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Topic: Minkowski space

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	Minkowski space - Wikipedia, the free encyclopedia
		In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime.
		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_space (965 words)

	Hermann Minkowski - Wikipedia, the free encyclopedia
		Hermann Minkowski (June 22, 1864 - January 12, 1909) was a Jewish German mathematician who developed the geometrical theory of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.
		Hermann Minkowski was born in Aleksotas (a suburb of Kaunas, Lithuania), and educated in Germany at the Universities of Berlin and Königsberg, where he achieved his doctorate in 1885.
		Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions.
	en.wikipedia.org /wiki/Hermann_Minkowski (307 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).
		The mathematical space used by mathematical physicists to represent physical spacetime is four dimensional and in that space, the space of places is a 3-d sub-space and time is another 1-d sub-space.
		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 (12294 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Minkowski space is a four dimensional space with the metric whose signature is.
		Around 1907 Hermann Minkowski realized that the special theory of relativity, introducted by Albert Einstein in 1905 and based on previous work of Hendrik Lorentz and Henri Poincaré, could be mathematically described using a four-dimensional spacetime, which combines the dimension of time with the three space dimensions.
		The Lorentz transformations of special relativity can be represented as generalized rotations of the Minkowski space.
	www.informationgenius.com /encyclopedia/m/mi/minkowski_space.html (135 words)

	Minkowski (Site not responding. Last check: 2007-11-06)
		Minkowski's doctoral thesis, submitted in 1885, was a continuation of this prize winning work involving his natural definition of the genus of a form.
		Minkowski presented Räumliche Anschauung und Minima positiv definiter quadratischer Formen (Spatial visualization and minima of positive definite quadratic forms) which was not published at the time but in 1991 the lecture was published in [11].
		Minkowski developed a new view of space and time and laid the mathematical foundation of the theory of relativity.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Minkowski.html (1496 words)

	Minkowski space (Site not responding. Last check: 2007-11-06)
		In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional spacetime.
		Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” – Hermann Minkowski, 1908
		Nevertheless, even in such cases, Minkowski space is still a good description in a infinitesimally small region surrounding any point (barring gravitational singularities).
	www.sciencedaily.com /encyclopedia/minkowski_space (1009 words)

	The non-Euclidean style of Minkowskian relativity (Site not responding. Last check: 2007-11-06)
		Although Minkowski did not bother to unfold the geometry of velocity vectors, in the hypersurfaces (1) and (2), we have the premises of an explanation for Minkowski's description of the world as being-in a certain sense-a four-dimensional non-Euclidean manifold.
		Minkowski's preference for circular functions may be understood in relation to his project to express the laws of physics in four-dimensional terms.
		Minkowski was probably aware of the relatively rudimentary level of mathematical skills possessed by most physicists, and may have considered that non-Euclidean geometry would stand in the way of the acceptance by physicists of his space-time formalism.
	www.univ-nancy2.fr /DepPhilo/walter/papers/nesh.html (14219 words)

	The World of Minkowski (Site not responding. Last check: 2007-11-06)
		Minkowski specialized in the study of invariants, constants unaffected by designated mathematical procedures (such as the transformation of coordinates), and wrote a famous paper on the decomposition of integers into the sum of five squares.
		The Minkowski universe is a 4 dimensional continuum in which the history of a single space-point in the course of time must be a curved line, where an event in time represents a point.
		Minkowski's view of space-time had a large effect on the theories of relativity, but as it was the correct view for the theory of special relativity, the theory of general relativity required the use of another theory.
	www.montgomerybell.com /~clarkb/leland.htm (776 words)

	Relativity
		In classical Newtonian mechanics space the three-dimensional "world" is a place where all the events occur and time is absolute and the same for everybody.
		Space and time are separate and independent of each other and they cannot be mixed in any way.
		In special relativity, however, space and time are just different coordinates of the so-called space-time, i.e., space and time merge together into a four-dimensional "world".
	nobelprize.org /physics/educational/relativity/transformations-7.html (254 words)

	2.4 Example: Minkowski space (Site not responding. Last check: 2007-11-06)
		The standard conformal diagram for Minkowski space [109] is shown in Figure 2.
		This is, in particular, a property of the space-like hyperboloids in Minkowski space.
		In the specific case of Minkowski space-time we use the coordinates T and R on the Einstein cylinder.
	relativity.livingreviews.org /Articles/lrr-2000-4/node6.html (1449 words)

	Minkowski space-time (Site not responding. Last check: 2007-11-06)
		A Class of Metric Theories of Gravitation on Minkowski Spacetime, A. Nairz...
		Poincaré gauge invariance and gravitation in Minkowski spacetime...
		The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the S...
	www.scienceoxygen.com /phys/231.html (172 words)

	The Speed of Light - A Limit on Principle? (Site not responding. Last check: 2007-11-06)
		Obviously, the covariant framework imposed by Minkowski space-time is the most attractive one to describe matter in electromagnetical and gravitational fields.
		While time and space appear somehow "on equal rights" in the Lorentz transformation equations, this is not the case within the formalism of quantum mechanics.
		John G. Cramer: "Space Drives": A collection of articles published in Analog, amongst a well-done discussion of Miguel Alcubierre's paper on the warp drive and as a followup the Krasnikov tube: a subway to the stars.
	homepage.sunrise.ch /homepage/schatzer/space-time.html (4215 words)

	Hermann Minkowski
		Since the [relativity] postulate comes to mean that only the four-dimensional world in space and time is given by phenomena, but that the projection in space and in time may still be undertaken with a certain degree of freedom, I prefer to call it the postulate of the absolute world.
		It seems Minkowski believed the theory of relativity implied that our world and all objects are four-dimensional (4D) since he introduced the unification of space and time into an indivisible 4D entity (which he called "the World") in a rather substantival manner:
		Minkowski, Hermann (1952) "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 /~scol/seminars/conference/minkowski.html (923 words)

	In memory of Nikolay Alexandrovich Kozyrev who saw in time the vital basis of the Universe (Site not responding. Last check: 2007-11-06)
		The vector length notion is not intrinsic in the theory of Minkowski space because the root extracting operation is not used in a vector space.
		Space and time form a unified four-dimensional substance; it is endowed with Minkowski space geometry and possesses certain physical properties due to which it interacts with matter, physical fields and processes occurring in it.
		An orientation of space may be introduced either by a direct choice of the class of bases, or by setting a mirror asymmetric geometric figure, i.e., by putting into correspondence the classes of oppositely oriented bases to the figure and to its enantiomorphous modification by a certain rule.
	www.chronos.msu.ru /EREPORTS/shikhobalov_what/shikhobalov_what.htm (15135 words)

	2.4 Example: Minkowski space (Site not responding. Last check: 2007-11-06)
		In the case of Minkowski space-time, the metric can be extended in a regular way to three points representing future and past time-like infinity and space-like infinity, but this is not generally so.
		These are surfaces of constant Minkowski time, which implies that the signal is again a pure sine wave.
		Even in Minkowski space-time we could choose space-like hypersurfaces that are not surfaces of constant Minkowski time but which nonetheless are asymptotically Euclidean.
	relativity.livingreviews.org /Articles/lrr-2004-1/articlesu4.html (1381 words)

	June Lester- mathematical publications
		Geometrically interesting in their own right (as Euclidean n-space or Minkowski spacetime, for example), they are also invaluable as coordinate spaces: it's quite extraordinary just how many classical geometries can be coordinatized by n-tuples subject to some indefinite scalar product.
		And looking at these geometries through their coordinate spaces often makes obvious the isomorphisms between different models of the same geometry, or even between different geometries: the same coordinate space implies the same or related geometries.
		The Beckman-Quarles Theorem in Minkowski Space for a Spacelike Square-Distance.
	www.cecm.sfu.ca /~jalester/WebCV/publications.html (921 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Points in Minkowski space are given in homogeneous coordi- nates.
		A frame in hyperbolic n-1 space is represented by an expression of the form Minkowski[Frame[{x1,...
		The isometry is given by an orthogonal n by n transformation matrix for Minkowski space.
	library.wolfram.com /infocenter/MathSource/1872/hypintro.txt (620 words)

	Publisher description for Library of Congress control number 95046491 (Site not responding. Last check: 2007-11-06)
		Minkowski geometry is a type of non-Euclidean geometry in a finite number of dimensions in which distance is not 'uniform' in all directions.
		This is followed by a treatment of two-dimensional spaces and characterisations of Euclidean space among normed spaces.
		The central three chapters present the theory of area and volume in normed spaces, a fascinating geometrical interplay among the various roles of the ball in Euclidean space.
	www.loc.gov /catdir/description/cam027/95046491.html (194 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)

	TGD_spacetime (Site not responding. Last check: 2007-11-06)
		This means that the symmetries of the empty Minkowski space are lost as are lost also the corresponding conservation laws, in particular the conservation of energy.
		The generalized imbedding space results when the real H and all p-adic versions H_p of the imbedding space are glued together along rational points.
		Generalized imbedding space is union of all p-adic imbedding spaces H_p and real imbedding space H intersecting along rational points common to all (Q denotes rationals, R for reals, and R_p for p-adic numbers).
	www.emergentmind.org /tgdillu/illua.html (3405 words)

	Citebase - An alternative to Minkowski space-time
		Authors: Almeida, Jose B. The starting point of this work is the principle that all movement of particles and photons must follow geodesics of a 4-dimensional space where time intervals are always a measure on geodesic arc lengths.
		The last part of the presentation is dedicated to electromagnetic interaction and Maxwell's equations, showing that there is a particular solution where one of the space dimensions is eliminated and the geodesics become equivalent to light rays in geometrical optics.
		The paper is based on the recently proposed 4-dimensional optical space theory and draws some of its consequences for gravitation.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:gr-qc/0104029 (969 words)

	Learn more about Spacetime in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Hint: Play with putting spaces before and after your words to see the different results you get.
		In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional manifold called spacetime (alternatively, space-time; see below).
		The central lesson of general relativity is that spacetime cannot be a fixed background, but is rather a network of evolving relationships.
	www.onlineencyclopedia.org /s/sp/spacetime.html (714 words)

	Minkowski, Hermann -- Encyclopædia Britannica
		His idea of combining the three dimensions of physical space with that of time into a four-dimensional “Minkowski space”—space-time—laid the mathematical foundations for Albert Einstein's special...
		Based on Euclidean geometry, the Cartesian coordinate system is designed to identify any point (event) in space by its reference to three mutually perpendicular lines or axes meeting at an...
		Since the classic interpretation of Einstein's special theory of relativity by Hermann Minkowski, a Lithuanian-German mathematician, it has been clear that physics has to do not with two entities, space and time, taken separately, but with a unitary entity space–time, in which, however, timelike and spacelike directions can be distinguished.
	www.britannica.com /eb/article-9052860 (716 words)

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