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Topic: General covariance

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Estimation of covariance matrices

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	The Field of All Fields
		The reluctance to discard the special theory as a separate theory may be due largely to the fact that it represents a simple and widely- applicable special case of the general theory, and it's convenient to have a name for this limiting case.
		The basic point is that although special relativity serves as the local limiting case of the general theory, it is not able to stand alone, because it cannot identify the applicability of its premises, which renders it incapable of yielding definite macroscopic conclusions about the physical world.
		We can compare this to the general theory of relativity, which is compelled by the equivalence principle to represent the metric of spacetime as (so to speak) "the field of all fields".
	www.mathpages.com /rr/s4-06/4-06.htm (2166 words)

	General covariance - Wikipedia, the free encyclopedia
		In theoretical physics, general covariance (also known as diffeomorphism covariance) is the invariance of the form of physical laws under arbitrary coordinate transformations.
		The general principle of relativity as used in Einstein's general theory of relativity is that the laws of physics must take the same form in all reference frames.
		General covariance and the foundations of general relativity: eight decades of controversy, by J. Norton
	en.wikipedia.org /wiki/General_covariance (345 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.
		In general relativity, the elegance of a flat spacetime and the ability to use a preferred coordinate system are lost (due to stress-energy curving spacetime and the principle of general covariance).
		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 (5032 words)

	Early Philosophical Interpretations of General Relativity
		From this vantage point, general covariance is but the most recent refinement of the methodological principle of "unity of determination" governing the constitution of objects of physical knowledge, completing the transposition in physics from concepts of substance into functional and relational concepts.
		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 Council of the Church - Hutchinson encyclopedia article about General Council of the Church (Site not responding. Last check: 2007-10-09)
		At the turn of the 15th century there were two pressing needs for a General Council: to resolve the Great Schism (the period of rival popes with seats in Rome and Avignon) and to deal with the new threat of the Hussite heresy.
		There was a tension, then, between the Catholic Church's need to appoint a single pope through a General Council and the inevitable concern of the elected pope to avoid further Councils and their conciliarist threat.
		In the early 16th century there was an abortive attempt to call a General Council in defiance of Pope Julius II – the ‘schismatic’ Council of Pisa (1511).
	encyclopedia.farlex.com /General+Council+of+the+Church (490 words)

	Notes to The Hole Argument
		Passive general covariance allows use of all coordinate charts of the differential manifold and is conferred automatically on theories formulated by modern methods.
		The notion of general covariance has been the subject of a protracted dispute that extends to the earliest years of general relativity, when Kretschmann argued that the requirement of general covariance was merely a mathematical condition with no physical content.
		Once the formulation is fixed, however, then the general covariance of the theory, like most of its formal properties, will have some physical meaning.
	www.science.uva.nl /~seop/entries/spacetime-holearg/notes.html (616 words)

	Reference.com/Encyclopedia/Physics
		General relativity is the geometrical theory of gravitation published by Albert Einstein in 1915/16.
		In general relativity, the curvature of space-time is produced by the energy of matter and radiation.
		General relativity is distinguished from other metric Theories of gravitation by its use of the Einstein field equations to relate space-time content and space-time curvature.
	www.reference.com /browse/wiki/Physics (4258 words)

	General Regression Models (GRM)
		True to its description as general, the general linear model can be used to analyze designs with effects for categorical predictor variables which are coded using either parameterization method.
		In many uses of the general linear model, it is arbitrary whether categorical predictors are coded using the sigma-restricted or the overparameterized coding.
		Some general ways in which designs might differ can be suggested, but keep in mind that any particular design can be a "hybrid" in the sense that it could have combinations of features of a number of different types of designs.
	www.statsoft.com /textbook/stgrm.html (4699 words)

	The Hole Argument
		In general relativity, there is a further problem with arguing that the metric field properly belongs to spacetime since it carries essential spatiotemporal properties.
		Over the previous year, he had been determined to find a gravitation theory that was generally covariant, that is, whose equations were unchanged by arbitrary transformation of the spacetime coordinates.
		One view is that this goes too far, that general relativity is distinct from many other spacetime theories in that its spacetime geometry has become dynamical and it is only in such theories that the hole argument should be mounted.
	plato.stanford.edu /entries/spacetime-holearg (5340 words)

	QuantumGeneralRelativity
		On the other hand, these quantum geometries incorporate into their very structure a fundamental length in such a manner that, in the non-relativistic and the special relativistic regime, conventional results can be recovered in the sharp-point limit that is obtained as that length is allowed to tend to zero.
		A very general quantum-geometric framework for general relativity was expounded in a recent monograph of the present author.
		In general, considerable emphasis is set on the kind of precise notation and rigorous argument that is long overdue in some of the fields with which this monograph is dealing.
	individual.utoronto.ca /prugovecki/QuantumGeneralRelativity.html (1742 words)

	Citebase - General Covariance in Quantum Gravity (Site not responding. Last check: 2007-10-09)
		Authors: Kazakov, Kirill A. The question of general covariance in quantum gravity is considered in the first post-Newtonian approximation.
		It is found that the one-loop contributions violate the principle of general covariance, in the sense that the quantities which are classically invariant under such deformations take generally different values in different reference frames.
		Quantization of gravitation theory as gauge theory of general covariant transformations in the framework of Batalin-Vilkoviski (BV) formalism is considered.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0201246 (1033 words)

	Canonical quantization and covariance (Site not responding. Last check: 2007-10-09)
		Lorentz covariance is broken by the fact that one has to\nchoose a Lorentz frame for the t variable.
		The very foundation of general covariant\nphysics is the idea that the notion of a simultaneity surface all over the\nuniverse is devoid of physical meaning.
		The very foundation of general covariant\nandgt; physics is the idea that the notion of a simultaneity surface all over the\nandgt; universe is devoid of physical meaning.
	www.physicsforums.com /showthread.php?threadid=79720 (6006 words)

	SparkNotes: Albert Einstein: General Relativity
		In contrast, his general theory of relativity accounts not only for these, but also for bodies that accelerate (i.e., change their velocity).
		Based on this principle, Einstein formulated the principle of general covariance, which forms the basis of his general theory of relativity.
		Basically, with his general covariance principle, Einstein applied the equivalence principle to special relativity: given that the laws of physics are the same in all inertial reference frames and that inertial and gravitational masses are equivalent, the laws of physics are the same in all accelerating frames as well.
	www.sparknotes.com /biography/einstein/section7.rhtml (944 words)

	Re: General covariance, background independence
		> > > >The term "general covariance" has a long and convoluted history, > >since it goes back at least to Einstein's early work, where he > >had strong intuitions about what made general relativity different > >from all previous theories, but said some misleading stuff, because > >he hadn't yet figured everything out.
		I have written a long paper with this argument, and Lubos nevertheless manage to interpret the phrase "spacetime diffeomorphisms" to mean "a membrane generalization of Weyl scalings".
		So general covariance without background structure is a very strong statement.
	www.lns.cornell.edu /spr/2001-10/msg0036153.html (919 words)

	ESF Network for Philosophical and Foundational Problems of Modern Physics
		In his general theory of relativity Einstein sought to generalize his special relativistic principle of relativity to a principle according to which all frames of reference, regardless of their motion, are equivalent.
		He thought to have achieved this aim through the general covariance of the equations of GRT: because the equations take the same form in every frame of reference, all frames are physically equivalent.
		The fact-like character of the differences between frames in general relativity justifies regarding them as equivalent in the same sense as inertial frames in special relativity are equivalent.
	www.philphys.nl /conferencesoxford.html (2133 words)

	Abstract_Belot_13_Apr_2004 (Site not responding. Last check: 2007-10-09)
		Abstract: The general covariance of general relativity is supposed to lead to a problem of time for classical and quantum theories of gravity.
		Often all of this is tied up with a feeling that the source of confusion must lie in the 3+1 decompositions of spacetime from which presentations of the problem of time typically depart.
		I will discuss formulations of general relativity in which no 3+1 decomposition is made—the space of state is a space of solutions, rather than a space of initial data.
	www.pitt.edu /AFShome/p/i/pittcntr/public/html/Abstracts/2004/Apr_2004/Abstract_Belot_13_Apr_2004.htm (207 words)

	General Relativity:Does it Prove the Cause and Strength of Gravity?
		Mass (which by the well-checked equivalence principle of general relativity is identical for inertial and gravitational forces), arises not from the fundamental core particles of matter themselves, but by a miring effect of the spacetime fabric, the ‘Higgs bosons’.
		General relativity is a mathematical accounting system and this factor of two comes into it from the energy considerations ignored by Newtonian physics, due to the light speed of the gravitational field itself.
		In his essay on general relativity in the book ‘It Must Be Beautiful’, Penrose writes: ‘… when there is matter present in the vicinity of the deviating geodesics, the volume reduction is proportional to the total mass that is surrounded by the geodesics.
	www.wbabin.net /physics/cook4.htm (4920 words)

	Einstein-Heuristics (Site not responding. Last check: 2007-10-09)
		Also, Einstein promoted the use of a formal mathematical principle, covariance, as a natural way to define what one meant by a "(general) law of physics," and then used it gain a considerable unification of physical theories.
		Having thus elevated covariance to a fundamental heuristic of his "formal point of view," it was a quick succession of steps to the unification of Galileo-Newtonian mechanics and Maxwell's optics.
		Thus, either Lorentzian covariance subsumes Galilean, or vice versa, or they are both related to each other by being subcovariances of some third covariance.
	www.ajnpx.com /html/Einstein-analects/Einstein-Heuristics.html (1256 words)

	The Field Equations
		It strikes many people as ironic that Einstein found the principle of general covariance to be so compelling, because, strictly speaking, it's possible to express almost any physical law, including Newton's laws, in generally covariant form (i.e., as tensor equations).
		Now, if we consider partial derivatives (which in these special coordinates are the same as covariant derivatives) of this tensor, we see that the derivative of the quantity in square brackets still vanishes, because the product rule implies that each term is a Christoffel symbol times the derivative of a Christoffel symbol.
		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 (1713 words)

	General Relativity For Tellytubbys (Site not responding. Last check: 2007-10-09)
		This statement essentially says that "Relativity" was used as a backbone for General Relativity, but was mistakenly referred to as "covariance".
		Covariance is whether or not one can write general equations independent of coordinate systems, irrespective of whether or not these preferred characteristics actually exist in the universe.
		Historically, even Einstein confused "general covariance" with "general relativity", of which he is on record as acknowledging that he had such confusion.
	www.anasoft.co.uk /physics/gr/Postulate1/postulate1.html (971 words)

	What is the "ontology" of general relativity ? (Site not responding. Last check: 2007-10-09)
		I used to think that general covariance simply meant that we could choose arbitrary coordinates on this manifold, but it seems that this is not sufficient.
		On the other hand, we have seen that the general covariance of general relativity implies that G and G' are physically indistinguishable.
		To block the hole argument and retain general covariance, general relativity must forgo any concept of a spacetime without a gravitational field.
	www.physicsforums.com /showthread.php?p=856236#post856236 (870 words)

	References > P
		"A generalized inverse of matrices" PCPS 51 (1955) 405-413.
		"Generalized Kochen-Specker theorem" qp/9510018, FP 26 (1996) 807-812 [>qm-exp].
		"On the general covariance and strong equivalence principles in quantum general relativity" FP 24 (1994) 989-1076 [>covariance, ep, st].
	www.phy.olemiss.edu /~luca/Refs/p.html (7962 words)

	PhilSci Archive - General Covariance, Gauge Theories and the Kretschmann Objection.
		PhilSci Archive - General Covariance, Gauge Theories and the Kretschmann Objection.
		General Covariance, Gauge Theories and the Kretschmann Objection.
		How can we reconcile two claims that are now both widely accepted: Kretschmann's claim that a requirement of general covariance is physically vacuous and the standard view that the general covariance of general relativity expresses the physically important diffeomorphism gauge freedom of general relativity?
	philsci-archive.pitt.edu /archive/00000380 (130 words)

	Table of Contents
		High-level monograph focuses on an explicit treatment of the principle of general covariance as applied to electromagnetics, examining—among other subjects—the natural invariance of the Maxwell equations, general properties of the medium, nonuniformity, anisotropy and general coordinates in three-space, reciprocity and nonreciprocity, and matter-free space with a gravitational field.
		The covariant form of the linear constitutive equations 3.
		The covariant form of the wave equation 4.
	www.doverpublications.com /cgi-bin/toc.pl/0486654273 (195 words)

	Amazon.com: General Relativity: Books: Robert M. Wald (Site not responding. Last check: 2007-10-09)
		There have been many books written on general relativity from both a physical and mathematical viewpoint, but this one stands out as one that is a hybrid between mathematical rigor and physical insight.
		In the next chapter, the principle of general covariance is introduced as one that prohibits the existence of perferred vector fields in the laws of physics.
		The next chapter is devoted entirely to the Schwarzschild solution, which is used to discuss the four experimental verifications of general relativity, namely the gravitational redshift, the precession of Mercury's orbit, bending of light by the Sun, and the time delay of radar signals.
	www.amazon.com /exec/obidos/tg/detail/-/0226870332?v=glance (2512 words)

	A GENERAL SPATIAL COVARIANCE MODEL WITH HIERARCHICAL ORGANIZATION, AND APPLICATION TO PERMEABILITY IN CROSS-BEDDED ... (Site not responding. Last check: 2007-10-09)
		In such deposits, the correlation (which is negative) across different unit types is predominant in the scaling.
		At each higher level (larger scale) the cross-transition probabilities for units with greatest relative difference in permeability dictate the shape and range of the spatial covariance.
		The shape lies somewhere in the evolution between a perfectly periodic piece-wise linear structure and an a-periodic exponential structure, dependant largely on the coefficient of variation in length for the unit types and their pattern.
	gsa.confex.com /gsa/2002AM/finalprogram/abstract_41543.htm (303 words)

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