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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 spherically symmetric, vacuum solution of the Einstein 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.
The Schwarzschild solution is named in honour of its discoverer Karl Schwarzschild, who found the solution in 1915, only about a month after the publication of Einstein's theory of general relativity. It was the first exact solution of the Einstein field equations other than 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.
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 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.

Contents

The Schwarzschild metric

Singularities and black holes

Flamm's paraboloid

Orbital motion

Quotes

See also

References

The Schwarzschild metric

Main articles: Deriving the Schwarzschild solution

In Schwarzschild coordinates, the 'Schwarzschild metric' has the form:
:$$
c^2 {d au}^{2} =
left( 1 - rac{r_{s}}{r}
ight) c^{2} dt^{2} - rac{dr^{2}}{1 - rac{r_{s}}{r}} - r^{2} d heta^{2} - r^{2} sin^{2} heta , darphi^{2}

where
:''τ'' is the proper time (time measured by a clock moving with the particle) in seconds,
:''c'' is the speed of light in meters per second,
:''t'' is the time coordinate (measured by a stationary clock at infinity) in seconds,
:''r'' is the radial coordinate (circumference of a circle centered on the star divided by 2π) in meters,
:''θ'' is the colatitude (angle from North) in radians,
:''φ'' is the longitude in radians, and
:''r_s'' is the Schwarzschild radius (in meters) of the massive body, which is related to its mass ''M'' by
::$$
r_{s} = rac{2GM}{c^{2}}

:where ''G'' is the gravitational constant.Landau 1975.
The classical Newtonian theory of gravity is recovered in the limit as the ratio ''r''_''s''/''r'' goes to zero. In that limit, the metric returns to the Minkowski metric of special relativity , which has no curvature
:$$
c^{2} d au^{2} = c^{2} dt^{2} - dr^{2} - r^{2} d heta^{2} - r^{2} sin^{2} heta dphi^{2}

In practice, the ratio ''r''_''s''/''r'' is almost always extremely small. For example, the Schwarzschild radius ''r''_''s'' of the Earth is roughly 9 mm (³⁄₈ inch), whereas a satellite in a geosynchronous orbit has a radius ''r'' that is roughly four trillion times larger, at 42,164 km (26,200 miles). 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 holes and other ultra-dense objects such as neutron stars.
The Schwarzschild metric is a solution of 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 $r\; >\; R$. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at $r\; =\; R$.

Singularities and black holes

The Schwarzschild solution appears to have singularities at $r\; =\; 0$ and $r=r\_s$; 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 $R\; >\; r\_s$. 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.
One might naturally wonder what happens when the radius $R$ becomes less than or equal to the Schwarzschild radius $r\_s$. It turns out that the Schwarzschild solution still makes sense in this case, although it has some rather odd properties. The apparent singularity at $r\; =\; r\_s$ 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, Kruskal-Szekeres coordinates or Novikov coordinates.
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
:
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''.
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 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 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 $r\; =\; r\_s$ 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.

Flamm's paraboloid

The space-time curvature of the Schwarzschild solution for $r>r\_s$ can be visualized as follows. Consider a plane that passes through the equator of the mass and let the position of a particle moving in this plane be described in Cartesian coordinates (''x'', ''y'') or in polar coordinates (''r'', ''θ''). Imagine now that there is an additional Cartesian dimension called ''w'', but no human can see it; one can see only ''x'' and ''y''. (Perhaps they are looking straight down onto the ''x''-''y'' plane and see only the projection of the particle's motion.) Imagine further that this plane is ''not'' flat but dimpled downwards in ''w'' according to the equation (''Flamm's paraboloid'')
:$$
w = 2 sqrt{r_{s} left( r - r_{s}
ight)}.

A particle traveling along this dimpled plane will dip and curve around the central mass; if conditions are right, the particle may even orbit around the central mass. Yet there is no direct force between the central mass and the particle. Rather, the central mass bends the surface (think of a billiard ball dimpling a rubber sheet), whereas the particle just follows the surface, itself always trying to go as straight as possible. To the humans who cannot see the ''w'' direction, however, there appears to be a mysterious force (gravity) acting between the particle and the central mass. As an aside, this is why the Kepler orbits don't depend on the mass of the moving particle; the particle's orbit is determined only by the surface, which in turn is determined only by the central mass.
Flamm's paraboloid may be derived as follows. The Euclidean metric in the cylindrical coordinates (''r'', ''θ'', ''w'') is written
:$$
mathrm{d}s^2 = mathrm{d}w^2 + mathrm{d}r^2 + r^2 mathrm{d}phi^2.,

Letting the surface be described by the function $w=\; w(r)$, the Euclidean metric can be written as
:$$
mathrm{d}s^2 = left[ 1 + left(rac{mathrm{d}w}{mathrm{d}r}
ight)^2
ight] mathrm{d}r^2 + r^2mathrm{d}phi^2,

Comparing this with the Schwarzschild metric in the equatorial plane (''θ'' = π/2) at a fixed time (''dt''=0)
:$$
mathrm{d}s^2 = left(1-rac{r_{s}}{r}
ight)^{-1} mathrm{d}r^2 + r^2mathrm{d}phi^2,

yields an integral expression for ''w''(''r''):
:$$
w(r) = int rac{mathrm{d}r}{sqrt{rac{r}{r_{s}}-1}} = 2 r_{s} sqrt{rac{r}{r_{s}}- 1} + mbox{ a constant}.

whose solution is Flamm's paraboloid.

Orbital motion

A particle orbiting in the Schwarzschild metric can have a stable circular orbit with $r\; >\; 3r\_s$. Circular orbits with $r$ between $3r\_s/2$ and $3r\_s$ are unstable, and no circular orbits exist for . The circular orbit of minimum radius $3r\_s/2$ corresponds to an orbital velocity approaching the speed of light. It is possible for a particle to have a constant value of $r$ between $r\_s$ and $3r\_s/2$, 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 have exact solutions in simple form at your disposal.)'' – Karl Schwarzschild, 1916.

See also

★ Deriving the Schwarzschild solution

★ Reissner-Nordström metric (charged, non-rotating solution)

★ Kerr metric (uncharged, rotating solution)

★ Kerr-Newman metric (charged, rotating solution)

★ BKL singularity (interior solution)

★ Black hole, a general review

References

★ Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes nach der Einstein'schen Theorie. ''Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften'' '1', 189-196.

★ Schwarzschild, K. (1916). Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit. ''Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften'' '1', 424-?.

★ Beiträge zur Einstein'schen Gravitationstheorie, , L, Flamm, Physikalische Zeitschrift, 1916

★ 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''.