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Topic: Einstein tensor

	PlanetMath: Einstein field equations
		The Einstein Field Equations are the fundamental equations of Einstein's general theory of relativity.
		In order to adress this issue and to be able to treat the Einstein equations much as one would treat other differential equations, a common practise is to supplement the Einstein equations with auxiliary condidtions which serve to define a coordinate system and hence single out a particular element of an equivalence class in diffeomorphism.
		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.
	planetmath.org /encyclopedia/EinsteinFieldEquations.html (615 words)

	Einstein's field equation - Wikipedia, the free encyclopedia
		In physics, the Einstein field equation or Einstein equation is a differential equation in Einstein's theory of general relativity.
		is the stress-energy tensor, and the constant is given in terms of π (pi), c (the speed of light) and G (the gravitational constant).
		Einstein's equation reduces to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation.
	en.wikipedia.org /wiki/Einstein's_field_equation (993 words)

	Einstein's field equation - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-10)
		Here $R_{ab} is the Ricci$ tensor, $R is the Ricci scalar, g_{ab} is the metric$ tensor, $T_{ab} is the stress-energy$ tensor, and the constant is given in terms of $\pi (pi), c (the speed of light) and G (the gravitational constant).$
		The EFE is understood to be an equation for the metric tensor $g_{ab} (given a specified distribution of matter and energy in the form of a stress-energy$ tensor).
		Einstein's equation reduces to Newton's law of gravity by using both the weak-field approximation and the.
	www.bexley.us /project/wikipedia/index.php/Einstein_field_equation (1164 words)

	Einstein-Cartan theory - Wikipedia, the free encyclopedia
		The symmetry of the Einstein curvature tensor forces the momentum tensor to be symmetric.
		It has long been known that the spin angular momentum tensor Spin(a,b,^k} is the "Noether current" of rotational symmetry of spacetime, and the momentum tensor P{a,k} is the Noether current of translational symmetry.
		The momentum tensor P{a,^k} describe the flux of a-momentum through a flux box normal to the k-direction in spacetime, and the spin tensor Spin{a,b,^k} describes the flux of angular momentum in the a × b plane through a flux box normal to the k-direction in spacetime.
	en.wikipedia.org /wiki/Einstein-Cartan_theory (2424 words)

	black-holes.org—Einstein's Equations
		Einstein had these warped pieces of spacetime that he needed to describe in some quantifiable way.
		Einstein combined certain numbers describing the metric's changes from place to place into what is now called the Einstein tensor.
		But, the Einstein tensor represents the geometry of spacetime, so this is what the left side really represents.
	www.black-holes.org /numrel1.html (1745 words)

	Einstein
		Three of Einstein's fellow students, including Grossmann, were appointed assistants at ETH in Zurich but clearly Einstein had not impressed enough and still in 1901 he was writing round universities in the hope of obtaining a job, but without success.
		Einstein worked in this patent office from 1902 to 1909, holding a temporary post when he was first appointed, but by 1904 the position was made permanent and in 1906 he was promoted to technical expert second class.
		Einstein was not the first to propose all the components of special theory of relativity.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Einstein.html (2020 words)

	Einstein's Theory (Site not responding. Last check: 2007-10-10)
		Second, a frequent complaint about Einstein's law of gravitation is that it does not fit into the convenient and soluble structure of Fredholm's equation, or in the usual phrasing, that you can't solve for the metric tensor given the mass and momentum tensor.
		As the distance function is symmetric end to end, the metric tensor has n(n+1)/2 independent components, where n is the dimension of the space, which are further restricted by Ricci's theorem: the covariant derivative of the metric tensor (in any direction) is zero.
		The metric tensor is then differentiated and certain sums are performed, yielding a tensor of the same type, which is then set equal to the source term, which is a symmetric tensor having n(n+1)/2 independent components restricted by a conservation law.
	www.math.ucla.edu /~jimc/klein_h/einstein.html (522 words)

	General relativity:Einstein's equation - Wikibooks
		The Einstein field equation or Einstein equation is a dynamical equation which describes how matter and energy change the geometry of spacetime, this curved geometry being interpreted as the gravitational field of the matter source.
		is the Ricci curvature tensor * R is the Ricci scalar (the tensor contraction of the Ricci tensor) * g
		is a (symmetric 4 x 4) metric tensor * Λ is the Cosmological constant * π is pi=3.1415..., the ratio between a circle's circumference and diameter * G is the Gravitational constant * c is the speed of light in free space * T
	en.wikibooks.org /wiki/General_relativity:Einstein's_equation (971 words)

	Einstein tensor - Wikipedia, the free encyclopedia
		is the metric tensor and R is the Ricci scalar (or scalar curvature).
		Einstein tensor is sometimes referred to as the trace-reversed Ricci tensor.
		In general relativity, the Einstein tensor allows a compact expression of the Einstein equations:
	en.wikipedia.org /wiki/Einstein_tensor (109 words)

	Einstein's Field Equations
		to represent the energy-momentum tensor that is causing the spacetime curvature.
		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.
		Einstein then referred to the introduction of the cosmological constant as the ‘biggest blunder of his life’.
	scholar.uwinnipeg.ca /courses/38/4500.6-001/Cosmology/Field-Equations.htm (648 words)

	Yilmaz Theory of Gravitation (Site not responding. Last check: 2007-10-10)
		Therefore the two rightmost terms must be equal to zero by Einsteins postulate that the trajectory of a point particle in a gravitational field is a geodesic.
		This gravitational field stress-energy tensor is not to be confused with the Einstein pseudo-tensor.
		The problem however, is that it is not possible to have the active mass equivalent to the other two in Einstein's formulation because as we have seen after incorporating the Freud identity to avoid rest mass conservation, we are left with zero accelerations.
	members.tripod.com /jdbrown371/newtheory.html (4611 words)

	The Field Equations
		Let's follow Einstein's original presentation in his famous paper "The Foundation of the General Theory of Relativity", which was published early in 1916. He notes that for empty space, far from any gravitating object, we expect to have flat (i.e., Minkowskian) spacetime, which amounts to requiring that Riemann's curvature tensor R
		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.
		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 (1695 words)

	Einstein equation (Site not responding. Last check: 2007-10-10)
		Einstein was the first to point out that a consequence of the strong equivalence principle, which states that the results of local experiments in a frame of free fall are independent on the motion, and that the results of all such experiments should be the same, is that gravitational fields should affect radiation.
		Hence the stress-energy tensor is a second-rank symmetric divergenceless tensor which vanishes in empty space.
		Einstein identified the stress-energy tensor as the source of spacetime curvature and suggested the simplest possible relationship between it and the curvature, namely
	www.nikhef.nl /~henkjan/astro/node15.html (1030 words)

	General Relativity:Does it Prove the Cause and Strength of Gravity?
		To simplify the history, Einstein recognised that while this equation reduces to Newton’s law for low speeds, it is in error because it violates the principle of conservation of mass-energy, since a gravitational field has energy (i.e., ‘potential energy’) and vice-versa.
		A year later, Einstein force-fitted it to the assumed static universe of 1916 by inventing a new cosmic ‘epicycle,’ the cosmological constant, to make gravity weaken faster than the inverse square law, become zero at a distance equal to the average separation distance of galaxies, and to become repulsive at greater distances.
		Einstein was enraged and wrote to the editor [27 July 1936] that he objected to his paper being shown to colleagues before publication… Einstein… never published in the Physical Review again.’ – Abraham Pais, Subtle is the Lord, the Science and the Life of Albert Einstein, Oxford University Press, Oxford, 1982, p.
	wbabin.net /physics/cook4.htm (2795 words)

	Maxima Manual - Ctensor (Site not responding. Last check: 2007-10-10)
		DIM is the dimension of the manifold with the default 4.
		EINSTEIN computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions CHRISTOF and RICCICOM).
		As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor.
	www.ma.utexas.edu /maxima/maxima_28.html (1068 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)

	Einstein’s Field Equations
		The strong form of the equivalence principle states that the gravitational force is equivalent to an inertial force.
		This assumption is used to derive an expression for the gravitational force, which is expressible in terms of the components of metric tensor as
		From this it follows that the components of the metric tensor form a set of 10 independent gravitational potentials.
	www.geocities.com /physics_world/gr/einsteins_field_equations.htm (525 words)

	ModPhy1
		In the following equations, we will use Einstein’s summation convention such that whenever an index is repeated in a term, once as a subscript and once as a superscript, it signifies that that term should be summed over all possible values of the index.
		The Ricci tensor is the symmetrical second order tensor obtained by contracting the Riemann tensor on its first and third vectors.
		The Einstein tensor is a symmetrical second rank tensor whose divergence is identically zero.
	physics.tamuk.edu /~hewett/ModPhy1/Unit1/GeneralRelativity/EinsteinGravity/EinsteinT/EinsteinT.html (724 words)

	Amazon.com: Books: The Meaning of Relativity (Site not responding. Last check: 2007-10-10)
		Einstein argues that the hypothesis that the universe is
		Einstein complexified it a bit by using identities like E = - grad(phi) - dA/dt and B = curl A, but there it was, really a monument to his powers of analysis.
		Einstein's solution in Cartesian coordinates is very useful not only for general relativity's overthrow, but for figuring out the kind of a theory most likely to succeed.
	www.amazon.com /exec/obidos/tg/detail/-/0691023522?v=glance (2378 words)

	Appendix B
		The corresponding formulas for the Einstein tensor are obtained from Tolman [2], and were first prepared by Prof.
		The covariant and contravariant metric tensor elements in rectangular coordinates are
		Herbert Dingle for the Einstein tensor when the metric tensor is diagonal.
	www.olduniverse.com /appendix_b.htm (864 words)

	Gravitational Energy-Momentum Tensor in General Relativity - Fall Velocity & Energy Conservation
		The tensorial expression for the gravitational energy-momentum proposed by Nissani and Leibowitz [N & L, 91, 92] is thoroughly presented and used in calculating the integral gravitational energy in the surrounding of a star, and the fall velocity of a body in a gravitational field.
		Therefore, as a consequence of the global ordinary-divergencelessness of the energy-momentum tensor density valid on the non-rotating coordinates, which in turn derives from the covariant-divergencelessness imposed on T by Einstein's equation, the energy-momentum of the antenna must remain strictly unaltered.
		It would be in accordance with the Landau and Lifshitz association of the commutator of the affine connections appearing in the customary expression of the Ricci tensor, which vanishes in the local geodesic coordinates, with their proposed non-tensorial gravitational energy-momentum [Landau and Lifshitz, 51].
	www.liberal.org.il /physics/mach2.htm (3272 words)

	Abstract for Manuscript Number [103] (Site not responding. Last check: 2007-10-10)
		Einstein's equation G = kT specifies an open nonlinear system wherein inertial forces derived from mass tells space-time how to curve and the curvature (gravitational forces also derived from mass) tells masses how to move (geodetically).
		Here the structure (curvature) and function (stress) of the psychological space-time is explored using Einstein's nonlinear tensor dynamics and a tensor derived from Turing's morpho-dynamics of patterning.
		To express these internal space-time geometrodynamical effects, we develop the concept of a counterpart, internal field equation to Einstein's for external space-time: namely, G = kT (Einstein's equation) and H = k'T (psychodynamic equation) A Mahalanobis-Bose radius is defined for the space-time of dreams (230-260 m.
	www.interjournal.org /manuscript_abstract.php?3214 (263 words)

	Stuff about the Einstein tensor - Science Forums and Debate (Site not responding. Last check: 2007-10-10)
		As I see it the Ricci tensor and the Ricci scalar are both curvature-type quantities (which are contracted from Riemann) and it is OK to call them curvature.
		tensor is just a combination of Ricci tensor together with the scalar times the metric, so in a general sense I think of it as measuring curvature too.
		Anyway the basic notion of the einstein eqn is that the LHS measures spacetime curvature and the RHS involves a measure of energy density and that the two are related by a force
	www.scienceforums.net /forums/showthread.php?t=9529 (993 words)

	Science Forums and Debate - Relativity (Site not responding. Last check: 2007-10-10)
		This tensor does not explicitly appear in the GR field equations, although the Ricci tensor is a contraction of it.
		It should be readily seen that, if anything, GR states that the ratio of the Einstein tensor to the stress-energy tensor is a constant.
		And vectors are first rank tensors, and if two vectors have the same direction, I don't see any problem with dividing one by the other, and so here is another case where one tensor can be divided by another.
	www.scienceforums.net /forums/printthread.php?t=9322 (3608 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Due to the symmetry of the Einstein tensor and the energy momentum tensor the field equations represent 10 coupled, non-linear partial differential equations, which written explicitly may contain of the order of 100,000 terms in the general case.
		In analogy to the prediction of electromagnetic waves by the Maxwell equations of electrodynamics, the Einstein field equations admit radiative solutions with a characteristic propagation speed given by the speed of light.
		In fact the most promising sources of gravitational waves currently under consideration are the in-spiralling and merger of two compact bodies (neutron stars or fl holes) and complicated oscillation modes of neutron stars that increase in amplitude due to the emission of gravitational waves by extracting energy from the rotation of the star.
	gravity.phys.psu.edu /~sperhake/Research/GRGWaves/GRgwaves.html (853 words)

	Re: Tidal Tensor
		It is true that f_(,a) (the gradient of the function f) is a -covector- and thus a tensor, but to differentiate a covector to obtain a second rank tensor you need to use "covariant differentiation".
		The claim that E_(ab) plays the role of the tidal tensor of gtr follows from the "Jacobi geodesic deviation formula", which is derived in every gtr textbook, and which can be rewritten in terms of E_(ab).
		The only nonvanishing component of the Einstein tensor in the almost Cartesian chart (to first order in e) is proportional to Lap(phi).
	www.lns.cornell.edu /spr/2002-02/msg0039010.html (1061 words)

	Bibliography of Raina Ivanova
		On the decomposition of the skew-symmetric generalized curvature tensor of type (1,5) (joint with M. Yawata), Reports of C.I.T., {48} (2001), 7--10.
		Einstein manifolds with point-wise constant characteristical coefficients of the curvature operator, Mathematics and Education in Mathematics, (Proc.
		On a property of the 4-dimensional Einstein manifolds, Proc.
	www2.hawaii.edu /~rivanova/bibliography.html (1013 words)

	outline1
		TENSOR of "rank (0,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a number.
		A TENSOR of "rank (1,k)" at a point p of spacetime is a function that takes as input a list of k tangent vectors at the point p and returns as output a tangent vector at the point p.
		It is a tensor of rank (0,2), and it defined as follows: given any two tangent vectors u and v at a point p, the number T(u,v) says how much momentum-in-the-v-direction is flowing through the point p in the u direction.
	math.ucr.edu /home/baez/gr/outline1.html (1161 words)

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