Gravitation

Intuitive comprehension

To think, like Aristote, that on Earth (and with the assumption of the atmospheric Vide) more one body is heavy, more it quickly falls is to make a confusion between quantity and quality:

Quantity: let us take in hand a body attracted by the Ground, and break up it, by a fantastic notion, in a myriad of “ micro bricks of matière ”. Each “ brick of matière ”, being attracted by the Earth, exerts a force, named weight , on the hand and the great number of bricks exerting this weight gives the total weight. The total weight of an object depends on the quantity of matter.
Quality: let us release this body (which we will suppose makes of only one matter), it falls. Each microphone-brick falls because it is attracted by the Earth, only because of that and without taking account of the possible presence of other bricks around, and acquires a certain speed. Thus whatever the number of microphone-bricks, all fall simultaneously (because done everything of the same matter, therefore identical with a good microphone-brick cutting), at the same speed: it is the speed of the whole body, which thus does not depend on the number of bricks and thus does not depend on its weight. This speed is a quality of the body completely independent of the quantity of matter.

Thus, although they are closely associated in our current experiments and our feelings, the two concepts (weight and falling speed) are quite distinct.

The distinction above between quality and quantity does not explain why in the absence of air, from wood and metal fall exactly at the same speed. This experimental fact lets think that these two different matters (as all the others) have same quality in common. The experiments and the reflections on this subject gave the Principe of equivalence.

In more precise terms, more complete and especially more scientific than this intuitive introduction, general relativity studies the gravitation, and like “quality common” to the bodies in the problem arising above, it makes it possible to propose “energy”; although in any rigor this theory admits like assumption the existence of this “common quality” (by admitting the Principe of equivalence) and that it excludes any idea from attraction and gravitational force .

an experiment with the Gravity : while dropping simultaneously two compact objects from very different weights (for example a paper ball and a of the same metal ball diameter) since a breast height, one can think that there is equality falling speeds. But when the drop height is larger, of the perceptible differences appear, because of the frictions of the air. Galileo will be the first to understand that it is the only cause.

The modeling of Galileo Galilei (1564-1642)

Galileo is hardly worried problem of the fall in the Vide of objects of different nature: it has to make already much with the fall of metal spheres, of the serious bodies , on the spherical Ground itself; moreover the concept of vacuum is absent from its thought (the discovery of the physical concept of vacuum by Torricelli, raises of Galileo, takes place only in 1644).

By an experiment top of the Turn of Pisa, it would have noted that heavy balls and different weights have the same drop time, but when it affirms that the falling speed does not depend on the weight on the object, the reason which it exposes is due to experiments on pendulums with the different weights.

The modeling of Isaac Newton (1643-1727)

Mathematician as much as Physicist, Isaac Newton developed, between 1665 and 1685, its theory of the mechanics based on the study of acceleration, and not only of speed as did it Galileo and Rene Descartes.

Fundamental law of dynamics: starting from the principle of inertia of Descartes (which studied the conservation of the Quantité of movement), he concludes that the sum of the force S which are exerted on a body is equal to m has, where m is the mass “ inertielle ” (which makes difficult to push a heavy vehicle) and has is acceleration (the rate/rhythm of variation speed).

Newton sought to unify the laws known for the objects on ground and the laws observed at the Astre S, in particular the terrestrial gravitation and the movements of the Planets.

By considering two specific bodies exerting a gravitational force one on the other, a justification of the law of Newton is the following one:

From the Laws of Kepler, that this one had obtained by observing the movements of planets of the solar system, and of the law of Christiaan Huygens on the Centrifugal force, Newton concludes that the force acting between two bodies is exerted in straight line between the two bodies and is proportional to: 1 D ² , where D is the distance between the two bodies.

Considering that this force is proportional to the quantity of matter present in the body exerting this force (a body having twice more matter exerts a force equal to the sum of the forces of two bodies, therefore exerts a force twice larger), it supposes that the force is proportional to m_G , number called “ mass gravifique ”, proportional to the quantity of matter in the body and reflecting its capacity to exert this force (the “ charge ” gravitational in fact), undoubtedly depending on its nature (apple, lead, clay or gas…).

Under the terms of the principle of the action and the reaction, the force exerted by the other body on the first must be equal (and of opposite direction) and must also be proportional to m'_G , the gravific mass of the second body.

No other parameter not seeming to return in account, this force is expressed under the forme : F = G m_G m'_G / D ² where G is a constant, called gravitational Constante.

By writing the fundamental law of dynamics , one obtains m = G&thinsp has; m_G m'_G / D ² . It is noted that so that acceleration has (and thus speed) of a body in freefall on ground is independent of its inertial mass (like tried out it Galileo), it is necessary that m = m_G for this body, i.e. the “ mass gravifique ” is equal to the inertial mass, independently of the nature of the body. Newton tested this equality for many materials, and since the experiments never ceased, with more and more refinements (Balance of Eötvös, etc). Since, this equality was called the weak principle of equivalence.

The remote action (without contact, through the vacuum) and the instantaneous propagation of the force of gravitation also caused doubts, including Newton.

In the modern vectorial writing, the gravitational force is written:

$\ vec {F} _ {12} \; = - G \ frac {m_1 m_2} {d^2} \ vec {U} _ {12}$

^{$\ vec {F} _ {12}$} being the gravitational force exerted by body 1 on body 2 (in newton or m kg s^-2) ;
G , the Constant gravitational, which is worth 6,6742 N m² kg^-2 (or m³ kg^-1 s^-2) ;
m ₁ and m ₂ , masses of the two involved bodies (in Kilogram S) ;
D , the distance enters the 2 bodies (in Mètre S) ;
^{$\ vec {U} _ {12}$} is a unit Vecteur directed body 1 towards the body 2 ;
the sign – indicate that body 2 is attracted by body 1.

The Newtonian law of the gravitation makes it possible to find the law of Galileo, at first approximation: with D = ray terrestrial and m_T = mass of the Earth, one has G = G m_T / D ² = 9,81 .

The Newtonian theory is well checked in experiments. From a technical point of view, it is enough to make fly of the objects heavier than the air and to send men on the the Moon. The force of Pesanteur is the resultant of the force axifuges and force of gravity (the centrifugal force related to the rotation of the ground on itself, the law of the inertia of the movement, etc).

Contributions of Lagrange and Hamilton

Joseph-Louis Lagrange rewrote, as from 1762, the gravitation and the whole of physics by introducing there the Principe of less action which had been formulated by Pierre Louis Maupertuis about 1744.

William Rowan Hamilton, about 1830, substituted for the principle of less action the concept of energy, which is a constant for any isolated system (i.e.: without interaction with the outside) and which will be of the greattest importance for relativistic physics and in quantum Mécanique, at the 20th century.

These contributions were reformulations of physics, useful for certain applications, but not modifying the bases of the Newtonian gravitation.

The modeling of Albert Einstein (1879-1955)

After having stated the theory of the restricted Relativity in 1905, Einstein seeks to make it compatible with the gravitation whose force is propagated at an infinite speed, in the theory of Newton, whereas the Speed of light is maximum speed for restricted relativity.

About 1915, the solution will come from the assumption that the force of gravitation is not a force in the usual sense which one gives to this word in physics, but a manifestation of the deformation of the space time under the effect of the matter which is there. This assumption is a consequence of the observation that all the bodies fall in the same way into a field from gravitation, whatever their mass or their chemical composition. This observation, a priori fortuitous in Newtonian, and remarkably in experiments checked theory, is formalized under the name of Principe of equivalence and naturally brings to consider that the gravitation is a geometrical demonstration of space itself. The theory thus built, which bears the name of General relativity, incorporates the Principe of relativity, and the Newtonian theory is an approximation within the limit of the weak gravitational fields and small speeds in front of that of the light.

Gravitation and Astronomy

The Newtonian gravitation is sufficient to describe the majority of the phenomena observed on a scale star S. It is enough, for example, to describe the evolution of planets of the Solar system, more or less as the advances Mercury perihelion and the Effet Shapiro.

But the General relativity is necessary to model certain objects and particular astronomical phenomena: the neutron stars, the gravitational mirages, objects very compact such as the black holes…

Gravitation and Cosmology

The gravitation being the dominant force on an astronomical distance scale, the theories Newtonian and Einsteinian were confronted since their respective creations with the observations of the structure with large scales of the universe. So on the scales of the star S and the Galaxy S, the Newtonian gravitation is sufficient in many situations, the Newtonian theory is in difficulty. For example, it is unable to offer a coherent description of a homogeneous universe infinite. On the other hand, general relativity is perfectly able to describe such a situation.

The General relativity only is not enough however to describe the structure with large scales to the universe. It is necessary to associate assumptions to him on the space distribution of the matter. The observations indicate that with large scales, the universe is remarkably homogeneous (with more small scales, the matter is of course distributed in a nonuniform way: space between stars of the same galaxy is primarily empty, just like space between the galaxies). This observational fact had with the departure supposed by Einstein, which had given him the cosmological name of Principe. Provided with this assumption, general relativity allows, rather easily remainder, a coherent modeling of the universe. Its dynamics goes, it, to depend on the properties of the matter which composes it, in particular of sound equation of state. One can show that except particular case, the universe cannot be static: it is either in contraction, or in expansion. The observations confirm this prediction since one observes an apparent recession of the galaxy, those moving away from us all the more quickly as they are distant. The expansion of the universe was discovered by Edwin Hubble at the end of the Années 1920. It indicates that the universe such as we know it is resulting from an extraordinarily dense and hot phase: the Big Bang. The Destin of the universe is not known with certainty, because the long-term behavior of the matter is dubious. One observed a Accélération of the expansion of the universe which seems to be the probable sign that this one will last indefinitely without giving place to a phase of recontraction (Big Crunch).

Gravitation and Quantum physics

General relativity was conceived on the assumption of the continuity of the space time (and even its differentiability) and on the assumption of the continuity of the matter (inter alia building the tensor of density of energy-impulse). This second assumption is clearly an approximation in comparison with the quantum physics.

The quantum physics being the exploration of the infinitely small, the experimentation of the gravitation within this framework encounters an main issue: the three other forces which reign there are at least 10 25 stronger times, whereas it is already difficult to test on them; blow the effects of the gravitation are lost in the inevitable inaccuracies of measurements.

This experimental difficulty did not prevent the theoretical attempts from building a quantum Gravitation, without likely result to date of experimental checking.

One can however notice that:

the addition of the gravitational potential to the equation of Schrödinger makes it possible to find a known result: the particles fall.
the use of the integral of way of Feynman made it possible to envisage a dephasing of the function of wave due to the gravitation (galiléenne); these two effects correspond to a semi-traditional Approximation in quantum mechanics.
the equation of the gravitational waves can be interpreted like that of the propagation of a particle called Graviton, considered to be responsible for the gravitation, which one can deduce certain properties (in particular its mass, null, and its Spin, equal to 2). without that still being able to be checked in experiments.

Examples of quantum theories of the gravitation: Theory M, Supergravité, noncommutative Geometry, quantum Gravitation with loops.

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