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	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) (960 words)

	General relativity - Wikipedia, the free encyclopedia
		In general relativity, phenomena that in classical mechanics are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories) are taken in general relativity to represent inertial motion in a curved spacetime.
		Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.
		General relativity was developed by Einstein in a process that began in 1907 with the publication of an article on the influence of gravity and acceleration on the behavior of light in special relativity.
	en.wikipedia.org /wiki/General_relativity (5103 words)

	Learn more about General relativity in the online encyclopedia. (Site not responding. Last check: 2007-10-12)
		According to general relativity the force of gravity is a manifestation of the local geometry of spacetime.
		The fundamental idea in relativity is that we cannot talk of the physical quantities of velocity or acceleration without first defining a reference frame, and that a reference frame is defined by choosing particular matter as the basis for its definition.
		General relativity is distinguished from other theories of gravity by the simplicity of the coupling between matter and curvature, although we still await the unification of general relativity and quantum mechanics and the replacement of the field equation with a deeper quantum law.
	www.onlineencyclopedia.org /g/ge/general_relativity.html (1942 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.
		More generally, when the metric may give a negative value, the metric is called pseudo riemannian.
		is conventionally used for the components of the metric tensor (i.e., the elements of the matrix).
	en.wikipedia.org /wiki/Metric_tensor (985 words)

	General Relativity (Site not responding. Last check: 2007-10-12)
		The two theories that are used in contemporary physics to describe the structure of the universe are general relativity, that describes spacetime and gravitation, and quantum field theory, that describes the strong, weak, and electromagnetic interaction in the standard model.
		General relativity is a geometrical theory; whereas in the electroweak and strong interaction a quantum field in spacetime acts upon the particle fields, the spacetime itself is curved in the description of gravity in general relativity.
		The curvature tensor is applied in the Einstein equation of the gravitational field, and is used to describe the motion of particles in curved spacetime.
	www.nikhef.nl /~henkjan/astro/node13.html (1621 words)

	Early Philosophical Interpretations of General Relativity
		Relativity theory itself is a shining exemplar of this method for it has shown that the metric of spacetime describes an "objective property" of the world, once the subjective freedom to make coordinate transformations (the coordinating principle of general covariance) is recognized (1920, 86-7; 1965, 90).
		Thus, the theory of general relativity, on adoption of the coordinative definition of rigid rods ("universal forces = 0"), affirms that the geometry of spacetime in this region is of a non-euclidean kind.
		Despite the influence of this argument on the subsequent generation of philosophers of science, Reichenbach's analysis of spacetime measurement treatment is plainly inappropriate, manifesting a fallacious tendency to view the generically curved spacetimes of general relativity as stiched together from little bits of flat Minskowski spacetimes.
	plato.stanford.edu /entries/genrel-early (11439 words)

	General relativity/Metric tensor - Wikibooks, collection of open-content textbooks
		Recall that a tensor is a linear function which can convert vectors into scalars.
		The first problem comes in, in that tensors are linear functions, but we have some squares in our distance formula.
		And that is the equation of distances in Euclidean three space in tensor notation.
	en.wikibooks.org /wiki/General_relativity:Metric_tensor (260 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		The heuristic value of the concept of general covariance in seeking new laws of nature is acknowledged, in that any new tensor, which arises naturally within the development, is deemed as a candidate for representing a physical quantity.
		The natural appearance of this tensor shows that the field equations of General Relativity may be derived from a geometrical tensor identity.
		The metric tensor occurs in the separation formula for a pair of events in space-time and it is difficult to see why the quadratic terms should not be symmetric in the coordinate indices, so here the symmetry of the metric tensor itself is regarded as essential.
	www.ma.utexas.edu /mp_arc/html/papers/01-297 (2065 words)

	metric
		Well, in the world of general relativity, "metric" means no more and no less than a symmetric nondegenerate tensor of rank (0,2), or if you prefer, a dot product thingie.
		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.
		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)

	Generalized Electroform Field Equations of Vector-Boson Field Theory : The Mirror-Image Maxwell's Equations for Long- ...
		Instead, the GR grav-magnetic field is derived from artifactual off-diagonal components of the metric tensor in a way that injects the troublesome factor of 4 and spoils the Principle of Relativity from a forces point of view.
		This book replaces classical General Relativity with a unified quantum field theory of forces in flat space with gravitational time-dilation only that still yields the 2nd-order classical GR effects, such as the perihelion precession of Mercury, the curvature of light around the Sun, and the gravitational red-shift.
		According to the usual Principle of Relativity, motion against the background of absolute space cannot be determined by any measurement within the system, but in GR this not so, because the field equations do not apply to the force of gravitation, but to the components of the metric tensor.
	greenwdks.tripod.com /unifiedsummary.html (3185 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.
		It is commonly known that after the core development of general relativity Einstein continued to work on a unification theory for unification of the gravitational and electromagnetic "fields".
		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)

	WarpDrive_02.doc
		General Relativity does not forbid apparent FTL travel (as seen by a preferred topological observer) or communication, but it does require that the local restrictions of Special Relativity must apply (that is Special Relativity applies to ones local geometry).
		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 space-time situation.
		He has developed a “metric” for General Relativity, a mathematical representation of the curvature of space-time, that describes a region of at space surrounded by a warp or distortion that propels it forward at any arbitrary velocity, including FTL speeds.
	www.astrosciences.info /WarpDrive.htm (6809 words)

	3 The Problem of Hyperbolicity in General Relativity
		What follows are descriptions of the problem of hyperbolicity in general relativity and of the main approaches that have been proposed to deal with the problem.
		The object of the theory is not a metric tensor, but the whole equivalence class to which it belongs -all other metrics related to the first one by a smooth diffeomorphism.
		Since, as shown in § 2, hyperbolicity is equivalent to the existence of norms which are bounded under evolution, we see that for Einstein's equations there cannot be such norms on the space of metric tensors.
	relativity.livingreviews.org /Articles/lrr-1998-3/node9.html (519 words)

	Re: General Relativity
		At each point p in the manifold, the Riemann tensor field defines a multilinear mapping with four arguments (a multilinear mapping is called a "tensor"; (algebraic) tensors are generalizations of both linear operators and bilinear forms).
		You can think of it as defining a generalized "change in the direction of X" operator, where X is some vector field, and it is customary to write something like D_X Z for the change in Z in the direction X, where X,Z are two vector fields.
		The Riemann tensor field is of course a fourth rank tensor field, and it is called the "intrinsic curvature" because it does not depend on the way the slice bends in the higher dimensional manifold.
	www.lns.cornell.edu /spr/2000-03/msg0023241.html (1474 words)

	weyltheory.htm (Site not responding. Last check: 2007-10-12)
		and the tensor formalism of general relativity and differential geometry.
		is the covariant derivative of the metric tensor.
		The apparent similarity between a regauging of the metric and a gauge transformation of the 4-potential seemed to Weyl to represent a potential means of unifying general relativity with electromagnetism.
	www.weylmann.com /weyltheory.htm (4101 words)

	General Relativity and Spacetime (Site not responding. Last check: 2007-10-12)
		In general relativity, the same particle path is described by a world line in 4-dimensional spacetime.
		Now we will look at the metrics and metric tensors of some simple surfaces, and the curvature of the surfaces.
		The metrics discussed in the previous section, can be used to define a surface (the separation defines the allowed lines to move in).
	www.astro.ku.dk /~cramer/RelViz/text/geom_web/node2.html (900 words)

	Short introduction to general relativity
		In the general theory of relativity gravitation is regarded simply as curvature of spacetime.
		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.
		The general relativistic forms of the laws of physics can thus be obtained from the special relativistic ones by exchanging the ordinary derivatives with the covariant derivatives.
	www.astro.ku.dk /RelViz/peter/intro/intro.html (486 words)

	General Relativity
		The basic idea of general relativity (GR) is, that the spacetime is a four dimensional topological manifold M with a differentiable structure.
		One of the assumptions of GR is that all physical fields are described by tensorfields and the describing field equations are tensor equations and therefore independent of the co-ordinate patch used.
		This type of metric is not positive definite and therefore in disagreement with the definition of a metric by mathematicians.
	www.unet.univie.ac.at /~a9405544/pages/qg_gr/node2.html (966 words)

	Fizikai Szemle 1999/5 - Eugene P. Wigner: ON THE FUTURE OF PHYSICS
		Today the central problem of physics is that there is no common theory which could include the general relativity and quantum theory.
		The relation of the metric tensor to quantum theory is still puzzling.
		To know the metric tensor, the distance of two space-time points should be measured.
	www.kfki.hu /fszemle/archivum/fsz9905/wigner.html (1558 words)

	442 General Relativity and Cosmology
		Lecture 12: The symmetries of the Riemann tensor.
		There is another geometric interpretation of the Riemann tensor: geodesic deviation, I haven't covered this, but if you are interested it is treated in a usefully off-hand way in (W 148-149).
		Lecture 20: General relativity, the field equation, the cosmological constant, geodesic flow (d'I 142-143).
	www.maths.tcd.ie /~houghton/TEACHING/442/442-04-05/442lectures.php (421 words)

	Amazon.com: A Short Course in General Relativity: Books: James Foster,J. David Nightingale (Site not responding. Last check: 2007-10-12)
		Suitable for a one-semester course in general relativity for senior undergraduates or beginning graduate students, this text clarifies the mathematical aspects of Einstein's theory of relativity without sacrificing physical understanding.
		The originator of the general theory of relativity was Einstein, and in 1919 he wrote1 : The special theory, on which the general theory rests, applies to all physical phenomena with the exception of gravitation; the general theory provides the law of gravitation and its relation to other forces of nature.
		It is true that the physical motivation and meaning of general relativity are not treated in that much depth, but these can be picked up from other sources.
	www.amazon.com /exec/obidos/tg/detail/-/0387260781?v=glance (2014 words)

	Quantum Mechanics and General Relativity: Incomplete Working Notes
		One of the few General Relativity textbooks that uses differential manifolds rather than coordinates as the primary viewpoint for the subject.
		One of the key principles of General Relativity is that gravity cannot be distinguished from uniform acceleration.
		Since we have a differential-geometric metric (the Lorentz metric), we can use the standard definition of Hodge duality to define the Maxwell 2-form as Hodge-dual to the Faraday 2-form.
	www.zaimoni.com /Notes_GR_QM.htm (2841 words)

	[No title]
		Spacetime diagrams; World velocity; 4-momentum; 4-force; Spacetime of GR; Geodesic coordinates at P; Metric tensor around P; Curved spacetime of GR; Flat spacetime of STR; Generalization of 4-vectors in SR to those in GR; Motion of free particles and of photons in GR; Comparison between Newtonian mechanics and GR
		Generalization of the energy-momentum tensor to GR; Its divergence; Curvature tensor; Riemann tensor and its symmetry properties; Cyclic and Bianchi identities; Einstein's tensor and its divergence
		General theory of perihelion advance; Calculation of perihelion advance for the planet Mercury; General theory of bending of light near a massive object; Light bending near the limb of the sun
	www.tcnj.edu /~wick/GRAstronomy161LectureSummary.html (813 words)

	Does general relativity predict expansion? - Page 2 (Site not responding. Last check: 2007-10-12)
		Indeed, spaces with a negative metric component are referred to as "pseudo-Euclidean".
		One has to transform between coordinates to get the different metric components in different coordinate systems, but the actual value of the metric tensor, its trace or whatever, is coordainte invariant, as coordinate transformations are orthogonal.
		The components of the tensor themselves are not the same in every coordinate frame, its trace etc. is invariant.
	www.physicsforums.com /showthread.php?p=889412 (1285 words)

	Question about the metric tensor in Einstein's field equations. (Site not responding. Last check: 2007-10-12)
		I was wonder if some can explain to me what exactly are the 10 parameters for the metric tensors.
		If you remember that the metric tensor is symmetric and rank-2, then in a spacetime with T dimensions of time and S dimensions of space, the number of independent elements of the metric (or "parameters" as you call them) is (T+S)(T+S+1)/2.
		The i-j element in the metric tensor is the coefficient of the dx
	www.physicsforums.com /showthread.php?t=39093 (313 words)

	Minkowksi Metric Tensor (Site not responding. Last check: 2007-10-12)
		When we go to the curved space of general relativity in the presence of masses we will employ tensors which have off-axis components (although the tensors will still be symmetric about the diagonal).
		The notation which we use here is consistent with Einstein's original paper, and with most works, but you should be aware that some texts use a different notation.
		This material is part of an electronic text to support a university relativity course.
	www.mta.ca /faculty/Courses/Physics/4701/EText/MinkowskiMetric.html (222 words)

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