# General relativity

The general relativity is a theory Relativiste of the Gravitation, i.e. it describes the influence on the movement of the stars of the presence of matter and, more generally of energy, by taking account of the principles of the restricted Relativité. General relativity includes and supplants the theory of the universal gravitation of Isaac Newton which represents of it the limit at low the speeds (compared with the Speed of light) and with the weak gravitational fields.

General relativity is mainly the work of Albert Einstein, of which she is regarded as the major realization, which worked out it between 1907 and 1915. The names of Marcel Grossmann and David Hilbert are also associated for him, the first having helped Einstein to familiarize itself with the tools Mathématique S necessary to the comprehension of the theory (the differential Géométrie), the second having jointly reached with Einstein the last stages leading to the finalization of the theory after this last had presented to him in the current of the year the 1915 general ideas of its theory.

General relativity is based on concepts radically different from those of the Newtonian gravitation, while stipulating in particular that the Gravitation is not a force, but is the manifestation of the deformation of space (in fact of the Espace-temps) produced by the distribution of matter. This gravitation gives place for many purposes absent in the Newtonian theory but verifiable in experiments, commes the gravitational waves, the black holes, and the Expansion of the universe. None many the experimental tests to date carried out (2007) could put it at fault, except possible for the Anomalie Pioneer which could be the first indication of a difference between the phenomena observed and general relativity, though many other interpretations of this phenomenon are possible.

## General information

### Need for a relativistic theory of the gravitation

The theory of the Universal gravitation proposed by Newton at the end of the 17th century is based on the concept of force of gravitation acting according to the principle of remote action, i.e. the fact that the force exerted by a body (for example the Sun) on another (the Ground) is determined by their position relative to a given moment, and this whatever the distance separating them. This instantaneous character is incompatible with the idea of the restricted Relativité suggested by Einstein in 1905. Indeed, according to the latter, no information can be propagated more quickly than the Speed of light in the vacuum. In addition, the principle of the remote action rests on that of the Simultanéité of two events: the force that the Sun exerts on the Ground at a given moment is determined by their properties “ at this moment ”. Restricted relativity stipulates that the concept of simultaneity of two events is not defined, the concept of simultaneity differing from an observer to another for little that those are not animated a nonnull relative speed. These contradictions lead Einstein since 1907 to think of a theory of the gravitation which is compatible with restricted relativity. The result of its search is the general theory of relativity.

### Relativity of Galileo to restricted relativity

• At the 16th century, Galileo affirms, and explains, why the laws of physics are the same ones in reference frames in rectilinear translation and uniform the ones compared to the others. It is the principle of relativity (of Galileo).
• It will use also the additivity speeds which has as consequence which any speed can be reached: it is only one question of means. In a word: if a ball rolls to 10 km/h in a train (and in the direction of walk) which goes itself at 100 km/h compared to the ground, then the ball goes has 110 km/h compared to the ground.

In its Mechanics, Isaac Newton presupposed that the bodies were equipped an absolute velocity, in other words which they were either “really” at rest, or “really” moving. He also noticed that these absolute velocities were nonmeasurable differently than relative at the speeds of the other bodies (in the same way, the position of a body was measurable only relative with that of another body, etc). Consequently, all the laws of Newtonian mechanics were to operate with identical whatever the body considered and whatever its movement. However, Newton thought that its theory could not have direction without the existence of a reference frame fixes absolute in which the speed of any body could be measured, even if this one could not be detected. In fact, it is possible to in practice build a Newtonian mechanics without this assumption: the resulting theory (named besides Relativité galiléenne) does not have besides particular operational interest and does not have to be confused with the relativity of Einstein which implies in more constancy of the Speed of light in all the reference frames and in less the assumption galiléenne that relative speeds are added (these two axioms are indeed mutually incompatible). -->
• At the 19th century, the Scottish physicist James Clerk Maxwell formulated a whole of equations, the equations of the electromagnetic field, which resulted in predicting the wave propagation electromagnetic speed $c = \ tfrac \left\{1\right\} \left\{\ sqrt \left\{\ varepsilon_0 \ mu_0\right\}\right\}$ in an electrostatic of constant $\ epsilon_0$ and magnetostatic medium of constant $\ mu_0$. This phenomenally raised speed, even in a rarefied medium as the air had the same value as the speed of light propagation. He proposed that the light is anything else only one wave electromagnetic.
• the corpuscular theories of the light seemed compatible with the principle of relativity of Galileo; as well as the theory of Maxwell who leant in favor of the existence of a luminiferous Éther under consideration by Huygens. To measure the speed of the solar system compared to this springy medium was the object of the experiments of interferometry undertaken by Michelson and Morley. Their experiments showed that the apparent ether wind was null, whatever the period of the year. To suppose that the ether was constantly fixed on the ground would have been a too serious handing-over of cause of the principle of relativity of Galileo. In addition, the ether presented the disadvantage of being at the same time impalpable and very “rigid” since able to propagate the waves at a phenomenal speed.

• It was necessary to await Einstein in 1905 to radically call into question the concept of ether, to carry to highest the principle of relativity of Galileo while postulating than the Maxwell's equations obey themselves this principle, and to draw from them the revolutionary conclusions in an article remained famous: Of the electrodynamics of the bodies moving .

It is the birth of the restricted Relativité:

• the principle of relativity of Galileo is preserved.

• the invariance of the Maxwell's equations involves immediately the constancy of the Speed of light $c$ in all the reference frames galiléens: the additivity speeds is not true any more and speed of light is unattainable (except for the light).

• the measures of length, of time interval, (and speed) are not same the following the reference frame of the observer: to measure the length of the coach gives different results according to whether one is inside or that one is motionless on the ground (but it is not the case for the width of the coach, length perpendicular at the speed); in the same way for the flow of time; the electric field becomes magnetic and reciprocally… All these transformations of the frames of reference of the space-time continuum and the electromagnetic field are formalized by the transformations of Lorentz (paradoxically developed by Lorentz and Henri Poincaré to defend the existence of ether).

• the concept of absolute time disappears: two groups of two motionless clocks perfectly synchronized in a reference frame galiléen, and of two motionless clocks perfectly synchronized in another reference frame galiléen present defects of synchronization one compared to the other.

• By writing the expression of the kinetic energy of a body of mass $m$ in the simplest way respecting the principle of relativity, Einstein revealed an energy of rest $E = m c^2$ of which the direction will burst only one about thirty years later when Lise Meitner includes/understands the origin of the nuclear fission energy.

### Relativity restricted with General relativity

The restricted theory of relativity (1905) modified the equations used to compare the duration and measures of length made in various reference frames moving the ones compared to the others: that had as a consequence which physics could not treat any more time and space separately, but only like one space with four dimensions, the Espace-temps of Minkowski.

Indeed, at the time of movements at considerable speeds in front of C (speed of light in the vacuum), time and space deteriorate in a bound way, a little as two punctual coordinates in analytical Geometry deteriorate in a dependant way when the axes of the reference mark are swivelled.

For example, in usual Euclidean geometry the distance $\ Delta l$ between two points of coordinates $\left(X, there, Z\right)$ and $\left(x\text{'}, y\text{'}, z\text{'}\right)$ checks $\left(\ Delta L\right) ^2 = \left(\ Delta X\right) ^2 + \left(\ Delta there\right) ^2 + \left(\ Delta Z\right) ^2$ (with $\ Delta x=x\text{'}-x$, etc), but in the space of Minkowski two points are located by the coordinates $\left(T, X, there, Z\right)$ and $\left(you, x\text{'}, y\text{'}, z\text{'}\right)$, where $t$ and $t\text{'}$ are the coordinates time, and it $\ Delta l$ between these points “outdistances” checks $\left(\ Delta L\right) ^2 = \left(C. \ Delta T\right) ^2- \left(\ Delta X\right) ^2- \left(\ Delta there\right) ^2- \left(\ Delta Z\right) ^2$. This calculation gives a null “distance” between two points of the course of a luminous ray, it gives also all the material measures of length, time intervals, speeds in restricted Relativité which always cause the astonishment.

The space time of Minkowski being nevertheless of null Courbure (i.e. flat) one describes it as space pseudo Euclidean . C is always zero. -->

Such was to be, for Einstein, space without gravitation (and without acceleration for the observer). The Newtonian gravitation, being propagated instantaneously, was not compatible with. Einstein was put in search of a new theory of the Gravitation.

• Einstein admitted the equality between the gravific mass and the inertial mass like assumption, the famous formula $E=mc^2$ then authorizing to use the total energy of a body instead of its mass. It will be done thanks to the mathematical tool named Tenseur energy.

• Expert in experiments by the thought, it imagined a disc in rotation looked by an experimenter placed in his center and turning with: as for Huygens, there is a centrifugal force on the level of the perimeter which is perceived like a gravitational force (because the gravific mass and the inert mass are equal by assumption). Moreover, while wanting to remain within the framework of restricted relativity, it concluded that the observer must note the reduction of the perimeter but not ray: it is not possible in a flat space. Conclusion: the gravitation obliges to use a not-Euclidean Géométrie.

• Einstein imagined an experimenter locked up in an elevator with the opaque walls, undergoing a rise with constant acceleration: impossible for this person to know if there are constant acceleration or gravitational attraction (because the gravific mass and the inert mass are equal by assumption). Conclusion: local equivalence between accelerated movement and gravitation, which was to find in the differential equations new theory. It is its Principe of equivalence.

• Lastly, Einstein wanted to at the time find an expression of the natural laws (: dynamics, gravitation and electromagnetism) which is unchanged whatever the reference frame (accelerated or galiléen, etc…) : it is the relativity galiléenne generalized with all the reference marks (one names that the covariance).

• the great difficulty being of putting these principles in mathematical form, it discussed it with David Hilbert which, initially dubitative, failed to charm the high-speed motorboat at the same time to him by finding the theory as (see to him: Controversy on the paternity of relativity).

The general relativity added to restricted relativity that the presence of matter could deform the space time locally itself (and not just trajectories), in such a way that trajectories known as Géodésique S - i.e. intuitively minimal length - through the space time have properties of Courbure in space and time. The calculation of the “distance” in this curved space time is more complicated than in restricted relativity, makes the formula of the “distance of it” is created by the formula of the curve, and vice versa.

Geodetic are trajectories checking Principle of little action, followed by particles test (i.e. whose influence on the field of gravitation in which they move is negligible, which is the case for example of a Artificial satellite around the Earth or of a Photon passing beside the Sun but not of a star orbiting around an other in a Binary system oscillating quickly), they thus have an importance practices very important for the intuitive comprehension of a curved space.

### Theoretical consequences and observations

• Einstein immediately calculated (1915) the deviation of the apparent positions of stars by the sun: the May 29th 1919, measurements were made by Sir Arthur Eddington at the time of a solar eclipse, and in spite of some inaccuracies of measurement, that constituted the first confirmation of the theory.
• This theory envisages a slow rotation of the ellipse of revolution of Mercure which agrees perfectly with the observations.
• the gravitation (strong) of a planet must contract there the lengths observed since a remote position. That could not be observed directly to date.
• the gravitation must slow down time, therefore to modify the frequencies and the wavelengths of the emitted radiations: one can quote for example an experiment undertaken by Pound and Rebka to the Université Harvard (1959), which made it possible to detect a change of the Wavelength of a source Monochromatique of Cobalt caused by the terrestrial gravitational field on an altitude of 22,5 meters.
• Schwarzschild, while finding in 1916 an exact solution of the equations of the gravitation, showed that it could exist conditions where a phenomenon of Black hole appeared. Astronomy observes similar phenomena.
• Under certain conditions, of the gravitational, discrete waves, must be propagated in space. The Franco-Italian experiment Virgo seeks to detect some.
• Another consequence practices general relativity: the atomic clocks in Orbite around the Ground of the Système of positioning GPS ( Total Positioning System ) require a correction for the deceleration due to terrestrial gravity.

To summarize this theory, Einstein amused a public of journalists: “Imagine that you look at far, very far in front of you, and that you have very a good sight, one very very good sight, then you will manage to see… your back. ”

## Summary of the theory

### NonEuclidean geometries

The geometrical description of the physical theory due to Einstein finds its origins in the projections of the nonEuclidean Géométrie, which result from the various attempts during the centuries to show the fifth postulate of Euclide, which states that: “ by a point one can lead only one parallel to a given line ”. These efforts culminated at the 19th century with the discovery by the mathematicians Nicolaï Ivanovitch Lobatchevsky, János Bolyai and Carl Friedrich Gauss that this postulate could be replaced by another (several possible parallels, or not of parallel of the whole), and thus constituted only a arbitrary Axiome. None of these new geometries is more “true” that those of Euclide: they are simply different conceptual tools being able to be used as support with also different uses. The surface of a sphere, for example, can indifferently be regarded as the surface of an object in an Euclidean space with 3 dimensions or in a space nonEuclidean private individual with two dimensions, the second representation being able to prove more convenient in certain cases.

To illustrate, if the universe is characterized by such a geometry, that a physicist holds a stick vertically, and that at a certain distance, a cartographer measures his length by a technique of triangulation based on the Euclidean geometry, nothing guarantees that it will obtain the same result if the physicist brings the stick to him and that it measures it directly.

The generalization of these results, called nonEuclidean Geometry , was carried out by Bernhard Riemann, a pupil of Gauss, but she was regarded as simple mathematical curiosity until Einstein uses work of its professor Hermann Minkowski (who used Complex numbers to obtain nonEuclidean spaces easy to treat in analytical geometry… and expressed in 1907 in this description the Transformation of Lorentz!) to develop its general theory of relativity .

### Reference frames

The central idea of relativity is that one cannot speak about quantities such as speed or acceleration without to have chosen a framework of reference before, a Référentiel , defined in a given point. Any movement is then described relative with this reference frame. Restricted relativity postulates that this reference frame can be wide indefinitely in space and time. It treats only the case of the reference frames known as inertial, otherwise known as animated a constant speed and change of management. General relativity, it, treats the accelerated reference frames or not (with the vectorial direction). In general relativity, it is allowed that one can define a local reference frame with a precision given only over one finished period and in a finished area of space (in the same way, because of the curve of terrestrial surface, one can draw a chart without distortion only on one limited area). In general relativity, the laws of Newton are only valid approximations in an inertial local reference frame. In particular, the trajectory of free particles as of the Photon S is a straight line in an inertial local reference frame. As soon as these lines are extended beyond this local reference frame, they do not appear right any more, but are known under the name of Géodésique S. The first law of Newton must be replaced by the law of the geodetic movement.

The trajectory of a photon is for example geodetic null length…: the positive part of the square this length $\left(x^2+y^2+z^2\right)$ indeed equal and is opposed to its negative part $\left(- c^2t^2\right)$.

Let us reconsider the concept of inertial reference frame. We distinguish the inertial reference frames, in which a free body of any external action maintains a movement uniform, noninertial reference frames, in which a free body undergoes an acceleration whose origin is due to the acceleration of the reference frame itself. An example is the Centrifugal force which one feels when a vehicle which transports us carries out a rapid change of management, another example is the force known as of Coriolis, manifestation of terrestrial rotation. The centrifugal force is fictitious and is only one manifestation of inertia (first principle of Newton).

### Principle of equivalence

Because he forever be possible to highlight the least difference between the mass of Inertie (resistance of one body to acceleration) and the heavy mass (which determines its weight in a field of gravity), the Principe of equivalence in general relativity postulates that it is not necessary to locally distinguish a movement from freefall (without rotation) in a gravitational field, of a movement uniformly accelerated in the absence of gravitational field. In light, one locally does not observe gravitation in a reference frame in freefall, in as much as it is sufficiently small, compared to the means of detection, so that one cannot detect acceleration there. Around the Earth, the freefall can be for example a fall towards the ground or the movement of a satellite.

This result is only local , i.e. valid for a restricted space i.e. “small”. In a volume and with sensitive accelerometers, one will distinguish very well on the contrary a field from gravity (convergent forces), a simple acceleration (parallel forces) and a centrifugal effect (divergent forces). It is just a question of unifying what is similar in the phenomena in order to treat them by a single mechanism.

This equivalence is used in the training of the astronauts: those get into planes accomplishing a parabolic flight where the centrifugal force counterbalances a few minutes the forces of gravity, thus simulating the “freefall” of a put into orbit body (freefall which lasts indefinitely, since circular).

From this point of view, the gravitation observed on the terrestrial surface is the force observed in a reference frame defined in a point of the terrestrial surface which is not free, but on which acts all the rock which constitutes the core, and this force is of nature identical to the centrifugal force which would be felt in a spaceship sufficiently far away from the Earth to hardly undergo its attraction, and carrying out an operation of change of management. Or, the ground prevents an object from making its freefall by exerting a force to the top (called “reaction of the ground”); in Newtonian mechanics, one rather tends to consider that the freefall is an acceleration to the bottom, whereas here, the freefall is the state of reference and it is at-rest state compared to the ground which is an acceleration upwards.

The principle of equivalence amounts considering, to summarize, that the inertial mass and the gravitational mass represent two distinct things but which have the same value exactly.

### Tensor of energy and curve of space

Mathematically speaking, Einstein models the space time by a Variété pseudo-riemannienne four-dimensional, and its equation of the gravitational field connects the Courbure variety in a point, with the Tenseur energy-impulse in this point, this Tenseur being a measurement of the density of matter and energy (given that matter and energy are equivalent).

This equation is at the base of the famous formula which says that the curve of space defines the movement of the matter, and the matter defines the curve of space (both being equivalent). The best way of representing the geometry of the space time is to imagine than this one behaves like an elastic surface dug locally by the presence of a massive object, a ball for example.

The shortest way between two points - what remains the definition of the “straight line” - will then not be the same one as in the absence of deformation: if the trajectory passes too much close to the ball, indeed, the course “is lengthened” by the digging of the rubber sheet. Let us notice that we have to take into account in this analogy neither time nor gravity, which is normal since it is them whom we wish to describe at exit.

While transposing this image in physical space, the presence of a massive body will affect the curve of space, which will seem seen outside deteriorating the race of a luminous ray or an object moving which passes in its vicinity. To take again a famous expression due to John Archibald Wheeler: “The mass and energy say to the space time how to curve itself, and the curve of the space time called to the matter how to behave”.

That wrongly has as a consequence in astronomy the effect of gravitational mirage (sometimes named gravitational lens , because not having the properties nor of a convergent lens - what one sees immediately if one traces more than four rays! - nor those of a divergent lens).

This concept of curve of space explains the curve of the luminous rays in the vicinity of a massive star, which could not be due to the law of Newton if the photons do not have a Masse.

The equation of the field of Einstein is not a single solution and there is place for other models, if they are in agreement with the observations.

General relativity is distinguished from the other existing theories by the simplicity of the coupling between matter and geometrical curve, but it remains to carry out the unification between general relativity and the quantum Mécanique, and the replacement of the equation of the gravitational field by a more general quantum law.

Few physicists doubt that such a Théorie of All would give place to the equations of general relativity in some limiting of application, in the same way that the latter makes it possible to predict the laws of the gravitation of Newton within the limits low speeds (known as nonrelativistic speeds).

The equation of the field contains an “additional” parameter called the cosmological Constante $\ Lambda$ which was introduced at the origin by Einstein so that a static universe (i.e. a universe which is neither in expansion, nor in contraction) that is to say solution of its equation.

This effort showed a failure for two reasons: the static universe describes by this theory was unstable, and the observations of the astronomer Edwin Hubble ten years later showed that the Univers was in fact expanding. Thus $\ Lambda$ was given up, but recently, of the astronomical techniques showed that a nonnull value of $\ Lambda$ is necessary to explain certain observations.

The study of the solutions of the equation of Einstein (cf following paragraph) is a branch of named Physics Cosmologie. It in particular makes it possible to explain the excess of the advances Mercury perihelion, to predict the existence of the black holes, gravitational waves and to study the various scenarios of evolution of the Universe. Let us note that the well-known astrophysicist Stephen Hawking showed that a universe as ours comprised necessarily gravitational singularities.

More recently (October 2004), of the measurements taken by laser with satellites LAGEOS showed that the gravitational field of the Earth itself generates distortions of positioning of the Moon of two meters per annum compared to what would be envisaged by the only laws of Newton. This figure is in agreement with 1% near with what is envisaged by General relativity.

## Particular solutions of the equation of Einstein

• the Metric of Schwarzschild:
-->Dans the Vacuum and for an identically null cosmological constant, the equation of Einstein is reduced to:
$R_ \left\{\ driven \ naked\right\} \ - \ \ frac \left\{1\right\} \left\{2\right\} \, g_ \left\{\ driven \ naked\right\} \, R \ = \ 0$
Dans the particular case of a central field generated by a body with spherical symmetry, the Métrique of Schwarzschild (January 16th, 1916) provides an exact solution to this equation (which is valid only with the outside of the body):
$\ mathrm ds^2 \ = \ - \ \ left \left(1 \ frac \left\{2GM\right\} \left\{rc^2\right\} \ right\right) c^2 \ mathrm dt^2 \ + \ \ frac \left\{\ mathrm dr^2\right\} \left\{1 \ frac \left\{2GM\right\} \left\{rc^2\right\}\right\} \ + \ r^2 \ \ mathrm D \ Omega^2$
$M$ total mass of the body, and $\ mathrm D \ Omega^2$ the square of the elementary distance on the Euclidean sphere of ray unit in spherical coordinates:
$\ mathrm D \ Omega^2 \ = \ mathrm D \ theta^2 \ + \ \ sin^2 \ theta \ \ mathrm D \ varphi^2$

### Problem with two bodies & problem of the movement

In general relativity, the Problème with two bodies is not exactly soluble; only the “problem with a body” is. However, one can in general find a solution approached for what is called sometimes the “problem of the movement”.

#### Einstein & the problem of the movement (1915)

In its manuscript of at the end of 1915, Einstein starts by calculating the field of gravitation to spherical symmetry created by a star of mass $M$ when one places oneself far from the center of the star, the field being then of weak intensity. Einstein explores then the problem of the movement of a “particle test” of mass in this weak field. The particle test is thus supposed not to modify the field of gravitation created by the massive star.

The principle of equivalence had in addition led Einstein to to postulate the equations of the movement of the particle-test as being the equations whose solutions are unquestionable geodetic of the space time. Mathematically, the geodetic ones make the pseudo-distance extreme:

$\ delta \ \ int \ mathrm ds \ \ = \ 0$

In a Frame of reference locally inertial $X^ \left\{\ alpha\right\}$, these equations of the movement are written in components:

$\ frac \left\{\ mathrm d^2 X^ \left\{\ alpha\right\}\right\} \left\{\ mathrm D \ tau^2\right\} \ = \ 0$

where $\ tau$ is the clean time of the particle test (presumedly massive). In an unspecified frame of reference $x^ \left\{\ driven\right\}$, these equations of the movement take the following form:

$\ frac \left\{\ mathrm d^2 x^ \left\{\ driven\right\}\right\} \left\{\ mathrm D \ tau^2\right\} \ + \ \ Gamma^ \left\{\ driven\right\} _ \left\{~ \ rho \ sigma\right\} \ \ frac \left\{\ mathrm D x^ \left\{\ rho\right\}\right\} \left\{\ mathrm D \ tau\right\} \ \ frac \left\{\ mathrm D x^ \left\{\ sigma\right\}\right\} \left\{\ mathrm D \ tau\right\} \ = \ 0$

The solutions of these equations define the Géodésique S of the kind time of the space time.

#### Einstein & the problem of the movement (1938)

In its work of 1938 completed in collaboration with Infeld and Hoffmann, Einstein will show that equations of the movement of the particle-test:

$\ delta \ \ int \ mathrm ds \ \ = \ 0$

derives from the equations of the field. It is thus not necessary to introduce them by an additional postulate.

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