Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that 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 stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916.

According to Birkhoff's theorem, the Schwarzschild metric 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 is described by the 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 boundary, called the event horizon, which is situated at the Schwarzschild radius, often called the radius of a black hole. The boundary is not a physical surface, and if a person fell through the event horizon (before being torn apart by tidal forces), they would not notice any physical surface at that position; it is a mathematical surface which is significant in determining the black hole's properties. 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.

The Schwarzschild metric

In Schwarzschild coordinates, the line element for the Schwarzschild metric has the form

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),

where

The analogue of this solution in classical Newtonian theory of gravity corresponds to the gravitational field around a point particle.[2]

In practice, the ratio rs/r is almost always extremely small. For example, the Schwarzschild radius rs of the Earth is roughly 8.9 mm, while the Sun, which is 3.3×105 times as massive[3] has a Schwarzschild radius of approximately 3.0 km. 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,[4] such as the interior Schwarzschild solution.

History

The Schwarzschild solution is named in honor of Karl Schwarzschild, who found the exact solution in 1915 and published it in 1916,[5] a little more than 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 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.[6]

Johannes Droste in 1916[7] independently produced the same solution as Schwarzschild, using a simpler, more direct derivation.[8]

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.[9] In this paper he also introduced what is now known as the Schwarzschild radial coordinate (r in the equations above), as an auxiliary variable. In his equations, Schwarzschild was using a different radial coordinate that was zero at the Schwarzschild radius.

A more complete analysis of the singularity structure was given by David Hilbert[10] in the following year, identifying the singularities both at r = 0 and r = rs. Although there was general consent that the singularity at r = 0 was a 'genuine' physical singularity, the nature of the singularity at r = rs remained unclear.[11]

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 = rs. 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 = rs 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 = rs was not physical. In 1939 Howard Robertson showed that a free falling observer descending in the Schwarzschild metric would cross the r = rs singularity in a finite amount of proper time even though this would take an infinite amount of time in terms of coordinate time t.[11]

In 1950, John Synge produced a paper[12] that showed the maximal analytic extension of the Schwarzschild metric, again showing that the singularity at r = rs was a coordinate artifact and that it represented two horizons. A similar result was later rediscovered by George Szekeres,[13] and independently Martin Kruskal.[14] The new coordinates nowadays known as Kruskal-Szekeres 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.[11]

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 = rs singularity in the Schwarzschild metric as an event horizon (a hypersurface in spacetime that can only be crossed in one direction).[11]

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. 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 singularity at r = rs divides the Schwarzschild coordinates in two disconnected patches. The exterior Schwarzschild solution with r > rs is the one that is related to the gravitational fields of stars and planets. The interior Schwarzschild solution with 0 ≤ r < rs, which contains the singularity at r = 0, is completely separated from the outer patch by the singularity at r = rs. The Schwarzschild coordinates therefore give no physical connection between the two patches, which may be viewed as separate solutions. The singularity at r = rs 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 = rs and can extend the external patch to values of r smaller than rs. Using a different coordinate transformation one can then relate the extended external patch to the inner patch.[15]

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^{\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} \,.

At r = 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.

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. The surface r = rs 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.[16]

Alternative coordinates

The Schwarzschild solution can be expressed in a range of different choices of coordinates besides the Schwarzschild coordinates used above. Different choices tend to highlight different features of the solution. The table below shows some popular choices.

Alternative coordinates
Coordinates Line element Notes Features
Eddington-Finkelstein coordinates (ingoing) \left(1-\frac{r_s}{r} \right) dv^2 - 2 dv dr - r^2 d\Omega^2 regular at horizon
extends across future horizon
Eddington-Finkelstein coordinates (outgoing) \left(1-\frac{r_s}{r} \right) du^2 + 2 du dr - r^2 d\Omega^2 regular at horizon
extends across past horizon
Gullstrand–Painlevé coordinates \left(1-\frac{r_s}{r} \right)dT^2- 2\sqrt{\frac{r_s}{r}} dT dr - dr^2-r^2d\Omega^2 regular at horizon
Isotropic coordinates \frac{(1-\frac{r_s}{4R})^{2}}{(1+\frac{r_s}{4R})^{2}}{d t}^2 - \left(1+\frac{r_s}{4R}\right)^{4}(dx^2+dy^2+dz^2) R = \sqrt{ x^2 + y^2 + z^2 }[17] isotropic lightcones on constant time slices
Kruskal-Szekeres coordinates \frac{4r_s^3}{r}e^{-r/r_s}(dT^2 - dR^2)- r^2 d\Omega^2, T^2 - R^2 = \left(1-\frac{r}{r_s}\right)e^{r/r_s} regular at horizon
Maximally extends to full spacetime
Lemaitre coordinates dT^{2} - \frac{r_{s}}{r} dR^{2}- r^{2}d\Omega^{2} r = \left[\frac{3}{2}(R-T)\right]^{2/3}r_{s}^{1/3} regular at horizon

In table above, some shorthand has been introduced for brevity. The speed of light c has been set to one. The notation d\Omega^2= d\theta^2+\sin(\theta)^2 d\phi^2 is used for the metric of a two dimensional sphere. Moreover, in each entry R and T denote alternative choices of radial and time coordinate for the particular coordinates. Note, the R and/or T may vary from entry to entry.

Flamm's paraboloid

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 r>r_s 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)

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. 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

\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.

Orbital motion

For more details on this topic, see Schwarzschild geodesics.

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.

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

"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 & Liftshitz 1975).
  2. ↑ Ehlers, J. (1997). "Examples of Newtonian limits of relativistic spacetimes". Classical and Quantum Gravity 14: A119–A126. Bibcode:1997CQGra..14A.119E. doi:10.1088/0264-9381/14/1A/010.
  3. ↑ Tennent, R.M., ed. (1971). Science Data Book. Oliver & Boyd. ISBN 0-05-002487-6.
  4. ↑ Frolov, Valeri. Introduction to Black Hole Physics. Oxford. p. 168.
  5. ↑ Schwarzschild, K. (1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 7: 189–196. Bibcode:1916AbhKP......189S. For a translation, see Antoci, S.; Loinger, A. (1999). "On the gravitational field of a mass point according to Einstein's theory". arXiv:physics/9905030 [physics].
  6. ↑ O'Connor, John J.; Robertson, Edmund F., "Karl Schwarzschild", MacTutor History of Mathematics archive, University of St Andrews.
  7. ↑ Droste, J. (1917). "The field of a single centre in Einstein's theory of gravitation, and the motion of a particle in that field" (PDF). Proceedings of the Royal Netherlands Academy of Arts and Science 19 (1): 197–215. Bibcode:1917KNAB...19..197D.
  8. ↑ Kox, A. J. (1992). "General Relativity in the Netherlands:1915-1920". In Eisenstaedt, J.; Kox, A. J. Studies in the History of General Relativity. Birkhäuser. p. 41. ISBN 978-0-8176-3479-7.
  9. ↑ Brown, K. (2011). Reflections On Relativity. Lulu.com. Chapter 8.7. ISBN 978-1-257-03302-7.
  10. ↑ Hilbert, David (1924). "Die Grundlagen der Physik". Mathematische Annalen (Springer-Verlag) 92 (1-2): 1–32. doi:10.1007/BF01448427.
  11. 1 2 3 4 Earman, J. (1999). "The Penrose–Hawking singularity theorems: History and Implications". In Goenner, H. The expanding worlds of general relativity. Birkhäuser. p. 236-. ISBN 978-0-8176-4060-6.
  12. ↑ Synge, J. L. (1950). "The gravitational field of a particle". Proceedings of the Royal Irish Academy 53 (6): 83–114.
  13. ↑ Szekeres, G. (1960). "On the singularities of a Riemannian manifold". Publicationes Mathematicae Debrecen 7 7: 285. Bibcode:2002GReGr..34.2001S. doi:10.1023/A:1020744914721.
  14. ↑ Kruskal, M. D. (1960). "Maximal extension of Schwarzschild metric". Physical Review 119 (5): 1743–1745. Bibcode:1960PhRv..119.1743K. doi:10.1103/PhysRev.119.1743.
  15. ↑ Hughston, L.P.; Tod, K.P. (1990). An introduction to general relativity. Cambridge University Press. Chapter 19. ISBN 978-0-521-33943-8.
  16. ↑ Brill, D. (19 January 2012). "Black Hole Horizons and How They Begin". Astronomical Review.
  17. ↑ Eddington, A. S. (1924). The Mathematical Theory of Relativity (2nd ed.). Cambridge University Press. p. 93.

References

Schwarzschild, K. (1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 7: 189–196. Bibcode:1916AbhKP1916..189S. 

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