Schwarzschild metric
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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 Schwarzchild solution is the most general static, spherically symmetric, vacuum solution of Einstein's field equations.
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.
Schwarzschild's solution showed how the predictions of general relativity would deviate from the predictions obtained from Newtonian gravity. Using his solution for the gravitational field of the Earth and the Sun, the outcome of three classical tests of general relativity has been predicted. For about half a century they were the only experimental verification of general relativity. The classical tests are the gravitational redshift, the gravitational deflection of light and the perihelion shift of the planet Mercury. The perihelion shift of Mercury was one of the major problems that astronomers were trying to understand; when Einstein used Schwarzschild's solution to calculate the observed shift, he found that it was exactly (within experimental errors) the observed shift. For Einstein, this was the first major triumph of general relativity.
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The Schwarzschild metric
In spherical coordinates, the Schwarzschild metric can be put into the form (see deriving the Schwarzschild solution)
- <math>ds^{2} = -\left(1-\frac{2GM}{c^2 r} \right) dt^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2+ r^2 d\Omega^2<math>
where <math>G<math> is the gravitational constant, <math>M<math> is interpreted as the mass of the gravitating object, and
- <math>d\Omega^2 = d\theta^2+\sin^2\theta d\phi^2\,<math>
is the standard metric on the 2-sphere (i.e. the standard element of solid angle). The constant
- <math>r_s = \frac{2GM}{c^2} <math>
is called the Schwarzschild radius and plays an important role in the Schwarzschild solution.
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 <math>R<math> the solution is valid for <math>r > R<math>. (Although, if <math>R<math> is less then the Schwarzschild radius <math>r_s<math> 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 <math>r = R<math>.
Note that as <math>M\to 0<math> or <math>r \rightarrow\infty<math> one recovers the Minkowski metric:
- <math>ds^{2} = -c^2dt^2 + dr^2 + r^2 d\Omega^2.\,<math>
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.
Singularities and black holes
The Schwarzschild solution appears to have singularities at <math>r = 0<math> and <math>r=r_s<math>; 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 <math>R<math> of the gravitating body, there is no problem as long as <math>R > r_s<math>. For ordinary stars and planets this is always the case. For example, the radius of the Sun is approximately 700,000 km, while it's Schwarzschild radius is only 3 km.
One might naturally wonder what happens when the radius <math>R<math> becomes less than or equal to the Schwarzschild radius <math>r_s<math>. It turns out that the Schwarzchild solution still makes sense in this case, although it has some rather odd properties. The apparent singularity at <math>r = r_s<math> 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.
This case <math>r = 0<math> is different, however. If one asks that the solution be valid for all <math>r<math> 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^{abcd}R_{abcd}= \frac{12 r_s^2}{r^6}<math>
At <math>r=0<math> 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 <math>r > 0<math>, 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 <math>r < r_s<math> the Schwarzschild radial coordinate <math>r<math> becomes timelike and the time coordinate <math>t<math> becomes spacelike. A curve at constant <math>r<math> is no longer a possible worldline of a particle or observer. Causality requires a particle to fall inwards. The surface <math>r = r_s<math> 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.
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 <math>t=t_0<math> and <math>\theta=\pi/2<math> and map this into three dimensions with the Euclidean metric (in cylindrical coordinates):
- <math>ds^2 = dz^2 + dr^2 + r^2d\phi^2.\,<math>
We will get a curved surface <math>z= z(r)<math> by writing the Euclidean metric in the form
- <math>ds^2 = \left(1 + \left(\frac{dz}{dr}\right)^2 \right)dr^2 + r^2d\phi^2<math>
where we have made the identification
- <math>dz = \frac{dz}{dr}dr.<math>
We can then relate this to the Schwarzschild metric for the equatorial plane at a fixed time:
- <math>ds^2 = \left(1-\frac{2GM}{c^2 r} \right)^{-1} dr^2 + r^2d\phi^2<math>
Which gives the following expression for z(r):
- <math>z(r) = \int \frac{dr}{\sqrt{\frac{c^2 r}{2GM}-1}} = 4GM\sqrt{\frac{c^2 r}{2GM}- 1} + \mbox{ a constant}.<math>
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.