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General relativity
G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}\,
Einstein field equations
Introduction to...
Mathematical formulation of...
Solutions
Schwarzschild
Reissner-Nordström · Gödel
Kerr · Kerr-Newman
Kasner · Milne · Robertson-Walker
pp-wave · ADM · BSSN
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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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity (GR is a Theory of Gravitation that was developed by Albert Einstein between 1907 and 1915 The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational The Gödel metric is an exact solution of the Einstein field equations in which the Stress-energy tensor contains two terms the first representing the In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged The Kasner metric is an exact solution to Einstein 's theory of General relativity. The Milne model was a special relativistic cosmological model proposed by Edward Arthur Milne. In General relativity, the pp-wave spacetimes, or pp-waves for short are an important family of Exact solutions of Einstein's field equation The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation of General relativity. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 A gravitational field is a model used within Physics to explain how gravity exists in the universe A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or Sun. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Sun (Sol is the Star at the center of the Solar System. According to Birkhoff's theorem, the Schwarzschild solution is the most general spherically symmetric, vacuum solution of the Einstein field equations. In General relativity, Birkhoff's theorem states that any spherically symmetric solution of the Vacuum field equations must be stationary and Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. In General relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e In Physics, a charge may refer to one of many different quantities such as the Electric charge in Electromagnetism or the Color charge in In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position 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. Karl Schwarzschild ( October 9, 1873 - May 11, 1916) was a German Jewish Physicist and Astronomer. Year 1915 ( MCMXV) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar (or a Common year It was the first exact solution of the Einstein field equations other than the trivial flat space solution. In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity 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. World War I (abbreviated WWI; also known as the First World War, the Great War, and the War to End All

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. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic Radius associated with every Mass. 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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Contents

The Schwarzschild metric

Main article: Deriving the Schwarzschild solution

In Schwarzschild coordinates, the Schwarzschild metric has the form:

c^2 {d \tau}^{2} = \left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \frac{dr^2}{1-\frac{r_s}{r}} - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right)

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
rs is the Schwarzschild radius (in meters) of the massive body, which is related to its mass M by
r_{s} = \frac{2GM}{c^{2}}
where G is the gravitational constant. The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see General relativity) In the theory of Lorentzian manifolds Spherically symmetric spacetimes admit a family of nested round spheres. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock In Spherical coordinates, colatitude is the Complementary angle of the Latitude, i Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic Radius associated with every Mass. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass [1]

The classical Newtonian theory of gravity is recovered in the limit as the ratio rs/r goes to zero. In that limit, the metric returns to the Minkowski metric of special relativity , which has no curvature

c2dτ2 = c2dt2 − dr2 − r2dθ2 − r2sin2θdφ2

In practice, the ratio rs/r is almost always extremely small. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial An asymptotically flat spacetime is a Lorentzian manifold in which roughly speaking the curvature vanishes at large distances from some region so that at large distances For example, the Schwarzschild radius rs 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 billion times larger, at 42,164 km (26,200 miles). EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Millimetre ( American spelling: millimeter, symbol mm) is a unit of Length in the Metric system, equal to Inches redirects here To see the Les Savy Fav album see Inches. This article is about artificial satellites For natural satellites also known as moons see Natural satellite. A geosynchronous orbit is an Orbit around the Earth with an Orbital period matching the Earth's sidereal rotation period The kilometre ( American spelling: kilometer) symbol km is a unit of Length in the Metric system, equal to one thousand A mile is a unit of Length, usually used to measure Distance, in a number of different systems including Imperial units United States 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. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type

The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the 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 = rs; some of the metric components blow up at these radii. 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 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. The Sun (Sol is the Star at the center of the 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, Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates or Novikov coordinates. Lemaitre metric is a spherically symmetric solution to vacuum Einstein equation apparently obtained by Georges Lemaître in 1938 by a coordinate transformation In General relativity Eddington-Finkelstein coordinates, named for Arthur Stanley Eddington and David Finkelstein, are a pair of Coordinate systems In General relativity Kruskal-Szekeres coordinates, named for Martin Kruskal and George Szekeres, are a Coordinate system for the Schwarzschild

The 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. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become 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

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. In the theory of Lorentzian manifolds, particularly in the context of applications to General relativity, the Kretschmann scalar is a quadratic scalar invariant 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 theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e

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 because spacetime has been curved so much that the direction of cause and effect (the particle's future light cone) points into the singularity. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. In Special relativity, a light cone (or null cone) is the pattern describing the temporal evolution of a flash of Light in Minkowski spacetime The surface r = rs demarcates what is called the event horizon of the black hole. In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot 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. Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity.

Flamm's paraboloid

A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a gravity well.
A plot of Flamm's paraboloid. It should not be confused with the unrelated concept of a gravity well. In Physics, a gravity well is the Gravitational potential field around a massive body (a particular kind of Potential well)

The spatial curvature of the Schwarzschild solution for r > rs can be visualized as follows. 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)

w = 2 \sqrt{r_{s} \left( r - r_{s} \right)}.

This surface has the property that distances measured within it match distances in the Schwarzschild metric, because with the definition of w above,

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

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. In Physics, a gravity well is the Gravitational potential field around a massive body (a particular kind of Potential 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 all particles moving across it must have infinite velocity). SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Physics, velocity is defined as the rate of change of Position. 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. A tachyon (from the Greek, takhyónion, from, takhýs, ie swift fast is any hypothetical particle that travels at Faster-than-light See the gravity well article for more information. In Physics, a gravity well is the Gravitational potential field around a massive body (a particular kind of Potential well)

Flamm's paraboloid may be derived as follows. The Euclidean metric in the cylindrical coordinates (r, φ, 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(\frac{\mathrm{d}w}{\mathrm{d}r}\right)^2 \right] \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,

Comparing this with the Schwarzschild metric in the equatorial plane (θ = π/2) at a fixed time (t = constant, dt = 0)

\mathrm{d}s^2 = \left(1-\frac{r_{s}}{r} \right)^{-1} \mathrm{d}r^2 + r^2\mathrm{d}\phi^2,

yields an integral expression for w(r):

w(r) = \int \frac{\mathrm{d}r}{\sqrt{\frac{r}{r_{s}}-1}} = 2 r_{s} \sqrt{\frac{r}{r_{s}}- 1} + \mbox{constant}

whose solution is Flamm's paraboloid. The cylindrical coordinate system is a three-dimensional Coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually

Orbital motion

For more details on this topic, see Kepler problem in general relativity. The Kepler problem in general relativity involves solving for the motion of two spherical bodies interacting with one another by Gravitation, as described by the theory of

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 have exact solutions in simple form at your disposal. ) – Karl Schwarzschild, 1916.

See also

Notes

  1. ^ Landau 1975. The Schwarzschild solution is one of the simplest and useful solutions of the Einstein field equations (see General relativity) In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body The Kerr-Newman metric is a solution of Einstein's General relativity field equation that describes the spacetime geometry in the region surrounding a charged A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e In the theory of Lorentzian manifolds Spherically symmetric spacetimes admit a family of nested round spheres. In General relativity Kruskal-Szekeres coordinates, named for Martin Kruskal and George Szekeres, are a Coordinate system for the Schwarzschild In General relativity Eddington-Finkelstein coordinates, named for Arthur Stanley Eddington and David Finkelstein, are a pair of Coordinate systems

References

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