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Topic: Curvature of space

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	Curvature - Wikipedia, the free encyclopedia
		For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center.
		Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a cylinder and a plane are locally isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection.
	en.wikipedia.org /wiki/Curvature (930 words)

	Curvature -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		The magnitude of curvature at points on physical curves can be measured in (A unit of measurement of the refractive power of a lens which is equal to the reciprocal of the focal length measured in meters) diopters (also spelled dioptre); a diopter has the dimension one per meter.
		Unlike Gauss curvature, the mean curvature depends on the embedding, for instance, a (A surface generated by rotating a parallel line around a fixed line) cylinder and a plane are locally (A line connecting isometric points) isometric but the mean curvature of a plane is zero while that of a cylinder is nonzero.
		A space without curvature is called a "flat space" or (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space.
	www.absoluteastronomy.com /encyclopedia/c/cu/curvature.htm (1037 words)

	Spacetime - Wikipedia, the free encyclopedia
		Treating space and time on the same footing and as two aspects of a unified whole was devised by Hermann Minkowski shortly after the theory of special relativity was developed by Albert Einstein.
		This unification is further exemplified by the common practice of expressing time in the same units as space by multiplying time measurements by the speed of light.
		In general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the Riemann tensor.
	en.wikipedia.org /wiki/Space-time (1708 words)

	PlanetMath: curvature (space curve)
		be a parameterized space curve, assumed to be regular and free of points of inflection.
		Physically, curvature may be conceived as the ratio of the normal acceleration of a particle to the particle's speed.
		This is version 7 of curvature (space curve), born on 2002-02-02, modified 2005-11-02.
	planetmath.org /encyclopedia/CurvatureOfACurve.html (258 words)

	Geometry (Site not responding. Last check: 2007-10-21)
		of space is equivalent to gravity and is a measure of the amount of matter in the universe.
		In practice, the curvature is measured by the size of the spots in the microwave background maps.
		By measuring the curvature of the universe in the size of spots in the microwave background temperature maps, we determine the ultimate fate of the universe.
	background.uchicago.edu /~whu/beginners/curvature.html (463 words)

	Space
		Space is that which separates bodies of matter, whether these bodies be atoms, galaxies, or any component part of either.
		Since the curvature of Space is considered to be determined by the amount of density of the matter present in it, at that time the Universe was very small.
		If the radius of Space were increased, it is obvious that the matter did not follow the curvature of Space, but, actually moved perpendicularly to it (or perhaps at a tangent).
	www.timestar.org /space.htm (1388 words)

	Space Curvature Reduction - Bad Astronomy and Universe Today Forum
		What causes the curvature is another question but mass and time and gravity which are inter-related in a such marvelous way are the leading candidates in modern physics.
		This is the state of the curvature of space at the time of the Big Bang.
		tan a is the measure of curvature of the space occupied by the universe.
	www.bautforum.com /showthread.php?t=25384 (1445 words)

	Curvature - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-21)
		Curvature is the amount by which an geometric object deviates from being flat.
		The magnitude of curvature at points on physical curves can be measured in diopterss (alternative spelling: dioptre); a diopter is one per meter.
		This last formula also gives the mean curvature of an hypersurface in Euclidean space.
	encyclopedia.learnthis.info /c/cu/curvature.html (781 words)

	Lecture Notes for Day 14
		Space is curved near a massive object; the greater the mass the greater the curvature.
		Another test for the curvature of space is simply the sum of the interior angles of a triangle.
		Because of the curvature of space, which is always curved in the direction of the mass, (concave toward the mass) it is difficult to escape from a mass: space is always curving you back toward it.
	www.colorado.edu /APAS/APAS1010/lecture/day.14.html (829 words)

	PhysOrgForum Science, Physics and Technology Discussion Forums -> No ether - no curvature of space!
		The curvature of space, as I understand it, is the effect gravity has on objects.
		Space warps from massive gravitational pulls, I assume this warping of space is this "curvature".
		Space may well be curved and the idea of curved space has absolutly nothing to do with ether.
	forum.physorg.com /index.php?showtopic=2222&view=getlastpost (2055 words)

	Direction of space curvature - Physics Help and Math Help - Physics Forums
		The first one describes how the geodesics in space move with respecto to each other and the second one describes the rate of change of the tangent vector to the geodesics in space with respect to the normal vector to the spatial 'hypersurface'.
		It is possible to visualize intrinsic curvature as a surface imbeded as a higher dimensional space, like the Earth's 2d surface being the surface of a 3d object.
		The curvature of space-time, though (as opposed to the curvature of space) is easily measured.
	www.physicsforums.com /showthread.php?threadid=55955 (1455 words)

	Papers on Morphological Scale-Spaces
		Properties of this scale space pattern are investigated and contrasted with those of Gaussian scale space.
		The notion of curvature of planar curves has emerged as one of the most powerful for the representation and interpretation of objects in an image.
		Although curvature extraction from a digitized object contour would seem to be a rather simple task, few methods exist that are at the same time easy to implement, fast, and reliable in the presence of noise.
	www.lems.brown.edu /vision/people/leymarie/Refs/CompVision/ScaleSpace/Morphological.html (1112 words)

	Space and Time
		In the early universe the Weyl curvature was probably zero, but in a dying universe the large number of fl holes, Penrose argues, will give rise to a high Weyl curvature.
		PHASE SPACE A phase-space diagram is a mathematical volume of many dimensions formed when coordinate axes are assigned to each of the distance and momentum values of each particle.
		The Weyl tensor is that part of the curvature of space-time that is not locally determined by the matter through the Einstein equations.
	members.fortunecity.com /templarser/space.html (4186 words)

	sciforums.com - Gravity/Space Curvature
		The degree of space curvature is directly linked to the force of gravity.
		But, given the position of the 'anchor' body in relation to other bodies in space, the geodesic speed of the passing or orbiting body can be determined.
		The curvature of spacetime varies as the energy and matter within it moves around.
	www.sciforums.com /showthread.php?t=17238 (2347 words)

	Rudy Rucker's KappaTau Space Curve Paper (Site not responding. Last check: 2007-10-21)
		The size of the curvature is determined by the principle that a circle of radius R should have curvature of 1/R. The smaller the radius, the greater the curvature.
		The curvature of a space curve is essentially the same as the curvature of a plane curve: it measures how rapidly the curve is bending to one side.
		The sizes of the curvature and torsion on a helix with radius R and turn-height H are given by two nice equations.
	www.mathcs.sjsu.edu /faculty/rucker/kaptaudoc/ktpaper.htm (3020 words)

	Gravity & Curvature of Space - Bad Astronomy and Universe Today Forum
		If gravity bends or curves space, then it stands to reason that on the surface of this planet we are standing in curved space.
		But now plot the arcs in a mathematical four dimensional space, where the coordinates are x, y, z, and ct (t time of flight multiplied by the speed of light c).
		It's not space that is curved, it's spacetime.
	www.bautforum.com /showthread.php?t=2178 (1897 words)

	Does space curve?
		Moreover, “space-time” is a mathematical concept, which amounts to a fancy way of referring to proper time in relativity (the time kept by perfect clocks), and does not involve any curvature of space.
		Yet the fact that space does not curve just because "space-time" does would come as a surprise to many students of the history of relativity, who tend to think as Riemann did.
		If the curved path of a body through space is not caused by a curvature of space, then space remains Euclidean (flat) and an external force is still required to produce and explain any deviation from straight line motion.
	www.metaresearch.org /cosmology/gravity/spacetime.asp (970 words)

	Flames (Site not responding. Last check: 2007-10-21)
		Flames and sparks decrease the curvature of the Universe and increase its entropy.
		As in the "Quantum Space" text, if space is pictured as a quantum foam grid-like structure held together largely by electromagnetic forces, a flame might be a physical separation in the electromagnetic "glue" that holds the grid structure together.
		Therefore, matter (as a space-time curvature) and the forces of electricity in the form of sparks (space-time "cracks") are closely-related phenomena in the fifth dimension--the Higgs fields.
	www.johnkharms.com /flames.htm (5023 words)

	sciforums.com - curvature of space-time
		This is giving me problems because in a 2d model the curvature seems to be a fuction of the distance from the center of the gravitational plane, in a more complex model you need to represent the curvature at a distance perpendicular to the gravitational plane.
		The function for curvature over distance is no longer valid as the curvature tends to decrease with an increase in perpendicular distance for any fixed distance from the center in the gravitational plane.
		Space-time is a 4 dimensional vactor space, and unless you can visualise 4 dimensions(which no one can), then you are confined to construct at most 3d analogies.
	www.sciforums.com /showthread.php?t=22950 (199 words)

	Space/Time Continuum - a thesis
		The curvature occurs as a result of the influence of mass against movement in time.
		Einstein stated that space is finite but time is infinite.
		A perceived fixed object (a stone lying on the ground, a building, a mountain) does move as a consequence of movements through space (Earth orbital, geologic, etc.) and the force moving the object through time.
	www.west.net /~ke6jqp/spacetime/spacetime.html (1057 words)

	Tuning In To The Curvature Of Space-Time
		Scientists have uncovered sets of oscillating X-ray signals from three neutron stars that may tell the story of the bending of the very fabric of space around these objects, broadcast to us from the stars much like the details of a science talk show buried within oscillating AM radio waves.
		The signals that the Rossi Explorer is capturing are high-frequency oscillations in the X-ray emission, likely produced by clumps of gas in the accretion disk that are hurtling around the neutron star just above its surface at nearly the speed of light.
		The Rossi X-ray Timing Explorer is operated from Goddard Space Flight Center in Greenbelt, Md. Launched in 1995, the spacecraft was developed by Goddard with Massachusetts Institute of Technology and University of California, San Diego.
	www.spacedaily.com /news/gravity-00c.html (1185 words)

	Hydrophobic compaction, curvature of space and deciphering protein sequences
		In 2D space, the ideal, uniform curvature of this line is the circle (in which the two ends of the chain – canonically called N and C – touch).
		This strategy thus makes use of this curvature of space of sequences and also takes into account a suitable separator of information [the "connectivity distance", equal to the number of amino acids, for a helix of given pitch, separating amino acid i from its farthest near neighbor j (e.g.
		Proteins are compact, and the first HCA curvature of space of the sequence, which exploits the intrinsic properties of helical compaction (e.g.
	www.europhysicsnews.com /full/19/article4/article4.html (3308 words)

	Curvature
		But when it comes to us, and our curved space, we cannot see it since this would entail our standing outside space and looking down on it.
		The curvature of space is real and is generated by the mass of the bodies in it.
		Correspondingly the curvature of space determines the trajectories of all bodies moving in it.
	phyun5.ucr.edu /~wudka/Physics7/Notes_www/node94.html (383 words)

	Curious About Astronomy: If gravity is a "curvature of space" rather than a force, why do a ball and bullet follow ...
		If gravity can be defined as the curvature of space rather than a force of attraction, why does not a bullet shot out of a gun, say perpendicular to the Earth's crust, and a ball thrown by me on the same trajectory (but obviously at a much slower speed) follow the same curve?
		Einstein's idea (discussed further on our relativity page) was that there is no such thing as a "force" of gravity which pulls things to the Earth; rather, the curved paths that falling objects appear to take are an illusion brought on by our inability to perceive the underlying curvature of the space we live in.
		So if we agree that space-time, and not space, is the proper arena in which to consider the question, then we can understand why the ball and the bullet don't wind up in the same place at the end of their trip.
	curious.astro.cornell.edu /question.php?number=649 (1039 words)

	Dynamic theory of gravity Article, Dynamictheorygravity Information (Site not responding. Last check: 2007-10-21)
		Since action and reaction are coexistent, it followsthat the supposed curvature of space is entirely impossible - But even if it existed it would not explain the motions of thebodies as observed.
		This theory is a logicalextension of the rotating magnetic field model.Dynamic theory of gravity does not include the "curvature of space".
		The theory claims four dimensions (x, y, z, t) andthe field of force is not commutative.Einstein's theory of general relativity claims that space itself is curved as well as claiming that space-time is curved(the two according to Einstein are equivalent).
	www.anoca.org /tesla/space/dynamic_theory_of_gravity.html (1257 words)

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