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Topic: Metric tensor

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Riemannian metric

Contravariant

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	Tensor Information - tensor lamps
		In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain.
		Physicists and engineers are among the first to recognise that vectors and tensors have tensor lamps a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated.
		Examples of physical tensors are the tensor analysis tensor fasciae latae energy-momentum tensor, the inertia tensor and the polarization tensor.
	www.inanot.com /Ina-Electronics_Topics_T-/Tensor.html (1961 words)

	Metric tensor - Wikipedia, the free encyclopedia
		In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space.
		is conventionally used for the components of the metric tensor (i.e., the elements of the matrix).
		In tensor analysis, the metric tensor is often used to provide a canonical isomorphism from the tangent space to the cotangent space: given a manifold M, v ∈ T
	en.wikipedia.org /wiki/Metric_tensor (1216 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)

	[No title]
		Tensor operations such as contraction or covariant differentiation are carried out by actually summing over repeated (dummy) indices with DO statements.
		Tensor operations such as contraction or covariant differentiation are performed by manipulating the indices themselves rather than the components to which they correspond.
		When used to assign values to the metric tensor wherein the components contain dummy indices one must be careful to define these indices to avoid the generation of multiple dummy indices.
	www.unf.edu /public/cap4630/kmartin/gradfall94/maxima/tensor/manual.txt (4543 words)

	Metric tensor (general relativity) - Wikipedia, the free encyclopedia
		In general relativity, the metric tensor (or simply the metric) is the fundamental object of study.
		Mathematically, spacetime is represented by a 4-dimensional differentiable manifold M and the metric is given as a covariant, second-rank, symmetric tensor on M, conventionally denoted by g.
		One of the core ideas of general relativity is that the metric (and the associated geometry of spacetime) is determined by the matter and energy content of spacetime.
	en.wikipedia.org /wiki/Metric_tensor_(general_relativity) (968 words)

	9.4.1 Some rules of tensor analysis on manifolds
		One of the key reasons that tensor analysis is so useful in physics is that an equation written in a form which respects a basic set of tensor rules remains form invariant under changes in coordinates.
		The purpose of this section is to summarize salient aspects of tensor analysis.
		The metric for spherical coordinates on a sphere is diagonal.
	www.gfdl.noaa.gov /~smg/MOM/web/guide_parent/s2node107.html (738 words)

	Mathematics: N-Dimensional Numbers (Site not responding. Last check: 2007-11-07)
		Another way to underline the fact that vectors (tensors) carry their identity is to widen the meaning of the term "invariant" by stating that vectors (tensors) are the invariants (it does not matter that their components are changing) with respect to an arbitrary transformation of coordinates of a point in n-d space.
		They have to be calculated in new coordinates from the metric tensor and this metric tensor in new coordinates can be obtained only by transformation from the old (original) coordinates where the metric tensor was given.
		The coordinate system remains the same, the metric tensor remains the same, the original point P with original coordinates remains the same, but the transformed real numbers are the coordinates of another point P1 in the same space.
	www.wbabin.net /yuri/keilman8.htm (5206 words)

	[No title]
		Indicial tensor manipulation is implemented by representing tensors as functions of their covariant, contravariant and derivative indices.
		TENSORS 13 [5] Indicial Tensor Manipulation In ITENSR a tensor is represented as an "indexed object".
		TENSORS 29 11 Indicial Tensor Manipulation- Indexing Functions COUNTER determines the numerical suffix to be used in generating the next dummy index.
	www.unf.edu /public/cap4630/kmartin/gradfall94/maxima/tensor/tensor.doc (4926 words)

	Metric engineering NASA Modern Relativity modernrelativity special black hole mass energy wormhole time Schwarzschild ...
		Spacetime's differential geometry given by the metric tensor is responsible for how all things in the universe tend to age and move.
		The stress energy tensor diverges on the event horizon (where we have (1 - A)f(r) = 1) where there is a singularity in the contravariant metric tensor.
		The metric tensor gives rise to the affine connections that we interpret as a gravitational field and it also gives rise to the components of the density tensor that we interpret as electric and magnetic fields.
	www.geocities.com /zcphysicsms/chap12.htm (5816 words)

	Tensors, Contravariant and Covariant
		The key attribute of a tensor is that it's representations in different coordinate systems depend only on the relative orientations and scales of the coordinate axes at that point, not on the absolute values of the coordinates.
		We should note that when dealing with a vector (or tensor) field on a manifold each element of the field exists entirely at a single point of the manifold, with a direction and a magnitude, rather than imagining each vector to actually extends from one point in the manifold to another.
		In the special case where the determinant of the metric tensor is 1, the scale factor drops out and we can say that the contravariant and covariant versions of a vector are really both just ordinary contravariant representations of the same vector based on mutually dual coordinate systems.
	www.physics.uq.edu.au /people/ross/phys2100/tensors.htm (3207 words)

	2.1 Geometry
		An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.
		The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
		There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (Newton-Cartan geometry, cf.
	relativity.livingreviews.org /Articles/lrr-2004-2/articlesu3.html (2680 words)

	Origins of the metric tensor - Advanced Physics Forums (Site not responding. Last check: 2007-11-07)
		However, in relativity, g can have the value 0 for widely-separated points: the interval between two events E and F is defined to be zero when the two events are the emission and detection of a beam of light, ie when A lies on the boundary of the light cone.
		Metric tensor is a bilinear form and as such requires the algebraic structure of a vector space.
		The metric tensor is indeed used to caracterize a certain manifolds in terms of angles and distances.
	www.advancedphysics.org /forum/showthread.php?t=554 (947 words)

	DRAFT: Deducing a Unified Field Theory from Electromagnetism (Site not responding. Last check: 2007-11-07)
		By including a symmetric field strength tensor in the GEM action, it becomes possible for the action to account for the changes in metric tensor, and thus remove the metric from the background structure.
		The idea is not to treat the metric as an active field, but instead to use a symmetry of the action to provide a constraint on a metric that varies in spacetime.
		For the GEM approach, one could choose a dynamic metric which would be equivalent to first order PPN accuracy to the metric from general relativity, or with the practical power of a diffeomorphism, choose to work in a flat spacetime and have a 4-potential completely characterize the gravitational field.
	theworld.com /~sweetser/quaternions/gravity/em2gem/em2gem.html (2641 words)

	Tensors, Contravariant and Covariant
		One of the most important examples of a second-order tensor is the metric tensor.
		Notice that each component of the new metric array is a linear combination of the old metric components, and the coefficients are the partials of the old coordinates with respect to the new.
		Thus the metric of a polar coordinate system is diagonal, just as is the metric of a Cartesian coordinate system, and so the contravariant and covariant forms at any given point differ only by scale factors (although these scale factor may vary as a function of position).
	www.mathpages.com /rr/s5-02/5-02.htm (2503 words)

	Metric and Metric Tensor
		Related to the metric is the very important idea of a metric tensor, which is usually represented by the symbol g, sometimes with the indicies explicitly shown (e.g.
		The metric tensor is one of the most important concepts in relativity, since it is the metric which determines the distance between nearby points, and hence it specifies the geometry of a spacetime situation.
		In the flat spacetime of special relativity, g is represented by the Minkowski metric tensor.
	www.mta.ca /faculty/Courses/Physics/4701/EText/MetricTensor.html (552 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)

	Cerebellar Coordination via a Metric Tensor, 1980
		Beyond this, the question arises as to whether the CNS hyperspace is endowed with a geometry determined by a metric tensor.
		The tensorial treatment of motor coordination suggests that the problem is solved by the embedding of the external three-space into the multidimensional CNS space and that this hyperspace is endowed with a metric tensor represented, for movements, by the cerebellar neuronal network.
		This transformation is the proposed role of the cerebellar metric tensor, resulting in a coordinated unique implementation of movements.
	usa-siliconvalley.com /inst/pellionisz/80_metric/80_metric.html (631 words)

	metric
		In other realms, "metric" means other things, and then to be specific we say that the metric in GR is a "Lorentzian metric" if it has that minus sign in front of the time part, or a "Riemannian metric" if it's just for space.
		If you have a tiny rigid rod moving along a path in space-time, the points in space-time occupied by its ends, which are simultaneous in the locally inertial frame for the rod, are a distance apart equal to the length of the rod.
		The metric is assumed to be one which at any one point can be transformed by a change of coordinates into a Minkowski metric.
	math.ucr.edu /home/baez/gr/metric.html (994 words)

	Riemannian Geometry
		Therefore, the metrical relations on the manifold over any sufficiently small region approach arbitrarily close to flatness to the first order in the coordinate differentials.
		In general, however, the metric components need not be constant to the second order of changes in position.
		If there exists a coordinate system at a point on the manifold such that the metric components are constant in the first and second order, then the manifold is said to be totally flat at that point (not just asymptotically flat).
	www.mathpages.com /rr/s5-07/5-07.htm (2194 words)

	TTC:Tools of Tensor Calculus (tutorial)
		If t is a (p, q) tensor, the covariant derivative of t in the direction of v is also a (p, q) tensor.
		g is the metric used to raise and lower indexes and basisname is the name of the basis used.
		is the symbolic Curvature of scalar of the Riemann tensor
	baldufa.upc.es /xjaen/ttc/tutorial/cref.htm (2689 words)

	Chapter 2
		In the vacuum of space outside the star, the energy momentum tensor is identically zero, and so the Einstein gravitational field equation requires that the Einstein tensor must be zero for the Einstein theory.
		These metric tensor elements were obtained from Tolman [2] on p.
		The metric equation for the Schwartzschild solution is not isotropic, and it is diagonal only in spherical coordinates.
	www.olduniverse.com /chapter_2.htm (2361 words)

	Determining metric tensor
		the contravariant or covariant metric tensor of GM?
		We throw in a\nnormalization of the metric as a tacit assumption.
		The 10th is given by the normalization.\nThis is a kind of reducibility of the metric and was mentioned here by\nA.
	www.physicsforums.com /showthread.php?t=61125 (1126 words)

	Short introduction to general relativity (Site not responding. Last check: 2007-11-07)
		be called a mixed tensor of rank 3 with contravariant 1st and 3rd indices and covariant 2nd index.
		From equations (3) and (4) it is obvious that if a vector (or tensor) vanishes in one coordinate system, it will vanish in any coordinate system.
		The same is true for more complicated tensor equations and therefore if a law of physics can be expressed as a tensor equation in a given coordinate system it will have the same tensor form in any other coordinate system.
	www.astro.ku.dk /RelViz/peter/intro/intro.html (486 words)

	[No title]
		Section 2: Tensors and Tensor Operations A tensor T is defined as an ordered collection of its components: EMBED Equation.DSMT4 where each of the i and each of the j can assume any integer value from 1 to N and all components are real numbers.
		When a quantity with proper components is defined in multiple coordinate systems or in terms of tensors, the quantity is said to be a tensor if it always obeys the transformation equation.
		Tensors are usually preferred over “nontensors” because a tensor needs to be defined in only one coordinate system and because change of tensor components under coordinate transformations is predictable and depend only on the coordinates and the nature of the tensor.
	web.mit.edu /dmytro/www/GR_theory.doc (1538 words)

	Raising and lowering indices with the metric tensor
		Raising and lowering indices with the metric tensor
		The metric tensor plays the extremely important role in tensor analysis, of allowing one to lower or raise indices on vectors and tensors.
		It is not difficult to show also that the metric tensor inverse has zero covariant derivative, since the Kronecker delta is a scalar having the same values (0 or 1) in all reference frames.
	www.colorado.edu /physics/phys7840/phys7840_fa02/GENREL2/node3.html (179 words)

	Metric Tensor (Site not responding. Last check: 2007-11-07)
		The metric perturbation can be further broken up into the normal modes of scalar, vector and tensor types as in §IIC.
		Scalar and vector modes exhibit gauge freedom which is fixed by an explicit choice of the coordinates that relate the perturbation to the background.
		Note that tensor fluctuations do not exhibit gauge freedom of this type.
	background.uchicago.edu /~whu/tamm/webversion/node15.html (83 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)

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