and its Certain Consequences (Site not responding. Last check: 2007-10-24)
		Authors: Srivastava, S. Ricci scalar is the key ingredient of non-Newtonian theory of gravity, where space-time geometry has a crucial role.
		A scalar is a mathematical concept representing a spinless particle.Here, particle concept, manifesting the physical aspect of the Ricci scalar, is termed as riccion.It is a scalar particle with (mass)
		One-loop renormalization of riccion indicates fractal geometry at high energy.Homogeneous and inhomogeneous models of the early universe are derived using dual role of the Ricci scalar.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0510086 (285 words)

	1.8 Motion of a scalar charge in curved spacetime
		The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the electromagnetic theory.
		This phenomenon is linked to the fact that a scalar field has zero spin: The particle can radiate monopole waves and the radiated energy can come at the expense of the rest mass.
		In generic situations the mass of a point scalar charge will vary with proper time.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-6/articlesu8.html (357 words)

	The Volume of Bitnets
		I believe that to hold in general, i.e., that the scalar curvature always depends on the variance of the central nodes.
		The average scalar curvature, < R >, is computed by integrating R given by (28), with respect to the volume element of [E\vec]
		Being a true geometric invariant, the information contained in the Ricci scalar holds in all possible descriptions (reparametrizations, markov equivalence transformations, etc..) of the model and it must be telling us something significant about the difficulty of estimation at each point.
	omega.albany.edu:8008 /bitnets1/bitnets1-l.html (2477 words)

	Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
		The key idea in Finsler geometry is to consider the projectivized tangent bundle PTM (i.e., the bundle of line elements) of the manifold M. The main reason is that all geometric quantities constructed from F are homogeneous of degree zero in y and thus naturally live on PTM, even though F itself does not.
		It is a scalar function on PTM and is defined as $g^{ik} \, (\ell^j \, R_{jikl} \, \ell^l)$.
		[4]) is the Ricci tensor $Ric_{jk} := (\frac{1}{2} \, F^2 \, Ric)_{y^j y^k}$.
	www.math.iupui.edu /~zshen/Finsler/history/chern.html (2855 words)

	:: My Research::
		Right now, I'm working on analysing the stability of a universe modelled with a gravity that has the Modified Gauss-Bonnet Scalar correction made to the Einstein-Hilbert Action with prof.
		where, g is the determinant of the metric R is the Ricci Scalar, f(G) is the modified gauss-bonnet scalar, G is the Gauss Bonnet scalar which is essentially a combination of the Ricci Scalar, Ricci Tensor and the Reimann Curvature Tensors.
		This work is with the idea that all theories which have their lagrangian depending on derivatives of order greater than 2 are unstable according to the ostrogradski instability condition.
	www.phys.ufl.edu /~sridhar/research.html (365 words)

	[No title]
		Computing the Ricci scalar, the Total curvature and the Mean curvature of a dag:
		In the case of complete dags is constant but in general it returns a formula (often very complicated) that depends on the parameters for the model.
		To obtain the integral of the Ricci scalar and the mean curvature (i.e.
	omega.math.albany.edu:8008 /bitnets/how2use-vTool.mw (344 words)

	Question on Phenomenology of Einsteins Field Equations
		In GR it seems that the starting point is not the Lagrangian but instead the metric - but that the end result, the stress energy tensor, is again, the equations of motion of the field.
		That is the Ricci tensor is the lagrangian density.
		The Ricci scalar is equivalent to the lagrangian density
	www.physicsforums.com /showthread.php?t=91796 (1061 words)

	Gravity Theory Seminars Abstracts (Site not responding. Last check: 2007-10-24)
		We consider theories of gravity where the Lagrangian is a nonlinear function of the Ricci scalar, in the Palatini variational formalism where the connection and metric are independent.
		We show that such theories are equivalent to scalar-tensor theories in which the scalar field kinetic energy term is absent from the action.
		A more general class of theories, stable under loop corrections, is given by taking the Lagrangian to be some function of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection.
	www.physics.umd.edu /grt/abstracts04.html (2127 words)

	Higher-derivative gravity in brane world models
		We investigate brane world models in higher-derivative gravity theories where the gravitational Lagrangian is an arbitrary function of the Ricci scalar.
		We solve for a gravity model that has corrections quadratic in the Ricci scalar and find it is possible to have a vanishing bulk cosmological constant.
		An analysis of tensor and scalar perturbations shows that gravity is localized on the brane, the Newtonian limit is recovered and the brane–bulk system appears to be stabilized.
	stacks.iop.org /1475-7516/2005/i=04/a=014 (267 words)

	Einstein-Hilbert action - Wikipedia, the free encyclopedia
		is the determinant of the Lorentz metric, R is the Ricci scalar, k is a constant which depends on the units chosen (see below), the Lagrangian being
		In the rival field theory of gravitation called Brans-Dicke theory, k is replaced by a scalar field.
		To derive the full field equations, it is natural to assume that an extra term - a matter Lagrangian (density) L
	en.wikipedia.org /wiki/Einstein-Hilbert_action (505 words)

	oz1
		In 2d it only takes one number to describe the Riemann curvature at each point, so there is the same amount of information in the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar.
		For example, a round sphere of radius r has Ricci scalar curvature R = 2/r^2 at every point.
		Well, okay, so we take the metric and feed it into this machine..." He scurries behind a curtain; loud banging noises ensue, followed by a deafening explosion and a puff of smoke; he returns somewhat flened but smiling.
	math.ucr.edu /home/baez/gr/old/oz1.bac (774 words)

	Real Algebraic and Analytic Geometry (Site not responding. Last check: 2007-10-24)
		Curvature bounds which play the role of Ricci, scalar curvature and Einstein tensor bounds are introduced for subanalytic topological manifolds.
		It is shown, using metric properties of subanalytic sets, that an upper (lower) bound on the sectional curvature in the sense of Alexandrov implies an upper (lower) bound on the Ricci curvature and on the Einstein tensor.
		In the same way, an upper (lower) bound on the Ricci curvature or on the Einstein tensor implies an upper (lower) bound on the scalar curvature.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG/preprints/0072.html (101 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of the Universe at present time.
		We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini formalism.
		It is striking that both Ricci scalar and Ricci squared theories are described in the same mathematical framework and both the generalized Einstein equations and generalized Friedmann equations have the same structure.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20697974 (335 words)

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