Schwarzschild metric

Schwarzschild metric

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 conditions became sufficiently favorable to allow for its formation.

The Schwarzschild metric

In Schwarzschild coordinates, the Schwarzschild metric has the form:

:c^2 {d au}^{2} = left(1 - frac{r_s}{r} ight) c^2 dt^2 - frac{dr^2}{1-frac{r_s}{r - r^2 left(d heta^2 + sin^2 heta , dvarphi^2 ight)

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.Landau 1975.]

The analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle. [Citation|last=Ehlers|fist=J.|title=Examples of Newtonian limits of relativistic spacetimes|journal=Classical and Quantum Gravity|year=1997|pages=A119–A126|doi=10.1088/0264-9381/14/1A/010 ]

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 billion 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.

ingularities 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, Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates or Novikov coordinates.

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. 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. 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 r < r_s 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 spatial curvature of the Schwarzschild solution for r>r_s 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} ight)}.

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 dvarphi^2 = -c^2 d au^2 = frac{dr^2}{1 - frac{r_s}{r + r^2 dvarphi^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. 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). 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

: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} 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 ("t" = constant, "dt" = 0)

:mathrm{d}s^2 = left(1-frac{r_{s{r} ight)^{-1} mathrm{d}r^2 + r^2mathrm{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.

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 r<3r_s/2. 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.)" &ndash; 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
* Schwarzschild coordinates
* Kruskal-Szekeres coordinates
* Eddington-Finkelstein coordinates

Notes

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-?.
*

* 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".


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • de Sitter–Schwarzschild metric — In general relativity, the de Sitter–Schwarzschild solution describes a black hole in a causal patch of de Sitter space. Unlike a flat space black hole, there is a largest possible de Sitter black hole, which is the Nariai spacetime. The Nariai… …   Wikipedia

  • Schwarzschild — is a German surname meaning black sign or black shield and may refer to: * Karl Schwarzschild, (1873 1916), physicist and astronomer * Martin Schwarzschild, (1912 1997), astronomer * Roger Schwarzschild, linguistNamed after Karl Schwarzschild: *… …   Wikipedia

  • Metric tensor (general relativity) — This article is about metrics in general relativity. For a discussion of metrics in general, see metric tensor. Metric tensor of spacetime in general relativity written as a matrix. In general relativity, the metric tensor (or simply, the metric) …   Wikipedia

  • Schwarzschild radius — The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic radius associated with every mass. It is the radius for a given mass where, if that mass could be compressed to fit within that radius,… …   Wikipedia

  • Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia

  • Schwarzschild black hole — noun An uncharged (q = 0), nonrotating (L = 0) black hole. See Also: Schwarzschild circumference, Schwarzschild metric, Schwarzschild radius …   Wiktionary

  • Schwarzschild coordinates — In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres . In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical… …   Wikipedia

  • Kerr metric — In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body. According to this metric, such rotating bodies should exhibit frame dragging, an unusual prediction of general relativity; …   Wikipedia

  • Deriving the Schwarzschild solution — The Schwarzschild solution is one of the simplest and most useful solutions of the Einstein field equations (see general relativity). It describes spacetime in the vicinity of a non rotating massive spherically symmetric object. It is worthwhile… …   Wikipedia

  • Lemaitre metric — is a spherically symmetric solution to vacuum Einstein equation apparently [L.D.Landau and E.M.Lifshitz Course of Theoretical Physics, vol. 2: The Classical Theory of Fields .] [Andre Gsponer, More on the early interpretation of the Schwarzschild …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”