General relativity

The general theory of relativity (short ART) describes the interaction between matter (including fields) on the one hand and space and time on the other. She points 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 on 25 November 1915, the Prussian Academy of Sciences. To describe the curved space- time, he used the differential geometry.

The general theory of relativity extends the special theory of relativity and goes for sufficiently small regions of space-time in this over. It can also be understood as an extension of the Newton 's law of gravitation, because it provides this in the limiting case of sufficiently low mass densities and speeds. The general theory of relativity has been confirmed experimentally in many cases (see tests of general relativity ), so that it is generally recognized as a theory of gravity. In particular, it has been able to prevail in the form formulated by Einstein against all alternatives proposed later. Open questions have focused on the relationship of quantum mechanics.

3.1 Principle of Relativity
3.2 Mach 's Principle
3.3 Principle of Equivalence
3.4 spacetime curvature

4.1 Basic concepts
4.2 Einstein's field equations
4.3 Equations of Motion

5.1 Gravitational time dilation and redshift
5.2 deflection of light and light delay
5.3 perihelion
5.4 gravitational waves
5.5 Black Holes
Lense- Thirring effect 5.6
5.7 cosmology

6.1 Classical Physics
6.2 Quantum Physics

Introduction

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

Energy and momentum of matter affect the geometry of space-time in which they are located. This influence can be formulated on a general curvature term, and in the ART space and time are described by the concept of space-time curvature.
Matter on which no force is applied, moving in space and time according to the classical notion along a geodesic line. However, a geodesic of space-time is usually not a straight line. Straight lines are geodesics uncurved spaces, such as the 3- dimensional space of classical mechanics. The influence of matter on this motion, the classical mechanics describes the help of gravity, the ART describes only about the geometry of spacetime. Here, a movement of an object along a certain path is interpreted in the room as in the special theory of relativity as a way into the four dimensions of space-time and called its world line.

The first statement describing an effect of the material to the space-time, the second describes the effect of the space-time to the movement of the material. The presence of matter thus changes the geometric relationships of space-time, which give rise to the equations of motion of matter. The ART considered here the spatial and temporal coordinates as equals and treats all changes over time as a geometric problem.

History

Generalization of the equivalence principle

The classical equivalence principle, sometimes referred to as a weak equivalence principle is based on considerations of Galileo Galilei (1636 /38) and experiments back in the field of kinematics. The original formulation of the equivalence principle of Galileo states that all bodies have regardless of their properties in the vacuum case, the same behavior. That is, two bodies under the influence of gravity, leaving the same place at successive times, behave in the sense identical, that they go through the same path, independent of all other properties of the body 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 principle of equivalence as equality of inertial mass and gravitational mass. This means that the law of gravity and the law of inertia occurs the same mass.

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

The preparation of the field equations

The foundations of the general theory of relativity were developed mainly by Albert Einstein. He used the developed by Carl Friedrich Gauss, Bernhard Riemann, Elwin Bruno Christoffel, Gregorio Ricci and Tullio Levi -Civita Curbastro differential geometry, as they learned from Marcel Grossmann, a fellow mathematician. He used this differential geometry to had formulated the special theory of relativity in space-time, with the Hermann Minkowski to formulate gravity as a property of proportions. Considerations of Ernst Mach influenced Einstein's conviction that even with only gravitational motion to other bodies is relatively physically significant.

The first release, which can be attributed to the general theory of relativity, is a 1908 published work of Einstein on the influence of gravity and acceleration on the behavior of light in special relativity theory. In this work he already formulated the principle of equivalence and the gravitational time dilation and says redshift and the deflection of light by massive bodies before. The bulk of the theory was worked out but only in the years 1911 to 1915 by Einstein. The beginning of his work marked it a second publication on the effect of gravity on light in 1911, in which Einstein worked up his publication of 1908.

Before he completed the work in 1913 Einstein published a draft of the theory of relativity, which already used a curved spacetime. Due to problems with the principle of general covariance, which ultimately proved to be correct, however, Einstein pursued in the period following a wrong approach before he finally was able to solve the problem in 1915. He held during his work also lectures about it and exchanged ideas with mathematicians, notably Marcel Grossmann and David Hilbert, from.

In October 1915, Einstein published a paper on the perihelion advance of Mercury, in which he still went out of the wrong field equations which were not compatible with the local conservation of energy and momentum. In November 1915 Einstein found the correct field equations and published them of the meeting notes 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 gave his work a five- days before the Göttingen Royal Society of Sciences for publication. However contain the proofs of Hilbert's work, unlike the later published version, not the field equations - the proofs, however, are not obtained completely. Einstein's later article, the basis of the general theory of relativity may be considered as the first review article of ART. It was published on 20 March 1916 in the Annals of Physics, two months after Einstein had argued that the native of Schwarzschild solution of his field equations of the Prussian Academy of Sciences.

On the Hilbert action functional of ART goes back, from which he derived the field equations in his 1916 published article.

Exact solutions of the field equations

After the formation of the field equations searched for solutions to this under various boundary conditions. The first exact solution of the field equations is found in 1916 by Karl Schwarzschild and named after him Schwarzschild metric, which is used for the description of black holes. It was further developed in 1916 by Hans Reissner and Gunnar Nordström in 1918 by the Reissner - Nordström named after them - metric that can be described electrically charged black holes.

1963 Roy Kerr was named after him with the Kerr metric can be described near a rotating black hole spacetime. The extension on electrically charged and rotating black holes is 1965 found Kerr-Newman metric.

When Einstein realized that the field equations do not allow for a cosmological model of a static universe, he led in 1917 the cosmological constant. Alexander Friedmann in 1922 found a solution of the field equations without cosmological constant, which allowed an expanding or contracting universe forming and 1927 Georges Lemaître found an exact solution for an expanding universe. When Edwin Hubble in 1929 published his observations on the redshift and thus provided a receipt for the expansion of the universe, Einstein discarded the cosmological constant, calling it George Gamow According as his " greatest blunder ". In modern astronomy, however, the possibility of a non-vanishing cosmological constant is taken into account.

The Robertson - Walker metric is a further development of Lemaître's solution, the Howard Percy Robertson and Arthur Geoffrey Walker 1935, 1936 formulated independently. She is also an exact solution of the field equation and describes an expanding, homogeneous and isotropic universe is therefore used as a model for the description of our universe. It is therefore in the cosmology of very great importance.

Basic Concepts

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

The theory does not necessarily follow from these premises, and at least in the Mach 's principle is not clear whether the ART there ever met. The three principles explain but which physical problems led Einstein to formulate the ART as a new theory of gravitation.

The description of the space-time curvature builds logically on the equivalence principle, so she is also treated in this chapter.

Principle of relativity

In the general theory of relativity in relation to the special theory of relativity extended principle of relativity is assumed: The laws of physics have not only in all inertial frames of the same shape, but also with respect to all coordinate systems. This applies to all coordinate systems that assign four parameters each event in space and time, these parameters being sufficiently differentiable functions of its locally defined Cartesian coordinates are on small space-time regions that obey the special theory of relativity. This requirement of the coordinate system is necessary for the methods of differential geometry for the curved space-time can be applied at all. A curved space-time is generally not more globally to describe in a Cartesian coordinate system. The extended principle of relativity is also called general coordinate covariance.

The coordinate covariance is a requirement for the formulation of equations ( field equations, equations of motion ) to have the validity of ART. However, the special theory of relativity can already formulate generally covariant. For example, represented on a rotating chair 's position that he himself is at rest and the cosmos rotate around it self an observer. Here, the paradox that the stars and the light emitted by them in the coordinate system of the rotating observer move computationally faster than light, which contradicts apparently the special theory of relativity arises. The resolution of this paradox is that the generally covariant description is by definition local. This means that the constancy of the speed of light must apply only near the world line of the observer, which is satisfied for the rotating observer as well as for any other observer. The covariant, ie in the sense of the general principle of relativity, written equations arise for the stars so faster than light circular movements, but are consistent with the principles of special relativity. This is also made clear that it is impossible for an observer at rest in the vicinity of a star in the rotating coordinate system and therefore meets the star faster than light. So this observer has forced a different coordinate system than the rotating observers and measures the "right" speed of light.

Although it is possible to describe the cosmos from the perspective of a rotating observer correctly, are the equations of a reference system in which to rest most of the objects or move only slowly, mostly simple. The condition of a non- rotating coordinate system for inertial systems and the distinction in their view, requires the classical physics, but is omitted in principle.

In the case of a multi-body system in a confined space, the spacetime is highly curved and the curvature in each coordinate system also varies in time. Therefore no candidate for an excellent coordinate system is identified from the outset, which is suitable for the description of all phenomena. The principle of relativity states for this general case that it is not necessary to look for it, because all coordinate systems are equal. So you can depending on which phenomenon you want to describe, choose different coordinate systems and select the computationally simplest model.

Therefore, the ART can also dispense with the classical astronomical concept of illusoriness of movements, required the still arrested in the Newtonian view heliocentric worldview.

Mach 's Principle

Einstein was greatly influenced in the development of the theory of relativity by Ernst Mach. In particular, the assumption that the forces of inertia of a body does not depend on its motion relative to an absolute space, but of its movement relative to the other masses in the universe, which he described as mach cal principle, was an important basis for Einstein. The inertial forces are on this view, so result of the interaction of the masses with another, and one independent of these masses of existing space is denied. Thus, for example, centrifugal force of rotating bodies should disappear when the rest of the universe " co-rotates ".

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

Einstein saw the Lense- Thirring effect, the ART predicted, as a confirmation of his version of the mach 'principle. Consequence of this effect is that within a rotating reference systems experience involving mass hollow sphere precession, which Einstein interpreted so that the mass of the ball has an influence on the inertial forces. However, since a " dormant " reference system was adopted in the form of fixed stars in the account and the interpretation, even this interpretation is controversial.

The commonly held version of the mach 'principle, which Einstein formulated, so is too vague to be able to decide whether it is compatible with the ART.

Equivalence principle

Already in classical mechanics, the principle of equivalence of inertial and gravitational mass was known. It states in its classical form, which is also known as weak equivalence principle that the gravitational mass, which indicates how strong is the force generated by a gravitational field on a body and the inertial mass, which determines by the law of force, how much a the body is accelerated by a force equivalent. This means in particular that every body is independent of its mass in a gravitational field moves the same ( in the absence of other forces ). ( Charged bodies are excluded from it due to the synchrotron radiation. ) So for example, fall in a vacuum all ( uncharged ) body at the same speed, and the geostationary orbit is for heavy satellites like for light satellites always the same. Consequence of the classical equivalence principle is that an observer in a closed laboratory, without information from the outside, from the mechanical behavior of objects in the laboratory can not read, whether it is in microgravity or in free fall.

This principle was generalized by Einstein: Einstein's strong equivalence principle states that an observer can determine in a closed laboratory without interaction with the environment through absolutely no experiment if he far away is located in the weightlessness of masses or in free fall near a mass. This means in particular that a ray of light for an observer in free fall - is parabolically curved - like in an accelerated frame of reference. On the other hand, has an observer at rest in the gravitational field, for example by standing on the earth's surface, perceive a light beam curved because it is accelerated all the time against the free fall upwards.

It should be noted, however, that this principle is only locally:

Thus, a "down" (closer to the Gravizentrum ) object located is attracted more strongly than a more "up" befindliches. Is the free-falling space in the vertical direction is large enough, the observer will therefore find that objects that are above, from those who are further down, remove.
Conversely, will differ with sufficient horizontal extent of the space, the direction of attraction to two horizontally away from each other noticeably objects, since they both are accelerated towards the center of gravity. Therefore, the free-falling observer will note that far apart preferred body move towards each other. An extensive body is thus experience a force that pulls apart it in one direction and compresses in the directions perpendicular thereto.

In ART, the equivalence principle follows directly from the description of the motion of bodies: Since all bodies move along geodesics of spacetime, an observer who moves along a geodesic, only then determine a curvature of space- time that it could be interpreted as gravitational field when the observable space it time piece is significantly curved. In this case, he observed the above tidal forces as a relative proximity or distance of neighboring freely falling body. The curvature also ensures that charged bodies is not locally interact with their own field and therefore the principle of equivalence to this principle is not applicable because their electromagnetic field is basically ensures long range.

Space-time curvature

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

A parallel conveyor is a shift in a direction in which the alignment is maintained, that is, a local coordinate system is carried. A shift in direction in space is clearly evident in a spacetime without masses. The definition of time is dependent upon the special theory of relativity on the motion of the coordinate system. A constant time direction is given only for unaccelerated coordinate systems. In this case means a shift in the time direction in a space-time without the masses, that an object at rest relative to the coordinate system. It then moves along the time axis of this coordinate system.

According to the equivalence principle can thus be understood in a gravitational field of the parallel transport along the time axis. The equivalence principle states that a free-falling observer in a gravitational field is equivalent to an unaccelerated observer far from a gravitational field. Therefore, a parallel transport along the time axis corresponds to a time interval T to a free fall of the time t. This means that a parallel shift in time also has a movement in the room result. However, since the direction of the free fall of the place depends, it now makes a difference whether an observer is moved parallel first 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 transport is significant.

So far, large displacements were considered, where obviously the order of the parallel transport is significant. However, it is useful to be able to make statements about any small regions of space-time in order to describe the behavior of bodies, even for short times and distances can. If one makes the parallel transport over ever shorter distances and times, the end points for different sequences of transport are still different, but the difference decreases accordingly. With the help of discharges can be an infinitesimal parallel transport at a point describe. The measure of the deviation of the endpoints in interchanging the order of two parallel transport is then given by the so-called curvature tensor.

Due to the space-time curvature Also, the above mentioned tidal forces can be explained. Two spheres in free fall in a freely falling laboratory are both moving along the time axis, ie parallel lines. The fact that the parallel transports are not commutative, is equivalent to the fact that parallel lines do not have a constant distance. The trajectories of the balls can move closer to each other or away from each other. In the Earth's gravity field approximation for very long case is very small. If the time will be treated similarly to a spatial dimension, the time intervals are multiplied by the speed of light. The space-time curvature is so tiny and ever recognizable only for long durations case. This is similar to a clothesline, which viewed from the side that seems right, but if you look her along a curvature disclosed.

To describe the curvature so it is not necessary to embed the spacetime into a higher-dimensional space. The curvature is not to be understood as curvature in a fifth dimension, or as a curvature of the space in the fourth dimension, rather than noncommutativity of parallel transport. It is also necessary for this representation to treat space and time as a four-dimensional space-time, because the three-dimensional space alone does not need to be curved.

How does the space-time is curved, is defined in the ART by the Einstein's field equations.

Mathematical Description

Basic concepts

The mathematical description of space-time and its curvature is using the methods of differential geometry, which replaces the Euclidean geometry of our familiar "flat" three-dimensional space of classical mechanics. The differential geometry used to describe curved spaces such as the space-time of ART, known as manifolds. Important properties are described with so-called tensors, the pictures on the manifold.

The curved space-time is described as a Lorentzian manifold.
Particular importance is attached to the metric tensor. If you use two vector fields in the metric tensor, we obtain a real number for each point of spacetime. In this respect, one can understand the metric tensor as a generalized, point -dependent scalar product of vectors of spacetime. With his help, distance and angle are defined and it is therefore referred to briefly as metric.
Equally important is the Riemannian curvature tensor describing the curvature of the manifold, which is a combination of first and second derivatives of the metric tensor. If a tensor in any coordinate system is non-zero at a point can be found no coordinate system, so that it is at this point is zero. This applies accordingly for the curvature tensor. Conversely, the curvature tensor is zero in all coordinate systems, if it is in a coordinate system is zero. It is thus independent of the coordinate system to decide whether diversity is curved at a certain point or not.
The relevant parameter for the description of the energy and momentum of matter is the energy-momentum tensor. As this tensor determines the curvature properties of space-time, shows the following section.

Einstein's field equations

The Einstein's field equations establish a connection between some curvature properties of space-time and the energy-momentum tensor, which contains the local mass density, or on the energy density and thus characterizes the relevant properties of matter.

These basic equations of general relativity included 10 independent components, similar to a vector equation of the Euclidean space of three components:

Here is the Ricci curvature tensor -, the Ricci Krümmungsskalar, the metric tensor, the cosmological constant, the speed of light, the gravitational constant and the energy-momentum tensor. Since all tensors are symmetric in this equation (for example ), only 10 of these 16 equations are independent.

The aim is to specify the components of the energy-momentum tensor of the right side of the equations, and then to use the field equations to determine the metrics. The expression on the left side of equation consists of sizes which are derived from the curvature. They therefore contain derivatives of this metric. Therefore obtained 10 differential equations for the components of the metric. However, the metric and its derivatives are usually found on the right hand side of the equations in the energy - momentum tensor. To make matters worse, that the sum of two solutions in general is not a solution of the field equations, the solutions are not so superponierbar. This is due to the non-linearity of the field equations, which is considered a major feature of the ART. Because of this complexity of the equations, it is often not possible to find an exact solution for the field equations. In such cases, may be used for part of a method for finding an approximate solution.

In the field equations is not the curvature tensor, but only derived from it Ricci curvature tensor and the Ricci - Krümmungsskalar. These two terms are collectively referred to as the Einstein tensor, and this does not contain all information about the curvature of spacetime. Part of the space-time curvature, called the Weyl curvature, so is not directly dependent on the energy -momentum tensor and thus on the mass and energy density. However, the Weyl curvature tensor, is not arbitrary, as it is defined in part due to the geometrical Bianchi identities by the Ricci curvature tensor -.

Einstein at first believed that the universe its size does not change with time, so he led the cosmological constant in order to allow such a universe theory. The balance, which he reached it, but turned out to be unstable equilibrium. has formally the importance of a kind of integration constants, and first thus has no specific numerical value, which would follow directly from the theory. So she has to be determined experimentally. An alternative view on the cosmological constant summarizes the corresponding term on as part of the energy-momentum tensor and sets. This means that the cosmological constant presents itself as a perfect fluid with negative pressure and is regarded as an exceptional form of matter or energy. In today's cosmology in this context the term " dark energy " has prevailed.

The field equations indicate how the matter and energy content affects the curvature of space- time. However, they also contain all information about the effect of spacetime curvature on the dynamics of particles and fields, so on the other direction of the interaction. Nevertheless, it does not directly use the field equations to describe the dynamics of particles or fields, but redirects to the equations of motion ago. The equations of motion are therefore " technically " of importance, although their information content is conceptually included in the field equations.

A particularly elegant derivation of Einstein's field equation provides the principle of least action, which plays an important role in the Newtonian mechanics. A suitable formula for the action whose variation leads here in the calculus of variations to these field equations, the Einstein - Hilbert action, which was first stated by David Hilbert.

Equations of motion

In order to formulate the equations of motion, an arbitrary world line of a body must be parameterized. This can be done by a zero and a positive direction are determined, and then each point on the world line is assigned from zero up to this point with the sign corresponding to the arc length. How do you make sure that each point is uniquely determined on the world line. A very similar parameterization is the parameterization by the proper time. The two are identical when the equations simplified by ignoring all C by formally so is the speed of light. The following formulas are to be understood in Bogenlängenparametrisierung.

Hereinafter the term never designated 'force' the gravity, but, for example, electromagnetic or mechanical forces, since the gravity is interpreted as a geometric effect. Referring now to a body to which a force, as the equations of motion

In the event that no force acts on a body, its world line is described by the Geodätengleichungen the curved space-time. They are obtained by putting the power in the above power law

Where m is the mass of the body and are the four space-time components of the world-line of the body; represents the time component. Points above sizes are derivatives with respect to arc length and not according to the time component. is a Christoffel symbol that characterizes the dependence of the metric tensor of the space-time point, ie, the space-time curvature. They are components of the comet generic tensor inverse to the metric tensor.

In the formula also short notations are used: For the differentials, and the summation convention, which states that over indices that appear below each standing at top and one is automatically summed from 0 to 3.

The power law is a generalization of the classical action principle () on four dimensions of a curved space-time. The equations can be solved only if the metric tensor is known.

In principle, the equations of motion for a particle cloud and the Einstein's field equations can now be considered as a system of equations which describes the dynamics of a cloud of massive particles. Due to the aforementioned difficulties in the solution of the field equations but this is not practical, so that always calculated for Mehrteilchensysteme with approximations.

The forces acting on a body, thereby calculated in general slightly different from the special theory of relativity. Since the formulas must be written koordinatenkovariant in the ART, in the formulas for the forces, for example in the Maxwell equations, now to use the covariant derivative instead of the partial derivative with respect to space-time components. Since the derivatives with respect to space -time components describe the changes in size, this means that the changes of all fields (ie, position-dependent variables ) must be now described in the curved space-time. The Maxwell equations arise to order

The use of the covariant derivatives thus affects only the inhomogeneous Maxwell equations, while the homogeneous equations do not change compared to the classic form. The definitions of the covariant derivatives of tensors are given in the article Christoffel symbols.

Physical Effects

For experimental verification of ART it is not enough to conduct experiments, with which you can decide between the ART and the Newtonian mechanics, because there are competing theories to ART. It is therefore necessary to experimentally decide between the ART and other theories of gravitation. Deviations from the predictions of ART could also be a new impetus to the development of a consistent and experimentally verifiable quantum theory of spacetime.

The general theory of relativity predicts the experimental results within the accuracy of measurement correctly predicted. The equivalence principle is confirmed with an accuracy of up to 10-13. for other phenomena of ART up to 10-5. The following are some physical phenomena are explained, the precise experimental testing well so far confirms the ART and has the scope for alternative theories very reduced. In addition, the good agreement between experiment and prediction can be expected that quantum effects of gravity are very small, since they would be visible from the predictions of ART as deviations.

Gravitational Time Dilation and Redshift

The gravitational time dilation follows from the special theory of relativity and the equivalence principle of ART. It was predicted by Einstein in 1908. If you look at a stationary clock in a gravitational field, it must be held by an opposing force in peace, like a person standing on the earth's surface. It is therefore continually accelerated, so that you can use the formula for time dilation in an accelerated reference system from the special theory of relativity. This has the consequence that the effect is not symmetrical, as it is known from two uniformly moving reference frames in special relativity theory. An observer in the universe therefore provides the clocks on earth go slower than his own clock. Conversely, an observer sees on earth watches in space go faster than his own clock. With very accurate optical atomic clocks, the gravitational time dilation can even at a height difference of only a few centimeters measure.

A direct consequence of time dilation, the gravitational redshift. It was predicted by Einstein in 1911 before the completion of the general theory of relativity. Since both effects can be derived already from the equivalence principle, their experimental verification in itself does not confirm the validity of the ART. However, if one deviates from the prediction of behavior observed, this would be the ART refute so that the experimental confirmation of the effects is necessary for the validity of the theory.

Light (that is, the center of gravity away) emitted from a light source with a given frequency according to above is therein measured at a lower frequency, similar to the Doppler effect. Thus, this means in particular that for a light signal having a predetermined number of oscillations of the time interval between the beginning and the end of the signal at the receiver is greater than the sender. This can be understood by the gravitational time dilation.

Due to the gravitational time dilation, the time interval between the beginning and end of the light wave is longer, the further up you move in a gravitational field, because the time passes faster and faster. This means that the shaft is always measured as it moves more upwardly. Therefore, the distance between wave peaks is always more to grow, so that the light that is always long-wavelength ( red shifted direction ), thus appears less energy.

Deflection of light and light delay

Light near a large mass moves slower than the vacuum speed of light as seen by a distant observer. This phenomenon is called after its discoverer as Shapiro delay. In addition, a distant observer perceives a deflection of light near large masses. These two effects are due to the same explanation. The real time, the so-called closing time, near the ground is different from the concept of time of the remote viewer. In addition, the mass has an impact on the behavior of the space, similar to a Lorentz contraction, which can be only in the context of ART and not classically explain. These two effects are approximately equal for small masses and add up. An observer positioned himself near the ground is, accordingly measure the vacuum speed of light as the speed of the light beam. However, the distant observer perceives a reduced speed, it can be described as position-dependent refractive index. This description also provides an explanation for the deflection of light, which can be interpreted as a kind of aperture.

The above explanation is based on an analogy. The abstract interpretation in the context of ART is that the null geodesics on which moves light, appear curved near large masses in space. It should be borne in mind that the light also moves in time, so that there actually exists a space-time curvature and not a pure curvature of the three-dimensional space.

The angle of deflection is dependent on the mass of the Sun, the distance from the sun the next point on the path to the center of the Sun and the speed of light. It may according to the equation

Be calculated. Is the gravitational constant.

On deflection of light in the gravitational field also is the observed gravitational lensing in astronomy.

Perihelion

The perihelion shift is an effect of (eg Jupiter ) is produced mostly by the gravitational force of other planets. When Mercury is measured 571 " per century, of which 43.3 " not the result of these disorders. The theory of relativity could explain this value, which was a first success of the theory. The perihelion of the Earth is " even greater than that of Mercury per century, the relativistic loss amounts but by the earth only 5" 1161. The measured failure contributions to the perihelion advance of other planets as well as the minor planet Icarus agree with the predictions of the theory of relativity. Is currently in planning Europe-Japan Mercury probe BepiColombo will make it possible to determine with unprecedented accuracy the motion of Mercury and thus to test Einstein's theory in more detail.

In binary systems of stars or pulsars orbiting each other in a very short distance, the perihelion with several degrees per year is significantly larger than the planets of the solar system. The indirectly measured in these star systems values of the perihelion are consistent with the predictions of ART.

Gravitational waves

ART allows the description of fluctuations of space-time curvature, which propagate at the speed of light. In a first approximation, these fluctuations are comparable with transverse waves, therefore they are called gravitational waves. A description of this phenomenon without approximations not exist yet ( 2007). Gravitational waves would thus be observed that expands periodically transverse ( transverse) to their direction of propagation of the space and contracts. Since there is no positive and negative charge as in electromagnetism are in gravitation, gravitational waves can not appear as dipole radiation, but only as a quadrupole. In addition, the coupling of the gravity of matter is very much weaker than the electromagnetism.

It follows a very low intensity of the gravitational wave, which is very difficult to prove. The expected ratio of change in length of the link concerned is of the order of 10-21, which is about one thousandth of proton diameter per kilometer. Because of these difficulties is still no direct detection of gravitational waves succeeded.

However, there is indirect evidence of gravitational waves, because at each other orbiting stars, the gravitational waves lead to a loss of energy of the star system. This energy loss is manifested in a decrease in the rotation speed, which has been observed for example in the binary system PSR 1913 16.

Black Holes

The ART predicts that a highly compact body bends the space-time so much that a region of space forms, from which it can escape no light and therefore no longer matter. Such an object is called a black hole and was first described by the Schwarzschild metric. The surface above which a light beam can not escape is referred to as the event horizon. Since a black hole can emit or reflect light, it is invisible and can only be observed indirectly through the effects of massive spacetime curvature.

The existence of the black hole is now regarded as empirical secured although there is no direct observation of such objects. So is now accepted that there are supermassive black holes at the centers of most galaxies. The observation of so-called matter - jets in galaxies and the measurement of orbital periods close to the center stars are clear indications of such black holes.

Lense- Thirring effect

In 1918 it was predicted theoretically by the mathematician Josef Lense and Hans Thirring physicist named after them, the Lense- Thirring effect ( also frame - dragging effect). The effect describes the influence of the local inertial frame by a rotating mass, which can be thought of simplified so that the rotating mass of the spacetime around themselves like a viscous fluid drags slightly and thus twisted.

Currently, is still debated whether the scientists to Ignazio Ciufolini of the University of Lecce and Errico's Pavlis of the University of Maryland at Baltimore in 2003, the experimental evidence of the effect has been achieved. You the exact measurements for the orbits of geodetic satellites LAGEOS 1 and 2 precisely because their position and orientation of the mass of the rotating earth should be influenced. Due to possible sources of error by the non-uniform gravitational field of the earth is disputed whether the centimeter-level position provisions of the LAGEOS satellites sufficient to establish this relativistic effect.

The NASA Gravity Probe B satellite, launched in April 2004, is equipped with multiple -precision gyroscopes, which can measure the effect much more accurate. To measure the effect of the changes in the directions of rotation of four gyroscopes are determined at this satellite.

Cosmology

Cosmology is the branch of astrophysics that deals with the origin and evolution of the universe. Since the evolution of the universe is largely determined by the gravitation, cosmology is one of the main applications of ART. In the standard model of cosmology, the universe is assumed to be homogeneous and isotropic. Using these symmetries, the field equations of ART simplify the Friedmann equations. The solution of these equations for a universe with matter imply a phase of expansion of the universe. The sign of the scalar curvature is on a cosmic scale crucial for the development of an expanding universe.

For a positive scalar curvature, the universe will initially expand and then contract again, at zero scalar curvature, the expansion rate will assume a fixed value, and for a negative scalar curvature, the universe is expanding faster.

Einstein added the cosmological constant Λ in 1917 in the field equations in order to allow a model of a static universe. The cosmological constant can increase the cosmic expansion or antagonize depending on the sign.

Astronomical observations have now greatly refined the relativistic model of the world and brought accurate quantitative measurements of the properties of the universe. Observations of distant supernovae of type 1a have shown that the universe is expanding faster. Measurements of the spatial structure of the background radiation with WMAP show that the scalar curvature vanishes within the error limits. These and other observations lead to a positive, non-zero cosmological constant. The current knowledge about the structure of the universe are summarized in the Lambda -CDM model.

Relation to other theories

Classical Physics

The ART has to move slowly and not too large masses the Newtonian law of gravitation included as a limiting case. This is in fact well confirmed for slow- moving and not too large masses. Large masses activation result in large gravitational acceleration at its surface, which lead to relativistic effects such as time dilation and Lorentz contraction. It is therefore necessary for this not apply to the Newtonian law of gravitation.

On the other side must also include the special theory of relativity in spacetime regions where gravity is negligible, be included in the ART. This means that the special theory of relativity must be reproduced in the limit of a vanishing gravitational constant G. In the vicinity of masses it applies only in differentially small regions of space at small time intervals.

The requirement that the equations of ART must meet the two limiting cases mentioned above, is called the correspondence principle. This principle states that the equations of obsolete theories that provide good results in a certain scope, must be included for this scope as a limiting case in the new theory. Some authors go under this term in relation to the ART only one of the two limiting cases, mostly with respect to the Newtonian theory of gravitation, a.

The classical equations of motion, ie not quantum mechanical, field theories change over classical mechanics, as described above. Thus it is possible without problems to describe gravitational and electromagnetic interaction of charged objects simultaneously.

Quantum physics

The ART is not compatible with quantum physics at very high particle energies in the range of the Planck scale or equivalent with very small space-time regions with high curvature. Although there is no observation which contradicts the ART and their predictions are well confirmed, therefore stands to reason that there is a more comprehensive theory, in the framework of the ART is a special case. So this would be a quantum field theory of gravitation, which is an association of ART with quantum field theory.

However, the formulation of a quantum field theory of gravity raises problems that are not solvable with the previously known mathematical methods. The problem is that the ART is not renormalizable as a quantum field theory. The sizes that can be calculated from it, so are infinite. These infinities can be understood as a fundamental weakness in the formalism of quantum field theories, and they can be with other theories usually by renormalization of the physically meaningful results separate. In the ART but that is not possible with the conventional method, so it is not clear how to make physically meaningful predictions.

The currently (2007) the most discussed approaches to solving this problem are string theory and loop quantum gravity. There are a number of other models, however, are not so well known.

50311