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Topic: Schwarzschild solution

	Deriving the Schwarzschild solution - Wikipedia, the free encyclopedia
		The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see general relativity).
		The solution is assumed to be spherically symmetric, static and vacuum.
		In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static.
	en.wikipedia.org /wiki/Deriving_the_Schwarzschild_solution (689 words)

	Schwarzschild metric - Wikipedia, the free encyclopedia
		In Einstein's theory of general relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the gravitational field outside a spherical, non-rotating mass such as a (non-rotating) star, planet, or fl hole.
		The Schwarzschild solution is named in honour of its discoverer Karl Schwarzschild who found the solution in 1916, only a few months after the publication of Einstein's theory of general relativity.
		The Schwarzschild fl hole is characterized by a surrounding area, called the event horizon which is situated at the Schwarzschild radius, often called the radius of a fl hole.
	en.wikipedia.org /wiki/Schwarzschild_solution (1391 words)

	Ernie's Web Pages - Black Holes
		Karl Schwarzschild saw an early paper on the theory and (while he was dying in an army hospital from a skin infection he had picked up in the trenches during World War I) solved Einstein's equation for a spherical mass.
		The explanation was that the solution was "incomplete," that is it did not contain the whole space.
		The solution was one that modeled, in a very simple way, a star made out of an ideal fluid.
	home.comcast.net /~ernie1001/blackhole.htm (1028 words)

	Schwarzschild Modern Relativity modernrelativity special general black hole mass energy Einstein wormhole time modern ...
		The Schwarzschild solution is the solution for the case of a non-rotating uncharged fl hole.
		The photon sphere of a Schwarzschild fl hole is a sphere given by a radius at which a photon can make a complete closed orbit around the fl hole.
		The second is the correct solution for the photon sphere.
	www.geocities.com /zcphysicsms/chap10.htm (1869 words)

	S
		The Schwarzschild solution is of practical importance as the outlying regions of the corresponding model universe describe the space-time distortion around all kinds of objects that are spherically symmetric, or nearly so, such as the sun or the earth (cf.
		The Schwarzschild radius for an object the mass of the earth is 9 millimeters, for an object with the mass of the sun, 2.95 kilometers.
		In the context of general relativity: a solution or, more precisely, a solution of the Einstein equations is a model universe that follows the law of gravity as prescribed by general relativity.
	www.einstein-online.info /en/navMeta/dictionary/s/index.html (3087 words)

	Schwarzschild Black Hole -- from Eric Weisstein's World of Physics
		The exterior solution for such a fl hole is known as the Schwarzschild solution (or Schwarzschild metric), and is an exact unique solution to the Einstein field equations of general relativity for the general static isotropic metric (i.e., the most general metric tensor that can represent a static isotropic gravitational field),
		An exact solution turns out to also be possible for a spherical body with constant density; see Schwarzschild fl hole--constant density.
		Weinberg, S. "The Schwarzschild Solution." §8.2 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity.
	scienceworld.wolfram.com /physics/SchwarzschildBlackHole.html (372 words)

	Joel Smoller \| Research - General Relativistic Fluids
		In [26] we construct the simplest solution of the Einstein equations that incorporates a shock-wave into a standard Friedmann-Robertson-Walker metric whose equation of state accounts for the Hubble constant and the cosmic microwave background temperature.
		This produces a new solution of the Einstein equations from which we are able to derive estimates for the shock position at present time.
		In [11] we derive closed form expressions for solutions to our ode's, when the TOV solution is the Schwarzschild solution, and also for arbitrary TOV metrics when the FRW metric is restricted to the case of "critical expansion": (k=0).
	www.math.lsa.umich.edu /~smoller/research/research_grf.html (1203 words)

	Event Horizon: Conceiving Black Holes
		Briefly following the release of Einstein's General Relativity, Karl Schwarzschild, a German physicist, discovered a mathematical solution to Einstein's field equations that described the gravitational field of a point mass while fighting for the German army in World War I. Schwarzschild died several months later from a rare disease contracted during the war.
		This solution, known as Schwarzschild Geometry, describes the space and time around any spherical mass including the distance from the center of a sphere at which light cannot escape.
		Kerr's solution was later proven to actually describe the space and time around spinning fl holes and all fl holes that exist in nature.
	library.thinkquest.org /25715/discovery/conceiving.htm (813 words)

	Tidal Forces in a Schwarzschild Spacetime
		The Schwarzschild solution is one of the best-known exact solutions of the Einstein equations and was derived a few months after the theory was proposed.
		Impose the constraints that the metric is spherically symmetric and static (i.e.
		Note that despite your intuition and the familiar concept of ``mass'', the Schwarzschild metric is a solution of the Einstein equations in vacuum, i.e.
	www.sissa.it /~rezzolla/lnotes/virgo/node6.html (460 words)

	FAQ to SCI.PHYSICS on Black Holes by Matt McIrvin
		What such a solution really looks like is a "metric," which is a kind of generalization of the Pythagorean formula that gives the length of a line segment in the plane.
		Schwarzschild expressed his metric in terms of coordinates which, at large distances from the object, resembled spherical coordinates with an extra coordinate t for time.
		Therefore the fatal r goes as the cube root of the mass, whereas the Schwarzschild radius of the fl hole is proportional to the mass.
	antwrp.gsfc.nasa.gov /htmltest/gifcity/bh_pub_faq.html (3321 words)

	EST ( Expanding Spacetime Theory )
		Schwarzschild's [Schwarzschild, 1916] solution to Einstein's field equations assumes that all components of the energy-momentum tensor for vacuum is equal to zero.
		Schwarzschild's external solution crucially depends on the assumption that the vacuum energy density is zero.
		I derive a spherically symmetric solution to Einstein's General Relativity relations assuming that the energy-momentum tensor for vacuum is as proposed by the [Masreliez, 1999].
	www.estfound.org /gravitation.htm (2764 words)

	Sources in Motion
		The preceding sections focused on the spherically symmetrical solution of Einstein's field equations represented by the Schwarzschild geometry, since nearly all of the experimentally observable effects of general relativity are actually just tests of Schwarzschild geometry combined with the geodesic hypothesis.
		Part of the reason that people such as Einstein have occasionally doubted the reality of the wave solutions is that all gravitational waves imply a singularity (as does the Schwarzschild solution), albeit "merely" a coordinate singularity.
		It might seem as though there ought to be spherically symmetrical "pulsating" solutions that radiate gravitational waves, but this is not the case, as is clear from Birkhoff's proof that the Schwarzschild solution is the unique (up to transformation of coordinates) spherically symmetrical solution of the field equations, even without the "static" assumption.
	www.mathpages.com /rr/s6-08/6-08.htm (1978 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		Recall that solutions to Maxwell's equation can be classified according to the nature of the eigenvectors of the EM field tensor (the eigenvectors are always null vectors).
		The Petrov classification of vacuum solutions to Einstein's field equation is the analogue of the classification of vacuum solutions to Maxwell's field equation according to whether or not they represent pure radiation fields.
		Recall that for a spherically symmetric solution, "space" should be filled with nested euclidean spheres, and a static solution can be written in coordinates for which no time dependence appears in the metric.
	math.ucr.edu /home/baez/PUB/exactsolutions (6943 words)

	Schwarzschild solution
		This solution is the ``corner stone'' of GR.
		Because the Newton's gravitational law is recovered, it is not a surprise that investigation of motion along geodesics in Schwarzschild space-time gives the known planetary motion in the ``0'th order''.
		Thus, this solution describes a ``fl hole'': not even light (and therefore any information) can get out from ``under the horizon''; the only way to observe it is via some indirect gravitational effects.
	www.pha.jhu.edu /~belyaev/holes/node4.html (501 words)

	Imposing a 4-Dimensional Background on General Relativity
		For example, the temporal coordinate of the Schwarzschild Solution is a background time: It is based on the observations of a distant observer and its intervals can differ from those of the local clock.
		As shown later, the background spacetimes for curved EFE solutions usually are not flat (as required by Postulate 1).
		The Schwarzschild Solution for the external vacuum spacetime surrounding a spherically symmetric non-rotating massive object is[101984Wald,111994Ohanian and Ruffini]:
	mysite.verizon.net /ems57fcva/Distortion/article_full.html (1553 words)

	Black Holes
		In the Schwarzschild solution there is always a natural lower limit on the radius coordinate for a star.
		In Kruskal coordinates both the r and the t coordinates of the Schwarzschild solution are transformed in a transcendental way as follows.
		The Kruskal solution is called the Maximal Extension solution since it takes in regions of the manifold that are not covered by other coordinate systems.
	scholar.uwinnipeg.ca /courses/38/4500.6-001/Cosmology/Black_Holes.htm (968 words)

	Strings and Quantum Gravity
		62] who observed that one may obtain a Schwarzschild solution to the non-linear Einstein equations (with a point source of mass m) as two different expansions.
		One expansion is valid in the large-r regime and is obtained by linearizing the field equations; the other expansion (leading to the Schwarzschild solution) is valid in the small-r regime.
		Middleton tackled the problem of a point souce in the DGP model (Schwarzschild solution) which seems to be considerably more complicated than a cosmological solution.
	aesop.phys.utk.edu /research/node16.html (293 words)

	Black Holes Made Simple
		Shortly after Einstein developed the general theory of relativity, Karl Schwarzschild found a solution for the equations of general relativity in empty space.
		In fact, one of the first tests of general relativity was the prediction of the motion of Mercury using the Schwarzschild solution.
		The Schwarzschild solution has a problem at a certain distance, called the Schwarzschild radius or the Event Horizion, at which light cannot escape.
	www.geocities.com /autotheist/Physics/bh.htm (918 words)

	riding4
		This is the solution for a distribution of matter that is a nonrotating, nonchanging sphere of matter.
		The Schwarzschild radius is only a feature of the comparison between the point of view that we have far from the star and the point of view near the star.
		Other solutions of Einstien's equations, though, do offer the possibility of creating spacetime singularities not enclosed by an event horizon (the event horizon is the generalization of the Schwarzschild radius to these other solutions).
	home.pacbell.net /stevepur/physics/riding/riding.4.html (2664 words)

	Schwarzschild's solution to Einstein's field equations (Site not responding. Last check: 2007-11-04)
		This was the first solution to the field equations, and it's the simplest.
		Schwarzschild chose to look for spherically symmetric solutions for the sake of simplicity, so we naturally deal with one time co-ordinate, c.t, one spatial radial co-ordinate, r, and two spatial angular co-ordinates, n and m (measured in radians).
		Thus Schwarzschild's solution takes E = 1 and h = 0, so that Ricci(D) is zero; and interprets k as −2.G.m/c/c for a mass m, since the resulting metric then matches up with the weak field limit obtained by approximating Newtonian gravitation.
	www.chaos.org.uk /~eddy/physics/Swarzchild.html (1266 words)

	Schwarzschild solution
		The Scwarzschild solution works inside as well as outside a body, in the sense that it satisfies the field equations rigorously.
		Its not called the Schwarzschild solution, but the derivation of the weak field solution for steller interiors is touched on in MTW's Gravitation.
		The solution is consistent with an ideal gass.
	www.physicsforums.com /showthread.php?p=286333 (551 words)

	White Holes and Wormholes
		The complete Schwarzschild geometry consists of a fl hole, a white hole, and two Universes connected at their horizons by a wormhole.
		The trapped region between the two horizons is the Schwarzschild bubble encountered on the trip into the fl hole.
		We are at 0.35 Schwarzschild radii from the central singularity.
	casa.colorado.edu /~ajsh/schww.html (1197 words)

	Schwarzschild Solution and Friends [Was: Schwartzchild solution]
		I think Ilja is talking about matching the Schwarzschild static spherically symmetric exterior vacuum to a static spherically symmetric perfect fluid across the surface r = r0 of the fluid ball.
		The simplest example of this kind of solution is due to Schwarzschild himself, who matched an his "incompressible fluid ball" across its surface (sphere of zero pressure) to a Schwarzschild vacuum exterior.
		By the way, there are also exact solutions representing Schwarzschild or Reissner-Nordstrom objects sitting in a cylindricaly symmetric stationary tube of magnetic flux or in a static spherically symmetric perfect fluid (the McVittie fluid).
	www.lns.cornell.edu /spr/2001-03/msg0031563.html (624 words)

	Scalar field solution
		Discussion of any solution in GR starts with adopting some coordinate system (amongst infinite number of possible ones) which has the clearest physical interpretation.
		It is analogous to ``central singularity'' in Schwarzschild, but it is not covered by a horizon and can be ``seen'' by outside observers.
		In fact, it is indistinguishable from Schwarzschild BH for the present day astrophysics and says that what is observed and interpreted as a BH may in reality be very different from it in the strong gravity region.
	www.pha.jhu.edu /~belyaev/holes/node8.html (532 words)

	Not-So-Cosmic Censorship and Black Holes
		Equivalents of the Schwarzschild metric that are static and that do not exhibit fl holes are presented for the first time and discussed.
		An exact solution for this class of problems was found years ago by Schwarzschild.
		Presented here are static solutions to Einstein’s field equations for an uncharged, non-rotating, spherically symmetric mass – the problem treated by Schwarzschild many years ago – that exemplify this present-day counter-productive development.
	www.angelfire.com /empire/intensity/brunstein.htm (2782 words)

	The Birth of Wormholes
		In general relativity, a point mass curves spacetime around it in a way that was calculated by Karl Schwarzschild in 1916 [1].
		The Schwarzschild solution has mathematical singularities both at zero and at the so-called Schwarzschild radius.
		Viewed from afar, either part of this solution represents the gravitational effect of a mass because spacetime is strongly curved, but no physical body is present.
	focus.aps.org /story/v15/st11 (771 words)

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