Schwarzschild metric
static  rotating  

uncharged  Schwarzschild metric  Kerr metric 
loaded  ReissnerNordström metric  KerrNewman metric 
: Electric charge ; : Angular momentum 
The Schwarzschild metric (named after Karl Schwarzschild , also known as the Schwarzschild solution ) is a solution to Einstein's field equations , which describes the gravitational field of a homogeneous, noncharged and nonrotating sphere , especially in the context of general relativity .
The complete Schwarzschild model consists of the outer Schwarzschild solution for the space outside the mass distribution and the inner Schwarzschild solution, with which the field equations inside the mass distribution are solved under the additional assumption that the mass is a homogeneous fluid. The solutions are constructed in such a way that they are continuous and differentiable at the limit of the mass distribution.
Outer solution
The outer Schwarzschild solution is the static vacuum solution of the field equations for the outer space of a spherically symmetrical distribution of matter. It also applies to dynamic mass distributions, provided that the masses only move radially and spherical symmetry is retained. It was found in 1916 by the German astronomer and physicist Karl Schwarzschild (and independently of Johannes Droste ) and was the first known exact solution of Einstein's field equations.
Line element
The Einstein field equations set the geometry of space, described by the metric tensor , on the proportionality of Einstein's gravitational constant in relation to the energymomentum tensor .
Under the boundary conditions that apply in this case, the field equations can be fundamentally integrated . With the time coordinate and the spherical coordinates as well as the replacement of the Schwarzschild radius by , the line element is:
These coordinates are known as Schwarzschild coordinates. The signs correspond to the spacetime signature mostly used in the theory of relativity .
In a natural system of units with the line element becomes
 .
In contrast to spherical coordinates in a Euclidean space , the coordinate differentials and prefactors that are dependent on have here. They are the components of the twostage metric tensor in Schwarzschild coordinates. corresponds to the gravitational central mass except for constant factors.
The physical distance between and is then not , but has the greater value
For distances that are large compared to the Schwarzschild radius , this can be developed around and results in:
As a result, a spherical shell of given circumference has a larger volume in the presence of a central mass than in the absence of the mass.
Geometric interpretation
The line element can be interpreted in two ways:
The one interpretation
If the radial coordinate line is interpreted as a real walkable path, the metric tensor contained in the line element represents a spin 2 field. The fact that this field obeys equations that can be derived from Riemannian geometry is only seen as incidental in this case.
The interface at the Schwarzschild radius is called the event horizon , the latter term also being used as a synonym for the Schwarzschild radius. At this point the radial part of the metric has a coordinate singularity , an artifact of the Schwarzschild coordinates. By choosing suitable coordinates, such as the KruskalSzekeres coordinates , this problem can be eliminated. Within the Schwarzschild radius, space and time coordinates exchange their meaning, since the radial line element becomes timelike and the previously timelike line element becomes spacelike. Movement through space becomes movement through time and vice versa.
An event horizon only exists when a large mass, such as the core of a heavy star, has contracted to an area within its Schwarzschild radius  mass outside a radius of is irrelevant. Such an object is called a black hole , which now contains a physical singularity.
The KruskalSzekeres coordinates contain solutions for a possible link to a white hole where matter can escape but not penetrate. Connections of this kind are called wormholes and the transition from a black to a white hole is called the EinsteinRosen Bridge . The Schwarzschild wormhole is a mathematical solution to Einstein's field equations, but it cannot exist because the connection is never created. Even in the case of an open connection, it collapses when approaching the singularity. It would only be stable using a speculative negative energy density.
The other interpretation
The other interpretation, which serves to illustrate the spatial curvature in the Schwarzschild solution, is based on Einstein's original conception of understanding gravity as the curvature of spacetime. The curvatures of spacetime determine the gravitational effects. For reasons of better understanding, one can imagine spacetime embedded in a higherdimensional space in order to then illustrate its curvature. For the spatial part of the Schwarzschild model, the underlying geometry can be revealed quite easily. The radial line element is an element on the (lying) parabola , with the additional dimension denoting the embedding space.
A section is considered at (and thus ) and and the metric in the remaining spatial coordinates:
A comparison of the coefficients results in the parabolic equation given above.
The guideline of the parabola lies on and on its apex. If you rotate the upper branch of the parabola around the guideline through the angle , you get a 4th order surface, the Flammian paraboloid , omitting the third spatial dimension .
In the context of this consideration, the coordinate is not a walkable path, but an auxiliary variable. This model cannot make any statements within the Schwarzschild radius, the variable has the range of values . The "hole" for , which arises on the flaming paraboloid , is covered with another surface that can be derived from the inner Schwarzschild solution.
LeviCivita context and Christoffel symbols
The LeviCivita connection , which belongs to the Schwarzschild metric, can be described by the Christoffel symbols . The Christoffel symbols different from 0 are in Schwarzschild coordinates and the natural units mentioned above ( ):
Equations of motion
The equation of motion for a particle under the influence of the central mass is the geodesic equation
The curve is parameterized by the affine parameter . For particles with a mass, this parameter can be equated with the proper time of the particle. Due to the spherical symmetry of spacetime, the motion of a particle can be investigated in the plane without loss of generality . From the expressions given above for the Christoffel symbols and the aforementioned restriction of movement to the plane, the following two equations result in addition to the equation for the second derivative of the coordinate :
Solution for particles with mass
As already described, in the case of nonvanishing mass, the derivative according to the affine parameter may be replaced by the derivative according to the proper time of the particle. This is marked with a point in the following. In addition, one can derive the useful equation from the formula for the line element that the square is the speed of four . From this one can transform the equation
derive. If this expression for the first derivative of the timelike coordinate according to the proper time of the test body is inserted into the formula for the second derivative of the coordinate according to the proper time, the result is:
This equation differs from the classical Newtonian equation by the additional term in the equation for the radial component. This has the effect that particles do not move in closed orbits around the stellar object, with the exception of the case in which the speed of revolution is the exact orbit speed :
 At distances , the relativistic centrifugal force is reduced compared to classical physics. As a result, the curvature of the path away from the exact circular path is greater than that of an elliptical path and the path does not close after one revolution. This effect has been experimentally confirmed in studies of the perihelion rotation of Mercury's orbit .
 In the case of distances , the relativistic centrifugal force has an opposite sign compared to classical physics. Seen clearly, the centrifugal force has an attractive effect in this area instead of a repulsive one. Orbits that penetrate the area of 1.5 Schwarzschild radii around the mass lead to the particle falling directly into the mass.
Conservation quantities and equations of motion
The spherical symmetry (three spacelike killing fields ) present in the Schwarzschild metric leads to the conservation of the angular momentum, the timelike killing field leads to the conservation of energy.
In order to use clear parameters, a local velocity can be introduced for each point on the particle's orbit . This speed is measured by a stationary observer with his local clock and his local scale. Due to the restriction to one movement in the plane, this speed can be broken down into a radial and an azimuthal component . These components can be calculated as follows:
 and .
The following also applies:
 .
With the help of these definitions, the constants of motion ( conservation quantities ), such as total energy and angular momentum, can be expressed as follows:
 and
The total energy of the test particle is exposed
So the rest, the kinetic and the potential energy together, whereby
 and .
The equations of motion are given by the term
as a function of the conserved quantities and the local triple velocity to:
regarding proper time : as a function of : regarding the coordinate time :
In order to continue the orbit beyond the event horizon to the central singularity, only the form with respect to proper time is suitable, since the coordinate time at diverges.
Solution for particles without mass
The solution of the equation of motion for particles without mass differs from the solutions for particles with mass in that the square is the speed of four . The conservation quantities and equations of motion therefore have the following different form in this case.
Conservation quantities and equations of motion
The motion for particles without mass, such as photons, can also be described using the conservation quantities energy and angular momentum of the particle. The local triple speed described above always corresponds to the speed of light in this case. Therefore applies here . The conserved quantities can again be expressed by the components of as follows:
stands for Planck's quantum of action and for the frequency.
The equations of motion are given by the term
as a function of the conserved quantities and the components of the local threeway velocity now to:
regarding the parameter : as a function of :
Shapiro delayed speed
The components and the shapedelayed speed in the system of a sufficiently distant stationary observer are:
It is again radial and azimuthal component of the local triplespeed. The radial component contains the square of the root term, since in addition to the gravitational time dilation there is also a radial length contraction of .
Applications
The outer Schwarzschild metric describes the gravitational field of a stellar object to a good approximation. Applied to our solar system, the calculated values for the deflection of light at the sun agree with the observations. The deviation of Mercury's perihelion rotation from the value determined with classical mechanics can also be explained using the Schwarzschild metric. For the physics inside and outside of stars, one uses the complete Schwarzschild model with the inner Schwarzschild solution for the area inside the star.
Coordinate systems
The line element in a Schwarzschild map for a static, spherically symmetric spacetime has the general form
and that for an isotropic map of a static, spherically symmetric spacetime
with the solid angle element and the coordinates
Here and are any functions of the radial coordinate . In addition to the Schwarzschild coordinates, there are therefore a number of other coordinate systems that are useful when examining different aspects of the Schwarzschild solution. In the following table, all coordinates that differ from the Schwarzschild coordinates are marked with a tilde:
Coordinates Line element comment properties Schwarzschild Areas with constant time and constant radius
are spheres (matching curvature and area).EddingtonFinkelstein
(incoming)regular at , expanded into the future, for incident light:
EddingtonFinkelstein
(expiring)regular at , extended in the past, for failed light:
GullstrandPainlevé regular at isotropic
(sphere)Cones of light for planes of constant time are isotropic. isotropic
(Cartesian)Cones of light for planes of constant time are isotropic. KruskalSzekeres regular at , extended to the entire spacetime Lemaître regular at , for incident particles:
Inner solution
The inner Schwarzschild solution describes the metric of an imaginary homogeneous sphere of liquid. The integration of the field equations is reduced to the simple linear summation of a potential (from to for a body with a radius ). For the two solutions to belong together, it is a prerequisite that the metric and its first derivatives match at the interface.
Line element
For a static, ideal fluid with constant density in the inner area of the stellar object, one obtains for the line element
a strict solution of Einstein's field equations. is the value of the radial variable at the interface of the inner solution and the outer solution, thus the value at the surface of the stellar object.
By substituting the line element in the form
write.
Geometric interpretation
The geometric methods introduced by Einstein into gravitational physics suggest that the above line element should also be interpreted geometrically. Through the coordinate transformation
you get
 .
This shows that the spatial part of the metric is the line element on a threedimensional spherical dome in fourdimensional flat space with the radius and the opening angle .
Complete Schwarzschild solution
In order to get an idea of how the complete Schwarzschild solution can be embedded in a flat space with the help of an extra dimension, one restricts oneself to the first two terms of the line elements. The outer solution is visualized by the flame paraboloid. This surface is cut off at a suitable point and a spherical cap is adjusted from below so that the tangential surfaces of both Schwarzschild surfaces coincide.
The addition of the third term in the metric results in a repetition of this consideration for a further subarea. The time part of the metric can only be understood if the factor 3 contained therein is traced back to a basic property of the parabola as the determining curve of the external solution. If the curvature vector of the parabola is extended to its guideline, the sections of the resulting line have the ratio 1: 2. Since the distance between the parabola and the guideline is at the interface , the curvature vector has the length and the entire distance there . The projection in the direction of the extra dimension is . The radius vector to any point on the spherical cap has the projection . The two lines are rotated around the imaginary angle . Two concentric imaginary (open) circles are created, the pseudoreal images of which are hyperbolas. (Imaginary circles are also called hyperbolas of constant curvature.) The distance between the circles corresponds to the expression in brackets in the above metric. As you progress on the circles around this distance sweeps over an area that is proportional to the time passed.
Conservation Law
The energymomentum tensor of the ideal static fluid has the form in Cartesian coordinates
 .
increases inward, which corresponds to the attraction of the liquid sphere to its outer parts. A look at the denominator of the pressure function shows that if the critical angle is too large, the pressure becomes infinite or the sign changes and is directed outwards. Thereby the stability of the celestial body is lost. On the other hand, the pressure function has such a steep slope that one can also use the inner Schwarzschild solution to describe exotic objects whose internal pressure is so high that the atomic or even the elementary structure of matter collapses. Under no circumstances can a hemisphere be adapted to the event horizon . Therefore, no black holes can be described in the context of the complete Schwarzschild solution .
The energy density
corresponds to the factor of the matter density and is constant, which expresses the incompressibility of the liquid. With the continuity equation
where stands for the covariant derivative , it can be shown that pressure and energy density are conserved covariantly. From the structure of one obtains
 .
The pressure increase inwards is due to the gravity of the gravitational field
certainly. Pressure and energy density are constant over time. The inner Schwarzschild solution is therefore an attempt to geometrize matter.
Generalizations to Other Metrics
The Schwarzschild metric can be generalized by adding further phenomena such as electrical charge , angular momentum or extra dimensions .
An exact solution of Einstein's field equations for the addition of angular momentum is the Kerr metric , which represents a vacuum solution of rotating, but uncharged black holes. If one continues to consider static (vanishing angular momentum), but electrically charged black holes, the exact solution is the ReissnerNordström metric . The KerrNewman metric is an exact solution for both rotating and electrically charged black holes in four dimensions.
The simplest exact solution of Schwarzschildlike black holes in extra (spatial) dimensions (so that overall dimensions are used) is the SchwarzschildTangherlini metric . It also represents the solution to the electrically neutral, static problem.
Another generalization for the case of temporally nonconstant mass (e.g. due to Hawking radiation ) is the Vaidya metric .
literature
 Original work
 Karl Schwarzschild : About the gravitational field of a mass point according to Einstein's theory . Session reports of the Royal Prussian Academy of Sciences. Reimer, Berlin 1916, pp. 189196.
 Karl Schwarzschild: About the gravitational field of a ball made of incompressible liquid . Session reports of the Royal Prussian Academy of Sciences. Reimer, Berlin 1916, pp. 424434.
 Ludwig Flamm : Contributions to Einstein's theory of gravity. In: Physikalische Zeitschrift. Volume 17, 1916, p. 448.
 further reading
 Arthur Stanley Eddington : The mathematical theory of relativity. Chelsea Publications, New York 1975, ISBN 0828402787 .
 Stephen Hawking & George FR Ellis : The large scale structure of spacetime. CUP, Cambridge 2006, ISBN 0521099064 (reprinted Cambridge 1973 edition).
 Pascual Jordan : Gravity and Space. Vieweg, Braunschweig 1955.
 Max von Laue : The general theory of relativity (= The theory of relativity. 2). Vieweg, Braunschweig 1965.
 Christian Møller : The theory of relativity. OUP, Oxford 1972, ISBN 0198512562 .
 Wolfgang Rindler : Essential relativity. Special, general and cosmological. Springer, Berlin 1997, ISBN 3540100903 .

Ya. B. Zel'dovich , ID Novikov : Relativistic Astrophysics. The University of Chicago Press, Chicago, Ill.
 1. Vol. Stars and relativity. 1978, ISBN 0226979563 .
 2. Vol. The structure and evolution of the universe. 1983, ISBN 0226979571 .
 John Lighton Synge : Relativity. The general theory. NorthHolland Publishing Company, Amsterdam 1976, ISBN 072040066X .
 Richard C. Tolman : Relativity, thermodynamics and cosmology. Dover Publications, New York 1987, ISBN 0486653838 (reprinted Oxford 1934 edition).
 Kip S. Thorne : Black Holes and Time Warps: Einstein's Outrageous Legacy.
Individual evidence
 ↑ Torsten Fließbach : General Theory of Relativity . 7th edition. Springer Spectrum, 2016, ISBN 9783662531051 .
 ↑ UE Schröder: Gravitation . Introduction to general relativity. 4th edition. Harri Deutsch, 2007, ISBN 9783817117987 , pp. 103 .
 ↑ Sean Carroll : Lecture Notes on General Relativity. Chapter 7, Equation 7.29.
 ↑ Sean Carroll : Lecture Notes on General Relativity. Chapter 7, equation 7.33.
 ↑ Leonard Susskind : General Relativity Lecture October 4th, 15, 2012 (Youtube timestamp 34m18s ).
 ^ David Lerner: Geodesics and curvature.
 ↑ Misner, Thorne & Wheeler: Gravitation. P. 166, § 6.2 ff. , Accessed on April 29, 2017 .
 ^ ^{A } ^{b} Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics 10.
 ↑ GM Clemence: The Relativity Effect in Planetary Motions . In: Reviews of Modern Physics . 19, No. 4, 1947, pp. 361364. bibcode : 1947RvMP ... 19..361C . doi : 10.1103 / RevModPhys.19.361 .
 ^ Marek Abramowicz: Centrifugalforce reversal near a Schwarzschild black hole . In: Monthly Notices of the Royal Astronomical Society . 245, No. 4, August 15, 1990, p. 720. bibcode : 1990MNRAS.245..720A .
 ↑ LD Landau, EM Lifschitz, Volume 2, Classical Field Theory, 12th Edition, 1997, page 298 f.
 ↑ ^{a } ^{b} Torsten Fließbach : General Theory of Relativity. S. 139, 4th edition, Spektrum Akademischer Verlag, 2003, ISBN 3827413567 .
 ^ Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics 09.
 ↑ Jose Wudka: precession of the perihelion of Mercury. (No longer available online.) Archived from the original on August 13, 2011 ; accessed on March 20, 2017 (English). Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.
 ↑ WeiTou Ni (Ed.): One Hundred Years of General Relativity. From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity . tape 1 . World Scientific, ISBN 9814635146 , pp. I126 ( limited preview in Google Book search).
 ^ Mike Georg Bernhardt: Relativistic stars. (PDF, 751kB) Equation 2.58. (No longer available online.) In: Homepage of the author. Max Planck Institute for Extraterrestrial Physics , October 18, 2010, p. 22 , archived from the original .
 ↑ Torsten Fließbach : General Theory of Relativity. P. 238, 4th edition, Spektrum Akademischer Verlag, 2003, ISBN 3827413567 .
 ^ Mei Xiaochun: The Precise Inner Solutions of Gravity field Equations of Hollow and Solid Spheres and the Theorem of Singularity . doi : 10.4236 / ijaa.2011.13016 .