# General relativity theory

**general relativity theory** describes the reciprocal effect between space and time on the one hand and subject (inclusive fields) on the other hand. In its core statement it leads back the gravitation to a geometrical phenomenon in a curved 4-dimensionalen space-time . It was developed by Albert Einstein and published 1916.

General relativity theory represents an extension of the special Relativitätstherie and changes for sufficiently small areas of space-time into these.

Although general relativity theory is experimentally not as easily accessible as the special, there is a sufficient number of experimental vouchers for it. In particular it could itself intersperse so far in the form formulated by Einstein against all alternatives suggested later.

The following article develops on the remarks of the article relativity theory and has to the goal of deepening the understanding concerning the phenomena and structures mentioned there.

## the reciprocal effect between subject and space-time

a remarkable result of general relativity theory is one the naive opinion inaccessible reciprocal effect between the subject and space-time with the two following characteristics:

- Energy and impulse of the subject curve space-time in its environment.
- An article, on which no Kraft is exerted, always moves between two places in space-time along a way straight-line in certain sense. More exactly regarded it concerns geo DATE in such a way specified, i.e. a line, which connect all points on it by a extremal way. Usually this does not mean however that the movement follows geo DATE of the area.

The first characteristic describes an effect of energy and impulse on space-time, and second in reverse. It concerns therefore a reciprocal effect in the sense of word.

To the curvature contributes thereby not only the mass, which correspond to E=mc 2 ^{of an energy} over the relationship, and its impulse, but all forms of energy. So are for example also possibly. existing electromagnetic fields to consider, since they represent also a form of energy, as well as a field impulse to likewise have can. The relevant size is the energy impulse tensor in such a way specified. In which way it curves space-time, by the Einstein Feldgleichungen specified (see below).

The second characteristic describes the gravitation. The movement of an article is interpreted along a certain way in the area as line in the 4-dimensionalen space-time and called its world line. That is described, as for example by the example of a system of mass points a globular star cluster. Since an observer can notice the usual three-dimensional area in each moment only, and not the entire 4-dimensionale space-time, he can recognize the geo data of the individual stars not directly as such. On its own way by space-time he observes instead in the area bent trajectories of the stars around the center of the heap, from which he judges after the Newton's mechanics forces, which he calls gravitation forces. The underlying cause is however the curvature of space-time. Each star flies in certain sense in space-time so well straightforward, as it is at all possible in view of the curvature. In the context of general relativity theory there are in the long run no gravitation forces. In this sense also the force-freeness, from which above the speech is, refers only to the absence of not gravitativen forces.

## The inclined throw as consequence of a curved space-time

a curvature of space-time entails generally also a curvature of the area embedded into it. For the explanation of the gravitation the view of the bent area is not sufficient alone however. Like that the area, in which we live on earth, is naturally not so strongly curved that he could explain a trajectory parabola. To understand the trajectory parabola one must consider that for example a ball, which a juggler of a hand throws into the other one on its way puts a way back by the area of for instance 1m also by the time of for instance 1s. In the framework this corresponds to mathematics of space-time one second in certain way of a kind distance toward the time axis of approximately 300,000 km. One obtains this value, by x=ct the time t over a way x in space-time assigns, whereby C is the speed of light. Which we see in the long run, thus a tiny curvature of space-time is in an area of astronomical extent. The situation is comparable with a tautly strained Wäscheleine. If one regards it from the side, then she appears straight. If one regards it however from an end and looks with an eye in its direction, then one notices also a relatively weak dip clearly. The circumstance that we accompany the ball on its way of 300.000 km, shows us similar to the curvature space-time clearly.

That equalsneeze with the Wäscheleine is plausible and supplies qualitatively the correct result, it is strictly taken not completely applicable however. In contrast to the Wäscheleine the ball flies along geo DATE and thus actually straightforward, so „well “it in curved space-time can. We against it take the more bent way, since we are accelerated upward by a counter acting force, with that to us the soil, on which we stand, upward press. It is the counter acting force, which compensates the gravitation strength, with the favourable consequence that we do not fall into the depth. More exactly regarded thus the curvature of space-time in this situation expresses itself by the fact that we do not come from the place, although we are exposed to a permanent Kraft from downside. The argument that here two forces would compensate themselves, is groundless by the fact that the gravitation strength directed downward has only a geometrical cause. The situation is comparable with that of the apparent equilibrium of centrifugal and centripetal force during a rotation, which is present for the rotary observer. For the resting observer however the centrifugal energy is a fictitious force, so that an acceleration is actually present.

## Gravitative red shift and space-time curvature

the space-time curvature can be demonstrated very beautifully at the gravitativen red shift: Light, which is radiated upward a frequency given by a source of light with (thus away from the gravitation center), is measured there with a smaller frequency (similarly Doppler effect). That means thus in particular that with a light signal with a certain number of oscillations the temporal distance between the beginning and the end of the signal is larger with the receiver than with the transmitter.

Now however in the time in the arrangement nothing changed, therefore the end of the light signal must have been just as for a long time on the way as the beginning (of it, like the way of the light in detail looked independent!). In a ungekrümmten space-time thus, since the ways of the light-beam beginning and light-beam end ran parallel, (temporal) the distance of the receipt of the beginning to the receipt of the end would be equal that of sending the beginning for sending the end, a red shift would thus not take place. The measured red shift (see below) can be regarded thus as proof of the space-time curvature.

## Equivalence of carrier and heavy mass

in the classical mechanics was well-known already the principle of the equivalence of carrier and heavy mass. It means in its classical form the fact that the heavy mass, which indicates, how strongly Kraft produced by a gravitational field at a body is, and the slow-acting mass, which says, how strongly a body is accelerated by a Kraft, equivalent is. This means in particular that each body moves independently of its measures in a gravity field (with absence of other forces) directly. Thus fall for example in the vacuum all equal body fast, and the geostationary course (the course, in which a satellite needs straight one day for an orbit around the earth, so that the satellite over the earth's surface seems to stand still) always is for heavy satellites as for light satellites the same.

Consequence of the classical equivalence principle is also that an observer in a closed area, without observation of the environment, from the movement cannot read off from articles in the area, whether it is in weightlessness or in the free case. This principle was generalized by Einstein. The Einstein equivalence principle means that an observer in a closed area without information can determine *from the outside by* at all no experiment, whether it is in weightlessness or not.

It must be however noted that this principle applies *only* locally: Thus far down (more near on the earth) object present by the earth more strongly one tightens, than far present above. If the freely falling area is large enough, then the observer will therefore state that objects, which are further above remove from those, which further down are. During sufficient horizontal expansion of the area the direction of the gravitational pull of the earth turns around noticeably to change, so that the freely falling observer will determine to move that far apart convenient bodies one on the other. An expanded body will thus experience a Kraft, which pulls apart it in a direction and in in addition the senkrechten directions squeezes together. On the basis these Kraft, Gezeitenkraft mentioned, he can state that he is in a gravitational field. Therefore the area must be sufficiently small, so that this effect remains below the detection limit (more exact measuring instruments to cause according to a still smaller area).

## Curvature of space-time without 5. Dimension

one first to assume that for the curvature of the 4-dimensionalen space-time is necessary a fifth dimension, into which space-time is embedded, as in the everyday life bent surfaces only in the area is conceivable. Such a fifth dimension would be however inaccessible in principle, and the kind of the imbedding of space-time would not be clear. Since it is possible to describe the curvature mathematically without a purchase to a fifth dimension you also no reality are assigned. Thus for example a curvature of the area can be measured over the determination of the relationship of diameter and circumference of a circle or control of the Winkelsumme of the triangle, without having to analyze this area from a further dimension.

## The mathematical description of the gravitation

the mathematical description of a bent space-time takes place with the methods of Riemann geometry, which Euclidean geometry us trusted flat area replaces. The curvature is described over the so-called curvature tensor. The Einstein Feldgleichungen manufacture the connection with the energy impulse tensor in such a way specified , in particular the local mass density and/or over <math> E=mc^2< /math> the power density contains. These principal equations of general relativity theory contain 10 independent components, similarly as a vector equation from 3 components exist. They read:

<math> R_ {IC} - {g_ {IC} R \ more over 2} + \ Lambda g_ {IC} = 8 \ pi {G \ over c^4} T_ {IC}< /math>.

Math <R_> is {IC}< /math> the Ricci Krümmungstensor, <math> R< /math> the Ricci Krümmungsskalar, <math> g_ {IC}< /math> the metric tensor, <math> \ Lambda< /math> the cosmological constant, <math> T_ {IC}< /math> the energy impulse tensor, <math> C< /math> the speed of light, <math> G< /math> the gravitation constant and <math> \ pi< /math> the circle number. The cosmological constant of Λ was first only introduced by Einstein, in order to ensure a temporally stable universe. The equilibrium, which it reached thereby, proved however as an unstable. <math> \ Lambda< /math> has formally the value of a kind integration constants, and has therefore first no certain numerical value, which would follow directly from the theory.

## The relativity principle in general relativity theory

a bent space-time is not any more with cartesian coordinates recordably. Instead the coordinate system, for which one wants to set up the Einstein Feldgleichungen, can be selected almost at will. It must assign only in space and time 4 somehow parameters to each event. Exactly taken they must be only on small spaces, which obey special relativity theory, sufficiently differentiable functions of the there locally definable cartesian coordinates, so that the methods of the Differentialgeometrie for bent space-time can be used at all.

Thus a clearly extended relativity principle applies in general relativity theory. The laws of physics do not only have thereafter in all inertial systems the same form, as it is in special relativity theory the case, but in arbitrary coordinate systems.

This result has consequences, which are not understandable at first attempt. Thus it means for example that even an observer on a rotary turntable can hold the point of view, it is in peace and the cosmos rotates around it. Indeed the Einstein Feldgleichungen themselves describe this situation correctly. In this rotary coordinate system the curvature tensor takes values, which entail actually the enormous centripetal forces, which the stars hold on their circular path around the observer on their course. The fact that thereby the stars from view of the rotary observer with multiple speed of light move does not contradict the theory, since the speed of light applies only in special relativity theory as border, i.e. for sufficiently small space-time ranges, which fulfill the criteria for inertial systems. From the view of the rotary observer however no stars can be distance in peace in some light-years perpendicularly to the rotation axle, so that anywhere stars cannot meet locally with trans-light velocity. An information and/or a subject transport from a star to another with trans-light velocity further not possibly remains thereby.

Although it is possible to describe the cosmos from the view of a rotary observer the equations non-rotary of a reference system, in which most objects rest or move only slowly, are usually simpler. Generally case as for example a globular star cluster from neutron stars and black holes, which on all-closest area circle themselves, so that space-time high-grade curved and besides dynamic is, are from the beginning no candidate for an excellent coordinate system recognizably. The relativity principle means for this general case that it is also not necessary to look for it.

## General relativity theory and the Mach principle

Einstein were affected with the development of relativity theory strongly by Ernst Mach and its, by Einstein in such a way designated, machschen principle. This principle means that the forces of inertia of a body depend not on its movement relative to an absolute area, but on its movement relative to the other masses in the universe. The forces of inertia are after this view result of the reciprocal effect of the masses among themselves and an independently of these masses existing area are thus answered in the negative. Therefore for example centrifugal forces of rotary bodies should disappear, if the remaining universe corotates „“. The treatment of the problem is however mathematically very requirementful and to today article of research. It turned out that this principle follows only on the assumption of certain cosmological boundary conditions from the Einstein Feldgleichungen. Thus Gödel found 1949 a global solution of the Feldgleichungen, the Gödel universe in such a way specified , which contradicts short the machschen principle. D.R. Brill and J.M Cohen could indicate however 1966 for a slowly rotating thin-walled hollow ball as the diameter of its black sign radius an approximation solution of the Einstein Feldgleichungen, which the machsche principle fulfills.

## Experimental examination of general relativity theory

### the classical tests and their modern variants

The Periheldrehung of planet courses and the diverson and the red shift of light in the gravitational field are forecasts of general relativity theory, on which the three classical tests so mentioned of the KIND are based.

Also Periheldrehung of the courses of planets is forecast by relativity theory around the suns. Already 1854 were recognized by Urbain Jean Joseph Le Verrier that the course exhibits Merkur a Periheldrehung of approximately 0.1 Bogensekunden per circulation, what is not be due alone to the disturbance by other planets and be explained by relativity theory thus could, what a first success for this theory were. Also the Periheldrehung of other planet as well as for example also the small planet Icarus, confirmed meanwhile , agrees with theoretical computations in accordance with relativity theory. The European-Japanese Merkursonde BepiColombo which is under the planning is it to make possible to determine and still more exactly test thus Einstein's theory the movement Merkur with unequalled accuracy.

The first purposeful experimental examination of the general relativity theory, which excited large attention in the public and which famous general relativity theory made, became 1919 accomplished (see. F. W. Dyson, A. S. Eddington, C. Davidson, 1920: Philos. Trans. Royal Soc. London, volume. 220A, pp. 291-333) and the forecast of general relativity theory examined that light, as each electromagnetic radiation is diverted, in a gravitational field. A solar eclipse was used, in order to measure the apparent shift of the position of a star close of the sun disk, since the effect should be strongest here. The forecast of the Einstein' theory that star light, which touches the edge of the sun disk on its way to the earth should be diverted around 1,75 Bogensekunden, was confirmed with this original measurement with a deviation from 20%. Similar measurements were accomplished later with improved instruments. Later measured into the 1960ern the position of quasars, with which an accuracy was reached of 1,5%, during similar measurements with the VLBI (Very Long cousin LINE Interferometry) the accuracy on 0,2% increased. Also the positions were measured by^{ 10} 5 stars by the ESA satellite Hipparcos, with which the forecasts of the KIND could be examined for 0.1% exactly. Been based on diverson of light in the gravitational field also the gravitation lens effect observed in the astronomy. The ESA space probe Gaia, which is to be started until 2012, is to measure the position of over a billion stars and to determine thus the space curvature still more accurately.

The gravitative red shift was already forecast by Einstein 1911 before completion of general relativity theory and can be already deduced from the energy conservation, so that its experimental confirmation is a necessary condition for the validity of the KIND, but on the other hand not very large force of expression has. Of W. S. Adam the red shift at the white dwarf Sirius B was proven to 1925. The measurement of the gravitativen red shift at white dwarves is to be differentiated however with difficulty from the red shift by the independent movement, and those accuracy is limited. Robert Pound and Glen Rebka proved with the help of the Moessbauer effect the gravitative red shift of the radiation of a gamma source in the earth gravitational field to 1962 with a difference in height of only 25 m with sufficient accuracy. Later improvements (Pound Rebka Snider experiment) reached an accuracy of approximately 1.5%. The gravitative red shift was proven by means of space probes also for the sun and the Saturn. The planned satellite OPTIS is, beside other tests to special and general relativity theory, which test gravitative red shift with an accuracy of^{ 10} -5.

Fourth classical test the Shapiro test is often called, that of I.I. Shapiro was accomplished for the first time 1970. Here the Zeitverschiebung was measured by at the Venus reflected radar signals, while this was from the earth behind the sun, so that the radar waves close at the Sonnenrand had to happen. The accuracy of the measurements amounted at the beginning of still to several per cent. During repeated measurements and later also by measurements by space probes (marine, Viking) in place of the Venus the accuracy on 0,1% could be increased.

### It

made further examinations the development of atomic clocks possible to measure the influence of the gravitation also directly on the time. In principle this measurement is a variation of the proofs of the gravitativen red shift. 1971 became by J. Hafele and R. Keating with cesium clocks in airplanes the Gangunterschied caused by the gravitation proven by clocks in differently heights in accordance with general relativity theory with approximately 10% accuracy clearly. By a similar experiment by C. Alley (Maryland experiment) could be increased the accuracy 1976 on 1%. R. Vessot and M. Levine published 1979 results of a similar experiment by rockets and indicated an accuracy of 0,02%. With the today's satellite-based government inspection department navigation system both corrections must be considered in accordance with the special and general relativity theory, whereby effects outweigh by general relativity theory. Turned around this can be regarded also as confirmation of these theories.

Direct test of the equality of heavier and carrier mass by Eötvös starting from 1890 before the development of relativity theory were already accomplished. Since the Einstein' equivalence principle is based on this equality, such tests are suitable, in order to disprove general relativity theory. Not least because the equality of heavier is relevant and carrier mass also for the possible proof of a fifth natural force, is this topic also today still very up-to-date, and many appropriate experiments became accomplished. Eötvös could increase the accuracy of its experiments in the course of the time in such a way that he could prove the equality with an accuracy^{ of} 10 -9. By experiments with the laser reflectors, which had been set up with Apollo missions on the moon, Shapiro 1976 could prove the validity of the equivalence principle with an accuracy of^{ 10} ,-12. Adelberger et al. published 1999 a work, which confirms this principle with an accuracy of^{ 10} ,-13. New experiments are planned, the accuracy on 10^{,-15} (TEPEE/GREAT: General Relativity Accuracy test) or up to 10^{,-18} (STEPS: Satellite test OF the Equivalence Principle) to increase are.

The gravitation waves predicted by relativity theory could not be proven despite intensive research since beginning of the 1960er (for example gravitation wave receivers of weber with a swinging cylindrical aluminum mass) yet directly. 1969 one stated signals from the center of the Milky Way to have received which could not be confirmed however. Gravitation waves in the meantime indirectly by measurement of the slowing down of the course period of the pulse acre PSR 1913+16, the part of a double star system with another neutron star or a white dwarf than partners is, proven. Similarly this in a long-term observation until March 2005 (together with 3 further statements of the KIND) at could be confirmed the double star system J0737-3093 consisting of two pulse arene, which approximates after resulting computations by sending of gravitation waves daily by 7 mm. The appropriate slowing down agrees accurate in both cases with the delay value computed by general relativity theory, if one assumes that energy is radiated in the form of gravitation waves.

Although the original technology with oscillationable masses was strongly improved in the meantime and today is many more sensitive, many newer experiments use interferometric techniques (Michelson interferometers) for the proof of gravitation waves. A terrestrially based system is the GermanBritish system GEO 600 close Hanover with an expansion of 600 M. A satellitengestützes system is the Esa/NASA project LISA (laser interferometer space Antenna, starting date: 2010) become. LISA consists of three individual space probes, which in a triangle in the distance of several millions kilometers in the universe are to be stationed. Other projects to the proof are TAMA (Japan), LIGO (the USA) and VIRGO (Italy).

The NASA satellite *Gravity sample B*, started in April 2004, is equipped with several precise gyroscopes, which are to measure „the dragee Force unüberprüfte predicted by general relativity theory so far and “around rotary bodies like the earth. In accordance with this forecast space-time should be practically „twisted around rotary bodies “(gravitomagnetic effect). For the measurement of this effect the changes of the directions of rotation are intended by four gyroscopes.

The KIND existed all past direct experimental tests. Also the existence of black holes, forecast by the KIND, is in the meantime considered as empirically secured. Measurements of the movements of objects such as stars or galaxies, which stand under the influence of a gravitational field of galactic and intergalactic dimensions, however generally show a deviation from the movement, which is expected alone by by the visible subject in accordance with the KIND computed gravitational field. This is so far however generally attributed to presence by dark subject and not to a failure of the KIND, although there are also suggestions to explain these discrepancies by alternative Gravitationstheorien. Also as for instance Pioneer 10 and 11, which lie within the outside ranges of the solar system, became small with space probes, but unexplainable deviations of the courses discovers.

The Einstein Feldgleichungen do not follow compellingly from the equivalence principle, but they are only the simplest form of a Gravitationstheorie, which develops on the equivalence principle. There are mathematically more complicated theories, which fulfill also the equivalence principle. They result for example, by adding the Einstein equations kovariante terms with higher derivatives of the Metrik. Well-known alternative theory is also the Brans thickness theory. For the confirmation of the KIND it is not sufficient therefore to accomplish experiments with which one can decide between the KIND and the Newton's mechanics. It is in the long run also necessary to decide experimentally between the KIND and other Gravitationstheorien. Deviations from the forecasts of the KIND could lead also a new impact to the development of a conclusive and experimentally examinable quantum theory of space-time. Finally general relativity theory and the present quantum theory , lose two Grundpfeiler of today's physics within very small length ranges (Planck length) its applicability. In order to unite both theories, on a quantum theory of the Gravition one works already for some time (see also TOE).

## See also

## literature

### popular scientific

- to Harald Fritzsch:
*Bent space-time*, Piper, 1997. ISBN 3-492-22546-2 - Marcia Bartusiak:
*Einstein's legacy*, European publishing house, 2005. ISBN of 3-4345-0529-6

### text books

- Torsten flow brook:
*General relativity theory*, 4. Edition, Elsevier - spectrum academic publishing house, 2003. ISBN 3-8274-1356-7. - Charles Misner; Kip S. Thorne, John. A. Wheeler:
*Gravitation*, W. H. Freeman, San Francisco, 1973. ISBN 0-7167-0344-0. - Hans Stephani:
*General relativity theory*, 4. Edition, Wiley VCH, 1991. ISBN 3326000839. - Steven vineyard:
*Gravitation and Cosmology: Principles and Applications OF the general Theory OF Relativity*, New York 1972. ISBN 0471925675.

### Scientific paper

- Klaus P. Summer:
*Who discovered general relativity theory? Priority controversy between Hilbert and Einstein*, Physik in our time 36 (5), S. 230 - 235 (2005), ISSN 0031-9252 - Clifford M. Wants:
*The Confrontation between general Relativity and experiment*, Living Rev. Relativity 9 (2006) 3, http://www.livingreviews.org/lrr-2006-3