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Einstein field equations

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	Einstein field equations - Wikipedia, the free encyclopedia
		The EFE collectively form a tensor equation and equate the curvature of spacetime (as expressed using the Einstein tensor) with the energy and momentum within the spacetime (as expressed using the stress-energy tensor).
		Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations.
		The study of exact solutions of Einstein's field equations is one of the activities of cosmology.
	en.wikipedia.org /wiki/Einstein_field_equations (1319 words)

	General relativity - Wikipedia, the free encyclopedia
		The solutions of the Einstein field equations that call for this behavior for the current universe, which may require the reintroduction of the cosmological constant, are for a stress-energy which is at least 70% dark energy.
		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.
		Einstein did not believe in an expanding universe, and so he added a cosmological constant to the field equations to permit the creation of static universe solutions.
	en.wikipedia.org /wiki/General_relativity (5428 words)

	ScienceDaily: Physics (Site not responding. Last check: 2007-10-31)
		Condensed matter physics, by most estimates the largest single field of physics, is concerned with how the properties of bulk matter, such as the ordinary solids and liquids we encounter in everyday life, arise from the properties and mutual interactions of the constituent atoms.
		The field of particle physics, also known as "high-energy physics", is concerned with the properties of submicroscopic particles much smaller than atoms, including the elementary particles from which all other units of matter are constructed.
		Since the 20th century, the individual fields of physics have become increasingly specialized, and nowadays it is not uncommon for physicists to work in a single field for their entire careers.
	www.sciencedaily.com /encyclopedia/physics (5014 words)

	Gravity - Facts, Information, and Encyclopedia Reference article
		The relationship between the presence of mass/energy/momentum and the curvature of spacetime is given by the Einstein field equations.
		Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light.
		Nikola Tesla challenged Albert Einstein's theory of relativity, announcing he was working on a Dynamic theory of gravity (which began between 1892 and 1894) and argued that a "field of force" was a better concept and focused on media with electromagnetic energy that fill all of space.
	www.startsurfing.com /encyclopedia/g/r/a/Gravity.html (4566 words)

	Einstein's Field Equations
		Einstein considered this equation but rejected it since it does not reduce down to the Newtonian gravitational equations in the form of the Poisson's equation, the necessary 2nd order form when matter is present in Newton's theory.
		The field equations (14.9) reduce properly to the Newtonian Poisson equation.
		Einstein then referred to the introduction of the cosmological constant as the ‘biggest blunder of his life’.
	io.uwinnipeg.ca /~vincent/4500.6-001/Cosmology/Field-Equations.htm (648 words)

	Encyclopedia :: encyclopedia : Equations (Site not responding. Last check: 2007-10-31)
		In mathematics, the theory of equations comprises a major part of traditional algebra.
		Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.
		From the point of view of abstract algebra, the material is divided between symmetric function theory, field theory, Galois theory, and computational considerations including numerical analysis.
	www.hallencyclopedia.com /Equations (152 words)

	Einstein Field Equations - Search Results - MSN Encarta
		James Clerk Maxwell, a 19th-century Scottish physicist, further developed the field concept in his electromagnetic theory.
		Equations, Theory of, branch of mathematics concerned with the nature of the roots of polynomial equations and with the methods of finding the...
		For the rest of his life Einstein devoted considerable time to generalizing his theory even more.
	encarta.msn.com /encnet/refpages/search.aspx?q=Einstein+Field+Equations (157 words)

	THE EINSTEIN EQUATIONS
		For a mathematician, the difficulties lie in the fact that the equations are nonlinear and coupled also and in their sheer number of terms.
		After he published his famous paper in 1916, Einstein later conceded that the mathematical difficulties of his General Theory of Relativity were a "very serious" impediment to its further development.
		Physicists are motivated to grapple with the complexity of Einstein's field equations because they can be used, theoretically at least, to describe all possible spacetime scenarios, from the collisions of fl holes to the gravit ational interactions of irregular chunks of matter.
	archive.ncsa.uiuc.edu /Cyberia/NumRel/EinsteinEquations.html (1024 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.
		A contact-force representation for the weak part of the field is also derivable from the electroform field equations by a method described by Cottingham and Greenwood, in Introduction to Nuclear Physics.
		Excerpt from the 11th edition deriving the factor of 4 in the solutions of the GR field equations.
	greenwdks.tripod.com /unifiedsummary.html (3233 words)

	Einstein's field equations
		Therefore the equations for the gravitational field in the vacuum of
		No, the gravitational field which is the Riemann tensor a field of spacetime curvature in modern relativity, not the Newtonian acceleration field, can not be transformed away.
		So when there is matter present, Enstein's "field" equations imply that the Riemann tensor is not zero in the presence of it.
	www.physicsforums.com /showthread.php?t=36769 (1911 words)

	INI : Abstracts : GMR : Integrable reductions of Einstein's field equations: monodromy transform and the linear ... (Site not responding. Last check: 2007-10-31)
		In principle, the monodromy data functions can be calcul ated also from some boundary, or initial, or characteristic initial data for the fields, and many physical properties of solutions are simply ``encoded'' in the analytical structures of these functions.
		In the introduction we give a short survey of various integrable symmetry reductions of Einstein's field equations and mention some interrelationships between various developed linear integral equation methods.
		The structure of the direct problem of the monodromy transform and general construction of the linear singular integral equation solving the inverse problem will be considered and some applications of this approach for construction of infinite hierarchies of exact solutions will be presented.
	www.newton.cam.ac.uk /programmes/GMR/alekseev.html (301 words)

	Schwarzschild's solution to Einstein's field equations
		This was the first solution to the field equations, and it's the simplest.
		Because we're dealing with the free-field equation with zero electromagnetic field, the field equation reduces to: the Ricci tensor is a scalar field multiplied by the metric, and its trace is zero.
		Einstein's field equation insists on Ricci(D)/g having trace equal to a particular (dimension-specific, but non-zero – unless dimension is two) multiple of the energy density (specifically, the trace of the energy-momentum-stress tensor, whose spatial diagonal entries are pressure terms); since this is zero in the vacuum solution, −3.h/2 must be zero, i.e.
	www.chaos.org.uk /~eddy/physics/Swarzchild.html (1266 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).
		Alas, neither Einstein nor anyone since has been able to make further progress in determining the true form of the right hand side of (2), although it is at the heart of current efforts to reconcile quantum mechanics with general relativity. At present we must be content to let T
		A slight (but significant) extension of the field equations was proposed by Einstein in 1917 based on cosmological considerations, as a means of ensuring stability of a static closed universe.
	www.mathpages.com /rr/s5-08/5-08.htm (1687 words)

	Solutions of the Einstein Equations
		These are 10 equations that must be solved simultaneously; they are notoriously difficult, and only a few solutions are known.
		When Einstein first realized that the solution of his equations subject to the constraints of the cosmological principle led to universes that were not static, he was dismayed because at the time (the period between 1915 and 1920) the expansion of the Universe had not yet been discovered by Hubble.
		If he had possessed sufficient confidence in his original equations, he would have predicted that either his theory was wrong or the Universe was expanding or contracting, well before there was experimental evidence of the expansion.
	csep10.phys.utk.edu /astr162/lect/cosmology/solutions.html (590 words)

	Time travel for beginners
		They are not, as yet, building TARDIS lookalikes in their laboratories; but they have realised that according to the equations of Albert Einstein's general theory of relativity (the best theory of time and space we have), there is nothing in the laws of physics to prevent time travel.
		Encouraged to look at the equations of the general theory of relativity in a new light, Thorne and his colleagues first found that there is nothing in those equations to prevent the existence of such wormholes, and then realised that any tunnel through space is also, potentially, a tunnel through time.
		The equations said that in order for an artificial wormhole to be held open, its throat must be threaded by some form of matter, or some form of field, that exerts negative pressure, and has antigravity associated with it.
	www.lifesci.sussex.ac.uk /home/John_Gribbin/timetrav.htm (8295 words)

	PlanetMath: Einstein field equations
		The Einstein Field Equations are a system of second order coupled nonlinear partial differential equations containing over one thousand terms when written out explicitly.
		Cross-references: ricci scalar, Ricci tensor, Einstein tensor, densities, mass, information, tensor, constant, representation, terms, partial differential equations, second order
		This is version 2 of Einstein field equations, born on 2005-02-16, modified 2005-02-16.
	planetmath.org /encyclopedia/EinsteinFieldEquations.html (106 words)

	Gemtry (Site not responding. Last check: 2007-10-31)
		In 1917 Einstein applied his newly developed static-spacetime general theory of relativity to cosmology [12], and introduced a cosmological constant to maintain the universe in what was then thought to be a static condition.
		In it the Hubble redshifts are now interpreted solely in terms of relativistic Doppler and Einstein gravitational redshifts, all cast within the framework of a finite, non-homogeneous, vacuum-gravity universe with Universal center (C) near the Galaxy.
		The NRI framework assumes the widely dispersed galaxies of the visible universe are enclosed by a thin, outer shell of hot hydrogen at a distance R from the Galaxy.
	www.evolution-facts.org /Gentry.htm (2715 words)

	2 Einstein’s Field Equations
		The field equations of general relativity form a system of ten second-order partial differential equations obeyed by the space-time metric
		For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner post-Newtonian field and the outer multipolar decomposition in the source’s exterior near zone.
		The gravitational field has been independent of time (stationary) in some remote past, i.e.
	relativity.livingreviews.org /Articles/lrr-2006-4/articlese2.html (639 words)

	First Step (Site not responding. Last check: 2007-10-31)
		So it is with the systematic solution of Einstein field equations, the equations that Einstein developed in 1916 to describe the nature of space and time.
		Unlike the equations that describe classical electrodynamics and even quantum theory, the fundamental equations of general relativity are nonlinear, a feature that has made them almost impossible to solve, except in approximate form.
		The approach that Hauser and I employed to prove the Geroch conjecture for stationary axisymmetric spacetimes, where elliptic differential equations were involved, was eventually generalized by us to cover the more difficult case of colliding gravitational plane waves, where hyperbolic differential equations are involved.
	members.localnet.com /~atheneum/exact/preface.html (436 words)

	Linearized Einstein Equations
		Looking at the Einstein equations (59) as a set of second-order partial differential equations it is not easy to predict that there exist solutions behaving as waves.
		These considerations suggest that the search for wave-like solutions to Einstein equations should be made in a spacetime with very modest curvature and with a metric line element which is that of flat spacetime but for small deviations of nonzero curvature, i.e.
		Before writing the linearized version of the Einstein equations (59) it is necessary to derive the linearized expression for the Christoffel symbols.
	www.sissa.it /~rezzolla/lnotes/virgo/node9.html (741 words)

	The Kerr-Newman solution to the Einstein-Maxwell field equations
		The Schwarzschild solution to Einstein's free-field equation for gravity describes the field around a spherically symmetric body with no electrodynamic field.
		Thus the body has mass but no charge, magnetic dipole or higher-order distribution of electromagnetic properties; and it is not even spinning (which would break spherical symmetry by having a preferred axis).
		Space-time's metric and the electromagnetic field around the Kerr-Newman fl hole with charge, mass and angular momentum encoded by lengths Q, M and an area J
	www.chaos.org.uk /~eddy/physics/Kerr-Newman.html (830 words)

	R-separable solutions of Einstein's field equations
		R-separable coordinate systems are introduced as a new class of systems in which the equations of general relativity may be solved.
		The static, axially symmetric vacuum and electrovac Einstein field equations are solved for two such systems, tangent spheres and bispherical.
		The relatively simple nature of the individual bispherical eigensolutions allows explicit integration of the field equations for a completely general, static, two-body source.
	stacks.iop.org /0305-4470/15/2401 (286 words)

	Re: R-T symmetry in Einstein's field equations
		In other words what the field equation actually says in this model is that at each particle there is a cusp in space-time of net curvature proportional to the proper time between particle interactions, which is itself proportional to mass in the reference frame of the particle, and to the stress-energy tensor generally.
		This is not a cusp viewed from a macroscopic reference frame because a particle does not have precise position, but the classical field equation should be found as the expectation of all such "cusps".
		To show that this actually works, it is interesting to look at a simple scalar argument which gives the Newtonian approximation.
	www.lns.cornell.edu /spr/2000-02/msg0022439.html (662 words)

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