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{{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
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In [[Albert Einstein|Einstein]]'s theory of [[general relativity]], the '''Schwarzschild solution''' (or the '''Schwarzschild vacuum'''),  is a solution to the [[Einstein field equations]] which describes the [[gravitational field]] outside a spherical mass, on the assumption that the [[electric charge]] of the mass, [[angular momentum]] of the mass, and universal [[cosmological constant]] are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many [[star]]s, [[planet]]s and [[black hole]]s, including [[Earth]] and the [[Sun]]. The solution is named after [[Karl Schwarzschild]], who first published the solution in 1916.
 
According to [[Birkhoff's theorem (relativity)|Birkhoff's theorem]], the Schwarzschild solution is the most general [[rotational symmetry|spherically symmetric]], [[Vacuum solution (general relativity)|vacuum solution]] of the [[Einstein field equations]]. A '''Schwarzschild black hole''' or '''static black hole''' is a [[black hole]] that has no [[Charge (physics)|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.
 
The Schwarzschild black hole is characterized by a surrounding spherical surface, 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 its 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 conditions became sufficiently favorable to allow for its formation.
 
{| class="wikitable" style="margin: 1em auto"
|+ Single mass solutions to Einstein's field equations
|
! scope="col"| Non-rotating (''J'' = 0)
! scope="col"| Rotating (''J'' ≠ 0)
|-
! scope="col"| Uncharged (''Q'' = 0)
| Schwarzschild metric
| [[Kerr metric]]
|-
! scope="col"| Charged (''Q'' ≠ 0)
| [[Reissner–Nordström metric]]
| [[Kerr–Newman metric]]
|}
 
where ''Q'' represents the body's [[electric charge]] and ''J'' represents its spin [[angular momentum]].
 
==The Schwarzschild metric==
{{see also|Deriving the Schwarzschild solution}}
 
In [[Schwarzschild coordinates]], the [[line element]] for the Schwarzschild metric has the form
:<math>
c^2 {d \tau}^{2} =
\left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right),
</math>
where
*<math>\tau</math> is the [[proper time]] (time measured by a clock moving along the same [[world line]] with the [[test particle]]),
*''c'' is the [[speed of light]],
*''t'' is the time coordinate (measured by a stationary clock located infinitely far from the massive body),
*<math>r</math> is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),
*''θ'' is the [[colatitude]] (angle from North, in units of [[radian]]s),
*''φ'' is the [[longitude]] (also in radians), and
*<math>r_s</math> is the [[Schwarzschild radius]] of the massive body, a [[scale factor]] which is related to its mass ''M'' by ''r<sub>s</sub>''&nbsp;=&nbsp;2''GM''/''c''<sup>2</sup>, where ''G'' is the [[gravitational constant]].<ref name="landau_1975">{{Harv|Landau|Liftshitz|1975}}.</ref>
 
The analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle.<ref>
{{Cite journal
|last=Ehlers |first=J.
|year=1997
|title=Examples of Newtonian limits of relativistic spacetimes
|journal=[[Classical and Quantum Gravity]]
|volume=14 |issue= |pages=A119–A126
|bibcode=1997CQGra..14A.119E
|doi=10.1088/0264-9381/14/1A/010
}}</ref>
 
In practice, the ratio ''r''<sub>''s''</sub>/''r'' is almost always extremely small. For example, the Schwarzschild radius ''r''<sub>''s''</sub> of the [[Earth]] is roughly {{convert|8.9|mm|in|2|sp=us}}, while the sun, which is 3.3×10<sup>5</sup> times as massive<ref>
{{cite book
|editor-last=Tennent |editor-first=R.M.
|year=1971
|title=Science Data Book
|publisher=[[Oliver & Boyd]]
|page=
|isbn=0-05-002487-6
}}</ref> has a Schwarzschild radius of approximately {{convert|3.0|km|mi|abbr=on}}.
Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion.  The ratio only becomes large close to [[black hole]]s and other ultra-dense objects such as [[neutron star]]s.{{citation needed|date=September 2012}}
 
The Schwarzschild metric is a solution of [[Einstein's field equation#Vacuum field equations|Einstein's field equations]] in empty space, meaning that it is valid only ''outside'' the gravitating body. That is, for a spherical body of radius ''R'' the solution is valid for ''r''&nbsp;>&nbsp;''R''.  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''&nbsp;=&nbsp;''R''.{{citation needed|date=September 2012}}
 
==History==
The Schwarzschild solution is named in honor of [[Karl Schwarzschild]], who found the exact solution in 1916,<ref name="Schwarzschild1916">
{{Cite journal
|last=Schwarzschild |first=K.
|year=1916
|title=Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
|url=http://www.archive.org/stream/sitzungsberichte1916deutsch#page/188/mode/2up
|journal=[[Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften]]
|volume=7 |issue= |pages=189–196
|bibcode=1916AbhKP......189S
|ref=harv
}} For a translation, see {{cite arXiv
|last1=Antoci |first1=S.
|last2=Loinger |first2=A.
|year=1999
|title=On the gravitational field of a mass point according to Einstein's theory
|eprint=physics/9905030
|class=physics
}}</ref> a little more than a month after the publication of Einstein's theory of general relativity.
It was the first [[Exact solutions in general relativity|exact solution]] of the Einstein field equations other than the trivial [[Minkowski spacetime|flat space solution]]. Schwarzschild 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]].<ref name="MacTutorBio">{{MacTutor Biography|id=Schwarzschild|title=Karl Schwarzschild}}</ref>
 
Johannes Droste in 1916<ref>
{{cite journal
|last=Droste |first=J.
|year=1917
|title=The field of a single centre in Einstein's theory of gravitation, and the motion of a particle in that field
|journal=[[Proceedings of the Royal Netherlands Academy of Arts and Science]]
|volume=19 |issue=1 |pages=197–215
|url=http://www.dwc.knaw.nl/DL/publications/PU00012325.pdf
|bibcode=1917KNAB...19..197D
|ref=harv
}}</ref>
independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.<ref>
{{cite book
|last=Kox |first=A. J.
|year=1992
|chapter=General Relativity in the Netherlands:1915-1920
|chapter-url=http://books.google.nl/books?id=vDHCF_3vIhUC&lpg=PA39
|page=41
|editor1-last=Eisenstaedt |editor1-first=J.
|editor2-last=Kox |editor2-first=A. J.
|title=Studies in the History of General Relativity
|publisher=[[Birkhäuser]]
|isbn=978-0-8176-3479-7
}}</ref>
 
In the early years of general relativity there was a lot of confusion about the nature of the singularities found in the Schwarzschild and other solutions of the [[Einstein field equations]]. In Schwarzschild's original paper, he put what we now call the event horizon at the origin of his coordinate system.<ref>
{{cite book
|last=Brown |first=K.
|year=2011
|title=Reflections On Relativity
|at=Chapter 8.7
|url=http://mathpages.com/rr/s8-07/8-07.htm
|publisher=[[Lulu.com]]
|isbn=978-1257033027
}}</ref>  An auxiliary variable ("Hilfsgröße") ''R'' was introduced which was related to his radial coordinate ''r'' by {{nowrap begin}}''R'' = (''r''<sup>3</sup> + ''r<sub>s</sub>''<sup>3</sup>)<sup>1/3</sup>{{nowrap end}}. However in this article and most modern treatments, we use ''r'' for what he called ''R''.
 
A more complete analysis of the singularity structure was given by [[David Hilbert]] in the following year, identifying the singularities both at ''r''&nbsp;=&nbsp;0 and ''r''&nbsp;=&nbsp;''r''<sub>s</sub>. Although there was general consent that the singularity at ''r''&nbsp;=&nbsp;0 was a 'genuine' physical singularity, the nature of the singularity at ''r''&nbsp;=&nbsp;''r''<sub>s</sub> remained unclear.
 
In 1921 [[Paul Painlevé]] and in 1922 [[Allvar Gullstrand]] independently produced a metric, a spherically symmetric solution of Einstein's equations,  which we now know is coordinate transformation of the Schwarzschild metric, [[Gullstrand-Painlevé coordinates]], in which there was no singularity at ''r''&nbsp;=&nbsp;''r''<sub>s</sub>. They, however, did not recognize that their solutions were just coordinate transforms, and in fact used their solution to argue that Einstein's theory was wrong. In 1924 [[Arthur Eddington]] produced the first coordinate transformation ([[Eddington–Finkelstein coordinates]]) that showed that the singularity at ''r''&nbsp;=&nbsp;''r''<sub>s</sub> was a coordinate artifact, although he also  seems to have been unaware of the significance of this discovery. Later, in 1932, [[Georges Lemaître]] gave a different coordinate transformation ([[Lemaître coordinates]]) to the same effect and was the first to recognize that this implied that the singularity at ''r''&nbsp;=&nbsp;''r''<sub>s</sub> was not physical. In 1939 [[Howard Percy Robertson|Howard Robertson]] showed that a free falling observer descending in the Schwarzschild metric would cross the  ''r''&nbsp;=&nbsp;''r''<sub>s</sub> singularity in a finite amount of [[proper time]] even though this would take an infinite amount of time in terms of coordinate time ''t''.<ref name=earman>
{{cite book
|last=Earman |first=J. 
|year=1999
|chapter=The Penrose-Hawking singularity theorems: History and Implications
|chapterurl=http://books.google.com/books?id=5mGZno8CvnQC&pg=PA236
|page=236
|editor-first=H. |editor-last=Goenner
|title=The expanding worlds of general relativity
|isbn=978-0-8176-4060-6
|publisher=[[Birkäuser]]
}}</ref>
 
In 1950, [[John Lighton Synge|John Synge]] produced a paper<ref>
{{cite journal
|last=Synge |first=J. L.
|year=1950
|title=The gravitational field of a particle
|journal=[[Proceedings of the Royal Irish Academy]]
|volume=53 |issue=6 |pages=83–114
|bibcode=
|doi=
}}</ref> that showed the maximal [[analytic continuation|analytic extension]] of the Schwarzschild metric, again showing that the singularity at ''r''&nbsp;=&nbsp;''r''<sub>s</sub> was a coordinate artifact and that it represented  two horizons. A similar result was later rediscovered by [[Martin Kruskal]],.<ref>
{{cite journal
|last=Kruskal |first=M. D.
|year=1960
|title=Maximal extension of Schwarzschild metric
|journal=[[Physical Review]]
|volume=119 |issue= 5|pages=1743–1745
|bibcode=1960PhRv..119.1743K
|doi=10.1103/PhysRev.119.1743
}}</ref> His coordinates were much simpler than  Synge's but both provided a single set of coordinates that covered  the entire spacetime. However, perhaps due to the obscurity of the journals in which the papers of Lemaître and Synge were published their conclusions went unnoticed, with many of the major players in the field including Einstein believing that singularity at the Schwarzschild radius was physical.<ref name=earman/>
 
Progress was only made in the 1960s when the more exact tools of [[differential geometry]] entered the field of general relativity, allowing more exact definitions of what it means for a [[Lorentzian manifold]] to be singular. This led to definitive identification of the ''r''&nbsp;=&nbsp;''r''<sub>s</sub> singularity in the Schwarzschild metric as an [[event horizon]] (a hypersurface in spacetime that can only be crossed in one direction).<ref name=earman/>
 
==Singularities and black holes==
 
The Schwarzschild solution appears to have [[mathematical singularity|singularities]] at ''r''&nbsp;=&nbsp;0 and ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub>; 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''&nbsp;>&nbsp;''r''<sub>''s''</sub>. For ordinary stars and planets this is always the case. For example, the radius of the [[Sun]] is approximately 700,000&nbsp;km, while its Schwarzschild radius is only 3&nbsp;km.
 
The singularity at ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub> divides the Schwarzschild coordinates in two [[connectedness|disconnected]] [[coordinate patch|patches]]. The outer patch with ''r''&nbsp;>&nbsp;''r''<sub>''s''</sub> is the one that is related to the gravitational fields of stars and planets. The inner patch 0&nbsp;<&nbsp;''r''&nbsp;<&nbsp;''r''<sub>''s''</sub>, which contains the singularity  at ''r''&nbsp;=&nbsp;0, is completely separated from the outer patch by the singularity  at ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub>. The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub> is an illusion however; it is an instance of what is called a ''[[coordinate singularity]]''. As the name implies, the singularity arises from a bad choice of coordinates or [[coordinate conditions]]. When changing to a different coordinate system (for example [[Lemaitre coordinates]], [[Eddington-Finkelstein coordinates]], [[Kruskal-Szekeres coordinates]], [[Novikov coordinates]], or [[Gullstrand–Painlevé coordinates]]) the metric becomes regular at ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub> and can extend the external patch to values of ''r'' smaller than ''r''<sub>''s''</sub>. Using a different coordinate transformation one can then relate the extended external patch to the inner patch.<ref>
{{cite book
|last1=Hughston |first1=L.P.
|last2=Tod |first2=K.P.
|year=1990
|url=http://books.google.com/books?id=2q5Rdjn0qfgC&lpg=PA126
|title=An introduction to general relativity
|at=Chapter 19
|publisher=[[Cambridge University Press]]
|isbn=978-0-521-33943-8
}}</ref>
 
The case ''r''&nbsp;=&nbsp;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
:<math>R^{\alpha\beta\gamma\delta} R_{\alpha\beta\gamma\delta} = \frac{12 {r_s}^2}{r^6} = \frac{48 G^2 M^2}{c^4 r^6} \,.</math>
 
At ''r''&nbsp;=&nbsp;0 the curvature 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 hole]]s''.
 
The Schwarzschild solution, taken to be valid for all ''r''&nbsp;>&nbsp;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''&nbsp;<&nbsp;''r''<sub>''s''</sub> the Schwarzschild radial coordinate ''r'' becomes [[Spacetime#Time-like interval|timelike]] and the time coordinate ''t'' becomes [[Spacetime#Space-like interval|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 because spacetime has been curved so much that the direction of cause and effect (the particle's future [[light cone]]) points into the singularity{{Citation needed|date=November 2010}}. The surface ''r''&nbsp;=&nbsp;''r''<sub>''s''</sub> 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.<ref>
{{cite journal
|last=Brill |first=D.
|year=2012
|title=Black Hole Horizons and How They Begin
|url=http://astroreview.com/issue/2012/article/black-hole-horizons-and-how-they-begin
|journal=[[Astronomical Review]]
|volume= |issue= |pages=
|bibcode=
|doi=
}}</ref>
 
==Alternative (isotropic) formulations of the Schwarzschild metric==
 
The original form of the Schwarzschild metric involves [[anisotropic]] coordinates, in terms of which the coordinate velocity of light (rate of change of the coordinate location with time) is not the same for the radial and transverse directions (pointed out by [[Arthur Stanley Eddington|A S Eddington]]).<ref name=eddntn1923>
{{cite book
|last=Eddington |first=A. S.
|year=1924 
|title=The Mathematical Theory of Relativity
|edition=2nd
|publisher=[[Cambridge University Press]]
|page=93
|lccn=
|isbn=
}} Note: In the formulae taken from the Eddington source, symbol usage has been adapted to the conventions used in the main section above.</ref> Eddington gave alternative formulations of the Schwarzschild metric in terms of [[isotropic coordinates]] (provided r ≥ r<sub>s</sub> <ref>
{{cite journal
|last=Buchdahl |first=H. A.
|year=1985
|title=Isotropic coordinates and Schwarzschild metric
|journal=[[International Journal of Theoretical Physics]]
|volume=24 |issue= 7|pages=731–739
|bibcode= 1985IJTP...24..731B
|doi= 10.1007/BF00670880
}}.</ref>). The coordinate velocities of course have no physical significance since they are arbitrary, artificial constructs.
 
In isotropic spherical coordinates, one uses a different radial coordinate, ''r''<sub>1</sub>, instead of ''r''. They are related by
:<math>r = r_1 {\left( 1 + \frac{r_s}{4 r_1} \right)}^{2}\, ,\quad r_1 =\frac{r}{2}-\frac{r_s}{4}+\sqrt{\frac{r}{4}\left(r-r_s\right)} \,.</math>
 
Using ''r''<sub>1</sub>, the metric is
:<math>
c^2 {d \tau}^{2} = \frac{(1-\frac{r_s}{4r_1})^{2}}{(1+\frac{r_s}{4r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{r_s}{4r_1}\right)^{4}\left(dr_1^2 + r_1^2 d\theta^2 + r_1^2 \sin^2\theta \, d\varphi^2\right)
\,.</math>
 
For isotropic rectangular coordinates ''x'', ''y'', ''z'', where
:<math>x = r_1 \, \sin\theta \, \cos\phi \,, \quad y = r_1 \, \sin\theta \, \sin\phi \,, \quad z = r_1 \, \cos\theta \,,</math>
 
and
:<math>r_1 = \sqrt{ x^2 + y^2 + z^2 } \,,</math>
 
the metric then becomes
:<math>
c^2 {d \tau}^{2} = \frac{(1-\frac{r_s}{4r_1})^{2}}{(1+\frac{r_s}{4r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{r_s}{4r_1}\right)^{4}(dx^2+dy^2+dz^2)
\,.</math>
 
In terms of these coordinates, the inferred radius of the event horizon past which light cannot escape lies at ''r''<sub>1</sub> = ''r''<sub>s</sub>/4 = ''GM''/2''c''<sup>2</sup> rather than ''r'' = ''r''<sub>s</sub> using Schwarzschild coordinates, and the coordinate velocity of light at any point is the same in all directions, but it varies with radial distance ''r''<sub>1</sub> (from the point mass at the origin of coordinates), where it has the value
:<math>\frac{(1-\frac{r_s}{4r_1})}{(1+\frac{r_s}{4r_1})^{3}} \, c \,.</math> &nbsp;&nbsp;&nbsp;<ref name=eddntn1923 />
 
==Flamm's paraboloid==
 
[[Image:Flamm.jpg|thumb|left|250px|A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a [[gravity well]].]]
 
The spatial curvature of the Schwarzschild solution for <math>r>r_s</math> can be visualized as the graphic shows.  Consider a constant time equatorial slice through the Schwarzschild solution (''θ'' = ''π''/2, ''t'' = constant) and let the position of a particle moving in this plane be described with the remaining Schwarzschild coordinates (''r'', ''φ'').  Imagine now that there is an additional Euclidean dimension ''w'', which has no physical reality (it is not part of spacetime). Then replace the (''r'', ''φ'') plane with a surface dimpled in the ''w'' direction according to the equation (''Flamm's paraboloid'')
:<math>
w = 2 \sqrt{r_{s} \left( r - r_{s} \right)}.
</math>
 
This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of ''w'' above,
:<math>dw^2 + dr^2 + r^2 d\varphi^2 = -c^2 d\tau^2 = \frac{dr^2}{1 - \frac{r_s}{r}} + r^2 d\varphi^2</math>
 
Thus, Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. It should not, however, be confused with a [[gravity well]]. No ordinary (massive or massless) particle can have a worldline lying on the paraboloid, since all distances on it are [[spacelike]] (this is a cross-section at one moment of time, so any particle moving on it would have an  infinite [[velocity]]). Even a [[tachyon]] would not move along the path that one might naively expect from a "rubber sheet" analogy: in particular, if the dimple is drawn pointing upward rather than downward, the tachyon's path still curves toward the central mass, not away. See the [[gravity well]] article for more information.
 
Flamm's paraboloid may be derived as follows.  The Euclidean metric in the [[cylindrical coordinates]] (''r'', ''φ'', ''w'') is written
:<math>
\mathrm{d}s^2 = \mathrm{d}w^2 + \mathrm{d}r^2 + r^2 \mathrm{d}\phi^2.\,
</math>
 
Letting the surface be described by the function <math>w= w(r)</math>, the Euclidean metric can be written as
:<math>
\mathrm{d}s^2 = \left[ 1 + \left(\frac{\mathrm{d}w}{\mathrm{d}r}\right)^2 \right] \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,
</math>
 
Comparing this with the Schwarzschild metric in the equatorial plane (''θ'' = π/2) at a fixed time (''t'' = constant, ''dt'' = 0)
:<math>
\mathrm{d}s^2 = \left(1-\frac{r_{s}}{r} \right)^{-1} \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,
</math>
 
yields an integral expression for ''w''(''r''):
:<math>
w(r) = \int \frac{\mathrm{d}r}{\sqrt{\frac{r}{r_{s}}-1}} = 2 r_{s} \sqrt{\frac{r}{r_{s}}- 1} + \mbox{constant}
</math>
 
whose solution is Flamm's paraboloid.
 
==Orbital motion==
 
{{Details|Schwarzschild geodesics}}
 
A particle orbiting in the Schwarzschild metric can have a stable circular orbit with <math>r > 3r_s</math>. Circular orbits with <math>r</math> between <math>3r_s/2</math> and <math>3r_s</math> are unstable, and no circular orbits exist for <math>r<3r_s/2</math>. The circular orbit of minimum radius <math>3r_s/2</math> corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of <math>r</math> between <math>r_s</math> and <math>3r_s/2</math>, 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.
 
==Symmetries==
The group of isometries of the Schwarzschild metric is the subgroup of the ten-dimensional [[Poincaré group]] which takes the time axis (trajectory of the star) to itself. It omits the spatial translations (three dimensions) and boosts (three dimensions). It retains the time translations (one dimension) and rotations (three dimensions). Thus it has four dimensions. Like the Poincaré group, it has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time reversed and spatially inverted.
 
==Quotes==
 
<blockquote>''"{{lang|de|Es ist immer angenehm, über strenge Lösungen einfacher Form zu verfügen.}}" (It is always pleasant to have exact solutions in simple form at your disposal.)'' – Karl Schwarzschild, 1916.</blockquote>
 
== See also ==
* [[Deriving the Schwarzschild solution]]
* [[Reissner–Nordström metric]] (charged, non-rotating solution)
* [[Kerr metric]] (uncharged, rotating solution)
* [[Kerr–Newman metric]] (charged, rotating solution)
* [[Black hole]], a general review
* [[Schwarzschild coordinates]]
* [[Kruskal–Szekeres coordinates]]
* [[Eddington–Finkelstein coordinates]]
* [[Gullstrand–Painlevé coordinates]]
* [[Lemaitre coordinates]] (Schwarzschild solution in [[synchronous coordinates]])
* [[Frame fields in general relativity]] (Lemaître observers in the Schwarzschild vacuum)
 
==Notes==
{{Reflist}}
 
==References==
{{Cite journal
|last=Schwarzschild |first=K.
|year=1916
|title=Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie
|url=http://www.archive.org/stream/sitzungsberichte1916deutsch#page/188/mode/2up
|journal=[[Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften]]
|volume=7 |issue= |pages=189–196
|bibcode=1916AbhKP......189S
|doi=
|ref=harv
}}
:* [http://de.wikisource.org/wiki/%C3%9Cber_das_Gravitationsfeld_eines_Massenpunktes_nach_der_Einsteinschen_Theorie Text of the original paper, in Wikisource]
:* Translation: {{cite arXiv
|last1=Antoci |first1=S.
|last2=Loinger |first2=A.
|year=1999
|title=On the gravitational field of a mass point according to Einstein's theory
|eprint=physics/9905030
|class=physics
}}
:* A commentary on the paper, giving a simpler derivation: {{cite arXiv
|last=Bel |first=L.
|year=2007
|title=Über das Gravitationsfeld eines Massenpunktesnach der Einsteinschen Theorie
|class=gr-qc
|eprint=0709.2257}}
*{{Cite journal
|last=Schwarzschild |first=K.
|year=1916
|title=Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit
|journal=[[Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften]]
|volume=1 |issue= |pages=424
|bibcode=
|doi=
}}
*{{cite journal
| last=Flamm | first=L.
| year=1916
| title=Beiträge zur Einstein'schen Gravitationstheorie
| journal=[[Physikalische Zeitschrift]]
| volume=17 |issue= | pages=448
| bibcode=
| doi=
}}
*{{cite book
|last1=Adler |first1=R.
|last2=Bazin |first2=M.
|last3=Schiffer |first3=M.
|year=1975
|title=Introduction to General Relativity
|edition=2nd
|at=Chapter 6
|publisher=[[McGraw-Hill]]
|isbn=0-07-000423-4
}}
*{{cite book
|last1=Landau |first1=L. D.
|last2=Lifshitz |first2=E. M.
|year=1951
|title=The Classical Theory of Fields
|edition=4th Revised English
|series=[[Course of Theoretical Physics]]
|volume=2
|publisher=[[Pergamon Press]]
|isbn=0-08-025072-6
|at=Chapter 12
|ref=harv
}}
* {{cite book
|last1=Misner |first1=C. W.
|last2=Thorne |first2=K. S.
|last3=Wheeler |first3=J. A.
|year=1970
|title=Gravitation
|publisher=[[W.H. Freeman]]
|isbn=0-7167-0344-0
|at=Chapters 31 and 32
}}
* {{cite book
|last=Weinberg |first=S.
|year=1972
|title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity
|at=Chapter 8
|publisher=[[John Wiley & Sons]]
|isbn=0-471-92567-5
}}
* {{cite book
|last1=Taylor |first1=E. F.
|last2=Wheeler |first2=J. A.
|year=2000
|title=Exploring Black Holes: Introduction to General Relativity
|publisher=[[Addison-Wesley]]
|isbn=0-201-38423-X
}}
*{{cite journal
|last=Heinzle  |first=J. M.
|last2=Steinbauer |first2=R.
|year=2002
|title=Remarks on the distributional Schwarzschild geometry
|journal=[[Journal of Mathematical Physics]]
|volume=43 |issue= 3|page=1493
|arxiv= gr-qc/0112047
|bibcode= 2002JMP....43.1493H
|doi=10.1063/1.1448684
}}
*{{cite arXiv
|last=Foukzon |first=J.
|year=2008
|title=Distributional Schwarzschild Geometry from nonsmooth regularization via Horizon
|class=physics.gen-ph
|eprint=0806.3026
}}
 
{{Black holes}}
{{Relativity}}
 
{{DEFAULTSORT:Schwarzschild Metric}}
[[Category:Exact solutions in general relativity]]
[[Category:Black holes]]

Latest revision as of 05:56, 15 December 2014

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