general theory of relativity

The general theory of relativity ( listen ^{? / I} ; short ART ) describes the interaction between matter (including fields ) on the one hand, and space and time on the other. She interprets gravity as a geometric property of the curved four-dimensional space - time . The foundations of the theory were largely developed by Albert Einstein , who presented the core of the theory to the Prussian Academy of Sciences on November 25, 1915 . To describe the curved spacetime he used differential geometry .

The general theory of relativity extends the special theory of relativity and Newton's law of gravity and goes over into these at sufficiently small space-time regions or mass densities and velocities. The ART has been experimentally confirmed in numerous tests and, in the form formulated by Einstein, is considered the only generally recognized theory of gravity .

Their relationship to quantum physics , the second cornerstone of modern physics in the 20th century, is not clear . Therefore, there is still no unified theory of quantum gravity .

introduction

Fundamental to the general theory of relativity is an interaction between all types of physical systems that can carry energy and momentum ("matter") and space-time with two properties:

• The energy and momentum of matter influence the geometry of space-time in which they are located. This influence can be formulated using a general term of curvature , and in the GRT space and time are described by the term spacetime curvature .
• Matter on which no force is exerted moves in space and time along a geodesic . In uncurved spaces (free of gravity), such geodesics are simple straight lines, such as in the 3-dimensional space of classical mechanics. However, while the influence of matter on movement in classical mechanics is described with the aid of a gravitational force , the ART refers exclusively to the now curved geometry of space-time. As in the special theory of relativity, the movement of an object along a certain path in space is interpreted more abstractly as a path in the four dimensions of space-time and referred to as the world line . If the movement is force-free (apart from gravitation), the world line is a time-like geodesic. However, a time-like geodesic of spacetime is generally not a straight line in three-dimensional space, but a connection between two events with a time-like distance, for which the elapsed proper time assumes an extreme value.

The first statement describes an effect of matter on space-time, the second describes the effect of space-time on the movement of matter. The presence of matter changes the geometrical relationships of space-time, from which the equations of motion of matter result. The ART considers the spatial and temporal coordinates to be equal and treats all temporal changes as a geometric problem.

history

Generalization of the equivalence principle

The classic equivalence principle, sometimes also referred to as the weak equivalence principle, goes back to the considerations of Galileo Galileo (1636/38) and experiments in the field of kinematics . The original formulation of the equivalence principle by Galileo states that all bodies, regardless of their properties, show the same fall behavior in a vacuum . That is, two bodies under the influence of gravity, which leave the same place at successive times, behave identically in the sense that they go through the same path, regardless of all other properties of the bodies such as chemical composition, size, shape and mass. The restriction to the vacuum results from the fact that otherwise friction effects and buoyancy forces play a role, which are dependent on the properties of the object. Isaac Newton formulated in his Philosophiae Naturalis Principia Mathematica (1687) the equivalence principle as equality of inertial mass and heavy mass. This means that the same mass occurs in the law of gravitation and the law of inertia.

Albert Einstein considered the equivalence principle, which was already confirmed in 1900 by the Eötvös experiment with an accuracy of 10 ⁻⁹ , to be a decisive property of gravity. Therefore Einstein extended the principle to non-mechanical phenomena and made it the starting point for his theory of gravity.

Establishing the field equations

The fundamentals of general relativity were essentially developed by Albert Einstein. He used the differential geometry developed by Carl Friedrich Gauß , Bernhard Riemann , Elwin Bruno Christoffel , Gregorio Ricci-Curbastro and Tullio Levi-Civita , as he had learned from Marcel Grossmann , a mathematician friend. Using this differential geometry, he formulated in space-time - which Hermann Minkowski had introduced for the special theory of relativity - gravity as a property of the proportions. Considerations of Ernst Mach influenced Einstein's belief that even under the influence of gravity only movement was physically considerably relative to other bodies.

The first publication that can be assigned to the general theory of relativity is a work by Einstein published in 1908 on the influence of gravitation and acceleration on the behavior of light in the special theory of relativity. In this work he already formulates the equivalence principle and predicts the gravitational time dilation and redshift as well as the deflection of light by massive bodies. The main part of the theory was only developed by Einstein between 1911 and 1915. The beginning of his work was marked by a second publication on the effect of gravity on light in 1911, in which Einstein revised his publication from 1908.

Before completing the work, Einstein published a draft for the theory of relativity in 1913 that already used a curved spacetime. However, due to problems with the principle of general covariance, which ultimately turned out to be correct, Einstein subsequently took a wrong approach before he was finally able to solve the problem in 1915. During his work he also gave lectures on this and exchanged ideas with mathematicians, namely with Marcel Grossmann and David Hilbert .

In October 1915 Einstein published a paper on the perihelion of Mercury, in which he still assumed incorrect field equations that were incompatible with the local conservation of energy and momentum. In November 1915 Einstein found the correct field equations and published them in the meeting reports of the Prussian Academy of Sciences on November 25, 1915 together with the calculation of the perihelion of Mercury and the deflection of light in the sun. Hilbert submitted his work to the Göttingen Royal Society of Sciences for publication five days earlier. However , unlike the version published later , the proofs of Hilbert's work do not contain the field equations - the proofs are not completely preserved. However, there never was a - occasionally claimed - priority dispute between Hilbert and Einstein, since Hilbert had only solved a computational aspect with the help of tensor analysis , which he had better mastered and which Einstein had to familiarize himself with.

Einstein's later article, The Basis of General Relativity, can be seen as the first review article of the GTR. It was published in the Annalen der Physik on March 20, 1916 , two months after Einstein presented Schwarzschild's solution to his field equations to the Prussian Academy of Sciences.

The action functional of the GTR goes back to Hilbert, from which he derived the field equations in his 1916 article.

Basic concepts

The starting points of the GTR can be formulated as three basic principles: the general principle of relativity , the principle of equivalence and the Mach principle .

The theory does not necessarily follow from these premises, and at least with Mach's principle it is unclear whether the GTR actually fulfills it. However, the three principles explain which physical problems prompted Einstein to formulate the GR as a new theory of gravity.

The description of the spacetime curvature is logically based on the equivalence principle, which is why it is also dealt with in this chapter.

Relativity Principle

→ Main article : Principle of relativity

In the general theory of relativity, an expanded relativity principle is assumed compared to the special theory of relativity : The laws of physics not only have the same form in all inertial systems , but also in relation to all coordinate systems . This applies to all coordinate systems that assign four parameters to each event in space and time, these parameters being sufficiently differentiable functions of the locally definable Cartesian coordinates in small space-time areas that obey the special theory of relativity . This requirement of the coordinate system is necessary so that the methods of differential geometry can be used for curved spacetime. A curved spacetime can generally no longer be described globally with a Cartesian coordinate system . The extended relativity principle is also called general coordinate covariance .

The coordinate covariance is a requirement for the formulation of equations (field equations, equations of motion) that should be valid in the ART. However, the special theory of relativity can already be formulated in a generally covariant manner. For example, even an observer on a rotating swivel chair can take the position that he himself is at rest and that the cosmos is rotating around him. This creates the paradox that the stars and the light they emit move arithmetically faster than light in the coordinate system of the rotating observer, which apparently contradicts the special theory of relativity. The resolution of this paradox is that the general covariant description is local by definition. This means that the constancy of the speed of light only has to apply close to the world line of the observer , which is just as true for the rotating observer as it is for every other observer. The covariant equations, in the sense of the general principle of relativity, result in circular movements faster than light for the stars, but are nevertheless in accordance with the principles of the special theory of relativity. This is also made clear by the fact that it is impossible for an observer to rest near a star in the rotating coordinate system and thus encounter the star at faster than light speed. This observer therefore necessarily has a different coordinate system than the rotating observer and measures the "correct" speed of light.

Although it is possible to correctly describe the cosmos from the perspective of a rotating observer, the equations of a frame of reference in which most objects rest or move slowly are usually simpler. The condition of a non-rotating coordinate system for inertial systems and the differentiation in their consideration, which is required by classical physics, is in principle omitted.

In the case of a multi-body system in a narrow space, space-time is significantly curved and this curvature is also variable over time in every coordinate system. Therefore, from the outset, no candidate for an excellent coordinate system that is suitable for describing all phenomena can be identified. The principle of relativity says for this general case that it is not necessary to search for it either, because all coordinate systems are equal. Depending on which phenomenon you want to describe, you can compare different coordinate systems and choose the most computationally simplest model.

Therefore, the GTR can also dispense with the classic astronomical concept of the appearance of movements, which the heliocentric worldview still entrenched in the Newtonian view required.

Mach's principle

→ Main article : Mach's principle

Einstein was strongly influenced by Ernst Mach in the development of the theory of relativity . In particular, the assumption, referred to by Einstein as Mach's principle , that the inertial forces of a body do not depend on its movement relative to an absolute space, but on its movement relative to the other masses in the universe, was an important working basis for Einstein. According to this view, the inertial forces are the result of the interaction of the masses with one another, and a space that exists independently of these masses is denied. According to this, for example, centrifugal forces from rotating bodies should disappear when the rest of the universe "rotates" with it.

This rather general formulation of Mach's principle, preferred by Einstein, is only one of many non-equivalent formulations. Therefore, the Mach principle and its relationship to ART is still controversial today. For example, in 1949 Kurt Gödel found a universe possible according to the laws of GART, the so-called Gödel universe , which contradicts some specific formulations of Mach's principle. However, there are other specific formulations of the principle that the Godel universe does not run counter to. Astronomical observations, however, show that the real universe differs greatly from Gödel's model.

Einstein saw the lens-thirring effect , which the ART predicted, as an affirmation of his version of Mach's principle. The consequence of this effect is that reference systems experience a precession within a rotating hollow sphere with mass , which Einstein interpreted to mean that the mass of the sphere has an influence on the inertial forces. However, since a “resting” reference system in the form of a fixed star sky was assumed for the calculation and the interpretation, this interpretation is also controversial.

The general version of the Mach principle that Einstein formulated is too imprecise to be able to decide whether it is compatible with the GTR.

Equivalence principle

→ Main article : Principle of equivalence (physics)

According to the principle of equivalence, one cannot decide within a windowless room whether it is in the gravitational field of a planet or whether it is accelerated like a rocket in space.

The principle of the equivalence of inert and heavy mass was already known in classical mechanics . In its classic form, which is also known as the weak equivalence principle, it says that the heavy mass, which indicates how strong the force generated by a gravitational field on a body, and the inertial mass, which is determined by the law of force, how strong a Body is accelerated by a force, are equivalent. This means in particular that every body moves in the same way regardless of its mass in a gravitational field (in the absence of other forces). (Charged bodies are excluded from this because of the synchrotron radiation .) For example, in a vacuum all (uncharged) bodies fall at the same speed, and the geostationary orbit is always the same for heavy satellites as for light satellites. The consequence of the classic equivalence principle is that an observer in a closed laboratory, without information from outside, cannot read from the mechanical behavior of objects in the laboratory whether he is in weightlessness or in free fall.

This principle was generalized by Einstein: Einstein's strong equivalence principle states that an observer in a closed laboratory without interaction with the environment can not determine by any experiment whether he is in weightlessness far away from masses or in free fall near a mass. This means in particular that a light beam for an observer in free fall is not curved in a parabolic shape, as in an accelerated reference system. On the other hand, an observer who rests in the gravitational field, e.g. B. while standing on the earth's surface, perceive a beam of light curved, as it is accelerated upwards against free fall all the time.

However, it must be noted that this principle only applies locally due to the tidal forces occurring in the gravitational field :

• An object located “below” (closer to the center of gravity ) is attracted more strongly than an object located further “above”. If the free falling space is large enough in the vertical direction, the observer will therefore notice that objects that are further up are removed from those that are further down.
• Conversely, with sufficient horizontal expansion of the space, the direction of the force of attraction on two horizontally distant objects will differ noticeably, since they are both accelerated in the direction of the center of gravity. Therefore, the freely falling observer will notice that bodies that are far apart are moving towards one another. An expanded body will experience a force that pulls it apart in one direction and compresses it in the perpendicular directions.

In the GTR, the equivalence principle follows directly from the description of the movement of bodies: Since all bodies move along geodesics of space-time, an observer who moves along a geodesic can only then determine a curvature of space-time, which he could interpret as a gravitational field , if the piece of space-time observable by him is significantly curved. In this case, he observes the above-mentioned tidal forces as a relative approach or distance of neighboring freely falling bodies. The curvature also ensures that charged bodies do not interact locally with their own field and therefore the principle of equivalence cannot in principle be applied to them, since their electromagnetic field is basically long-range.

Space-time curvature

See also : space curvature

Parallel transports near a massive sphere.
Blue arrows represent parallel transports in space along the x-axis.
Red arrows represent the movement in space during parallel transport along the time axis, which corresponds to a free fall.
The lengths of the parallel transports of the same type are in each case the same, i.e. Δx ₁  = Δx ₂ and Δt ₁  = Δt ₂ . With the first, upper path, the transport in the x direction is carried out first and then the transport in the time direction. In the second, lower path, the order of the parallel transports is reversed. The green double arrow illustrates the different end points when the parallel transports are swapped.

The curvature of spacetime discussed in this section is not an independent concept, but a consequence of the principle of equivalence. With the help of the equivalence principle, the concept of space-time curvature can therefore also be clearly explained. To do this, the concept of parallel transport along the time axis must first be explained.

A parallel transport is a shift in one direction in which the alignment of the person to be moved is retained, i.e. a local coordinate system is carried along. A mere shift in one spatial direction is clearly understandable in a space-time without masses. According to the special theory of relativity, the definition of time depends on the movement of the coordinate system. A constant time direction is only given for unaccelerated coordinate systems. In this case a shift in time direction in a space-time without masses means that an object is at rest relative to the coordinate system. It then moves along the time axis of this coordinate system. (The unmoved initial and final states are compared.)

According to the equivalence principle, the parallel transport along the time axis in a gravitational field can be understood. The equivalence principle states that a freely falling observer in a gravitational field is equivalent to an unaccelerated observer far away from a gravitational field. Therefore, a parallel transport along the time axis by a time interval t corresponds to a free fall of duration t. This means that a parallel shift in time also results in a movement in space. But since the direction of the free fall depends on the location, it makes a difference whether an observer is first shifted in space and then in time, or vice versa. It is said that the parallel transport is not commutative, that is, the order of the transports is important.

So far, large shifts have been considered, for which the sequence of the parallel transports is obviously important. However, it makes sense to be able to make statements about arbitrarily small areas of spacetime in order to be able to describe the behavior of bodies even for short times and distances. If the parallel transports are carried out over ever shorter distances and times, the end points for different sequences of transports are still different, but the difference is correspondingly smaller. With the help of derivatives, an infinitesimally small parallel transport at one point can be described. The measure for the deviation of the end points when the sequence of two parallel transports is reversed is then given by the so-called curvature tensor .

The above-mentioned tidal forces can also be explained by the curvature of space-time. Two spheres in free fall in a free falling laboratory both move along the time axis, i.e. on lines that are parallel to each other. The fact that the parallel transports are not commutative is equivalent to the fact that parallel lines do not have a constant distance. The paths of the balls can therefore approach or move away from one another. In the earth's gravitational field, the approach is only very small, even with a very long fall.

To describe the curvature, it is not necessary to embed space-time in a higher-dimensional space. The curvature is not to be understood as a curvature into a fifth dimension or as a curvature of space into the fourth dimension, but rather as a curvature without embedding or as a non-commutativity of parallel transports . (A premise of this representation is to treat space and time as four-dimensional spacetime. So space and time coordinates are largely analog, and there is only a subtle mathematical difference in the sign of the signature .)

The way in which space-time is curved is determined by Einstein's field equations .

Mathematical description

Basic concepts

The mathematical description of spacetime and its curvature is carried out with the methods of differential geometry , which includes and extends the Euclidean geometry of the familiar “flat” three-dimensional space of classical mechanics. Differential geometry uses so-called manifolds to describe curved spaces, such as the spacetime of GTR . Important properties are described with so-called tensors , which represent images on the manifold.

• The curved spacetime is described as a Lorentz manifold .
• The metric tensor is of particular importance . If you insert two vector fields into the metric tensor , you get a real number for each point in space-time. In this respect the metric tensor can be understood as a generalized, point-dependent scalar product for vectors of spacetime. With its help, distance and angle are defined and it is therefore briefly referred to as a metric .
• Equally important is the Riemann curvature tensor for describing the curvature of the manifold, which is a combination of first and second derivatives of the metric tensor. If any tensor in any coordinate system is not zero at a point, then one cannot find any coordinate system at all, so it becomes zero at that point. This also applies accordingly to the curvature tensor. Conversely, the curvature tensor is zero in all coordinate systems if it is zero in one coordinate system. So one will arrive at the same result in every coordinate system with regard to the question of whether a manifold is curved at a certain point or not.
• The decisive variable for describing the energy and momentum of matter is the energy-momentum tensor . How this tensor determines the curvature properties of spacetime is shown in the following section.

Einstein's field equations

→ Main article : Einstein's field equations

The Einstein field equations provide a link between certain curvature properties of space-time and the energy-momentum tensor ago, the local mass density or on the energy density contains and characterizes the relevant properties of matter. ${\displaystyle E=mc^{2}}$

These basic equations of general relativity are differential equations for the 10 independent components of the metric : ${\displaystyle g_{\mu \nu }}$

${\displaystyle R_{\mu \nu }-{\frac {R}{2}}\,g_{\mu \nu }+\Lambda \ g_{\mu \nu }={\frac {8\ \pi \ G}{c^{4}}}\ T_{\mu \nu }}$

It is the Ricci curvature , the Ricci Krümmungsskalar , the metric tensor , the cosmological constant that is also often omitted (see below), the speed of light , the gravitational constant and the energy-momentum tensor. Since all tensors in this equation are symmetrical (e.g. ), only 10 of these 16 equations are independent of each other. ${\displaystyle R_{\mu \nu }}$ ${\displaystyle R}$ ${\displaystyle g_{\mu \nu }}$ ${\displaystyle \Lambda }$ ${\displaystyle c}$ ${\displaystyle G}$ ${\displaystyle T_{\mu \nu }}$ ${\displaystyle R_{\mu \nu }=R_{\nu \mu }}$

The goal is to put the components of the energy-momentum tensor on the right hand side of the equations and then use the field equations to determine the metric . The expression on the left-hand side of the equation consists of quantities that are derived from the curvature tensor. They therefore contain derivatives of the metric you are looking for. So you get 10 differential equations for the components of the metric. However, the metric and its derivatives can usually also be found on the right-hand side of the equations in the energy-momentum tensor. To make matters worse, the sum of two solutions is generally not a solution to the field equations, so the solutions are not superposable . This is due to the non-linearity of the field equations, which is considered a major characteristic of the ART. Because of this complexity of the equations, it is often not possible to find exact solutions for the field equations. In such cases, methods for finding an approximate solution can in part be used.

The field equations do not contain the curvature tensor, but only the Ricci curvature tensor derived from it and the Ricci curvature scalar. These two summands are collectively referred to as the Einstein tensor, although this does not contain all information about the curvature of space-time. Part of the space-time curvature, the so-called Weyl curvature , is therefore not directly dependent on the energy-momentum tensor and thus on the mass and energy density. However, the Weyl curvature tensor cannot be chosen freely, since it is partly determined by the Ricci curvature tensor due to the geometric Bianchi identities . ${\displaystyle G_{\mu \nu }}$

Einstein initially believed that the universe did not change its size over time, so he introduced the cosmological constant to theoretically make such a universe possible. However, the equilibrium he achieved with this turned out to be an unstable equilibrium. Formally has the status of a kind of constant of integration, and therefore initially has no specific numerical value that would follow directly from the theory. So it has to be determined experimentally. An alternative view of the cosmological constant takes the corresponding term as part of the energy-momentum tensor and sets . This means that the cosmological constant presents itself as an ideal liquid with negative pressure and is understood as an extraordinary form of matter or energy. In today's cosmology, the term “ dark energy ” has established itself in this context . ${\displaystyle \Lambda }$ ${\displaystyle \Lambda }$ ${\displaystyle T_{\Lambda }^{\mu \nu }={\frac {c^{4}}{8\ \pi \ G}}\ \Lambda \ g^{\mu \nu }}$

The field equations indicate how the matter and energy content affects the curvature of spacetime. However, they also contain all information about the effect of space-time curvature on the dynamics of particles and fields, i.e. about the other direction of interaction. Nevertheless, one does not use the field equations directly to describe the dynamics of particles or fields, but derives the equations of motion for them. The equations of motion are therefore "technically" important, although their information content is conceptually contained in the field equations.

A particularly elegant derivation of Einstein's field equations is offered by the principle of the smallest effect , which also plays an important role in Newtonian mechanics . A suitable formula for the effect , the variation of which leads to these field equations in the context of the calculation of variations , is the Einstein-Hilbert effect , which was first given by David Hilbert .

Equations of motion

In order to be able to formulate the equations of motion, any world line of a body must be parameterized. This can be done, for example, by defining a zero point and a positive direction and then assigning the arc length from the zero point to this point with the appropriate sign to each point on the world line . This ensures that every point on the world line is clearly defined. A very similar parameterization is the parameterization according to the proper time . The two are identical if the equations are simplified by ignoring all factors c, i.e. by formally setting the speed of light . The following formulas are to be understood in arc length parameterization. ${\displaystyle c=1}$

In the following, the term “force” never describes gravity (which is understood as a geometric effect), but other forces, for example electromagnetic or mechanical forces. If one now looks at a body on which a force acts, the equations of motion are : ${\displaystyle K^{\mu }\ }$

${\displaystyle m\ {\ddot {x}}^{\mu }+m\ \Gamma _{\lambda \nu }^{\mu }\ {\dot {x}}^{\lambda }\ {\dot {x}}^{\nu }=K^{\mu }.}$

In the event that no force acts on a body, its world line is described by the geodesic equations of curved spacetime . It is obtained by setting the force in the above law of force : ${\displaystyle K^{\mu }\ =0}$

${\displaystyle {\ddot {x}}^{\mu }+\Gamma _{\lambda \nu }^{\mu }\ {\dot {x}}^{\lambda }\ {\dot {x}}^{\nu }=0.}$

Here m is the mass of the body and are the four space-time components of the world line of the body; stands for the time component. Points above the sizes are derivatives of the arc length and not the time component . ${\displaystyle \left(x^{\mu }\right)=(x^{0},\,x^{1},\,x^{2},\,x^{3})}$ ${\displaystyle x^{0}}$ ${\displaystyle x^{0}}$

${\displaystyle \textstyle \Gamma _{\lambda \nu }^{\mu }={\dfrac {g^{\mu \rho }}{2}}\left(\partial _{\lambda }\ g_{\nu \rho }+\partial _{\nu }\ g_{\lambda \rho }-\partial _{\rho }\ g_{\lambda \nu }\right)}$

is a Christoffel symbol that characterizes the dependence of the metric tensor on the space-time point, i.e. the space-time curvature. They are components of the cometric tensor, which is the inverse of the metric tensor . Abbreviations are also used in the formula: for the partial derivatives , as well as the summation convention , which means that indices that appear once at the top and once at the bottom are automatically added from 0 to 3. ${\displaystyle g^{\mu \rho }}$ ${\displaystyle g_{\nu \rho }}$ ${\displaystyle \textstyle \partial _{\mu }:={\tfrac {\partial }{\partial x^{\mu }}}}$

The law of force is a generalization of the classical principle of action ( ) on four dimensions of a curved spacetime. The equations cannot be solved until the metric tensor is known. Conversely, the metric tensor is only known when the equations have been solved for all orbits. This intrinsic requirement of self-consistency is one reason for the difficulty of the theory. ${\displaystyle {\vec {K}}=m\ {\ddot {\vec {x}}}}$

In principle, the equations of motion for a particle cloud and Einstein's field equations can now be viewed as a system of equations that describes the dynamics of a cloud of massive particles. Due to the above-mentioned difficulties in solving the field equations, however, this is not practically feasible, so that approximations are always used for multi-particle systems.

The forces that act on a body are generally calculated somewhat differently than in the special theory of relativity. Since the formulas in the ART must be written in a coordinate covariant manner, the covariant derivative must now be used in the formulas for the forces, for example in the Maxwell equations , instead of the partial derivative according to space-time components . Since the derivatives according to spacetime components describe the changes in a quantity, this means that the changes in all fields (i.e., position-dependent quantities) now have to be described in the curved spacetime. The Maxwell equations thus result in

${\displaystyle D_{\mu }\ F^{\mu \nu }=4\ \pi \ J^{\nu }}$

and

${\displaystyle D_{\mu }F_{\nu \rho }+D_{\nu }\ F_{\rho \mu }+D_{\rho }\ F_{\mu \nu }=\partial _{\mu }\ F_{\nu \rho }+\partial _{\nu }\ F_{\rho \mu }+\partial _{\rho }\ F_{\mu \nu }=0.}$

The use of the covariant derivatives therefore only affects the inhomogeneous Maxwell equations, while the homogeneous equations do not change compared to the classical form. The definitions of the covariant derivatives of tensors can be found in the article Christoffelsymbole . ${\displaystyle D_{\mu }\ }$

Metrics

After the publication of the general theory of relativity, some solution approaches or metrics with new coordinates have developed in the following decades, some of which are still used today.

Black Hole Metrics

Schwarzschild metric

The Schwarzschild metric was one of the first metrics developed after the general theory of relativity was published. Karl Schwarzschild introduces the Schwarzschild coordinates , with which the properties of special spacetime can be well described. For the first time, Schwarzschild was able to describe the gravitational field of an uncharged, non-rotating sphere whose mass was evenly distributed. The Schwarzschild metric is thus assumed to be the first description of a black hole .

Kerr metric

The Kerr metric describes rotating, uncharged objects in space-time, so it is well suited for describing rotating black holes. It was named after Roy Kerr , who developed it in 1963. In this metric there are two singular spacetime regions with rotating black holes, in the middle lies the ergosphere (described in more detail in the Kerr metric ). The special thing about this metric is that the singularity of a black hole is ring-shaped. ${\displaystyle r=0}$

Reissner-Nordström metric

The Reissner-Nordström metric describes electrically charged, static (i.e. non-rotating) black holes. Their line elements are similar to those of the Schwarzschild metric. There is a parameter  Q that describes the electrical charge.

Kerr-Newman metric

The Kerr-Newman metric describes electrically charged and rotating black holes. In the case of an electrically neutral black hole , the solutions to the simpler Kerr metric are simplified, if the angular momentum is missing, the Reissner-Nordström metric and the Schwarzschild metric are simplified . ${\displaystyle (Q=0)}$ ${\displaystyle J=0}$ ${\displaystyle J=0}$ ${\displaystyle Q=0}$

Other metrics

Godel metric

The Gödel metric was developed by Kurt Gödel in 1949. It describes a rotating spacetime on the basis of Einstein's field equations. The center of rotation is equally present at every point in space-time, this is required by the cosmological principle . One consequence of his rather mathematical model is that classical world lines can also run into the past with such a spacetime. His model caused quite a stir because it proved that Einstein's field equations allow the mathematical treatment of spacetime in which time travel is possible.

Kruskal solution

The Kruskal solution is a maximum extension of the Schwarzschild solution. It has intrinsic singularities, which is why it is not complete. The solution can be seen as a description of Einstein-Rosen bridges or wormholes.

Robertson Walker metric

The Robertson-Walker metric (also called " Friedmann-Lemaître-Robertson-Walker metric ") describes a homogeneous universe based on the cosmological principle. It is used as an approximation in some big bang theories. Since the exact model would assume that no structures such as galaxies and stars could form in the universe, a fast FLRM model is used, which can take into account small disturbances or density fluctuations.

De-sitter room

The de-sitter space is a maximally symmetrical vacuum solution of the field equations that contains a positive cosmological constant , so the space is positively curved. It can be seen as a submanifold to a higher-dimensional Minkowski space .

The de-sitter cosmos is a model that incorporates these considerations. If you choose a Friedmann solution with vanishing curvature ( ${\displaystyle k=0}$ in the Robertson-Walker metric) and without matter, the result is a flat, expanding De Sitter cosmos with a radius and the Hubble constant ${\displaystyle R(t)\sim e^{Ht}}$ ${\displaystyle H=c\cdot {\sqrt {\Lambda /3}}.}$

Therefore, most cosmologists assume that the universe in its initial phase was a de-sitter space that was expanding (see inflation ). In the distant future, however, the cosmos could again approach such a matter-free state .

Anti-de-sitter room

The anti-de-sitter space is the counterpart to the de-sitter space , so it has a negative cosmological constant and is therefore negatively curved. The room is so interesting because it has special physical properties and because it is often associated with the holographic principle and string theory .

Physical effects

→ Main article : Tests of general relativity

For the experimental verification of the GTR, it is not sufficient to carry out experiments with which one can decide between GTR and Newtonian mechanics, since there are competing theories on GTR. It is therefore necessary to decide experimentally between GTR and other theories of gravity. Deviations from the predictions of the GTR could also provide a new impetus for the development of a conclusive and experimentally verifiable quantum theory of spacetime.

The general theory of relativity correctly predicts the experimental results within the scope of the measurement accuracy. The equivalence principle is confirmed with a measurement accuracy of up to 10 ⁻¹³ , for other ART phenomena up to 10 ⁻⁵ . In the following, some physical phenomena are explained, the precise experimental verification of which has so far confirmed the GTR well and has greatly reduced the scope for alternative theories. In addition, the good agreement between experiment and prediction suggests that quantum effects of gravitation are very small, since they should be recognizable as deviations from the predictions of the GTR.

Gravitational time dilation and redshift

Gravitational redshift of a light wave

The gravitational time dilation already follows from the special theory of relativity and the equivalence principle of the GTR. It was predicted by Einstein in 1908. If you look at a clock resting in a gravitational field , it has to be kept at rest by an opposing force, like a person standing on the surface of the earth. It is thus continuously accelerated, so that one can use the formula for the time dilation in an accelerated reference system from the special theory of relativity. As a result, the effect is not symmetrical, as is known from two uniformly moving frames of reference in special relativity. An observer in space sees the clocks on earth moving more slowly than his own clock. Conversely, an observer on earth sees clocks in space moving faster than his own clock. With very precise optical atomic clocks, the gravitational time dilation can be measured even with a height difference of only a few centimeters.

A direct consequence of the time dilation is the gravitational redshift . It was predicted by Einstein as early as 1911 before the general theory of relativity was completed. Since both effects can already be derived from the principle of equivalence, their experimental confirmation in and of itself does not confirm the validity of the GTR. However, if behavior deviating from the prediction were observed, this would refute the ART. The experimental confirmation of the effects is therefore necessary for the validity of the theory, but not sufficient.

The redshift means that light that is emitted “upwards” (ie away from the center of gravity) from a light source with a given frequency is measured there with a lower frequency, similar to the Doppler effect . Accordingly, in the case of a light signal with a certain number of oscillations, the time interval between the beginning and the end of the signal is greater at the receiver than at the transmitter. The gravitational redshift was first detected in 1960 in the Pound-Rebka experiment . In 2018, the gravitational redshift of the star S2 at its closest approach to the black hole in Sagittarius A * in the center of the Milky Way was detected by the gravity collaboration.

Light deflection and light delay

→ Main articles : Gravitational lensing  and Shapiro delay

Simulation of the deflection of the light from a star (red) in the gravitational field of a neutron star (blue).

From the point of view of a distant observer, light near a large mass moves more slowly than the speed of light in a vacuum. This phenomenon is called Shapiro delay after its discoverer . In addition, a distant observer perceives a deflection of the light near large masses. These two effects go back to the same explanation. Real time, the so-called proper time , differs from the concept of time of the distant observer close to the mass. In addition, the mass also has an impact on the behavior of the space, similar to a Lorentz contraction , which can only be explained in the context of the GTR and not classically. An observer who is himself close to the mass will accordingly measure the vacuum speed of light as the speed of the light beam. However, the distant observer perceives a reduced speed, which he can describe as a location-dependent refractive index . This description also provides an explanation for the deflection of light, which can be interpreted as a type of refraction .

The above explanation is based on an analogy. The abstract interpretation within the framework of the ART is that the zero geodesics on which light moves appear curved near large masses in space. It must be taken into account that light also moves in time, so that there is actually a space-time curvature and not a pure curvature of three-dimensional space.

The deflection angle depends on the mass of the sun, the distance from the point of the orbit closest to the sun to the center of the sun and the speed of light . He can according to the equation ${\displaystyle \alpha _{\text{ablenk}}}$ ${\displaystyle M}$ ${\displaystyle r}$ ${\displaystyle c}$

${\displaystyle \alpha _{\text{ablenk}}\,=\,{\frac {4\ G\,M}{r\ c^{2}}}\,=\,{\frac {4\ r_{G}}{r}}}$

be calculated. Therein is the gravitational constant and the gravitational radius . ${\displaystyle G}$ ${\displaystyle r_{G}}$

The gravitational lensing effect observed in astronomy is also based on the deflection of light in the gravitational field .

Perihelion

→ Main article : Apsid rotation

The perihelion of the orbit of a planet. The eccentricity of the orbit and the amount of rotation are greatly exaggerated compared to the real perihelion of Mercury.

The perihelion of the planetary orbits - z. B. the orbit of the earth around the sun - is an effect that is largely caused by the gravitational force of other planets (e.g. Jupiter). Mercury measures 571 ″ per century, of which 43.3 ″ does not result from these disturbances. The theory of relativity could explain this value (by a different effective potential compared to Newtonian mechanics ), which was the first success of the theory. At 1161 ″ per century, the perihelion of the earth is even greater than that of Mercury, but the relativistic deficit for the earth is only 5 ″. The measured false contributions to the perihelion rotation of other planets as well as the minor planet Icarus agree with the predictions of the theory of relativity. The European-Japanese Mercury probe BepiColombo , which is currently being planned, will make it possible to determine the movement of Mercury with unprecedented accuracy and thus to test Einstein's theory even more precisely.

In binary systems of stars or pulsars that orbit each other at a very short distance, the perihelion rotation of several degrees per year is significantly greater than that of the planets of the solar system. The values ​​of perihelion rotation measured indirectly in these star systems also agree with the predictions of the ART.

Gravitational waves

→ Main article : Gravitational wave

A ring of test particles under the influence of a gravitational wave

Two-dimensional representation of gravitational waves that are emitted by two orbiting neutron stars .

The ART enables the description of fluctuations in space-time curvature that propagate at the speed of light. As a first approximation, these fluctuations are comparable to transverse waves, therefore they are called gravitational waves. A description of this phenomenon without approximations does not yet exist (2016). Gravitational waves would be observable in that the space periodically expands and contracts transversely to their direction of propagation. Since there is no positive and negative charge in gravitation as in electromagnetism , gravitational waves cannot occur as dipole radiation, but only as quadrupole radiation. In addition, the coupling of gravity to matter is much weaker than in electromagnetism.

This results in a very low intensity of the gravitational waves, which makes the detection extremely difficult. The expected ratio of change in length to the distance considered is in the order of magnitude of 10 ⁻²¹ , which corresponds to about a thousandth of a proton diameter per kilometer. Due to these difficulties, the direct detection of gravitational waves was only possible on September 14, 2015.

Indirect evidence of gravitational waves has been around for a long time, because when stars orbit each other, the gravitational waves lead to a loss of energy in the star system. This loss of energy manifests itself in a decrease in the mutual distance and the orbital time, as was observed, for example, in the double star system PSR 1913 + 16 .

Black holes

→ Main article : Black hole

One solution of the ART predicts that an extremely compact body bends space-time so strongly that a spatial region is formed from which no light and therefore no matter can escape. Such a spatial region has a singularity inside and was first described by Karl Schwarzschild in 1916 using the Schwarzschild metric . The surface, which a ray of light can no longer escape when crossed, is called the event horizon . Since a black hole cannot emit or reflect light, it is invisible and can only be observed indirectly via the effects of the enormous space-time curvature.

Lense thirring effect

→ Main article : Lense-thirring effect

In 1918, the mathematician Josef Lense and the physicist Hans Thirring theoretically predicted the Lense-Thirring effect (also known as the frame dragging effect), which was named after them. The effect describes the influence of a rotating mass on the local inertial system , which can be imagined in a simplified way that the rotating mass slightly pulls the space-time around itself like a viscous liquid and thus twists it.

It is currently still being discussed whether the scientists led by Ignazio Ciufolini from the University of Lecce and Erricos Pavlis from the University of Maryland in Baltimore succeeded in proving the effect experimentally in 2003. They measured the orbits of the geodetic satellites LAGEOS 1 and 2 precisely because their position and location should be influenced by the mass of the rotating earth . Due to possible sources of error due to the inconsistent gravitational field of the earth, it is controversial whether the centimeter-precise position determinations of the LAGEOS satellites were sufficient to prove this relativistic effect.

NASA's Gravity Probe B satellite , launched in April 2004, is equipped with several precise gyroscopes that can measure the effect much more precisely. To measure the effect, the changes in the directions of rotation of four gyroscopes are determined on this satellite.

cosmology

→ Main article : Cosmology

Cosmology is a branch of astrophysics that deals with the origin and development of the universe. Since the development of the universe is largely determined by gravity, cosmology is one of the main areas of application of GTR. In the standard model of cosmology, the universe is assumed to be homogeneous and isotropic. With the help of these symmetries, the field equations of the GR are simplified to the Friedmann equations . The solution of these equations for a universe with matter implies a phase of expansion of the universe . The sign of the scalar curvature on the cosmic scale is decisive for the development of an expanding universe.

If the scalar curvature is positive, the universe will expand and then contract again; if the scalar curvature disappears, the expansion speed will assume a fixed value, and if the scalar curvature is negative the universe will expand at an accelerated rate.

Einstein originally added the cosmological constant Λ to the field equations in 1917 in order to enable a model of a static cosmos, which he regretted after discovering the expansion of the universe. The cosmological constant can, depending on its sign, strengthen or counteract the cosmic expansion.

Astronomical observations have since significantly refined the relativistic world model and brought precise quantitative measurements of the properties of the universe. Observations of distant type 1a supernovae have shown that the universe is expanding at an accelerated rate. Measurements of the spatial structure of the background radiation with WMAP show that the scalar curvature disappears within the error limits. These and other observations lead to the assumption of a positive cosmological constant. The current knowledge about the structure of the universe is summarized in the Lambda CDM model .

Relation to other theories

Classical physics

The GTR must contain Newton's law of gravitation as a limit case, because this is well confirmed for slowly moving and not too large masses. Large masses, on the other hand, cause great gravitational accelerations on their surface, which lead to relativistic effects such as time dilation . Therefore Newton's law of gravity does not have to apply to them.

On the other hand, the special theory of relativity in spacetime regions, in which gravitation is negligible, must also be included in the GTR. This means that for the limiting case of a vanishing gravitational constant G, the special theory of relativity must be reproduced. In the vicinity of masses it is only valid in sufficiently small spacetime regions.

The requirement that the equations of the ART must fulfill the two borderline cases mentioned above is known as the correspondence principle . This principle says that the equations of outdated theories that give good results in a certain area of ​​validity must be included in the new theory as a borderline case for this area of ​​validity. Some authors use this term to only deal with one of the two borderline cases with regard to GTR, mostly with regard to Newton's theory of gravity.

The equations of motion of classical, i.e. not quantum mechanical, field theories change compared to classical mechanics, as described above . So it is possible without problems to describe gravitational and electromagnetic interactions of charged objects at the same time. In particular, it is possible to specify a non-relativistic (i.e. Newtonian, i.e. naturally incomplete) optimal approximation for the GTR. In addition, there is a post-Newtonian approximation to the general theory of relativity, which includes terms for the generation of gravitational fields according to Einstein's theory and differs in this from the post-Newtonian approximations of other metric theories of gravity and can thus serve to distinguish them experimentally.

Quantum physics

The ART is not compatible with quantum physics for very high particle energies in the area of ​​the Planck scale or correspondingly for very small spacetime regions with strong curvature . Therefore, while there is no observation that contradicts GTR and its predictions are well confirmed, it stands to reason that there is a broader theory in which GTR is a special case. So this would be a quantum field theory of gravity .

The formulation of a quantum field theory of gravitation raises problems that cannot be solved with the mathematical methods known up to now. The problem is that the GTR as a quantum field theory cannot be renormalized . The quantities that can be calculated from this are therefore infinite. These infinities can be understood as a fundamental weakness in the formalism of quantum field theories, and in other theories they can usually be separated from the physically meaningful results by renormalization processes. With ART, however, this is not possible with the usual methods, so that it is unclear how one should make physically meaningful predictions.

The currently (2015) most discussed approaches to solving this problem are string theory and loop quantum gravity . There are also a number of other models.

General theory of relativity and world formula

The diagram below shows the general theory of relativity in the structure of a hypothetical world formula .

Fundamental interactions and their descriptions
	Strong interaction	Electromagnetic interaction	Weak interaction	Gravity
classic		Electrostatics & magnetostatics , electrodynamics		Newton's law of gravitation , general relativity
quantum theory	Quantum ( standard model )	Quantum electrodynamics	Fermi theory	Quantum gravity  ?
		Electroweak Interaction ( Standard Model )
	Big Unified Theory  ?
	World formula ("theory of everything")?
Theories at an early stage of development are grayed out.

literature

Popular science:

• Harald Fritzsch : The bent space-time. Piper, 1997, ISBN 3-492-22546-2 .
• Marcia Bartusiak: Einstein's Legacy. European Publishing House, 2005, ISBN 3-434-50529-6 .
• Rüdiger Vaas: Tunnel through space and time. 2nd Edition. Franckh-Kosmos, 2006, ISBN 3-440-09360-3 .

Textbooks:

• Torsten Fließbach : General Theory of Relativity. 4th edition. Elsevier - Spektrum Akademischer Verlag, 2003, ISBN 3-8274-1356-7 .
• Charles Misner , Kip S. Thorne , John. A. Wheeler : Gravitation. WH Freeman, San Francisco 1973, ISBN 0-7167-0344-0 .
• Hans Stephani : General theory of relativity. 4th edition. Wiley-VCH, 1991, ISBN 3-326-00083-9 .
• Steven Weinberg : Gravitation and Cosmology. Principles and Applications of the General Theory of Relativity. New York 1972, ISBN 0-471-92567-5 .
• Hermann Weyl : Space, Time, Matter. Lectures on general relativity. 8th edition. Springer 1993, online (1919).
• Wolfgang Rindler : Relativity. Special, General and Cosmological. 2nd Edition. Oxford University Press, 2006, ISBN 0-19-856732-4 .
• Robert M. Wald : General Relativity. University of Chicago Press, ISBN 0-226-87033-2 .
• Rainer Oloff: Geometry of space-time. A mathematical introduction to the theory of relativity. 4th edition. Vieweg, 2008, ISBN 978-3-8348-0468-6 .
• Ray d'Inverno: Introduction to Relativity. 2nd Edition. Wiley-VCH, 2009, ISBN 978-3-527-40912-9 .
• MP Hobson, GP Efstathiou, AN Lasenby: General Relativity. An Introduction for Physicists. Cambridge University Press, Cambridge 2006, ISBN 0-521-82951-8 .
• Lewis Ryder: Introduction to General Relativity. Cambridge University Press, Cambridge 2009, ISBN 978-0-521-84563-2 .

Monographs:

• Thomas W. Baumgarte , Stuart L. Shapiro: Numerical Relativity: Solving Einstein's Equations on the Computer. Cambridge University Press, 2010, ISBN 978-0-521-51407-1 .
• Yvonne Choquet-Bruhat : General Relativity and the Einstein Equations. Oxford University Press, 2009, ISBN 978-0-19-923072-3 .
• Stephen W. Hawking , George FR Ellis : The Large Scale Structure of Space-time. Cambridge University Press, ISBN 0-521-09906-4 .
• Hans Stephani , Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt: Exact Solutions of Einstein's Field Equations. Cambridge University Press, ISBN 0-521-46702-0 .

History of ART:

• Abraham Pais : Subtle is the Lord.
  • "Lord God is clever ...". Albert Einstein, a scientific biography. Spectrum Academic Publishing House, Heidelberg / Berlin 2000, ISBN 3-8274-0529-7 .
• Thomas de Padova : Alone against gravity. Einstein 1914-1918. Hanser, Munich 2015, ISBN 978-3-446-44481-2 .
• Jürgen Renn , Hanoch Gutfreund: Albert Einstein. Relativity. The Special & the General Theory. Princeton University Press, Princeton, New Jersey, USA 2015, ISBN 978-0-691-16633-9 .
• Jürgen Renn, Hanoch Gutfreund: The Road to Relativity. The History and Meaning of Einstein's "The Foundation of General Relativity". Princeton University Press, Princeton, New Jersey, USA 2015, ISBN 978-1-4008-6576-5 .

Web links

Commons : General Theory of Relativity  - collection of images, videos and audio files

Wiktionary: General Theory of Relativity  - explanations of meanings, word origins, synonyms, translations

• einstein-online.info: General theory of relativity
• The general theory of relativity as a picture story
• Tutorial by John Baez (English)
• Scripts for the SRT and ART
• Spiegel Online : Turbulent spacetime: Laser measurement confirms Einstein's theory

Individual evidence

1. ^ Charles W. Misner, Kip S. Thorne, John Archibald Wheeler: Gravitation . WH Freeman and Company, San Francisco 1973, ISBN 0-7167-0334-3 , pp. 315 ff . (English).
2. ↑ ^{a b} Albert Einstein: About the principle of relativity and the conclusions drawn from it. In: Yearbook of Radioactivity and Electronics IV. 1908, pp. 411–462 ( facsimile , PDF ).
3. ↑ Albert Einstein: About the influence of gravity on the spread of light. In: Annals of Physics . 35, 1911, pp. 898-908 ( facsimile , PDF).
4. ↑ Albert Einstein, Marcel Grossmann: Draft of a generalized theory of relativity and a theory of gravitation. In: Journal of Mathematics and Physics . 62, 1913, pp. 225-261.
5. ↑ Albert Einstein: Explanation of the perihelion movement of Mercury from the general theory of relativity. In: Meeting reports of the Prussian Academy of Sciences. 1915, pp. 831-839.
6. ↑ Corry, Renn, Stachel in Science pointed this out . Volume 278, 1997, p. 1270
7. ^ F. Winterberg: On “Belated Decision in the Hilbert-Einstein Priority Dispute”, published by L. Corry, J. Renn, and J. Stachel. In: Journal of Nature Research A . 59, 2004, pp. 715–719 ( PDF , free full text). , in detail in Daniela Wuensch: Two real guys. 2nd Edition. Termessos Verlag, 2007. See also Klaus P. Sommer: Who discovered the general theory of relativity? Priority dispute between Hilbert and Einstein. In: Physics in Our Time. 36, No. 5, 2005, pp. 230-235, ISSN  0031-9252
8. ↑ Albert Einstein: The basis of the general theory of relativity . In: Annals of Physics . tape 354 , no. 7 , 1916, pp. 769-822 , doi : 10.1002 / andp.19163540702 .
9. ↑ David Hilbert: The basics of physics. In: Königliche Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Nachrichten (1915). Pp. 395-407.
10. ↑ Einstein himself saw these as the main aspects of the GTR: Albert Einstein: Principles for the general theory of relativity. In: Annals of Physics . 55, 1918, pp. 241-244 ( facsimile , PDF).
11. ↑ Øyvind Grøn, Sigurd Kirkevold Næss: An electromagnetic perpetuum mobile? In: General Relativity and Quantum Cosmology . June 3, 2008, arxiv : 0806.0464 (explanation on the “free fall” of the electron).
12. ^ GFR Ellis: Relativistic Cosmology. In: Proc. Int. School of Physics “Enrico Fermi” Course XLVIII - General Relativity and Cosmology (Varena, 1969). Ed. R. K. Sachs, Academic Press, New York 1971, pp. 104-182.
13. ↑ According to current observations of cosmology , the universe seems to expand at an accelerated rate, which speaks for a positive value of Λ.
14. ↑ Summary overview: Clifford M. Will: The Confrontation between General Relativity and Experiment. In: Living Rev. Relativity . 9, No. 3, 2006, ISSN  1433-8351 . Online document. ( Memento of June 13, 2007 in the Internet Archive ).
15. ^ S. Baeßler, BR Heckel, EG Adelberger, JH Gundlach, U. Schmidt, HE Swanson: Improved Test of the Equivalence Principle for Gravitational Self-Energy . In: Physical Review Letters . tape 83 , no. 18 , November 1, 1999, pp. 3585-3588 , doi : 10.1103 / PhysRevLett.83.3585 .
16. ↑ B. Bertotti, Iess L., P. Tortora: A test of general relativity using radio links with the Cassini spacecraft. . In: Nature. 425, 2003, pp. 374-376. (PDF, accessed on December 23, 2009; 199 kB).
17. ↑ Extremely accurate optical atomic clocks measure time dilation under everyday conditions (2010) .
18. ↑ Gravity Collaboration, R. Abuter et al. a., Detection of the gravitational redshift in the orbit of the star S2 near the Galactic center massive black hole, Astronomy & Astrophysics, Volume 615, 2018, L 15, Abstract
19. ↑ Ulrich E. Schröder: Gravitation: An introduction to the general theory of relativity . Harri Deutsch Verlag, Frankfurt am Main 2007, ISBN 978-3-8171-1798-7 , pp. 133 ( Page no longer available , search in web archives: limited preview in Google Book Search [accessed August 29, 2017]).
20. ↑ Torsten Fließbach: General Theory of Relativity. 3. Edition. ISBN 3-8274-0357-X , p. 171.
21. ↑ For example Misner, Thorne, Wheeler: Gravitation . Freeman, 1973, Chapter 39, p. 1068.

This article was added to the list of excellent articles on July 11, 2007 in this version .