Schwarzschild metric

Black Hole Metrics

	static ${\displaystyle (J=0)}$	rotating ${\displaystyle (J\neq 0)}$
uncharged ${\displaystyle (Q=0)}$	Schwarzschild metric	Kerr metric
loaded ${\displaystyle (Q\neq 0)}$	Reissner-Nordström metric	Kerr-Newman metric
${\displaystyle Q}$: Electric charge ; : Angular momentum ${\displaystyle \ J}$

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, non-charged and non-rotating 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 energy-momentum tensor . ${\displaystyle g_{\mu \nu }}$

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: ${\displaystyle x^{0}\mapsto t}$ ${\displaystyle (x^{1},x^{2},x^{3})\mapsto (r,\theta ,\phi )}$ ${\displaystyle 2GM/c^{2}}$ ${\displaystyle r_{\mathrm {s} }}$

${\displaystyle \mathrm {d} s^{2}=g_{\mu \nu }\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }=-c^{2}\left(1-{\frac {r_{\mathrm {s} }}{r}}\right)\mathrm {d} t^{2}+{\frac {1}{1-{\frac {r_{\mathrm {s} }}{r}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\mathrm {d} \phi ^{2}}$

These coordinates are known as Schwarzschild coordinates. The signs correspond to the space-time signature mostly used in the theory of relativity . ${\displaystyle \textstyle \left(-,+,+,+\right)}$

In a natural system of units with the line element becomes ${\displaystyle G=c=1,r_{\mathrm {s} }=2M}$

${\displaystyle \mathrm {d} s^{2}=-\left(1-{\frac {2M}{r}}\right)\mathrm {d} t^{2}+{\frac {1}{1-{\frac {2M}{r}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}}$.

In contrast to spherical coordinates in a Euclidean space , the coordinate differentials and pre-factors that are dependent on have here. They are the components of the two-stage metric tensor in Schwarzschild coordinates. corresponds to the gravitational central mass except for constant factors. ${\displaystyle \mathrm {d} t^{2}}$ ${\displaystyle \mathrm {d} r^{2}}$ ${\displaystyle r}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle M}$

The physical distance between and is then not , but has the greater value ${\displaystyle \Delta R}$ ${\displaystyle r_{1}}$ ${\displaystyle r_{2}}$ ${\displaystyle r_{2}-r_{1}}$

${\displaystyle \Delta R=\int _{r_{1}}^{r_{2}}{\frac {\mathrm {d} r}{\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}}=\left.\left(r{\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}+{\frac {r_{\mathrm {s} }}{2}}\ln {\frac {1+{\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}}{1-{\sqrt {1-{\frac {r_{\mathrm {s} }}{r}}}}}}\right)\right|_{\displaystyle {{r=}\ r_{1}}}^{\displaystyle {{r=}\ r_{2}}}.}$

For distances that are large compared to the Schwarzschild radius , this can be developed around and results in: ${\displaystyle r}$ ${\displaystyle r_{\mathrm {s} }}$ ${\displaystyle {\frac {r_{\mathrm {s} }}{r}}\approx 0}$

${\displaystyle \Delta R=r_{2}-r_{1}+{\frac {r_{\mathrm {s} }}{2}}\left(\ln {\frac {r_{2}}{r_{1}}}+{\mathcal {O}}\left({\frac {r_{\mathrm {s} }}{r_{2}}}\right)+{\mathcal {O}}\left({\frac {r_{\mathrm {s} }}{r_{1}}}\right)\right)}$

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

Outer Schwarzschild solution (flame paraboloid)

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. ${\displaystyle r}$

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 Kruskal-Szekeres coordinates , this problem can be eliminated. Within the Schwarzschild radius, space and time coordinates exchange their meaning, since the radial line element becomes time-like and the previously time-like line element becomes space-like. 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. ${\displaystyle 2M}$ ${\displaystyle r=0}$

The Kruskal-Szekeres 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 Einstein-Rosen 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.

Mathematical plot of a Schwarzschild wormhole

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 space-time. The curvatures of spacetime determine the gravitational effects. For reasons of better understanding, one can imagine space-time embedded in a higher-dimensional 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. ${\displaystyle w^{2}=4r_{\mathrm {s} }(r-r_{\mathrm {s} })}$ ${\displaystyle w}$

A section is considered at (and thus ) and and the metric in the remaining spatial coordinates: ${\displaystyle t={\text{const}}}$ ${\displaystyle \mathrm {d} t=0}$ ${\displaystyle \theta ={\tfrac {\pi }{2}}}$

${\displaystyle {\mathrm {d} s^{2}=\mathrm {d} w^{2}+\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \phi ^{2}=\left(1+\left({\frac {\mathrm {d} w}{\mathrm {d} r}}\right)^{2}\right)\,\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \phi ^{2}={\frac {\mathrm {d} r^{2}}{1-{\frac {r_{\mathrm {s} }}{r}}}}+r^{2}\,\mathrm {d} \phi ^{2}}}$

A comparison of the coefficients results in the parabolic equation given above. ${\displaystyle {\left({\frac {\mathrm {d} w}{\mathrm {d} r}}\right)}^{2}={\frac {r_{\mathrm {s} }}{r-r_{\mathrm {s} }}}}$

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 . ${\displaystyle r=0}$ ${\displaystyle w(r)}$ ${\displaystyle r=r_{\mathrm {s} }=2M}$ ${\displaystyle (w>0)}$ ${\displaystyle \theta }$

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. ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle \left[{r_{\mathrm {s} },\infty }\right[}$ ${\displaystyle r<r_{\mathrm {s} }}$

Levi-Civita context and Christoffel symbols

The Levi-Civita 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 ( ): ${\displaystyle \nabla }$ ${\displaystyle \Gamma _{ij}^{m}}$ ${\displaystyle G=c=1,r_{\mathrm {s} }=2M}$

${\displaystyle {\begin{array}{lll}\mathbf {t} &&&&&\mathbf {\theta } &&\\&\Gamma _{tr}^{t}=\Gamma _{rt}^{t}\!\!\!&=\;{\frac {M}{r(r-2M)}}&&&&\Gamma _{r\theta }^{\theta }=\Gamma _{\theta r}^{\theta }\!\!\!&=\;{\frac {1}{r}}\\\mathbf {r} &&&&&&\Gamma _{\phi \phi }^{\theta }&=-\sin \theta \,\cos \theta \\&\Gamma _{rr}^{r}&=-{\frac {M}{r(r-2M)}}&&&\mathbf {\phi } &&\\&\Gamma _{tt}^{r}&=\;{\frac {M(r-2M)}{r^{3}}}&&&&\Gamma _{r\phi }^{\phi }=\Gamma _{\phi r}^{\phi }\!\!\!&=\;{\frac {1}{r}}\\&\Gamma _{\phi \phi }^{r}&=-(r-2M)\,\sin ^{2}\theta &&&&\Gamma _{\theta \phi }^{\phi }=\Gamma _{\phi \theta }^{\phi }\!\!\!&=\;\cot \theta \\&\Gamma _{\theta \theta }^{r}&=-(r-2M)\end{array}}}$

Equations of motion

The equation of motion for a particle under the influence of the central mass is the geodesic equation

${\displaystyle {\frac {d^{2}x^{m}}{d\lambda ^{2}}}+\Gamma _{ij}^{m}{\frac {dx^{i}}{d\lambda }}{\frac {dx^{j}}{d\lambda }}=0.}$

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 : ${\displaystyle \lambda }$ ${\displaystyle (r,\phi )}$ ${\displaystyle (r,\phi )}$ ${\displaystyle t}$

${\displaystyle {\frac {\mathrm {d} ^{2}\phi }{\mathrm {d} \lambda ^{2}}}=-{\frac {2}{r}}\,{\frac {\mathrm {d} r}{\mathrm {d} \lambda }}{\frac {\mathrm {d} \phi }{\mathrm {d} \lambda }}}$

${\displaystyle {\frac {\mathrm {d} ^{2}r}{\mathrm {d} \lambda ^{2}}}={\frac {M}{r\,(r-2M)}}\left({\frac {\mathrm {d} r}{\mathrm {d} \lambda }}\right)^{2}\,-\,{\frac {M\,(r-2M)}{r^{3}}}\left({\frac {\mathrm {d} t}{\mathrm {d} \lambda }}\right)^{2}\,+\,(r-2M)\left({\frac {\mathrm {d} \phi }{\mathrm {d} \lambda }}\right)^{2}}$

Solution for particles with mass

Trajectory of a test body with mass
• left: solution of classical Newtonian mechanics
• right: Schwarzschild solution of relativity theory

As already described, in the case of non-vanishing 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 ${\displaystyle m\neq 0}$ ${\displaystyle \lambda }$ ${\displaystyle \tau }$ ${\displaystyle (\mathrm {d} s/\mathrm {d} \tau )^{2}=-1}$

${\displaystyle {\dot {t}}={\sqrt {\frac {1+r\ {\dot {r}}^{2}/(r-2M)+r^{2}\ {\dot {\phi }}^{2}}{1-2M/r}}}}$

derive. If this expression for the first derivative of the time-like 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: ${\displaystyle t}$ ${\displaystyle r}$

${\displaystyle {\ddot {r}}=\underbrace {-{\frac {M}{r^{2}}}+r{\dot {\phi }}^{2}} _{\text{klass. Term}}\underbrace {-3M{\dot {\phi }}^{2}} _{\text{rel. Zusatzterm}}}$

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 . ${\displaystyle \textstyle r>3M}$ ${\displaystyle \textstyle (r-3M){\dot {\phi }}^{2}}$ ${\displaystyle \textstyle {\dot {\phi }}^{2}={\frac {M}{r^{2}(r-3M)}}}$
• 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. ${\displaystyle \textstyle r<3M}$ ${\displaystyle \textstyle (r-3M){\dot {\phi }}^{2}}$

Conservation quantities and equations of motion

The spherical symmetry (three space-like killing fields ) present in the Schwarzschild metric leads to the conservation of the angular momentum, the time-like 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: ${\displaystyle v}$ ${\displaystyle (r,\phi )}$ ${\displaystyle v^{\parallel }}$ ${\displaystyle v^{\perp }}$

${\displaystyle v^{\parallel }=\left(1-2M/r\right)^{-1}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\right)\;}$and . ${\displaystyle \;v^{\perp }={\frac {r\;\left({\frac {\mathrm {d} \phi }{\mathrm {d} t}}\right)}{\sqrt {1-2M/r}}}}$

The following also applies:

${\displaystyle v^{2}=\left(v^{\parallel }\right)^{2}\;+\;\left(v^{\perp }\right)^{2}}$.

With the help of these definitions, the constants of motion ( conservation quantities ), such as total energy and angular momentum, can be expressed as follows: ${\displaystyle E}$ ${\displaystyle L}$

${\displaystyle L={\frac {mv^{\perp }r}{\sqrt {1-v^{2}}}}\;}$ and ${\displaystyle \;E=m\;{\sqrt {\frac {1-2M/r}{1-v^{2}}}}}$

The total energy of the test particle is exposed ${\displaystyle E}$

${\displaystyle E=m+E_{\rm {kin}}+E_{\rm {pot}}=m\;{\sqrt {\frac {1-2M/r}{1-v^{2}}}},}$

So the rest, the kinetic and the potential energy together, whereby

${\displaystyle E_{\rm {kin}}=m{\sqrt {1-2M/r}}\left({\frac {1}{\sqrt {1-v^{2}}}}-1\right)\;}$and . ${\displaystyle \;E_{\rm {pot}}=m\left({\sqrt {1-2M/r}}-1\right)}$

The equations of motion are given by the term

${\displaystyle \xi :={\sqrt {{\frac {E^{2}}{m^{2}}}-\left(1-{\frac {2M}{r}}\right)\left({\frac {L^{2}}{m^{2}\,r^{2}}}+1\right)}}}$

as a function of the conserved quantities and the local triple velocity to: ${\displaystyle v}$

regarding proper time : ${\displaystyle \tau }$	as a function of : ${\displaystyle v}$	regarding the coordinate time : ${\displaystyle t}$
${\displaystyle {\dot {r}}=\xi }$	${\displaystyle {\dot {r}}=v^{\parallel }{\sqrt {\frac {1-2M/r}{1-v^{2}}}}}$	${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} t}}={\frac {\xi m}{E}}\;(1-2M/r)}$
${\displaystyle {\dot {\phi }}={\frac {L}{m\;r^{2}}}}$	${\displaystyle {\dot {\phi }}={\frac {v^{\perp }}{r{\sqrt {1-v^{2}}}}}}$	${\displaystyle {\frac {\mathrm {d} \phi }{\mathrm {d} t}}={\frac {L}{E\;r^{2}}}\;(1-2M/r)}$
${\displaystyle {\dot {t}}={\frac {E}{m\;(1-2M/r)}}}$	${\displaystyle {\dot {t}}={\frac {1}{\sqrt {(1-v^{2})\left(1-{\frac {2M}{r}}\right)}}}}$	${\displaystyle {\frac {\mathrm {d} \tau }{\mathrm {d} t}}=1/{\dot {t}}={\frac {m}{E}}\;(1-2M/r)}$

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. ${\displaystyle r=2M}$

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. ${\displaystyle (\mathrm {d} s/\mathrm {d} \lambda )^{2}=0}$

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: ${\displaystyle E}$ ${\displaystyle L}$ ${\displaystyle v=1}$ ${\displaystyle v}$

${\displaystyle L=hfv^{\perp }r}$

${\displaystyle E=hf{\sqrt {1-2M/r}}}$

${\displaystyle h}$stands for Planck's quantum of action and for the frequency. ${\displaystyle f}$

The equations of motion are given by the term

${\displaystyle \xi :={\sqrt {E^{2}-\left(1-2M/r\right)\left({\frac {L^{2}}{r^{2}}}\right)}}}$

as a function of the conserved quantities and the components of the local three-way velocity now to: ${\displaystyle v}$

regarding the parameter : ${\displaystyle \lambda }$	as a function of : ${\displaystyle v}$
${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} \lambda }}=\xi }$	${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} \lambda }}=E\;v^{\parallel }}$
${\displaystyle {\frac {\mathrm {d} \phi }{\mathrm {d} \lambda }}={\frac {L}{r^{2}}}}$	${\displaystyle {\frac {\mathrm {d} \phi }{\mathrm {d} \lambda }}={\frac {E\;v^{\perp }}{r{\sqrt {1-2M/r}}}}}$
${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} \lambda }}={\frac {E}{1-2M/r}}}$

Shapiro delayed speed

The components and the shape-delayed speed in the system of a sufficiently distant stationary observer are: ${\displaystyle u_{\perp }}$ ${\displaystyle u_{\parallel }}$ ${\displaystyle u}$

${\displaystyle u_{\perp }=v^{\perp }{\sqrt {1-2M/r}}}$

${\displaystyle u_{\parallel }\;=v^{\parallel }\;\;(1-2M/r)}$

It is again radial and azimuthal component of the local triple-speed. 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 . ${\displaystyle v^{\parallel }}$ ${\displaystyle v^{\perp }}$ ${\displaystyle {\sqrt {1-2M/r}}}$

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

${\displaystyle \mathrm {d} s^{2}=-f(r)^{2}\,\mathrm {d} t^{2}+g(r)^{2}\,\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \Omega ^{2}}$

and that for an isotropic map of a static, spherically symmetric spacetime

${\displaystyle \mathrm {d} s^{2}=-f(r)^{2}\,\mathrm {d} t^{2}+h(r)^{2}\left(\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \Omega ^{2}\right)}$

with the solid angle element and the coordinates ${\displaystyle \mathrm {d} \Omega ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}}$

${\displaystyle -\infty <t<\infty ,\quad r_{1}<r<r_{2},\quad 0<\theta <\pi ,\quad -\pi <\phi <\pi .}$

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: ${\displaystyle f,g}$ ${\displaystyle h}$ ${\displaystyle r}$

Coordinates	Line element	comment	properties
Schwarzschild	${\displaystyle -\left(1-{\frac {r_{\rm {s}}}{r}}\right)\mathrm {d} t^{2}+{\frac {1}{1-{\frac {r_{\rm {s}}}{r}}}}\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \Omega ^{2}}$		Areas with constant time and constant radius are spheres (matching curvature and area).
Eddington-Finkelstein (incoming)	${\displaystyle -\left(1-{\frac {r_{\rm {s}}}{r}}\right)\,\mathrm {d} {\tilde {v}}^{2}+2\,\mathrm {d} {\tilde {v}}\,\mathrm {d} r+r^{2}\,\mathrm {d} \Omega ^{2}}$	${\displaystyle \textstyle {\tilde {v}}=t+\left(r+r_{\rm {s}}\ln \left|{\frac {r}{r_{\rm {s}}}}-1\right|\right)}$	regular at , expanded into the future, for incident light: ${\displaystyle r=r_{\rm {s}}}$ ${\displaystyle \mathrm {d} {\tilde {v}}=0}$
Eddington-Finkelstein (expiring)	${\displaystyle -\left(1-{\frac {r_{\rm {s}}}{r}}\right)\,\mathrm {d} {\tilde {u}}^{2}-2\,\mathrm {d} {\tilde {u}}\,\mathrm {d} r+r^{2}\,\mathrm {d} \Omega ^{2}}$	${\displaystyle \textstyle {\tilde {u}}=t-\left(r+r_{\rm {s}}\ln \left|{\frac {r}{r_{\rm {s}}}}-1\right|\right)}$	regular at , extended in the past, for failed light: ${\displaystyle r=r_{\rm {s}}}$ ${\displaystyle \mathrm {d} {\tilde {u}}=0}$
Gullstrand-Painlevé	${\displaystyle -\left(1-{\frac {r_{\rm {s}}}{r}}\right)\,\mathrm {d} {\tilde {t}}^{2}+2{\sqrt {\frac {r_{\rm {s}}}{r}}}\,\mathrm {d} {\tilde {t}}\,\mathrm {d} r+\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \Omega ^{2}}$	${\displaystyle \textstyle {\tilde {t}}=t+2{\sqrt {rr_{\rm {s}}}}-r_{\rm {s}}\ln \left|{\frac {{\sqrt {\frac {r}{r_{\rm {s}}}}}+1}{{\sqrt {\frac {r}{r_{\rm {s}}}}}-1}}\right|}$	regular at ${\displaystyle r=r_{\rm {s}}}$
isotropic (sphere)	${\displaystyle \textstyle -\left({\frac {1-{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}}{1+{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}}}\right)^{\!\!2}\mathrm {d} t^{2}+\left(1+{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}\right)^{\!4}\left(\mathrm {d} {\tilde {\rho }}^{2}+{\tilde {\rho }}^{2}\mathrm {d} \Omega ^{2}\right)}$	${\displaystyle r=\left(1+{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}\right)^{2}{\tilde {\rho }}}$	Cones of light for planes of constant time are isotropic.
isotropic (Cartesian)	${\displaystyle \textstyle -\left({\frac {1-{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}}{1+{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}}}\right)^{\!\!2}\mathrm {d} t^{2}+\left(1+{\frac {r_{\rm {s}}}{4{\tilde {\rho }}}}\right)^{\!4}\left(\mathrm {d} {\tilde {x}}^{2}+\mathrm {d} {\tilde {y}}^{2}+\mathrm {d} {\tilde {z}}^{2}\right)}$	${\displaystyle {\tilde {\rho }}={\sqrt {{\tilde {x}}^{2}+{\tilde {y}}^{2}+{\tilde {z}}^{2}}}}$	Cones of light for planes of constant time are isotropic.
Kruskal-Szekeres	${\displaystyle -{\frac {4r_{\rm {s}}^{3}}{r}}e^{-{\frac {r}{r_{\rm {s}}}}}\,\left(\mathrm {d} {\tilde {T}}^{2}-\mathrm {d} {\tilde {R}}^{2}\right)+r^{2}\,\mathrm {d} \Omega ^{2}}$	${\displaystyle {\tilde {T}}^{2}-{\tilde {R}}^{2}=\left(1-{\frac {r}{r_{\rm {s}}}}\right)e^{\frac {r}{r_{\rm {s}}}}}$	regular at , extended to the entire space-time ${\displaystyle r=r_{\rm {s}}}$
Lemaître	${\displaystyle -\mathrm {d} {\mathcal {\tilde {T}}}^{2}+{\frac {r_{\rm {s}}}{r}}\,\mathrm {d} {\mathcal {\tilde {R}}}^{2}+r^{2}\,\mathrm {d} \Omega ^{2}}$	${\displaystyle r=\left({\tfrac {3}{2}}({\mathcal {\tilde {R}}}-{\mathcal {\tilde {T}}})\right)^{\frac {2}{3}}r_{\rm {s}}^{\frac {1}{3}}}$	regular at , for incident particles: ${\displaystyle r=r_{\rm {s}}}$ ${\displaystyle \mathrm {d} {\tilde {R}}=0}$

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. ${\displaystyle r=0}$ ${\displaystyle r=R}$ ${\displaystyle R}$

Line element

For a static, ideal fluid with constant density in the inner area of the stellar object, one obtains for the line element ${\displaystyle \rho _{0}}$ ${\displaystyle \textstyle r<R}$

${\displaystyle \mathrm {d} s^{2}=g_{\mu \nu }\mathrm {d} x^{\mu }\mathrm {d} x^{\nu }=-\left({\frac {3}{2}}{\sqrt {1-{\frac {r_{\mathrm {s} }}{R}}}}-{\frac {1}{2}}{\sqrt {1-{\frac {r_{\mathrm {s} }r^{2}}{R^{3}}}}}\right)^{2}c^{2}\mathrm {d} t^{2}+{\frac {1}{1-{\frac {r_{\mathrm {s} }r^{2}}{R^{3}}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}}$

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. ${\displaystyle R}$

By substituting the line element in the form ${\displaystyle \textstyle {\mathcal {R}}^{2}={\frac {3c^{2}}{8\pi G\rho _{0}}}=R^{3}/r_{\mathrm {s} }}$

${\displaystyle \mathrm {d} s^{2}=-\left({\frac {3}{2}}{\sqrt {1-{\frac {R^{2}}{{\mathcal {R}}^{2}}}}}-{\frac {1}{2}}{\sqrt {1-{\frac {r^{2}}{{\mathcal {R}}^{2}}}}}\right)^{2}c^{2}\mathrm {d} t^{2}+{\frac {1}{1-{\frac {r^{2}}{{\mathcal {R}}^{2}}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\mathrm {d} \phi ^{2}}$

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

${\displaystyle r={\mathcal {R}}\sin \eta ,\qquad \mathrm {d} t=2{\mathcal {R}}\ \mathrm {d} \psi ,\qquad R={\mathcal {R}}\sin \eta _{\mathrm {g} }}$

you get

${\displaystyle \mathrm {d} s^{2}={\mathcal {R}}^{2}\mathrm {d} \eta ^{2}+{\mathcal {R}}^{2}\sin ^{2}(\eta )\,\mathrm {d} \theta ^{2}+{\mathcal {R}}^{2}\sin ^{2}(\eta )\,\sin ^{2}(\theta )\,\mathrm {d} \phi ^{2}+\left(3{\mathcal {R}}\cos \eta _{\mathrm {g} }-{\mathcal {R}}\cos \eta \right)^{2}\mathrm {d} \psi ^{2}}$.

This shows that the spatial part of the metric is the line element on a three-dimensional spherical dome in four-dimensional flat space with the radius and the opening angle . ${\displaystyle {\mathcal {R}}}$ ${\displaystyle \eta _{\mathrm {g} }}$

Complete Schwarzschild solution

Cross-section through the spherical cap and the flammable paraboloid ${\displaystyle r_{g}=R}$

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. ${\displaystyle r=R}$

The addition of the third term in the metric results in a repetition of this consideration for a further sub-area. 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 pseudo-real 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. ${\displaystyle {\mathcal {R}}}$ ${\displaystyle 2{\mathcal {R}}}$ ${\displaystyle 3{\mathcal {R}}}$ ${\displaystyle 3{\mathcal {R}}\cos \eta _{\mathrm {g} }}$ ${\displaystyle {\mathcal {R}}\cos \eta }$ ${\displaystyle \mathrm {i} \psi }$ ${\displaystyle \mathrm {d} \,\mathrm {i} \psi }$

Conservation Law

The energy-momentum tensor of the ideal static fluid has the form in Cartesian coordinates

${\displaystyle (T_{\mu \nu })={\begin{pmatrix}c^{2}\rho _{0}&0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{pmatrix}}}$.

The hydrostatic pressure

${\displaystyle p={1 \over \kappa }{\frac {3}{{\mathcal {R}}^{2}}}{\frac {\cos \eta -\cos \eta _{\mathrm {g} }}{3\cos \eta _{\mathrm {g} }-\cos \eta }},\qquad \kappa =8\pi G/c^{2}}$

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 . ${\displaystyle \eta _{\mathrm {g} }}$

The energy density

${\displaystyle c^{2}\rho _{0}={\frac {1}{\kappa }}{\frac {3}{{\mathcal {R}}^{2}}}}$

corresponds to the factor of the matter density and is constant, which expresses the incompressibility of the liquid. With the continuity equation ${\displaystyle c^{2}}$

${\displaystyle \nabla _{\mu }T^{\mu \nu }=0}$

where stands for the covariant derivative , it can be shown that pressure and energy density are conserved covariantly. From the structure of one obtains ${\displaystyle \nabla }$ ${\displaystyle T_{\mu \nu }}$

${\displaystyle \nabla _{\mu }p=\left(p+c^{2}\rho _{0}\right)E_{\mu },\qquad {\dot {p}}=0,\qquad {\dot {\rho }}_{0}=0}$.

The pressure increase inwards is due to the gravity of the gravitational field

${\displaystyle (E_{\mu })=\left(-{\frac {1}{\mathcal {R}}}{\frac {\sin \eta }{3\cos \eta _{\mathrm {g} }-\cos \eta }},0,0,0\right)}$

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 Reissner-Nordström metric . The Kerr-Newman metric is an exact solution for both rotating and electrically charged black holes in four dimensions.

The simplest exact solution of Schwarzschild-like black holes in extra (spatial) dimensions (so that overall dimensions are used) is the Schwarzschild-Tangherlini metric . It also represents the solution to the electrically neutral, static problem. ${\displaystyle n}$ ${\displaystyle D=4+n}$

Another generalization for the case of temporally non-constant 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. 189-196.
• 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. 424-434.
• 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 0-8284-0278-7 .
• Stephen Hawking & George FR Ellis : The large scale structure of space-time. CUP, Cambridge 2006, ISBN 0-521-09906-4 (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 0-19-851256-2 .
• Wolfgang Rindler : Essential relativity. Special, general and cosmological. Springer, Berlin 1997, ISBN 3-540-10090-3 .
• Ya. B. Zel'dovich , ID Novikov : Relativistic Astrophysics. The University of Chicago Press, Chicago, Ill.
  • 1. Vol. Stars and relativity. 1978, ISBN 0-226-97956-3 .
  • 2. Vol. The structure and evolution of the universe. 1983, ISBN 0-226-97957-1 .
• John Lighton Synge : Relativity. The general theory. North-Holland Publishing Company, Amsterdam 1976, ISBN 0-7204-0066-X .
• Richard C. Tolman : Relativity, thermodynamics and cosmology. Dover Publications, New York 1987, ISBN 0-486-65383-8 (reprinted Oxford 1934 edition).
• Kip S. Thorne : Black Holes and Time Warps: Einstein's Outrageous Legacy.

Individual evidence

1. ↑ Torsten Fließbach : General Theory of Relativity . 7th edition. Springer Spectrum, 2016, ISBN 978-3-662-53105-1 .
2. ↑ UE Schröder: Gravitation . Introduction to general relativity. 4th edition. Harri Deutsch, 2007, ISBN 978-3-8171-1798-7 , pp. 103 .
3. ↑ Sean Carroll : Lecture Notes on General Relativity. Chapter 7, Equation 7.29.
4. ↑ Sean Carroll : Lecture Notes on General Relativity. Chapter 7, equation 7.33.
5. ↑ Leonard Susskind : General Relativity Lecture October 4th, 15, 2012 (Youtube timestamp 34m18s ).
6. ^ David Lerner: Geodesics and curvature.
7. ↑ Misner, Thorne & Wheeler: Gravitation. P. 166, § 6.2 ff. , Accessed on April 29, 2017 . 
8. ^ ^{A b} Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics 10.
9. ↑ GM Clemence: The Relativity Effect in Planetary Motions . In: Reviews of Modern Physics . 19, No. 4, 1947, pp. 361-364. bibcode : 1947RvMP ... 19..361C . doi : 10.1103 / RevModPhys.19.361 .
10. ^ Marek Abramowicz: Centrifugal-force 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 .
11. ↑ LD Landau, EM Lifschitz, Volume 2, Classical Field Theory, 12th Edition, 1997, page 298 f.
12. ↑ ^{a b} Torsten Fließbach : General Theory of Relativity. S. 139, 4th edition, Spektrum Akademischer Verlag, 2003, ISBN 3-8274-1356-7 .
13. ^ Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics 09.
14. ↑ 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.  
15. ↑ Wei-Tou 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 981-4635-14-6 , pp. I-126 ( limited preview in Google Book search).
16. ^ 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 . 
17. ↑ Torsten Fließbach : General Theory of Relativity. P. 238, 4th edition, Spektrum Akademischer Verlag, 2003, ISBN 3-8274-1356-7 .
18. ^ 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 .