What is a gravitational soliton

Schwarzschild metric - Schröder – Bernstein property

or their disjoint union, the metric is not singular across the event horizon, as can be seen in suitable coordinates (see below). For is the Schwarzschild metric is asymptotic to the standard Lorentz metric in the Minkowski space. For almost all astrophysical objects, the ratio is extremely small For example, the Schwarzschild radius is Earth's approximately 8.9 mm, while the Sun, 3.3 × 10 times as massive, has a Schwarzschild radius of approximately 3.0 km. The ratio will only be in close proximity to black holes and other ultra-dense objects such as neutron stars.

large. The radial coordinate has physical meaning as "reasonable distance between two events" which occur simultaneously relative to the radially moving geodetic clocks, the two events lying on the same radial coordinate line. "

The Schwarzschild solution is analogous to a classical Newtonian theory of gravity, which corresponds to the gravitational field around a point particle. Even on the surface of the earth, Newtonian gravity corrections are only a fraction of a billion.


The Schwarzschild solution was named in honor of Karl Schwarzschild, who found the exact solution in 1915 and published it in January 1916, a little over a month after Einstein's general theory of relativity was published. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Schwarzschild died shortly after his work was published as a result of an illness that he contracted while serving in the German army during the First World War. Johannes Droste independently produced the same solution as Schwarzschild in 1916 using a simpler, more direct derivation.

In the early years of general relativity, there was much confusion about the nature of the singularities used in the Schwarzschild and other solutions to the Einstein field equations

A more complete analysis of the singularity structure was given by David Hilbert the following year, with the singularities at both r = 0 and r = r s have been identified. Although there was general consensus that the singularity at r = 0 was a "real" physical singularity, the nature of the singularity remained at r = r s not clear.

In 1921 Paul Painlevé and 1922 Allvar Gullstrand independently created a metric, a spherically symmetrical solution of Einstein's equations, which we now know to be the coordinate transformation of the Schwarzschild metric Gullstrand-Painlevé coordinates

In 1950, John Synge produced a paper showing the maximum analytical extent of the Schwarzschild metric, which again shows that the singularity at r = r s was a coordinate artifact and represented two horizons. A similar result was later rediscovered by George Szekeres and independently Martin Kruskal. The new coordinates, now known as Kruskal-Szekeres coordinates, were much simpler than Synge's, but both provided a single set of coordinates that covered all of space-time. However, perhaps due to the ambiguity of the journals in which Lemaître and Synge's articles were published, their conclusions went unnoticed, and many of the key players in the field, including Einstein, believed that the singularity at the Schwarzschild radius was physical.

Real advances were made in the 1960s when the more accurate tools of differential geometry entered the field of general relativity, allowing more precise definitions of what this meant for a Lorentzian manifold that was unique. This led to the final identification of the singularity of r = r s in the Schwarzschild metric as an event horizon (a hypersurface in spacetime that can only be crossed in one direction).

Singularities and black holes

The Schwarzschild solution seems to have singularities at r = 0 and r = r s to have; Some of the metric components "blow up" at these radii (division by zero or multiplication by infinity). Since it is expected that the Schwarzschild metric is only valid for radii that are larger than the radius R of the gravitational body, there is no problem as long as R> r s is. This is always the case with ordinary stars and planets. For example, the radius of the sun is approximately 700,000 km, while the Schwarzschild radius is only 3 km.

The singularity at r = r s splits the Schwarzschild coordinates into two separate patches. The outer Schwarzschild solution with r> r s is the one related to the gravitational fields of stars and planets. The inner Schwarzschild solution with 0 ≤ r <>sthat contains the singularity at r = 0 is replaced by the singularity at r = r s completely separated from the outer spot. The Schwarzschild coordinates therefore do not result in a physical connection between the two patches, which can be viewed as separate solutions. The singularity at r = r s however, it is an illusion; It is an example of what is known as coordinate singularity. As the name suggests, the singularity results from a poor choice of coordinates or coordinate conditions. When changing to another coordinate system (e.g. Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates, Novikov coordinates or Gullstrand– Painlevé coordinates) the metric is set at r = r s regularly and can extend the external patch to values ​​of r that are smaller than r s are. Another coordinate transformation can then be used to relate the extended external patch to the inner patch.

However, the case r = 0 is different. If one asks whether the solution is valid for all r, one encounters a real physical singularity or gravitational singularity at the origin. To see that this is a true singularity, one has to consider 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 becomes infinite, which indicates the presence of a singularity. At this point in time, the metric and spacetime itself are no longer precisely defined. For a long time it was believed that such a solution was not physical. However, a better understanding of general relativity led to the realization that such singularities were a generic feature of the theory rather than just an exotic special case.

The Schwarzschild solution, which is valid for all r> 0, is called Schwarzschild black hole designated. It is a perfectly valid solution to the Einstein field equations, although (like other black holes) it has rather bizarre properties. For r <>sthe Schwarzschild radial coordinate r becomes temporal and the time coordinate t becomes spatial. A curve at constant r is no longer a possible world line of a particle or observer, even when a force is exerted to try to keep it there; This is because spacetime has been bent so much that the direction of cause and effect (the particle's future cone of light) is pointing into the singularity. The surface r = r s delimits the so-called event horizon of the black hole. It represents the point at which light can no longer escape from the gravitational field. Every physical object whose radius R is less than or equal to the Schwarzschild radius has gone through a gravitational collapse and has become a black hole.

Alternative coordinates

The Schwarzschild solution can be expressed in a number of different coordinate options in addition to the Schwarzschild coordinates used above. Different choices highlight different functions of the solution. The following table shows some popular options.

Coordinates Line element Hints features
Eddington-Finkelstein coordinates
regularly on the future horizon
- past horizon is infinite at v = -
Eddington - Finkelstein coordinates
regularly on the past horizon
extends over the past horizon.
Future horizon at u = infinite
Gullstrand-Painlevé coordinates regularly on the (+ future / past) horizon
Isotropic coordinates
Only valid outside the event horizon:
isotr opic cones of light on constant time slices
Kruskal-Szekeres coordinates regularly on the horizon
Maximizes itself maximally to the full space-time
Lemaître coordinates regularly on the future / past horizon
Harmonic coordinates

In the table above, an abbreviation has been introduced for the sake of brevity. The speed of light c was set to one. The notation

is used for the metric of a two-dimensional sphere with a unit radius. In addition, designate in each entry and alternative choices for radial and time coordinates for the particular coordinates. Notice that the value and or can vary from entry to entry.

The Kruskal-Szekeres coordinates have the shape to which the Belinski-Zakharov transform can be applied. This implies that the Schwarzschild black hole is a form of gravitational soliton.

Flamm's paraboloid

is. A representation of Flamm's Paraboloid. It shouldn't be related to the unrelated concept of a gravity well.

be confused. The spatial curvature of the Schwarzschild solution for r> r s can be displayed as shown in the graphic. Consider an equatorial layer of constant time through the Schwarzschild solution (θ = ⁄ 2 , t = constant) and let the position of a particle moving in this plane be described with the remaining Schwarzschild coordinates (r, φ). . Now imagine that there is an additional Euclidean dimension w that has no physical reality (it is not part of spacetime). Then replace the (r, φ) -plane with a surface recessed in the w-direction according to the equation (flame paraboloid)

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

Also Flamm's paraboloid is useful for visualizing the spatial curvature of the Schwarzschild metric. However, it should not be confused with a gravity well. No ordinary (massive or massless) particle can have a world line lying on the paraboloid, since all the distances on it are space-like (this is a cross-section at a given point in time, so any particle moving on it would have an infinite speed). A tachyon could have a space-like world line that lies entirely on a single paraboloid. But even in this case, its geodesic path is not the trajectory obtained by a "rubber plate" analogy of gravitational drilling: In particular, if the dimple is pointing up rather than down, the tachyon's geodesic path is still curved in direction the central mass, not gone. For more information, see the article Gravity Drilling.

Flamm's paraboloid can be derived as follows. The Euclidean metric in the cylindrical coordinates (r, φ, w) is written

Let the surface be described by the function w = w (r), the Euclidean metric can be written as

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

gives an integer expression for w (r):

whose solution is Flamm's Paraboloid.

Orbital motion

Comparison between the orbit of a test particle in space-time according to Newton (left) and Schwarzschild (right); Note the apsidal precession on the right.

A particle orbiting in the Schwarzschild metric can have a stable circular orbit with r> 3r s to have. Circular paths with r between 1.5r s and 3r s are unstable, and for r <>s there are no circular paths. The circular path with a minimum radius of 1.5r s corresponds to a rotational speed that approaches the speed of light. It is possible that a particle has a constant value of r between r s and 1.5r s but only when a force acts to hold it there.

Non-circular orbits like that of mercury linger longer at small radii than would be expected with Newtonian gravity. This can be viewed as a less extreme version of the more dramatic case in which a particle passes the event horizon and dwells in it forever. Between the fall of Mercury and the fall of an object falling beyond the event horizon, there are exotic possibilities such as razor-sharp orbits where the satellite can make any number of nearly circular orbits, after which it flies back outward.


The group of isometries of the Schwarzschild metric is the subgroup of the ten-dimensional Poincaré group, which takes the time axis (flight path of the star) for itself. The spatial translations (three dimensions) and boosts (three dimensions) are omitted. The time translations (one dimension) and rotations (three dimensions) are retained. So it has four dimensions. Like the Poincaré group, it has four interrelated components: the component of identity; the time-reversed component; the spatial inversion component; and the component that is both inverted in time and inverted in space.


The Ricci curvature scalar and the Ricci curvature tensor are both zero. Are non-zero components of the Riemann curvature tensor

Components that can be obtained from the symmetries of the Riemann tensor are not indicated.

To understand the physical meaning of these quantities, it is useful to express the curvature tensor orthonormal. In an observer's orthonormal basis, the non-zero components are in geometric units

Here, too, components that are obtainable through the symmetries of the Riemann tensor are not displayed. These results are invariable for each Lorentz boost, so the components do not change for non-static observers. The equation for geodetic deviation shows that the tidal acceleration between two observers passing by are separated so that a body of length in the radial direction by an apparent acceleration and through in vertical directions .

See also



  • Flamm, L. (1916). "Contributions to Einstein's theory of gravity". Physical journal. 17: 448.
  • Adler, R .; Bazin, M .; S. Chiffer, M. (1975). Introduction to general relativity (2nd ed.). McGraw-Hill. Chapter 6. ISBN 0-07-000423-4.
  • Landau, L. D .; Lifshitz, E.M. (1951). The classical field theory. Theoretical Physics course. 2(4th revised English edition). Pergamon press. Chapter 12. ISBN 0-08-025072-6. CS1 maintenance: ref = harv (link)
  • Misner, CW; Thorne, KS; Wheeler, JA (1970). Gravity. WH Freeman. Chapters 31 and 32. ISBN 0-7167-0344-0.
  • Weinberg, S. (1972). Gravitation and Cosmology: Principles and Applications of General Relativity. John Wiley & Sons. Chapter 8. ISBN 0-471-92567-5.
  • Taylor, EF; Wheeler, JA (2000). Exploring Black Holes: Introduction to General Relativity. Addison-Wesley. ISBN0-201-38423- X.
  • Heinzle, JM; Steinbauer, R. (2002). "Remarks on the distributional Schwarzschild geometry". Journal of Mathematical Physics. 43(3): 1493. arXiv: gr-qc / 0112047. Bibcode: 2002JMP .... 43.1493H. doi: 10.1063 / 1.1448684.