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Encyclopedia > Schwarzschild metric
It has been suggested that Deriving the Schwarzschild solution be merged into this article or section. (Discuss)

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 black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or Sun. According to Birkhoff's theorem, the Schwarzschild solution is the most general static, spherically symmetric, vacuum solution of Einstein's field equations. A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole has a Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass. To meet Wikipedias quality standards, this article may require cleanup. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... The gravitational field is a field that causes bodies with mass to attract each other. ... The Pleiades star cluster A star is a massive body of plasma in outer space that is currently producing or has produced energy through nuclear fusion. ... A planet is generally considered to be a relatively large mass of accreted matter in orbit around a star that is not a star itself. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... Earth is the third planet from the Sun. ... The Sun is the spectral type fonduemeister star at the center of Earths solar system. ... In general relativity, Birkhoffs theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. ... In general relativity, a spacetime is said to be static if it admits a global, nowhere zero, timelike hypersurface orthogonal Killing vector field. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... For other topics related to Einstein see Einstein (disambig) Introduction In physics, the Einstein field equation or Einstein equation is a tensor equation in the Einsteins theory of general relativity. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ... Charge is a word with many different meanings. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ...

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. It was the first exact solution of Einstein's field equations (besides the trivial flat space solution). Schwarzschild had little time to think about his solution. He died shortly after his paper was published, as a result of a disease he contracted while serving in the German army during World War I. Karl Schwarzschild (October 9, 1873 - May 11, 1916) was a noted German Jewish physicist and astronomer, father of astrophysicist Martin Schwarzschild. ... 1916 (MCMXVI) is a leap year starting on Saturday (link will take you to calendar) // Events January-February January 1 - The Royal Army Medical Corps first successful blood transfusion using blood that had been stored and cooled. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Combatants Allies: â€¢ Serbia, â€¢ Russia, â€¢ France, â€¢ Romania, â€¢ Belgium, â€¢ British Empire and Dominions, â€¢ United States, â€¢ Italy, â€¢ ...and others Central Powers: â€¢ Germany, â€¢ Austria-Hungary, â€¢ Ottoman Empire, â€¢ Bulgaria Casualties Military dead: 5 million Civilian dead: 3 million Total: 8 million Full list Military dead: 3 million Civilian dead: 3 million Total: 6 million Full...

The Schwarzschild black hole is characterized by a surrounding area, called the event horizon which is situated at the Schwarzschild radius, often called the radius of a black hole. Any non-rotating and non-charged mass that is smaller than the Schwarzschild radius forms a black hole. The solution of the Einstein field equations is valid for any mass M, so in principle (according to general relativity theory) a Schwarzschild black hole of any mass could exist if nature is kind enough to form one. Event Horizon is a 1997 science fiction and horror film. ... The Schwarzschild radius (sometimes inappropriately referred to as the gravitational radius[1]) is a characteristic radius associated with every mass. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ...

In Schwarzschild coordinates, the Schwarzschild metric can be put into the form (see deriving the Schwarzschild solution) In General Relativity, Schwarzschild coordinates refers to the coordinate system of the Schwarzschild metric. ... The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see general relativity). ...

$ds^{2} = -c^2 left(1-frac{2GM}{c^2 r} right) dt^2 + left(1-frac{2GM}{c^2 r}right)^{-1}dr^2+ r^2 dOmega^2$

where G is the gravitational constant, M is interpreted as the mass of the gravitating object, and According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Mass is a property of a physical object that quantifies the amount of matter it contains. ...

$dOmega^2 = dtheta^2+sin^2theta dphi^2,$

is the standard metric on the 2-sphere (i.e. the standard element of solid angle). The constant For other uses, see sphere (disambiguation). ... A solid angle is the three dimensional analog of the ordinary angle. ...

$r_s = frac{2GM}{c^2}$

is called the Schwarzschild radius and plays an important role in the Schwarzschild solution. The Schwarzschild radius (sometimes inappropriately referred to as the gravitational radius[1]) is a characteristic radius associated with every mass. ...

The Schwarzschild metric is a solution to vacuum field equations, meaning that it is only valid outside the gravitating body. That is, for a spherical body of radius R the solution is valid for r > R. (Although, if R is less than the Schwarzschild radius rs then the solution describes a black hole; see below.) In order to describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at r = R. In physics, the Einstein field equation or Einstein equation is a differential equation in Einsteins theory of general relativity. ... A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ...

Note that as $Mto 0$ or $r rightarrowinfty$ one recovers the Minkowski metric: In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

$ds^{2} = -c^2dt^2 + dr^2 + r^2 dOmega^2.,$

Intuitively, this makes sense, as far away from any gravitating bodies we expect space to be nearly flat. Metrics with this property are called asymptotically flat. An asymptotically flat spacetime is a spacetime in which the geometry approaches that of Minkowski space at large distances from the source or sources of gravity. ...

## Singularities and black holes

The Schwarzschild solution appears to have singularities at r = 0 and r = rs; some of the metric components blow up at these radii. Since the Schwarzschild metric is only expected to be valid for radii larger than the radius R of the gravitating body, there is no problem as long as R > rs. For ordinary stars and planets this is always the case. For example, the radius of the Sun is approximately 700,000 km, while its Schwarzschild radius is only 3 km. In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ... The Sun is the spectral type fonduemeister star at the center of Earths solar system. ...

One might naturally wonder what happens when the radius R becomes less than or equal to the Schwarzschild radius rs. It turns out that the Schwarzschild solution still makes sense in this case, although it has some rather odd properties. The apparent singularity at r = rs is an illusion; it is an example of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates. By choosing another set of suitable coordinates one can show that the metric is well-defined at the Schwarzschild radius. See, for example, Eddington-Finkelstein coordinates or Kruskal coordinates. In general relativity, Eddington-Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry which are adapted to radial null geodesics (i. ... In general relativity, Kruskal-Szekeres coordinates are a coordinate system for a Schwarzschild geometry. ...

This case r = 0 is different, however. If one asks that the solution be valid for all r one runs into a true physical singularity, or gravitational singularity, at the origin. To see that this is a true singularity one must look at quantities that are independent of the choice of coordinates. One such important quantity is the Kretschmann invariant, which is given by This article is in need of attention from an expert on the subject. ... In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. ...

$R^{abcd}R_{abcd}= frac{12 r_s^2}{r^6}$

At r = 0 the curvature blows-up (becomes infinite) indicating the presence of a singularity. At this point the metric, and space-time itself, is no longer well-defined. For a long time it was thought that such a solution was non-physical. However, a greater understanding of general relativity led to the realization that such singularities were a generic feature of the theory and not just an exotic special case. Such solutions are now believed to exist and are termed black holes. A black hole is a concentration of mass great enough that the force of gravity prevents anything past its event horizon from escaping it except through quantum tunnelling behaviour (known as Hawking Radiation). ...

The Schwarzschild solution, taken to be valid for all r > 0, is called a Schwarzschild black hole. It is a perfectly valid solution of the Einstein field equations, although it has some rather bizarre properties. For r < rs the Schwarzschild radial coordinate r becomes timelike and the time coordinate t becomes spacelike. A curve at constant r is no longer a possible worldline of a particle or observer, not even if a force is exerted to try to keep it there; this occurs not just because the gravitational field is strong but because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone) points into the singularity. The surface r = rs demarcates what is called the event horizon of the black hole. It represents the point past which light can no longer escape the gravitational field. Any physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole. A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... Event Horizon is a 1997 science fiction and horror film. ... This article or section does not cite its references or sources. ...

## Embedding Schwarzschild space in Euclidean space

In general relativity mass changes the geometry of space. Space with mass is "curved", whereas empty space is flat (Euclidean). In some cases we can visualize the deviation from Euclidean geometry by mapping a 'curved' subspace of the 4-dimensional spacetime onto a Euclidean space with one dimension more.

Suppose we choose the equatorial plane of a star, at a constant Schwarzschild time t = t0 and θ = π / 2 and map this into three dimensions with the Euclidean metric (in cylindrical coordinates): This article describes some of the common coordinate systems that appear in elementary mathematics. ...

$ds^2 = dz^2 + dr^2 + r^2dphi^2.,$

We will get a curved surface z = z(r) by writing the Euclidean metric in the form

$ds^2 = left(1 + left(frac{dz}{dr}right)^2 right)dr^2 + r^2dphi^2$

where we have made the identification

$dz = frac{dz}{dr}dr.$

We can then relate this to the Schwarzschild metric for the equatorial plane at a fixed time:

$ds^2 = left(1-frac{2GM}{c^2 r} right)^{-1} dr^2 + r^2dphi^2$

Which gives the following expression for z(r):

$z(r) = int frac{dr}{sqrt{frac{c^2 r}{2GM}-1}} = 4GMsqrt{frac{c^2 r}{2GM}- 1} + mbox{ a constant}.$

## Orbital motion

A particle orbiting in the Schwarzschild metric can have a stable circular orbit with r > 3rs. Circular orbits with r between 3rs / 2 and 3rs are unstable, and no circular orbits exist for r < 3rs / 2. The circular orbit of minimum radius 3rs / 2 corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of r between rs and 3rs / 2, but only if some force acts to keep it there.

Noncircular orbits, such as Mercury's, dwell longer at small radii than would be expected classically. This can be seen as a less extreme version of the more dramatic case in which a particle passes through the event horizon and dwells inside it forever. Intermediate between the case of Mercury and the case of an object falling past the event horizon, there are exotic possibilities such as "knife-edge" orbits, in which the satellite can be made to execute an arbitrarily large number of nearly circular orbits, after which it flies back outward.

## Quotes

"Es ist immer angenehm, über strenge Lösungen einfacher Form zu verfügen." (It is always pleasant to avail of exact solutions in simple form.) – Karl Schwarzschild, 1916.

## References

• Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1, 189-196.
• Ronald Adler, Maurice Bazin, Menahem Schiffer, Introduction to General Relativity (Second Edition), (1975) McGraw-Hill New York, ISBN 0-07-000423-4 See chapter 6.
• Lev Davidovich Landau and Evgeny Mikhailovich Lifshitz, The Classical Theory of Fields, Fourth Revised English Edition, Course of Theoretical Physics, Volume 2, (1951) Pergamon Press, Oxford; ISBN 0-08-025072-6. See chapter 12.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.
• Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory or Relativity, (1972) John Wiley & Sons, New York ISBN 0-471-92567-5. See chapter 8.

Lev Davidovich Landau (Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist and winner of the Nobel Prize for Physics whose broad field of work included the theory of superconductivity and superfluidity, quantum electrodynamics, nuclear physics and particle physics. ... Evgeny Mikhailovich Lifshitz (Ð•Ð²Ð³ÐµÐ½Ð¸Ð¹ ÐœÐ¸Ñ…Ð°Ð¹Ð»Ð¾Ð²Ð¸Ñ‡ Ð›Ð¸Ñ„ÑˆÐ¸Ñ†) (February 21, 1915 â€“ October 29, 1985) was a Russian physicist. ... Steven Weinberg at Harvard University Steven Weinberg (born May 3, 1933) is an American physicist. ...

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 Schwarzschild metric - Wikipedia, the free encyclopedia (1332 words) 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.
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