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Topic: Ricci tensor

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Ricci curvature tensor

Scalar curvature

Einstein manifold

Tensor

Ricci curvature

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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.
		The Einstein field equations assert that the energy-momentum tensor is proportional to the Einstein tensor.
		scalar curvature, Einstein tensor, ricci scalar, Weyl tensor, Weyl curvature tensor
	planetmath.org /encyclopedia/RicciScalar.html (319 words)

	GRTensorII demonstrations-General Relativity & Geometry.
		Demonstration 1 (ss1): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in curvature coordinates.
		Demonstration 2 (ss2): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in comoving coordinates.
		Demonstration 3 (ss3): The Einstein tensor and Kretschmann scalar are calculated for the spherically symmetric self-similar metric in Bondi coordinates.
	grtensor.phy.queensu.ca /NewDemo/demo.html (1569 words)

	Ricci: A Mathematica package for doing tensor calculations in differential geometry
		Limitations: Ricci currently does not support computation of explicit values for tensor components in coordinates, or derivatives of tensors depending on parameters (as in geometric evolution equations or calculus of variations), although support for these is planned for a future release.
		Ricci also has no explicit support for general relativity, or for other mathematical physics or engineering applications, and none is planned.
		Tensors whose rank is a symbolic constant, such as n-forms or k-forms on an n-manifold.
	www.math.washington.edu /~lee/Ricci (783 words)

	EW - Advanced - The Ricci Tensor
		However, the particles generate their own gravitational field and hence they will be deflected from each other according to the Riemann tensor and the geodesic deviation equation, which defines, in terms of the Riemann tensor, how initially co-moving particles will accelerate away from each other.
		The Ricci tensor consists of ten components (half those of the complete Riemann tensor) and is dictated by the rate of change of the volume of our little ball of particles.
		The Ricci tensor tracks how the general paths of particles in space-time are deflected due to curvature.
	library.thinkquest.org /27608/scripts/aview.php3?id=69 (519 words)

	PlanetMath: Einstein field equations
		While such auxiliary equations should ideally single out a representative for each equivalence class, in practise, one is content with considerably less -- a particular choice auxiliary conditions might only work with some solutions or may only specify a subset of an equivalence class with more than one element.
		Throughout this entry, we shall use index notation for tensor fields because that is common in the literature (especially physics literature) and is convenient for computation of particular solutions.
		is the stress-energy tensor, which encodes information pertaining to the mass, energy, and momentum densities of the surrounding space.
	planetmath.org /encyclopedia/EinsteinFieldEquations.html (615 words)

	Springer Online Reference Works
		Thus, the Riemann tensor is a quantitative characteristic of the non-commutativity of the second covariant derivatives in a Riemannian space.
		Vanishing of the conformal curvature tensor is a necessary and sufficient condition for the space to coincide locally with a conformal Euclidean space.
		The conformal curvature tensor is used in the theory of formation of particles in a gravitational field.
	eom.springer.de /c/c027320.htm (2030 words)

	Chapter 5
		When an index in a tensor equation is repeated, with one index a subscript and the other a superscript, the corresponding term is to be summed over the four values (0, 1, 2, 3) of the index.
		The tensor density corresponding to the tensor A
		Hence it is reasonable to expect that the non-diagonal elements of the metric tensor should be much smaller then the diagonal elements when the velocities of matter are much less than the speed of light.
	www.olduniverse.com /chapter_5.htm (1491 words)

	Maxima Manual: 29. ctensor
		Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.
		Tensors of rank 2 are displayed as 2-dimensional matrices, while tensors of higher rank are displayed as a list of 2-dimensional matrices.
		Causes the contortion tensor to be included in the computation of the connection coefficients.
	maxima.sourceforge.net /docs/manual/en/maxima_29.html (2291 words)

	Maxima Manual - Tensor
		Tensor Package - will set to zero, in exp, all occurrences of the tensori that have derivative indices.
		Tensor Package - will set to zero, in exp, all occurrences of the differentiated object tensor that have n or more derivative indices as the following example demonstrates.
		Tensor package) This function first computes the covariant components LR[i,j] of the Ricci tensor (LR is a mnemonic for "lower Ricci").
	www.ma.utexas.edu /maxima/maxima_27.html (1080 words)

	[No title]
		Inserting all four indices yields the coefficients of the curvature tensor."; Curvature::usage = "Curvature[cn] is the curvature tensor associated to the connection cn.
		The symmetries are those of the Riemann curvature tensor, namely skew-symmetric in the first two indices, skew-symmetric in the last two indices, symmetric when the first two and last two are interchanged.
		Tensors are created by DefineTensor and removed by UndefineTensor."; TensorCancel::usage = "TensorCancel[x] attempts to simplify each term of x by canceling common factors, even when the factors have different names for their dummy indices.
	www.math.washington.edu /~lee/Ricci/Ricci.m (8565 words)

	How does matter couple to space-time so that space-time becomes curved?
		The components of the Riemann tensor identically satisfy a differential equation (the Bianchi identity), which is why the metric tensor (ten algebraically independent components at each event) can and does completely determine the Riemann curvature tensor (20 algebraically independent components at each event).
		The Riemann tensor R_(abcd) can be decomposed into two pieces, the Ricci tensor R_(ab) and the Weyl tensor C_(abcd), in a manner analogous to decomposing a matrix into trace and tracefree parts.
		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)

	The Ricci Tensor
		The Ricci tensor arises as a trace of the Riemann tensor which, in turn, is obtained by taking the second derivatives of tensor fields and antisymmetrising over the gradient-rank factors added to the tensor fields' ranks in so doing.
		Now, it so happens that the construction, starting with a differential operator and yielding its Ricci tensor, can actually be applied to a broad class of tensor operators, which I describe as Leibniz operators.
		By construction, this tensor is antisymmetric under interchange of its first two components: this slightly reduces the collection of options to be tried.
	utter.chaos.org.uk /~eddy/math/smooth/ricci.html (723 words)

	The Field Equations
		Of course, a tensor of rank four can be contracted in six different ways (the number of ways of choosing two of the four indices), and in general this gives six distinct tensors of rank two.
		In tensor calculus the divergence generalizes to the covariant derivative, so we expect that the covariant derivatives of the metrical field equations must identically vanish. The Ricci tensor R
		Noting that partial differentiation is commutative, and the metric tensor is symmetrical, we see that the sum of these three tensors vanishes at the origin of Riemann normal coordinates, and therefore with respect to all coordinates.
	www.mathpages.com /rr/s5-08/5-08.htm (1687 words)

	What exactly is a Warp in Spacetime? (Site not responding. Last check: 2007-10-11)
		The Ricci curvature because of its tensor signature is called a covariant field, if it is inversed it is called a contrivariant field and this is where we can input our knowledge of classical mechanics.
		The beautiful thing about the Ricci tensor is that it contracts (curves) this space in very appealing manner, and it is from this that we can mathematically state that gravity is a curvature in spacetime.
		There of course other tensor ranks and fields one popularly known tensor is the rank 4 Weyl tensor, the problem is that it is complicated to have it act in the general covariant field which general relativity is largely done in.
	members.tripod.com /da_theoretical1/Curves.html (1527 words)

	Physics: wormholes
		The Ricci tensor R_ab in and of itself does not satisfy this necessary requirement, but a new tensor can be created which does satisfy the requirement via a slight modification of the Ricci tensor, and, without disturbing the relation R_ab = 0 for the vacuum of empty space.
		The covariant energy-momentum tensor is T_ab, regarded as the cause, or the "source" of the metric curvature.
		One property of tensors is that if the tensor in question is not zero according to one frame, it is not zero according to any frame.
	en.allexperts.com /q/Physics-1358/wormholes.htm (1659 words)

	General Relativity:Does it Prove the Cause and Strength of Gravity?
		It is often asserted that Albert Einstein (1879-1955) was slow to apply tensors to relativity, resulting in the 10 years long delay between special relativity (1905) and general relativity (1915).
		In fact, you could more justly blame Ricci and Levi-Civita who wrote the long-winded paper about the invention of tensors (hyped under the name ‘absolute differential calculus’ at that time) and their applications to physical laws to make them invariant of absolute co-ordinate systems.
		The Ricci tensor is in fact a shortened form of a big Riemann rank 4 tensor (the expansions and properties of which are capable of putting anyone off science).
	www.wbabin.net /physics/cook4.htm (4920 words)

	Physics: the time space continuum
		Stephen 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 Heisenberg uncertainty principle.
		The general solution of Einstein's equations with the energy-momentum tensor of an ideal dust in a Friedman universe, which models the universe on a very large scale, has a zero Weyl curvature but a nonzero Ricci curvature at each event in the space-time.
		The Weyl tensor becomes analogous to the electromagnetic field tensor F_ab, which can be seen as an antisymmetric 4x4 matrix with 6 independent components at each event in the space-time.
	en.allexperts.com /q/Physics-1358/time-space-continuum.htm (1353 words)

	ricci.weyl (Site not responding. Last check: 2007-10-11)
		10 of these numbers are captured by the "Ricci tensor", while the remaining 10 are captured by the "Weyl tensor".
		Recall the definition of the Ricci tensor in terms of coffee grounds floating through outer space.
		We consider a bunch of initially comoving coffee grounds near a point P in spacetime, with the coffee ground that actually goes through P having velocity v at that instant.
	math.ucr.edu /home/baez/gr/ricci.weyl.html (560 words)

	15
		is a rank (0,2) tensor, and the covariant derivative acts differently on tensors of different rank (it includes Christoffel contractions on each tensor index).
		Calculating this noncommutation of covariant derivatives determines the Riemann tensor, the prime characterization of curvature for a manifold.
		These were the Ricci tensor and the Ricci scalar.
	www.emory.edu /PHYSICS/Faculty/Benson/380-04/notes/15/15.html (330 words)

	[No title]
		In the general case of a curved four-space, the infinitesimal parallel displacement of a vector is defined as a displacement in which the components of the vector are not changed in a system of coordinates, which is Galilean in the given infinitesimal volume element.
		We define the tensor EMBED Equation.3 (the Ricci tensor) as EMBED Equation.3 .
		It is a symmetric tensor, because both the Ricci tensor and the metric tensor are symmetric.
	www.physics.hku.hk /academic/courses/phys3033/Chapter8.doc (2225 words)

	Ricci Flow, 3-Manifolds and Geometry — Program
		The emphasis of this course is Perelman's works on Ricci flow in 3-dimensions and geometrization of 3-manifolds.
		Hamilton's 3-manifolds with positive Ricci curvature theorem: background and basic techniques used in its proof - linearization of the Ricci tensor, short time existence, basic evolution equations, maximum principles, curvature pinching estimates, convergence criteria.
		Hamilton's trace Harnack estimate for the Ricci flow on surfaces and its consequences
	www.claymath.org /programs/summer_school/2005/program.php (535 words)

	Riemannian Geometry & Tensor Calculus -- from Mathematica Information Center
		Each tensor is stored as a nested list under an appropriate global name.
		Several examples of the use of these functions on tensors computed using different metrics are given.
		Beginning with version 2.5, tensor components can be calculated with respect to an arbitrary frame, and approximate calculations (series expansions) can be carried out.
	library.wolfram.com /infocenter/MathSource/4484 (225 words)

	LORENE: Connection class Reference
		Indicates whether the connection is associated with a metric (in which case the Ricci tensor is symmetric, i.e.
		Pointer of the Ricci tensor associated with the connection.
		is a twice contravariant tensor, whose components w.r.t.
	www.lorene.obspm.fr /Refguide/classConnection.html (524 words)

	Tensor Analysis & Differential Geometry - Physics Forums Library
		What is a tensor and why is it useful?
		An extension of the concept of a tensor field
		proof that grad V is a (1,1) tensor
	www.physicsforums.com /archive/index.php/f-76.html (319 words)

	EW - Advanced - Einstein's Equation
		Well, to find that out we first need something called the Einstein Tensor (which is that G thing on the left hand side), and then we're going to go through quite a complicated set of Tensor manouvres.
		The Einstein Tensor is as follows, using the Riemann tensor and Ricci scalar:
		Well its actually very interesting as it states that the Ricci scalar equals the negative sum (because of our summation convention) of the diagonal components of the stress energy tensor.
	library.thinkquest.org /27608/scripts/tview.php3?id=71 (622 words)

	The Riemann tensor and the metric
		Contracting the tensor using the fact that the trace of the metric is three gives us the Ricci tensor given by
		Using this metric we calculate the diagonal components of the Ricci tensor as
		Calculating the connection coefficients we proceed to calculate the components of the Ricci tensor which come out to be
	members.tripod.com /surhudm/seminarhtml/node4.html (165 words)

	The Mathematical Details
		We can use this tensor to compute the relative acceleration of nearby particles in free fall if they are initially at rest relative to one another.
		The Ricci tensor only captures some of the information in the Riemann curvature tensor.
		The good thing about this version is that it gives a formula for the Ricci tensor, which has a simple geometrical meaning.
	math.ucr.edu /home/baez/einstein/node10.html (544 words)

	AMCA: Yamabe metrics with non parallel Ricci tensor by A. Raouf Chouikha (Site not responding. Last check: 2007-10-11)
		We descrive metrics with constant scalar curvature and with Ricci tensor non parallel.
		We show that their Ricci tensor are also non parallel except for the cylindric one.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/f/h/08.htm (197 words)

	Application to the scalar-tensor theories. (Site not responding. Last check: 2007-10-11)
		The class of theories that we consider here is described by the action
		is the energy-momentum tensor of the matter fields
		Indeed, inserting (78) in (67) yields the following expression for the curvature tensor
	www.obs-hp.fr /www/preprints/pp129/node5.html (545 words)

	Weak Field Limit
		That is to say that the metric tensor is written in terms of the sum of the Minkowski metric and correction term whose components are much less than unity.
		Since the Christoffel symbols are first order quantities, the only contribution to the Riemann tensor will come from the derivatives of
		Contract the Riemann tensor over m to obtain the Ricci tensor
	www.geocities.com /physics_world/gr/weak_field_limit.htm (204 words)

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