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	Minkowski space - Wikipedia, the free encyclopedia
		In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated.
		There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer.
		Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity previously worked out by Einstein and Lorentz could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.
	en.wikipedia.org /wiki/Minkowski_spacetime (965 words)

	Spacetime (Site not responding. Last check: 2007-10-18)
		A spacetime interval between two events is the frame-invariant quantity analogous to distance in Euclidean space.
		Events with a spacetime interval of zero are separated by the propagation of a light signal.
		Events with a positive spacetime interval are in each other's future or past, and the value of the interval defines the proper time measured by an observer travelling between them.
	hallencyclopedia.com /Spacetime (817 words)

	Bob Gardner's "Relativity and Black Holes" Special Relativity (Site not responding. Last check: 2007-10-18)
		Minkowski, in fact, was one of Albert Einstein's mathematics professors at the Zurich Politechnikum in 1900.
		As a sad footnote, Hermann Minkowski died of appendicitis in 1909 at the age of 45.
		Minkowski is honored today by the fact that his spacetime is often called Minkowski spacetime or the Minkowski vector space.
	www.etsu.edu /physics/plntrm/relat/spactim.htm (940 words)

	Time Supplement [Internet Encyclopedia of Philosophy]
		Minkowski meant it is fundamental in the sense that the spacetime interval between any two events is intrinsic to spacetime and does not vary with the reference frame, unlike a distance (space) or a duration (time).
		Minkowski diagrams are diagrams of a Minkowski space, which is a spacetime satisfying the Special Theory, and therefore it is falsely presupposed that physical processes such as gravitational processes have no effect on the structure of spacetime.
		Minkowski was the first person to construct such a mathematical space, and he was the first person to call time the fourth dimension, because it was the fourth dimension of his abstract space for spacetime.
	www.iep.utm.edu /ancillaries/time-sup.htm (12749 words)

	Minkowski space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		Minkowski space is named for the German mathematician (German mathematician (born in Russia) who suggested the concept of four-dimensional space-time (1864-1909)) Hermann Minkowski (See History).
		This terminology comes from the use of Minkowski space in the ((physics) the theory that space and time are relative concepts rather than absolute concepts) theory of relativity.
		More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional (A pipe that has several lateral outlets to or from other pipes) manifold for which the (additional info and facts about tangent space) tangent space to any point is a 4-dimensional Minkowski space.
	www.absoluteastronomy.com /encyclopedia/m/mi/minkowski_space.htm (1223 words)

	Abstracts
		In other words, the claim at the heart of Minkowski's analysis is, at the same time, extremely far- reaching and extremely modest: it is the claim that a world in which special relativity is true simply is a world with a particular spacetime structure.
		Minkowski revealed, in short, how the liberation of physical geometry from spatial intuition was not a separation of formalism from empirical content, but the simple result of shifting our attention from purely spatial principles (such as free mobility) to dynamical principles involving time.
		The fact that Minkowski spacetime is a fixed structure with global symmetries, made it impossible, of course, that it could be an appropriate global structure for general relativity.
	alcor.concordia.ca /~scol/seminars/conference/DiSalle.html (519 words)

	[No title]
		What is at stake in the debate on the ontological status of Minkowski spacetime constitutes perhaps the greatest intellectual challenge the human race has ever faced: if reality is a 4D world then the time dimension, like the space dimensions, is entirely given which means that all moments of time are given.
		The arguments that existence is absolute which means that Minkowski spacetime represents a 4D reality also shed light on the debate [16-19] on whether or not the simultaneity of distant events is a matter of convention.
		Most attacks against regarding Minkowski spacetime as adequately representing reality are based on the apparent contradiction between the equal existence of all events of spacetime and the fact the we realize ourselves only at the moment "now" and believe that it is without any doubt privileged.
	alcor.concordia.ca /~scol/seminars/absolute.html (5055 words)

	Minkowski Diagrams (Site not responding. Last check: 2007-10-18)
		One frequently used method of visualizing spacetime is the Minkowski Diagram.
		The defining feature of a Minkowski diagram is that light rays are drawn at a 45 degree angle to the line or plane respresenting space.
		One example of the use of Minkowski Daigrams is as follows (refer to Figure 3): A Square conveniently rests at the origin of spacetime.
	www.brown.edu /Students/OHJC/ma8/papers/minkowsk.htm (632 words)

	Minkowski Spacetime (Site not responding. Last check: 2007-10-18)
		Minkowski spacetime M is the model of spacetime geometry used in the theory of special relativity.
		Minkowski spacetime M is defined to be a four dimensional linear space with a nondegenerate, symmetric, bilinear form g:
		Lorentz transformations are used to relate the spacetime coordinates for an event observed by two observers in their respective frames of reference.
	bkocay.cs.umanitoba.ca /Students/Theory.html (2717 words)

	Quantum field theory on curved spacetime at the Erwin Schrödinger Institute
		Furthermore, theories that are renormalizable in Minkowski spacetime will also be renormalizable in curved spacetime, although additional ``counterterms'' corresponding to couplings of the quantum field to curvature will arise.
		The Bisognano-Wichmann theorem states that in Minkowski spacetime, the restriction of the vacuum state to a wedge region is a KMS state with respect to a 1-parameter subgroup of the Poincare group.
		In Anti-de Sitter spacetime the wedges are in one to one correspondence with double cones in the Minkowski spacetime at spacelike infinity.
	www.phys.lsu.edu /mog/mog20/node16.html (1098 words)

	In The Neighborhood
		It's customary to treat the relativistic spacetime manifold as an ordinary topological space with the same topology as a four-dimensional Euclidean manifold, denoted by R
		Minkowski spacetime gives us the opportunity to reconsider the famous "limit paradox" from freshman calculus in a new context. Recall the standard paradox begins with a two-part path in the xy plane from point A to point C by way of point B as shown below:
		To place this in the context of Minkowski spacetime, we can simply replace the y axis with the time axis, and replace the Euclidean metric with the Minkowski pseudo-metric.
	www.mathpages.com /rr/s9-01/9-01.htm (784 words)

	[No title]
		This conclusion is premature; a natural extension of Stein’s notion of becoming in Minkowski spacetime to accommodate the demands of quantum nonseparability yields such an account, an account that is in accord with a proposal which was made by Aharonov and Albert but which is dismissed by Albert as a ‘mere trick’.
		R is definable in terms of the geometry of Minkowski spacetime.
		On the global, foliation-relative notion, we have, for each spacetime point P, a multitude of choices for the present of P, one for each spacelike hypersurface containing P. We do not have a notion of an instantaneous present that is both spatially extended and independent of an arbitrary choice of foliation.
	philsci-archive.pitt.edu /archive/00000569/00/RQB.doc (2842 words)

	Independent Axioms for Minkowski Space-Time
		The primary aim of this monograph is to clarify the undefined primitive concepts and the axioms which form the basis of Einstein's theory of special relativity.
		Minkowski space-time is developed from a set of independent axioms, stated in terms of a single relation of betweenness.
		It is shown that all models are isomorphic to the usual coordinate model, and the axioms are consistent relative to the reals.
	www.ramex.com /ch/ch-3151.html (100 words)

	6.1 Perturbations around Minkowski spacetime
		The fact that the vacuum expectation value of the renormalized stress-energy operator in Minkowski spacetime should vanish was originally proposed by Wald [283], and it may be understood as a renormalization convention [100, 113].
		Note that other possible solutions of semiclassical gravity with zero vacuum expectation value of the stress-energy tensor are the exact gravitational plane waves, since they are known to be vacuum solutions of Einstein equations which induce neither particle creation nor vacuum polarization [107, 73, 104].
		Hence, even in the Minkowski background, there are quantum fluctuations in the stress-energy tensor and, as a result, the noise kernel does not vanish.
	relativity.livingreviews.org /Articles/lrr-2004-3/articlesu9.html (550 words)

	On the Ontological Status of Minkowski Space
		In 1908 H. Minkowski [1] gave a four-dimensional formulation of the special theory of relativity by uniting space and time into a single entity - the four-dimensional spacetime (sometimes called Minkowski space).
		The relativization of simultaneity can be explained either by assuming that the existence of the three-dimensional world is also relativized (observer-dependent) or by assuming reality to be a four-dimensional world whose existence remains absolute (observer-independent).
		Minkowski, "Space and Time" in Lorentz, Hendrik A., Albert Einstein, Hermann Minkowski, and Hermann Weyl, The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity.
	alcor.concordia.ca /~vpetkov/minkowski.html (645 words)

	Time Travel Portal :: View topic - Energy Conditions & Quantum Inequalities I (Site not responding. Last check: 2007-10-18)
		In a two-dimensional compactified Minkowski universe, we derive a covariant quantum inequality-type bound on the difference of the expectation values of the energy density in an arbitrary quantum state and in the Casimir vacuum state.
		When spacetime is curved and/or has boundaries, we argue that the bound should hold in regions small compared to the minimum local characteristic radius of curvature or the distance to any boundaries, since spacetime can be considered approximately Minkowski on these scales.
		These include spacetime averaged quantum inequalities in two-dimensional spacetime, the failure of generalizations of the averaged weak energy condition to piecewise geodesics, and the issue of when the local energy density is negative in the frame of all observers.
	timetravelportal.com /viewtopic.php?t=969 (2850 words)

	Special Relativistic Limit of the Homogeneous Evans Field Eqn (Site not responding. Last check: 2007-10-18)
		In general relativity the field is spacetime itself, so electromagnetism is spinning spacetime encapsulated in the torsion two-form.
		The standard model is inherently self-contradictory and self- inconsistent, because it uses non-Minkowski spacetime for gravitation and Minkowski spacetime for the other three sectors (electro-weak and strong).
		The Evans field theory uses non-Minkowski spacetime self consistently for all radiated and matter fields and does not use the experimentally refuted (and, to many, incomprehensible) Heisenberg Bohr philosophy.
	www.aias.us /Comments/comments08152004c.html (418 words)

	Segal Conformal Physics and GraviPhotons
		The physical 4-dimensional SpaceTime of the D4-D5-E6-E7-E8 VoDou Physics model is a 4-dimensional HyperDiamond lattice SpaceTime that is continuously approximated globally by RP1 x S3 and locally by Minkowski SpaceTime, with Gravity coming from the 15-dimensional Conformal Group Spin(2,4) by the MacDowell-Mansouri mechanism.
		The curved SpaceTime of General Relativity is not considered fundamental, but is produced by by starting with a linear spin-2 field theory in flat spacetime, and then adding higher-order terms to get Einstein-Hilbert gravity.
		In the D4-D5-E6-E7 physics model, 4-dimensional Physical Spacetime is the Shilov Boundary of an 8-real-dimensional Bounded Complex Homogeneous Domain corresponding to the Hermitian Symmetric Space Spin(2,4) / Spin(4) x Spin(2) = SU(2,2) / S(U(2)xU(2)), which symmetric space is a 15-6-1 = 8-real-dimensional space with Complex structure, or a 4-Complex-dimensional space.
	www.valdostamuseum.org /hamsmith/SegalConf.html (3933 words)

	The No-Curvature Interpretation of GR
		In the case of quantum mechanics there are alternative interpretations that do not explicitly involve a collapse of the wave-function, and that assume instead that the overall wavefunction always continues to evolve in accord with the purely linear Schrodinger equation.
		For purposes of expressing a physical theory of spacetime, we now face a task similar to the task in quantum mechanics of explaining the physical and phenomenological significance of the multiple linear branches, and accounting for the APPEARANCE of just a single branch.
		It's worth noting that the quantum wave functions of some particles have the property that they are returned to their original state not by a rotation of 360 degrees but by "going around twice", i.e., by a rotation of 720 degrees.
	www.mathpages.com /home/kmath178.htm (891 words)

	The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity
		This mathematically rigorous treatment examines Zeeman's characterization of the causal automorphisms of Minkowski spacetime and the Penrose theorem concerning the apparent shape of a relativistically moving sphere.
		Other topics include the construction of a geometric theory of the electromagnetic field; an in-depth introduction to the theory of spinors; and a classification of electromagnetic fields in both tensor and spinor form.
		Appendixes introduce a topology for Minkowski spacetime and discuss Dirac's famous "Scissors Problem." Appropriate for graduate-level courses, this text presumes only a knowledge of linear algebra and elementary point-set topology.
	store.doverpublications.com /0486432351.html (218 words)

	What ARE Clifford Algebras and Spinors?
		the Clifford algebra of spacetime is Cl(-+++) = Cl(1,3) and the signature of physical spacetime is -+++.
		In the full 8-dimensional theory, the 56 trivectors are related to the structure of 3+1=4-dimensional subspaces of 1+7=8-dimensional spacetime that are connected with the E8 HyperDiamond lattice links that are (normalized) sums of 4 of the basis octonions.
		To reduce the dimension of spacetime to 1+3=4 dimensions, an associative 3-form is used.
	www.valdostamuseum.org /hamsmith/clfpq2.html (5722 words)

	TEXAS TWISTORS
		While the background spacetime is flat, the gauge invariant predictions in physically observable circumstances are the same as in General Relativity.
		However, on an intuitive level, a geodesic in flat spacetime is determined by where it strikes the null cone of the event (0,0,0,0); and by its direction.
		(To an observer with timelike worldline in Minkowski spacetime.)
	etsuodt.tamu-commerce.edu /AcademicOrganizations/sigmaxi/twist.html (3438 words)

	[No title]
		Applying T-geometry to a construction of spacetime model, one obtains surprising results which could not be obtained in the degenerate geometry of Minkowski.
		The four-velocity and momentum of a particle are not parallel in the Minkowski spacetime associated with real spacetime constructed on the basis of T-geometry.
		Coupling the elementary length (distorion) with the quantum constant, one succeeds to choose the optimal uniform isotropic spacetime model in such a way that the statistical description of stochastic world tubes coincides with their quantum description by means of the Schroedinger equation.
	pavel.physics.sunysb.edu /~rylov/tgastm.htm (1481 words)

	How Gauge Bosons See Internal Space
		As Matti Pitkanen has noted, the Quaternions have both a natural Minkowski metric (given by the square root of the real part Re(ZZ) of Z times Z for Quaternion Z) and a natural Euclidean metric (given by the square root of ZZ* of Z times Z conjugate for Quaternion Z).
		The tilted lightcone spacetimes probably constitute a very small part of any quantum superposition describing physical spacetime, so that conformal symmetry and preservation of lightcone structure is probably a good enough approximation to use quantum conformal fluctuations to describe cosmology using the D4-D5-E6-E7-E8 model.
		However, even if only a very small fraction of the spacetime geometries in the superposition contain closed timelike loops resulting from tilted lightcones, non-computable operations could be performed by a human-brain quantum computer.
	www.valdostamuseum.org /hamsmith/See.html (4120 words)

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