David Hume

The restricted relativity is the formal theory worked out by Albert Einstein in 1905 in order to draw all the physical conclusions from the Relativité galiléenne and owing to the fact that the Speed of light in the vacuum with the same value in all inertial referential, which was implicitly stated in the Maxwell's equations (but interpreted well differently until there with “the absolute space” of Newton and the ether ).

the Relativité galiléenne stipulates, in modern language, that any experiment made in an inertial reference frame would proceed in a perfectly identical way in any other inertial reference frame. Become “Principle of relativity”, its statement will be then modified by Einstein to be wide with the not-inertial reference marks: of “restricted” relativity will become “  general   ”.

The restricted theory of relativity established news formulas making it possible to pass from a Référentiel galiléen to another. The corresponding equations lead to phenomena which run up against the common direction, one of most astonishing and most famous being known under the name of Paradoxe of the twins (a paradox which in addition was popularized in Science-fiction).

Restricted relativity had also an impact in Philosophie by eliminating any possibility of existence from a time and absolute durations as a whole of the universe. Following Henri Poincaré it forced the philosophers to be differently posed the question of time and space.

Origins of the theory

Short history

At the end of the 19th century, Maxwell establishes the equations governing the electromagnetic waves and in particular the light waves. According to this theory speed of light was to depend only on the properties electric and magnetic of the medium and not the speed of the reference mark of measurements, which posed a problem. Indeed, in Newtonian Mécanique speeds are added (to say the things quickly). So of a rocket moving at the speed of 7  km/s compared to the Earth one draws a ball from gun forwards at the speed of 1  km/s compared to the rocket, the speed of the projectile compared to the Earth will be of 8  km/s. If the ball is drawn backwards, its speed will be of 6  km/s.

The Maxwell's equations say on the contrary who if one emits a beam of light since the rocket forwards or backwards the speed of light measured on Earth will be the same one. The experiment was led by Michelson and Morley, the Earth itself playing the part of the rocket. As our planet moves around the Sun at the speed of 30  km/s, they wanted to see whether they could highlight a difference in speed of light between the direction of the movement of revolution and the opposite direction. Not having detected any such difference this experiment confirmed the validity of the Maxwell's equations.

formulas of transformation to pass from an observer to another were established by Lorentz; they were equations of compatibility whose significance was not clear. An explanation was then imagined to justify these strange formulas: the ether, medium considered to be previously necessary to the propagation of the light waves as the air is necessary to the propagation of the sound waves, would have the elastic properties which would lead to these equations. Poincaré published articles on the theory before Einstein.

In 1905, in its article entitled Of the electrodynamics of the bodies moving , Albert Einstein presented relativity as follows:

• the ether is an arbitrary concept which is not useful for the expression of the theory of relativity.

• the celerity of the light compared to the observers does not depend their speed.
• the laws of physics respect the Principe of relativity.

The equations of Lorentz which result from this are in conformity with physical reality. They have unexpected consequences. Thus an observer allots to a body moving a length shorter than the length allotted to this same body at rest and the duration of the phenomena which affect the body moving is lengthened compared to this “  même  ” lasted measured by motionless observers compared to this body.

Einstein also rewrote the formulas which define the Quantité of movement and the kinetic energy so as to make them invariant in a transformation of Lorentz.

The time and the three coordinates of space playing of the indissociable roles in the equations of Lorentz, Minkowski interpreted them in a Espace-temps with four dimensions. Let us notice however that time and space remain different natures and that one cannot thus compare one to the other. For example one can make half-turn in space whereas that is impossible in time. From this point of view, cosmologists as Wheeler defends the idea that it is not relevant to regard time as an imaginary coordinate and proposes to give up this practice. Of course it remains that time cannot be separate space and that relativity must develop in a space time with four dimensions, for time and three for space.

The distribution of the roles of such or such scientist in the emergence of the restricted theory of relativity is the object controversy, particularly since the years 2000.

Attitude of the Nobel committee

In 1912, Lorentz and Einstein were proposed for a Nobel Prize united for their work on the theory. The recommendation was of Wien, prize winner of 1911, who declares that “although Lorentz must be regarded as the first to have found the contents mathematical of the principle of relativity, Einstein succeeds in reducing it in a simple principle. One should consequently regard the merit of the two researchers as comparable”. Einstein never accepted the Nobel for relativity, the Nobel Prize, in theory, being never granted for a pure theory. The committee thus awaited an experimental confirmation. Time that the latter arises, Einstein had passed to other important work.

Einstein will be finally seen decreeing the Nobel Prize of physics in 1921, for its contributions with the theoretical physics, and in particular for its explanation of the photoelectric Effet.

Interpretation of time by Poincaré

It is Poincaré which gave to the whole of the formulas of transformation the name of “equations of Lorentz”. It indicates in its course of 1898 which local time that Lorentz presented as a fictitious parameter did not have reason not to be regarded as the very short time, which would be relative and not absolute. In June 1905, Poincaré also announces that the whole of the transformations into question forms a structure of Groupe on the space time, and that the term $(x^2+y^2+z^2-c^2t^2\; \backslash ,)$ constitutes an invariant of the group. In a text published in 1915, Lorentz approves the point of view of Poincaré.

We find in the book of T. Damour, a compared analysis of the concept of time at Poincaré and Einstein showing the value of what Einstein brings. Let us quote some sentences:

A crucial consequence of the limitation of the conceptual horizon of Poincaré is that the " time local" , he speaks in the text about 1904 cité above differs in an essential way of the " temps" that Einstein allots to a reference frame moving. Indeed, an attentive reading of the text of Poincaré of 1904, courses which it gave to the Faculty of Science of Paris during the winter 1906-1907, and of an article published in 1908, watch that the " temps" of which speaks Poincaré is always a time of which the " seconde" is beaten by clocks in " rest absolu".

In this respect, if one can discuss the fact that Einstein read or not Poincaré before June 1905, it agrees to wonder whether Poincaré had read the article of 1905 of Einstein thereafter.

Still let us quote the conclusion of T. Damour on the subject: “Like Lorentz and Poincaré always time in terms of absolute universal time of Newton thought, they never suggested, as Einstein did it, than a clock moving can beat a time different from that of a clock at rest. ”

The theory

Postulates of Einstein (1905)

1. the laws of physics have the same form in all the inertial referential . (Let us recall that a reference frame is known as inertial if it does not undergo any acceleration: a rocket in space far from any mass constitutes an inertial reference mark if no engine is ignited.)

2. the Speed of light in the vacuum with the same value in all the inertial referential .

The first postulate is the Principe of relativity itself, in its restricted design with the class of the inertial reference frames. It formalizes an intuition of Galileo according to which the uniform rectilinear motion is “like nothing” for the observer pertaining to the mobile reference frame.

The second postulate allows in practice a synchronization of the fixed clocks of a given reference frame, in any place of this reference frame: the use of the light signals allows the “guard time” to synchronize all the clocks of its reference frame. With this intention it suffices that this administrator emits a time signal, for midday for example. When the observer located at the distance R receives the signal it will take account of time r  /c that the signal will have put to reach him and will put its clock per hour “  midi+ r  /c   ”.

One can do without the second postulate to determine the equations of the transformations of Lorentz on the condition of introducing an additional assumption with the first postulate: the space time is homogeneous and isotropic . This fact was discovered since 1910 by Kunz and independently by Comstock. The additional assumption physically leads to a group of transformations to a parameter $c\; \backslash ,$, homogeneous at a speed. These transformations are identified:

• with the Transformations of Galileo if $c^2\; \backslash ,$ is infinite.

• with the Transformations of Lorentz if $c^2\; \backslash ,$ is finished positive.

The identification of $c\; \backslash ,$ with the speed of light, established as finished by the observations, results in the second postulate.

The reader interested by this aspect can consult, for example, or paragraph 2.17: Special Relativity Without the Second Postulate of.

The Transformations of Lorentz

One considers two reference frames $\backslash \; mathbb\; \{R\text{'}\}$ and $\backslash \; mathbb\; \{R\}$, the first reference frame $\backslash \; mathbb\; \{R\text{'}\}$ being animated speed $\backslash \; vec\; \{v\}$ compared to the reference frame $\backslash \; mathbb\; \{R\}$. To simplify calculation one works initially within the framework of transformations known as “special”, characterized by the fact that the systems of axes X, there, Z and x', y', z' are parallel and that the axes O  ' x  ' and OX are common and parallel at the speed $\backslash \; vec\; \{v\}$. This restriction harms the general information of the results by no means. One will write below the formulas relative at a speed pointing in an unspecified direction.

The assumptions of Einstein lead to the transformations known as “  of Lorentz   ”. The formulas of Lorentz make it possible to express the coordinates ( X , there , Z , T ) of an event given in the “fixed” reference mark (let us say the Earth) according to the coordinates ( x  ' , y  ' , z  ’ , t  ’  ) same event in the “mobile” reference mark (let us say a rocket). They écrivent :

$$

\ begin {boxes} ct = \ gamma (ct'+ \ beta x') \ \ X = \ gamma (x' + \ beta ct') \ \ there = y' \ \ Z = z' \ end {boxes}

where $\backslash \; beta$ and $\backslash \; gamma$ is factors without dimension defined by

$$

\ beta = v/c \ qquad \ gamma= \ frac {1} {\ sqrt {1 \ beta^2}} \.

These expressions are simplified and take the form of a rotation if one utilizes the hyperbolic functions of the angle θ defined by

$$

\ tanh \ theta = v/c \ equiv \ beta \ qquad \ text {is} \ qquad \ theta = \ mathrm {atanh} (v/c) \ equiv \ mathrm {atanh} \, \ beta

With these notations one obtains

$\backslash \; gamma\; =\; (1\; -\; \backslash \; beta^2)\; ^\; \{-\; 1/2\}\; =\; (1\; -\; \backslash \; tanh^2\; \backslash ,\; \backslash \; theta)\; ^\; \{-\; 1/2\}\; =\; \backslash \; cosh\; \backslash ,\; \backslash \; theta$

and

$$

\ begin {boxes} ct= ct' \ cosh \, \ theta + x' \ sinh \, \ theta \ \ X = ct' \ sinh \, \ theta + x' \ cosh \, \ theta \ end {boxes}

To obtain the formulas corresponding to the reverse transformation it is enough to change β in - β , and thus θ in - θ , which leads to:

$$

\ begin {boxes} ct'= ct \ cosh \, \ theta - X \ sinh \, \ theta \ \ x' = - ct \ sinh \, \ theta + X \ cosh \, \ theta \ end {boxes}

A trick: to find the sign to be put in front of sinh θ it is enough to consider a point at rest in one of the reference marks (say that of the rocket, with x  ’   =  0 for example) and to see which must be the sign of the space coordinate in another reference mark (let us say the reference mark fixes in which X grows if the rocket has a positive speed).

If the special transformations simplify the analytical study, they do not harm of anything the general information. One can easily pass if the reference frames moving are not parallel one with the other, and are of unspecified orientation compared to their relative speed $\backslash \; vec\; \{v\}$. It is always possible to break up the vector $\backslash \; vec\; \{R\}$ along two directions: that parallel with displacement $\backslash \; vec\; \{R\}\; \_\; \{//\}$ and that orthogonal with this one $\backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}$. One thus has: $\backslash \; vec\; \{R\}\; =\; \backslash \; vec\; \{R\}\; \_\; \{//\}\; +\; \backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}$

By posing

$\backslash \; vec\; \{\backslash \; beta\}\; =\; \backslash \; vec\; \{v\}\; /c$

the transformations of Lorentz give:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}$

ct'= \ gamma (ct- \ vec {\ beta} {\ cdot} \ vec {R}) \ \ \ vec {r'} _ {//} = \ gamma (- \ vec {\ beta} ct + \ vec {R} _ {//}) \ \ \ vec {r'} _ {\ club-footed} = \ vec {R} _ {\ club-footed} \ \ \end{matrix}\right. \. what leads to

$\backslash \; vec\; \{r\text{'}\}\; =\; \backslash \; vec\; \{r\text{'}\}\; \_\; \{\backslash \; club-footed\}\; +\; \backslash \; vec\; \{r\text{'}\}\; \_\; \{//\}\; =\; \backslash \; gamma\; (-\; \backslash \; vec\; \{\backslash \; beta\}\; ct\; +\; \backslash \; vec\; \{R\}\; \_\; \{//\})\; +\; \backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}\; =\; \backslash \; gamma\; (-\; \backslash \; vec\; \{\backslash \; beta\}\; ct\; +\; \backslash \; vec\; \{R\})\; -\; (\backslash \; gamma-1)\; \backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}\; \backslash .$

Like

$\backslash \; vec\; \{\backslash \; beta\}\; \backslash \; times\; \backslash \; vec\; \{R\}\; =\; \backslash \; vec\; \{\backslash \; beta\}\; \backslash \; times\; (\backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}\; +\; \backslash \; vec\; \{R\}\; \_\; \{//\})\; =\; \backslash \; vec\; \{\backslash \; beta\}\; \backslash \; times\; \backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}$

one has (while multiplying vectoriellement by $\backslash \; vec\; \{\backslash \; beta\}$)

$\backslash \; vec\; \{\backslash \; beta\}\; \backslash \; times\; (\backslash \; vec\; \{\backslash \; beta\}\; \backslash \; times\; \backslash \; vec\; \{R\})\; =\; \backslash \; beta^2\; \backslash \; vec\; \{R\}\; \_\; \{\backslash \; club-footed\}$

One thus obtains the expression of the general transformations of Lorentz in the form:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}$

ct'= \ gamma (ct- \ vec {\ beta} {\ cdot} \ vec {R}) \ \ \ vec {r'} = \ gamma (- \ vec {\ beta} ct + \ vec {R}) + \, \ vec {\ beta} \ times (\ vec {\ beta} \ times \ vec {R}) \ \ \end{matrix}\right.

Dilation of time and contraction lengths

The transformations of Lorentz lead to a revolutionary vision of physics and reveal phenomena which run up against the common direction.

In the examples which follow we will have to consider two successive events. One will thus rewrite the preceding formulas by replacing the X and the T by Δ X and of the Δ T representing the space or temporal variation enters the first event and the second.

Dilation of the durations

A physical phenomenon during an time interval in a reference frame lasts a different quantity in another reference frame.

Let us suppose an time interval $\backslash \; Delta\; t\text{'}$ corresponding to the interval between two palpitations of an individual, between two fakes of a clock, motionless in the reference frame $\backslash \; mathbb\; \{R\text{'}\}$ (let us say that of the rocket), which wants to say that in this reference frame the two events (1st beat, 2nd beat,…) take place at the same point of space of $\backslash \; mathbb\; \{R\text{'}\}$.

Since $\backslash \; Delta\; x\text{'}\; =\; 0$ the equation of Lorentz

$$

C \ Delta T = C \ Delta you \ cosh \, \ theta + \ Delta x' \ sinh \, \ theta show immediately that

$\backslash \; Delta\; T\; =\; \backslash \; Delta\; you\; \backslash \; cosh\; \backslash ,\; \backslash \; theta\; =\; \backslash \; gamma\; \backslash \; Delta\; you\; =\; \backslash \; frac\; \{\backslash \; Delta\; you\}\; \{\backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\}\; \backslash .$

This expression shows that Δ T is increasingly larger than Δ T   '. Thus, the same phenomenon during 1 second (for example) in the reference frame of the rocket is seen lasting $\backslash \; gamma$ seconds (γ >1) in the terrestrial reference frame: an embarked clock appears to slow down. It is this effect which is at the origin of the Paradoxe of the twins, the returning twin of an imaginary voyage at a speed close to that of the light (what in addition is impossible to realize) finding itself with the return younger than his/her brother remained on Earth.

It is necessary to insist on the significance of the concept of duration between two events occurring in the same point in a certain reference frame. According to the formula above it is in this reference frame that the measured duration is shortest. It is read via one only clock. One allots the name of to him lasted of clean time .

For any reference frame by report/ratio to which the traveller moves, the duration of the phenomenon requires, for its measurement, two clocks, with each point of the reference frame where the traveller at the initial moment and the final moment will be. This duration will be increasingly longer than the clean duration.

The experimental checks are numerous: lifespan of atmospheric muons, lifespan of particles in the accelerators, steps of the embarked clocks of the satellites (the phenomenon is used in this case to separate from the effects of the gravitation), etc

Contraction lengths

Let us suppose that in the reference frame $\backslash \; mathbb\; \{R\text{'}\}$ of the rocket is a fixed rule, length $L\text{'}$, along the axis $O\text{'}\; x\text{'}$. This length measured in the reference frame in which the rule is fixed is the clean length rule.

In the reference frame $\backslash \; mathbb\; \{R\}$ of the Earth, by report/ratio to which the rule is driven, one will be able to measure the length of the rule by considering the two following events: the first will be defined as that of the passage of the first end of the rule opposite such terrestrial observer when its clock indicates (let us say) midday, the second will be that of the passage of the second end of the rule opposite an other terrestrial observer per same hour , always midday. The distance from the two observers will be regarded as a measure of length from the rule in the reference mark fixes since it represents the distance between the points of $\backslash \; mathbb\; \{R\}$ which coincide with the ends of the rule at the same moment of $\backslash \; mathbb\; \{R\}$. Like Δ T   =  0, the equation of Lorentz

$\backslash \; Delta\; x\text{'}\; =\; \backslash \; Delta\; X\; \backslash \; cosh\; \backslash ,\; \backslash \; theta\; -\; C\; \backslash \; Delta\; T\; \backslash \; sinh\; \backslash ,\; \backslash \; theta$

is written

$L\text{'}=\; L\; \backslash \; cosh\; \backslash ,\; \backslash \; theta\; =\; \backslash \; gamma\; L\; \backslash .$

Consequently, the length L measured on Earth is

$L\; =\; the\; \backslash \; gamma\; =\; the\; \backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\backslash ,$

so that L is smaller than the .

It is however necessary to be defended to apply this phenomenon of contraction lengths in an unwise way. It is always necessary to pass by the careful definition of the events in question and to examine how their coordinates change from one reference mark to another. Otherwise one can fall on many paradoxes. One of most known relative to this relativistic contraction lengths is that of the supposed car to return in a garage shorter than it, on the condition of rolling rather quickly. All these paradoxes are studied in detail by Taylor and Wheeler.

Relativity of simultaneity

The modification of the value of the durations between two events at the time of the passage from one reference frame to another was often exploited in the first presentations of the theory of relativity, in particular by Einstein. In particular relativity limits the concept of simultaneity inside a reference frame galiléen.

Two simultaneous events in $\backslash \; mathbb\; \{R\}$, in two points different of $\backslash \; mathbb\; \{R\}$, are not simultaneous any more in another reference frame moving compared to $\backslash \; mathbb\; \{R\}$. Since the space-time interval within the meaning of the relativity restricted between two events is independent of the reference mark chosen, it is seen that the interval between two simultaneous events in a certain reference mark is necessarily of the type space , which means that the term $c^2\; \backslash \; Delta\; t^2\; -\; \backslash \; Delta\; x^2$ is negative. In other words such events are elsewhere one of the other, i.e. are not associated by a cause and effect link.

Examples

Simple illustration

The difference between the durations measured by two observers moving uniform relative constitutes a source of astonishment when restricted relativity is discovered. A popularizing work (see reference) gives a simple idea on this problem. The author considers a watch with photons in which a grain of light would carry out, at the speed C of the light, of the return tickets between two mirrors.

If this watch is fixed compared to the observer, the duration of a return ticket 2 T   'is equal to the quotient of the way carried out by the light. On the contrary, if the watch moves perpendicular to the way of the photon, this one follows the movement of the watch and the segment is replaced by a broken line longer than him. The celerity of the light remaining the same one, duration 2 T of the courses is higher than 2 T   ' : the watch moving delays.

The length of the hypothénuse of right-angled triangle ABH of the figure is ct , that the height is ct   'and that of the base is vt if one notes v the speed of traverse of the watch in the “fixed” reference mark. One thus has (Théorème of Pythagore):

$c^2t^2\; \backslash ,\; =\; \backslash ,\; c^2t\text{'}^2\; +\; v^2\; t^2\; \backslash ,$

from where one draws immediately

$t\; =\; \{you\; \backslash \; over\; \{\backslash \; sqrt\; \{1\; \{v^2\; /c\; ^2\}\}\}\}$

The celerity of the light being of: 300000  km/s, let us consider a flying plane with 0,3  km/s (either 1000  km/h). Its speed is millionth of that of the light and the error made by using the approximation galiléenne is lower than one millionth from millionth (either 10 -12 ), completely negligible in practice current. However for very precise measurements of time of ways used in the space experiments and also by GPS, it is imperatively necessary to take account of the relativistic corrections (at the same time those of restricted relativity and general relativity besides).

For a body moving at a speed equal to the tenth of that of the light, the relativistic effect is about a percent. Thus the relativistic effects become significant only for speeds close to the celerity of the light, impossible to reach in practice. It is one of the reasons for which we have difficulties in apprehend the operation of restricted relativity.

See also: Paradox of the train

See also: Paradox of the twins

The interval of space time between two events

The relativistic theory can give the impression (would be this only by its name) to make the things completely dependant in the way of measuring them. The first paragraphs of this article on the relativity lengths and durations illustrate this opinion and could support it. However this point of view is erroneous because more deeply restricted relativity sticks on the contrary to release what is invariant by change of coordinates . Accordingly the invariance of the interval of Espace-temps between two events is an element founder of the relativistic theory.

In a reference frame, an event is characterized by its coordinates spatio-temporelles : “such place, such moment”. Two events located respectively in X ₁, there ₁, Z ₁, T ₁ and in X ₂ there ₂, Z ₂, T ₂ are separated by an interval from space time whose square is

$(\backslash \; Delta\; s\_\; \{12\})\; ^\; \{2\}\; =-c^\; \{2\}\; (t\_\; \{2\}\; -\; t\_\; \{1\})\; ^\; \{2\}\; +\; (x\_\; \{2\}\; -\; x\_\; \{1\})\; ^\; \{2\}\; +\; (y\_\; \{2\}\; there\; \_\; \{1\})\; ^\; \{2\}\; +\; (z\_\; \{2\}\; -\; z\_\; \{1\})\; ^\; \{2\}\; \backslash .$

We will write more simply

This interval is a relativistic invariant : its value does not depend on the Référentiel galiléen in which one evaluates it. One can check it on the formulas of Lorentz (which were made besides to ensure this invariance !).

The sign of this invariant makes it possible to classify two events one compared to the other, and this classification is absolute .

• Two events can be elsewhere one compared to the other. It is the case when the square of the interval of space time defined above is positive. The distance Δ R is then “  too much grande  ” compared to the &Delta duration; T so that no light signal can connect events 1 and 2, that being true in all the reference frames. Two such events cannot thus be associated by a cause and effect link. This circumstance occurs in particular for two events which would be simultaneous in a certain reference mark (since Δ   T   =  0, Δ   S ² is inevitably positive).

• On the contrary if two events are bound by a cause and effect link that means that in reference frame signal connected them. Consequently a light signal, which is propagated more quickly than any other signal, can also connect them. In other words Δ R is smaller than C Δ T , which involves that the square Δ   S ² of the interval is negative (or no one if the interaction between the two events took place by the intermédaire of a luminous photon). In this last case the two events cannot be simultaneous since then we would fall down in the preceding case.

We note, and that is logical pledge of coherence, which one cannot reverse the course of the events. One cannot reverse an effect and a cause. One cannot in particular go up the wire of time and act on the past.

Finally in the borderline case where in a reference frame two events have same coordinates, they coincide (the interval of space time is then trivialement null). This coincidence is then preserved during a change of reference frame. That makes absolute the concept of shock of particles.

It will be still noticed that time and space play of the symmetrical roles and that it is thus logical in the same way to measure them. It is the point of view adopted by the new definition of the Speed of light, which, while being fixed in an arbitrary way, establishes an equivalence in fact between length and time, by redefining the Mètre starting from the second. Concretely one can measure a distance or a time indifferently in centimetres or seconds, according to the needs.

The definition of the square Δ   S ² of the space-time interval adopted above corresponds to the “  signature  ” (-, +, +, +) of what one calls metric space time, with a notation whose significance is obvious. According to this choice,

• the interval such as: $\backslash \; Delta\; s^2\; >\; 0$ is called “interval of kind spaces” (by expressing them in the same unit, the distance is larger than the duration).
• the interval such as: is called “interval of time kind” (the duration is larger than the distance).

In the latter case, if one needs to take the square root of the quantity in question, one will be able quite simply to change his sign and to consider the expression $c^2\; \backslash \; Delta\; t^2\; -\; \backslash \; Delta\; r^2\; \backslash ,$, which, it, is positive and in which one sees well that in fact the time is to some extent dominating (for this reason it is said that the interval is time kind). In practice, the definition of the square of the space-time interval is not fixed and corresponds either to the signature (-, +, +, +) or with (+, -, -, -) and it is necessary to refer about it to the context to know in which case the writer placed himself.

A definition of great importance related to the invariance of this interval of space time is that of time clean of a certain mobile. We for example inside a rocket place moving compared to the Earth at the speed v , which means that the X-coordinate of the rocket is given by

$x\; \backslash ,\; =\; \backslash ,\; v\; T$

in the terrestrial reference mark. But if we consider an internal clock with the rocket, its position does not change compared to this rocket and the square of the interval of space time is reduced to its temporal component which we will note $\backslash \; Delta\; \backslash \; tau^2\; \backslash ,$. As the interval of space time is independent of the reference mark in which one measures it, one has

$c^2\; \backslash \; Delta\; \backslash \; tau^2\; \backslash ,\; =\; \backslash ,\; c^2\; \backslash \; Delta\; t^2\; -\; \backslash \; Delta\; x^2\; =\; c^2\; \backslash \; Delta\; t\text{'}^2\; -\; \backslash \; Delta\; x\text{'}^2\; =\dots$

where ( T   “, X   ”) are the coordinates in another reference mark.

Time $\backslash \; tau\; \backslash ,$ defined by the relation is called the clean time mobile (or particle) considered.

Diagram of space time

In Newtonian mechanics, space is separated from time and one studies the movement of a particle according to an absolute time. Graphically one represents the trajectory in space (but not in the temps !). For example one traces the Ellipse which a planet around the Sun describes according to the Lois of Képler. The figure opposite watch the space way carried out by a certain number of particles has, B, C, D, E, etc animated a constant speed during the same amount of time, say one second, way proportional to the speed of the mobile. In the general case one can trace the trajectory of a M  point; ( X ,   there ,   Z ) in a Cartesian reference mark with three dimensions.

In restricted relativity one follows events in a space to 4 dimensions, three of space and one of time, and consequently it is impossible in the most general case to visualize the curve representing the succession of events representing the displacement of the particle at the same time in time and space . This curve is called Ligne of universe of the particle. To overcome the difficulty of the representation of 4 dimensions one often limits oneself to 2 dimensions, one of space and one of time. In other words one considers movements only along the axis of the X , the unchanged coordinates there and Z remainder. Do not remain whereas the variables X and T , which makes it possible to trace in a Cartesian reference mark with two dimensions the trajectory of a particle in space-temps : its line of universe.

The two figures opposite illustrate the passage from the Newtonian point of view from the relativistic point of view. In top one carried the space traversed into 1 second by various particles constant speed. While remainder at the same point has, B moves of a certain quantity, C goes more quickly and further goes, D even more while E moves in the other direction. In the figure of bottom, one carried in a space-time diagram the succession of the events constituting the movement of the particle. Since the speed of the particles is constant, their X-coordinate is obviously X   =  v   T so that their line of universe is a line. The slope of this one is proportional to the speed v .

The remarkable thing is that the line of universe of the particle at rest is not only one any more point but the segment of right-hand side OA. Indeed, if the particle does not move ( X   =  constant) time continues to run out for the period considérée !

If a segment of right-hand side represents in this diagram a movement at constant speed, in the general case it is an unspecified curve which will translate the movement of a particle. As example let us consider the line of universe opposite representing a mobile on the basis of the X-coordinate X   =  0 and returning a time T there later, measured time let us say on Earth. It could be a question of a rocket accomplishing a voyage intersidereal return ticket and we will continue to reason on this example.

The segment of right-hand side between “  départ  ” and “  arrivée  ” along the temporal axis the line of universe of the Earth represents, whose temporal coordinate, equalizes to 0, does not vary. The curved line represents the continuation of events constituting the voyage of the rocket. The curvilinear coordinate making it possible to locate a point on this curve is the clean time of the rocket, that which measurement the embarked clock.

The relativistic formulas show that clean time along the curvilinear way is shorter than clean time along the rectilinear way (here that which represents terrestrial time). This phenomenon is the base of the Paradoxe of the twins. One of the brothers makes a return ticket at a speed close to the light (what is impossible besides to realize, but it is about a imaginary experiment) while his/her brother remains with Ground. With the return the traveller finds himself young person than his brother.

Thus, while in Euclidean geometry ( X ,   ) the shortest way between two points has there and B is the straight line, in Lorentzian geometry ( X ,   T ) the temporal interval between two event has and B is maximum for the particle moving along rectilinear way AB. The longest voyage is that which corresponds to rectilinear way AB in the diagram space time. In this case this trajectory is that which a free mobile of any force follows and thus advancing at constant speed. This property is so capital that it allowed Taylor and Wheeler to found a simple presentation of general relativity by extending the principle of maximization of time of course to the free movement of a particle in a field of gravitation (in the vicinity of a black hole or Sun for example).

Law of composition speeds

In a rocket moving at the speed $v\; \backslash ,$ compared to the Earth one draws a ball from gun forwards at the speed $w\text{'}\; \backslash ,$ measured in the rocket. Which is the speed $w\; \backslash ,$ ball measured on Earth?

In kinematics galiléenne speeds are added and there would be

$W\; =\; w\text{'}\; +\; v\; \backslash .$

In relativistic kinematics the law of composition speeds is different.

In the rocket the distance Δ x  ' traversed by the ball during time Δ is to you

$\backslash \; Delta\; x\text{'}\; \backslash ,\; =\; \backslash ,\; w\text{'}\; \backslash \; Delta\; you$

By using the formulas of Lorentz

$\backslash \; begin\; \{boxes\}$

\ Delta X \, = \, \ gamma (\ Delta x' + v \ Delta you) \ \ \ Delta T \, = \, \ gamma you + (v/c^2) \ Delta x' \ end {boxes} and by replacing Δ x  ’ by its value one easily finds the speed of the ball in the terrestrial reference mark under the forme :

$w\; \backslash ,\; =\; \backslash ,\; \backslash \; frac\; \{\backslash \; Delta\; X\}\; \{\backslash \; Delta\; T\}\; =\; \backslash \; frac\; \{\backslash \; gamma\; (w\text{'}\; \backslash \; Delta\; you\; +\; v\; \backslash \; Delta\; you)\}\{\backslash \; gamma\; you\; +\; (v/c^2)\; w\text{'}\; \backslash \; Delta\; you\}\; \backslash ,$

that is to say

This relation shows that

• the law of composition speeds in restricted relativity is not any more one additive law
• speed C is a speed limit whatever the reference frame considered (when a speed is added to him, one falls down on C ).

There exists however a parameter setting making it possible to obtain an additive law. It is enough for that to pass from the speed v to the angular parameter speed θ introduced previously.

Let us show that in a composition speeds the angular parameters speed are added.

By posing

$\backslash \; theta\; \backslash ,\; =\; \backslash ,\; \backslash \; mathrm\; \{atanh\}\; (v/c)$

$\backslash \; alpha\text{'}\; \backslash ,\; =\; \backslash ,\; \backslash \; mathrm\; \{atanh\}\; (w\text{'}/c)$

$\backslash \; alpha\; \backslash ,\; =\; \backslash ,\; \backslash \; mathrm\; \{atanh\}\; (w/c)$

and by using the formula of addition of the hyperbolic functions

$\backslash \; tanh\; (\backslash \; theta\; +\; \backslash \; alpha\text{'})\; =\; \backslash \; frac\; \{\backslash \; tanh\; \backslash \; theta+\; \backslash \; tanh\; \backslash \; alpha\text{'}\}\; \{1\; +\; \backslash \; tanh\; \backslash \; theta\; \backslash ,\; \backslash \; tanh\; \backslash \; alpha\text{'}\}$

it is found immediately that

In other words the angular parameter speed of the ball compared to the Earth is the nap angular parameter speed of the ball compared to the rocket and the angular parameter speed of the rocket compared to the Earth. One cannot imagine simple  any more;!

With this formalism, the angular parameter corresponding at the speed C is infinite (since atanh ( X  ), the hyperbolic argument tangent of X , tends towards the infinite one when X tends towards 1) and consequently remains infinite whatever the quantity that one adds to him. The fact is thus found that C is a speed limit (in addition impossible to reach for a material particle) independent of the selected reference mark. Only the particles of null mass, like the photon, can move with speed of light.

This remarkable property of additivity of the angular parameter speed is exploited in the determination of the equations of an accelerated rocket.

Here a numerical application (speeds indicated are unrealizable from a pratique  point of view;!). Let us imagine that a shell is drawn with speed w'    =  0,75 C in the reference mark of a rocket moving itself at the speed v   =  0,75 C compared to the Earth. Which is the speed of the ball measured on Terre ? Clearly the value 1,5 C which the formula galiléenne would give us is false since speed obtained would exceed that of the light. The relativistic formulas invite us to proceed as follows. The parametric angle speed of the shell compared to the rocket is $\backslash \; alpha\text{'}\; =\; \backslash \; mathrm\; \{atanh\}\; (0,75)\; =\; 0,973\; \backslash .$ The parametric angle the speed of the rocket compared to the Earth with the same value $\backslash \; theta\; =\; 0,973\; \backslash .$ The angular parameter speed of the shell compared to the Earth is thus $\backslash \; alpha\; \backslash ,\; =\; \backslash ,\; 0,973\; +\; 0,973\; \backslash ,\; =\; \backslash ,\; 1,946$, which corresponds at the speed $w\; =\; C\; \backslash ,\; \backslash \; tanh\; (1,946)\; =\; 0,96\; \backslash ,\; C\; \backslash .$ In restricted relativity 0,75  +  0,75  =  0,96, if one can express ainsi !

One can obviously find this result directly on the formula giving W according to w  ’ and v .

The quadrivector speed

In Newtonian mechanics one studies the movement of a mobile while following his position $\backslash \; vec\; \{R\}$ according to time T , this time being supposed of absolute nature, independent of the clock which measures it. In relativity one gives up this vision of the things to regard the movement of a particle as a succession of events $\backslash \; mathcal\; \{P\}$, the curve followed by this event in a space with four dimensions (three for space, for time) then taking the name of “  line of univers  ”.

Just as in traditional mechanics one defines the speed of a particle by taking the derivative

$v\; \backslash ,\; =\; \backslash ,\; D\; \backslash \; vec\; \{R\}\; /dt$

position compared to time, in the same way in relativistic mechanics one defines the Flight Path Vector in four dimensions (or quadrivector speed)

$\backslash \; mathbf\; \{U\}\; \backslash ,\; =\; \backslash ,\; D\; \backslash \; mathcal\; \{P\}\; /d\; \backslash \; tau$

where $\backslash \; tau\; \backslash ,$ is the clean time of the particle, definite higher.

By clarifying the components of this quadrivector in a given reference mark one can write

$\backslash \; mathbf\; \{U\}\; =\; \backslash \; left\; (C\; \backslash \; frac\; \{dt\}\; \{D\; \backslash \; tau\},\; \backslash \; frac\; \{dx\}\; \{D\; \backslash \; tau\},\; \backslash \; frac\; \{Dy\}\; \{D\; \backslash \; tau\},\; \backslash \; frac\; \{dz\}\; \{D\; \backslash \; tau\}\; \backslash \; right)\; \backslash ,$

expression into which we introduced the factor C to work with homogeneous coordinates.

There exists a simple, and important relation, relative to this quadrivector. The definition of the square of the interval of definite space time above can spread with any quadrivector. One will thus define the square of the standard of a quadrivector like the difference between the square in his temporal part and that of his space part. And the capital result is that this standard is invariant by transformation of Lorentz. In other words it does not depend on the selected reference mark. In the case speed this result takes a particularly simple form. Indeed, in the clean reference mark of the particle, the space part of the quadrivector speed is null while the temporal part is worth quite simply C (D T /d τ   =  1 since time T is precisely time τ such as it is measured in the reference mark of the mobile). In other words in the clean reference mark of a particle, the quadrivector speed has as components (C, 0,0,0). Consequently in any reference mark galiléen there will be the relation

It is the invariance of this standard which makes it possible to speak about the quadrivector of a particle independently of any frame of reference .

The quadrivector energy-impulse

One can continue the reasoning. Just as the traditional impulse of a particle “$\backslash \; vec\; \{p\}\; was\; the\; product\; =\; m\; \backslash \; vec\; \{v\}$” of the mass by speed, in the same way the product “m$\backslash \; mathbf\; \{U\}$” of the quadrivector speed “$\backslash \; mathbf\; \{U\}$” by the mass “ m ” of the particle becomes a quadrivector impulse. It is often called vector “ energy-impulse ”, by thus expressing the fact that energy and impulse are joined together in a physical concept in an indissociable way, in the same way that space and time composes the space time. Indeed if the space components of this quadrivector are identified in an obvious way with those of a traditional impulse, the physicists were led by Einstein to identify the temporal component of this quadrivector with l’ energy of the particle considered. Although the reasons of this choice are multiple, it is not so easy to give true a demonstration of it, but this situation is completely current in physics, the assumptions being born at the same time as the theory develops and than the experimental confirmations follow one another. In truth tens of thousands of daily experimental confirmations of the theory thus imagined are a sufficient pledge of the accuracy of the assumptions which constitute the base of it.

In a reference mark of inertia (for example the terrestrial reference mark, named hereafter at first approximation reference mark of the laboratory ) the coordinates of the events related to the followed particle are ( T , X , there , Z ) and the components in this reference mark of the quadrivector energy-impulse of the mobile are

$\backslash \; mathbf\; \{p\}\; =m\; \backslash \; mathbf\; \{U\}\; \backslash ,\; =\; \backslash ,\; (E/c,\; p\_x,\; p\_y,\; p\_z)$

with

$E/c\; =\; mc\; \backslash \; frac\; \{dt\}\; \{D\; \backslash \; tau\}\; \backslash \; qquad\; p\_x\; =\; m\; \backslash \; frac\; \{dx\}\; \{D\; \backslash \; tau\}\; \backslash \; qquad\; p\_y\; =m\; \backslash \; frac\; \{Dy\}\; \{D\; \backslash \; tau\}\; \backslash \; qquad\; p\_z=m\; \backslash \; frac\; \{dz\}\; \{D\; \backslash \; tau\}\; \backslash .$

By taking account of the relation giving the relationship between clean time ( t  ’ of the rocket, or τ of the particle moving), one has

$dt\; \backslash ,\; =\; \backslash ,\; \backslash \; gamma\; D\; \backslash \; tau$

and one thus leads to the expression of the total energy of the particle in the reference mark of the laboratory , that by report/ratio to which the particle is animated speed $\backslash \; vec\; \{v\}$ (because energy depends on the reference frame in which one it calcule !) in the form

In addition like the components the speed of the particle in the reference mark of the laboratory are

$v\_x=dx/dt\; \backslash \; qquad\; v\_y=dy/dt\; \backslash \; qquad\; v\_z=dz/dt$

and while taking account of the factor of dilation of time between D T and D $\backslash \; tau$ , one arrives at the other important formula providing the value of the impulse in the reference mark of the laboratory

At the low speeds (i.e. small in front of that of the light) one obtains

$E\; \backslash ,\; =\; \backslash ,\; mc^2\; +\; (1/2)\; m\; v^2\; \backslash ,$

formulate showing that the energy of the particle is the sum of an energy at rest m  C ² unknown of Newtonian mechanics and the traditional kinetic energy (1/2) m  v ².

In restricted relativity, the total energy of a particle remains equal to the sum of energy at rest m  C ² contained in its mass and of the kinetic energy K . By taking account of the relativistic expression of energy, one sees that the kinetic energy of a particle is given by the expression :

$\backslash \; mathrm\; \{\backslash \; acute\; Energie~cin\; \backslash \; acute\; etic\}\; \backslash ,\; =\; \backslash ,\; K\; \backslash ,\; =\; E\; -\; m\; c^2\; \backslash ,\; =\; \backslash ,\; m\; c^2\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\}\; -\; 1\; \backslash \; right)\; \backslash .$

For speeds very close to that of the light, it is the 1  quantity; - β   =  who counts. One a :

$1\; -\; \backslash \; beta^2\; =\; (1\; +\; \backslash \; beta)\; (1\; \backslash \; beta)\; \backslash \; simeq\; 2\; (1\; \backslash \; beta)$

so that total energy is written

$E\; \backslash \; simeq\; PC\; =\; \backslash \; frac\; \{mc^2\}\; \{\backslash \; sqrt\; \{2\; (1\; \backslash \; beta)\}\}\; \backslash \; equiv\; \backslash \; frac\; \{mc^2\}\; \{\backslash \; sqrt\; \{2\}\}\; \backslash .$

Equivalence of energy and the rest mass

The quadrivector energy-impulse shows the characteristic to have his standard , or his scalar square (within the meaning of the square of interval of space time), invariant during a change of reference frame. In short the quantity

$E^2/c^2\; \backslash ,\; -\; \backslash ,\; p^2\; \backslash \; qquad\; \backslash \; text\; \{with\}\; \backslash \; qquad\; p^2\; \backslash ,\; =\; \backslash ,\; (p\_x^2\; +\; p\_y^2\; +p\_z^2)$

is independent of the reference frame in which it is calculated. However in the reference mark of the particle speed is null, just as the impulse, so that the standard of this invariant quantity is worth (m  C ) ². In any reference mark there is thus the following capital relation

$E^2/c^2\; -\; p^2\; \backslash ,\; =\; \backslash ,\; (m\; c)^2$

or (The factors C which are introduced into these formulas ensure their homogeneity, p with the size of m  v , E that of m  v   ².)

One can directly show this formula starting from those written above giving energy and the impulse. One can make several observations:

• the value of the total energy of the particle depends on the reference frame of the observer. However, the value of the energy of mass is identical in all the reference frames, and in particular in the clean reference frame of the particle. It is thus an intrinsic characteristic of the particle.
• When v tends towards C , $\backslash \; gamma$ tends towards the infinite one, which means that one needs an infinite energy to accelerate a particle until reaching the Speed of light. That is obviously impossible. One however manages to accelerate particles at speeds very close to C.
• restricted relativity appears in all the physical phenomena, even where speeds intervening are not relativistic. An obvious example is the Défaut of mass of the simplest atom: the mass of the hydrogen atom $H\_\; \{1\}\; ^\; \{1\}$ is lower than the sum of the masses of the electron and the proton of a quantity right equal to the equivalent in mass of the energy of ionization of the atom.

The equivalence of the mass and energy is given by famous the E=mc ². To pose this equivalence was a revolutionary step, because the concepts of matter and energy were distinct up to that point, although certain scientists, like Poincaré and Lorentz, had independently tried the bringing together in the field of electromagnetism. Nowadays, this equivalence should not either be over-estimated, because while the mass is the standard of the quadrivector energy-impulse, energy is only one of the component of this quadrivector. Mass given by

$m^2\; \backslash ,\; =\; \backslash ,\; (E^2\; -\; p^2c^2)\; /c^4$

invariant by change of reference mark is (it is the same one in any reference mark). Energy on the contrary depends on the reference mark chosen, it is obvious since speed changing, the kinetic energy changes too.

Speed of the particles of null mass

The relations above provide important results. The expressions giving E and p according to m and v lead immediately to the formula

$p\; \backslash ,\; =\; \backslash ,\; (v/c)\; (E/c)$

If the speed of the particle is equal to speed of light, then $p=E/c$ by calculating $E^2\; -\; p^2c^2$ one sees that the mass of the particle is inevitably null. In opposite direction, if the mass of the particle is null, then $p=E/c$ and consequently $v=c$.

We thus arrive at the important double conclusion according to which the material particles cannot reach speed of light and that only particles without mass move with speed of light.

The unit is perfectly coherent: any mechanism of propagation of energy to speed of light corresponds to a momentum p equal to energy and thus to a null “rest mass”. In opposite direction, a particle of null mass moves inevitably with speed of light.

Dilation of the time of the cosmic rays and the muons

One detects in Astronomie particles carrying an energy colossale : the cosmic rays. Although their mechanism of production remains still mysterious, one can measure their energy. The considerable numbers that one obtains show that their analysis requires the use of the formulas of relativity resteinte. The cosmic rays thus provide an ideal illustration of the theory of Einstein.

One detects particles until incredible energies of about 10 20 electronvolts, is hundred million TeV. Thus let us suppose that a cosmic ray is a Proton of 10 20   eV. Which is the speed of this particule ?

In the expression giving energy E , the term m  C ² represents the energy of the rest mass of the particle. That of the proton is of approximately 1  GeV, is 10 9   eV. The relationship between E and m  C ² is thus equal to 10 20 /10 9 =10 11 and is not other than the famous factor of stretching of time $\backslash \; gamma=\; -\; (v/c)^2^\; \{-\; 1/2\}$. Which is the speed of this proton ? By writing $1\; -\; (v/c)^2=\; [1\; -\; (v/c)]\; \backslash \; simeq\; 2\; -\; (v/c)$ one finds that

$1\; -\; (v/c)\; =\; 0,5\; \backslash \; times\; 10^\; \{-\; 22\}.$

In other words the speed of the proton considered is almost equal to speed of light. It remotely only per less than 10 -22 (but cannot in no case to equalize it).

Let us see what these figures imply for the relativistic factors existing between the clean reference mark of the particle and the terrestrial reference mark. Our clean Galaxie is crossed by the light in some hundred thousands of light-years. Consequently for a terrestrial observer the proton crosses this Galaxy in same time, approximately 100  000  years. Extraordinary it is that in the reference mark of the relativistic proton, corresponding time is 10 11 weaker time, and is worth thus 30 seconds (one year made 3× 10 seconds)  !

Our proton ultra-relativist and ultra-energetics crosses our Galaxy in 30 seconds of clean time but in: 100000 years of our terrestrial time.

When this cosmic ray runs up against an atom of oxygen or nitrogen of the terrestrial atmosphere at an altitude of about 20 to 50 kilometers above the ground, a sheaf of elementary particles starts container, in particular muons. A part of them move towards the ground with a speed practically equal to that of the light, of: 300000 kilometers a second in the terrestrial reference mark. These particles thus cross the few 30 kilometers of atmosphere into 10 -4 second (or 100 microseconds).

In the reference mark where it is at rest, a muon has a Demi-vie 2  μ S (2 microseconds, or 2× 10 -6   S). That means that among whole of muons produced at the top of the atmosphere, half will have disappeared at the end of 2 microseconds, transformed into other particles. A half of the remaining muons will disappear at the end of 2 more microseconds and so on. If the half-life were the same one (2 microseconds) in the terrestrial reference mark, in 10 -4 second of crossing of the atmosphere the muons would have counted 10 -4   /  2× 10 -6   =  50 half-lives. Consequently their number would be tiny room on arrival on the ground by a factor of (1/2) 50 that is to say approximately 10 -15 so that in practice no muon would reach it.

However measurements indicate that approximately 1/8, is (1/2) 3 , of the initial muons arrive at terrestrial surface, which proves that they underwent only 3 divisions of their number by 2 and not 15. In other words the time of crossing of the atmosphere in their own reference mark is of 3 half-lives and not 50, is 6 microseconds only (and not 100 microseconds). This result constitutes a strong proof of the accuracy of restricted relativity and in particular of the phenomenon of stretching of clean time (here that of the muon) when one takes the measures to an external reference mark (here that of the Earth). In the selected numerical example the factor of dilation of time $\backslash \; gamma\; =\; -\; (v/c)^2^\; \{-\; (1/2)\}$ is of 100/6.

One can deduce from it the speed and the energy of the muons. Indeed, there is as in preceding calculation

$1\; -\; (v/c)^2\; \backslash ,\; =\; \backslash ,\; 2\; \backslash \; times\; -\; (v/c)\; \backslash ,\; =\; \backslash ,\; (6/100)\; ^2\; \backslash ,$

what leads to

$1\; -\; (v/c)\; \backslash \; simeq\; 2\; \backslash \; times10^\; \{-\; 3\}\; \backslash .$

As the mass of a Muon is of approximately 100  MeV, the energy of the particle is 100/6 times larger, that is to say of approximately 2000  MeV or 2  GeV.

Conservation of the quadrivector energy-impulse of an isolated system

The great force of the theory of Einstein is to make it possible to the physicist to reason on quantities of geometrical nature having a reality transcending the language of the coordinates. Thus of the quadrivector energy-impulse which one can speak independently of his representation in term of coordinates. This vector with four dimensions gets the tool of choice to us to treat relativistic mechanics and in particular the problems of interaction and transformation of particles.

The analysis of a collision between a particle has and a particle B (or more generally of collisions between particles of a given system) rests on the following principle, independently of the details of the experiment  :

In other words écrire  can;:

To analyze an experiment, the physicist must proceed to measurements and thus translate this law in order to use the measurable quantities which are the coordinates of the quadrivector in the reference mark of his choice. Since the quadrivector is preserved, each one of its components in a frame of reference given (let us recall that the value itself of the components depends on the selected system) is also preserved in the collisions. The temporal component representing energy E system and the space component representing its impulse $\backslash \; vec\; \{p\}$, one thus ends in two laws of conservation, one for energy, the other for the momentum (or impulse).

By preoccupation with a clearness, let us state these laws in the case of two particles has and B undergoing a collision (one can immediately generalize with a system of several particles).

• the total energy of the system is preserved in a collision

In other words the sum of the energy of has and of the energy of B before the collision is equal to the sum of the energy of has and the energy of B after the collision.

One can formulate this law in the following way

$E\_1\; (A)\; +\; E\_1\; (B)\; \backslash ,\; =\; \backslash ,\; E\_2\; (A)\; +\; E\_2\; (B)$

• the total momentum of the system is preserved in a collision

In other words the sum of the momentum of the particle has and of the particle B before the collision is equal to the sum of the momentum of the particle has and the particle B after the collision.

Along each of the three axes one will thus have

$(p\_x)\; \_1\; (A)\; +\; (p\_x)\; \_1\; (B)\; \backslash ,\; =\; \backslash ,\; (p\_x)\; \_2\; (A)\; +\; (p\_x)\; \_2\; (B)$

$(p\_y)\; \_1\; (A)\; +\; (p\_y)\; \_1\; (B)\; \backslash ,\; =\; \backslash ,\; (p\_y)\; \_2\; (A)\; +\; (p\_y)\; \_2\; (B)$

$(p\_z)\; \_1\; (A)\; +\; (p\_z)\; \_1\; (B)\; \backslash ,\; =\; \backslash ,\; (p\_z)\; \_2\; (A)\; +\; (p\_z)\; \_2\; (B)$

An example (academic) of collision is represented in the figure opposite. A particle has of mass 8 (in arbitrary units) animated a speed v/c of 15/17 directed towards the line strikes a particle of mass 12 newcomer in opposite direction with a speed v/c of the 5/13 (digits were selected so that calculations " fall juste"). After the collision, has rebounds in the other direction while having communicated with B part of its momentum. Total energy, summons energies of the particles has and B is preserved, just as the total momentum. The sizes E and p indicated actually represent (E/c²) and (p/c) and are expressed in units of mass, arbitrary. With these sizes one with the relation E   ²  =  p   ²  +  m   ². The factor γ is always defined by γ   =  ^-1/2.

Elastic collision

In a Particle accelerator it happens that an electron of very high energy runs up against an electron at rest and communicates to this last part of its kinetic energy. If the only energy exchanges relate to this kinetic energy precisely, it is said that the shock is elastic. The formulas translating the conservation of the quadrivector of the system formed by these two electrons makes it possible to analyze the collision. In Newtonian mechanics the two electrons after the shock have directions forming a right angle. We will show that in relativistic mechanics it is not thus any more and that speeds of the electrons form an acute angle. This phenomenon is perfectly visible on the recordings of collisions carried out in bubble chambers.

Let us consider an electron of mass m and very high energy striking another electron intialement at rest. The vectors impulses of the two particles are traced on the figure opposite. Before the shock the impulse of the primary electron is $\backslash \; vec\; \{p\}$. After the shock, the impulses of the two electrons are $\backslash \; vec\; \{p\}\; \_1$ and $\backslash \; vec\; \{p\}\; \_2$. By writing the energy of an electron as the sum of its energy at rest mc 2 and of its kinetic energy K , one can write the total energy of the system before the collision comme :

$E\; \backslash ,\; =\; \backslash ,\; mc^2\; +\; mc^2\; +\; K$

In the same way,

$E\_1\; \backslash ,\; =\; \backslash ,\; mc^2\; +\; K\_1$

$E\_2\; \backslash ,\; =\; \backslash ,\; mc^2\; +\; K\_2\; \backslash .$

The law of conservation of energy says that E  =  E  +  E and consequently

$K\; \backslash ,\; =K\_1\; +\; K\_2\; \backslash ,$

formulate indicating well that the kinetic energy is preserved it also (elastic collision).

The law of conservation of the momentum says that

$\backslash \; vec\; \{p\}\; =\; \backslash \; vec\; \{p\}\; \_1\; +\; \backslash \; vec\; \{p\}\; \_2\; \backslash ,$

and consequently if we call θ the angle enters the two vectors $\backslash \; vec\; \{p\}\; \_1$ and $\backslash \; vec\; \{p\}\; \_2$, one with the relation

$p^2\; \backslash ,\; =\; \backslash ,\; p\_1^2\; +\; p\_2^2\; +\; 2\; p\_1p\_2\; \backslash \; cos\; \backslash \; theta$

from where one draws

$\backslash \; cos\; \backslash \; theta\; \backslash ,\; =\; \backslash ,\; \backslash \; frac\; \{p^2\; -\; (p\_1^2\; +\; p\_2^2)\}\{2p\_1p\_2\}\; \backslash .$

By expressing the square of the impulse of the various electrons according to their energy and their mass using the formulas indicated above one obtains

$c^2p^2\; \backslash ,\; =\; \backslash ,\; (mc^2\; +\; K)\; ^2\; -\; m^2c^4\; =\; K^2\; +\; 2Kmc^2$

for the primary electron and

$c^2p\_1^2\; \backslash ,\; =\; \backslash ,\; E\_1^2\; -\; m^2c^4\; =\; (mc^2\; +\; K\_1)\; ^2\; -\; m^2c^4=\; K\_1^2\; +\; 2K\_1mc^2$

$c^2p\_2^2\; \backslash ,\; =\; \backslash ,\; E\_2^2\; -\; m^2c^4\; =\; (mc^2\; +\; K\_2)\; ^2\; -\; m^2c^4=\; K\_2^2\; +\; 2K\_2mc^2$

for the electrons after the shock.

Like K  =  K  +  K one ends easily in the finally simple formula

$\backslash \; cos\; \backslash \; theta\; \backslash ,\; =\; \backslash ,\; \backslash \; frac\; \{K\_1K\_2\}\; \{(K\_1^2\; +\; 2K\_1mc^2)\; ^\; \{1/2\}\; (K\_2^2\; +\; 2K\_2mc^2)\; ^\; \{1/2\}\}\; \backslash \; equiv\; \backslash \; left\; (1\; +\; \backslash \; frac\; \{2mc^2\}\; \{K\_1\}\; \backslash \; right)\; ^\; \{-\; 1/2\}\; \backslash \; left\; (1\; +\; \backslash \; frac\; \{2mc^2\}\; \{K\_2\}\; \backslash \; right)\; ^\; \{-\; 1/2\}$

This formula shows that cos  θ is positive and thus that the directions of the electrons of the final state form between them an acute angle.

One easily finds in the literature the treatment of the case where the shock is symmetrical, two electrons having each one same energy K   =  K   =  K /2. In this typical location the general formula becomes

$\backslash \; cos\; \backslash \; theta=\; \backslash \; frac\; \{K\}\; \{K+4mc^2\}$       for a symmetrical collision.

• Within the Newtonian limit low speeds, the kinetic energies are much smaller than energy at rest mc 2 and consequently

$\backslash \; cos\; \backslash \; theta\; \backslash \; simeq\; \backslash \; frac\; \{\backslash \; sqrt\; \{K\_1K\_2\}\}\; \{2mc^2\}$

tends towards zero, which means that the angle θ tends towards π   /2. It is the nonrelativistic result.

• Within the limit contrary to very high energies, in fact the terms of kinetic energy are much larger than the term mc 2 and consequently

$\backslash \; cos\; \backslash \; theta\; \backslash \; simeq\; 1\; -\; \backslash \; frac\; \{mc^2\}\; \{K\_1\}\; -\; \backslash \; frac\; \{mc^2\}\; \{K\_2\}\; \backslash .$

In this case the cosine tends towards 1, which means that the angle between speeds of the electrons tends towards zero. That implies a behavior completely different from the Newtonian case.

The formulas apply obviously to the case of the collision between two protons.

Compton diffusion

A physical application of the formulas of conservation of energy and momentum of a system of particles are provided by the analysis of the collision between a photon of high energy and an electron at rest, shock constituent what one calls the Diffusion Compton.

See also: Diffusion Compton

Mass of a system of particles

In restricted relativity if energies are additive, if the impulses are additive, the masses are not it.

Physics in the space time with four dimensions of Einstein equips the matter with new characteristics which Newtonian physics could not consider. Thus for example the formulas of restricted relativity indicate that the total mass of a system of particles is higher than the sum of the rest masses of the individual particles .

If one considers a given whole of particles, without interaction between them, out the moment of the shocks, the quadrivector energy-impulse of a system of particles, isolated from any external action, is the sum of the individual energy-impulses, which one can écrire :

$\backslash \; mathbf\; \{p\}\; \_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; \backslash \; Sigma\_\; \backslash \; text\; \{J\}\; \backslash ,\; \backslash \; mathbf\; \{p\}\; \_\; \backslash \; text\; \{J\}$

If one wants to carry out numerical applications, one will translate this equation into terms of the components energy E and impulse $\backslash \; vec\; \{p\}$ :

$E\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; \backslash \; Sigma\_\; \backslash \; text\; \{J\}\; \backslash ,\; E\_\; \backslash \; text\; \{J\}$

$\backslash \; vec\; \{p\}\; \_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; \backslash \; Sigma\_\; \backslash \; text\; \{J\}\; \backslash ,\; \backslash \; vec\; \{p\}\; \_\; \backslash \; text\; \{J\}\; \backslash .$

Let us repeat well that the value of these components depends on the selected reference mark . Formulas allowing to pass from the coordinates of a reference mark $\backslash \; mathbb\; \{R\}$ to that of another reference mark $\backslash \; mathbb\; \{R\}\; \text{'}$ animated a speed v   =  β C compared to the first along the axis OX are the formulas of Lorentz already seen. They are written here under the forme :

$\backslash \; begin\; \{boxes\}$

E/c= \ gamma (E'/c + \ beta p'_x) \ \ p_x = \ gamma (\ E'/c beta + p'_x) \ \ p_y = p'_y \ \ p_z = p'_z \ end {boxes} for any quadrivector energy-impulse $(E/c,\; \backslash \; vec\; \{p\}).$

The capital property of the quadrivecteurs of the relativity and the transformations of Lorentz lies in the invariance of the “  norme  ” of the quadrivector in a change of coordinates, this standard thus translating an intrinsic property with the studied system, independent of the means of location. In the case of the quadrivector energy-impulse, this standard is equal to the mass M system (like in the case of a particle) and its square is given by the formule :

$M^2\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; (1/c^2)\; eme\}\; /c)^2\; -\; p^2\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; \backslash .$

(One finds easily the factors C to be introduced, with their power, by knowing that p is homogeneous with mc and E homogeneous with mc 2 .)

To evaluate this mass, one can place oneself in any reference mark, since it does not depend on the choice of this last. It is convenient to choose the reference mark in which the total impulse of the system is null, that thus for which

$\backslash \; vec\; \{p\}\; =0\; \backslash .$

Consequently one obtains

$M^2\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; (1/c^2)\; (serious\; E\_\; \backslash \; mathrm\; \{syst\; \backslash \; eme\}\; /c)^2\; \backslash \; equiv\; (1/c^4)\; \backslash \; left\; (\backslash \; Sigma\_jE\_j\; \backslash \; right)\; ^2\; \backslash ,$

that is to say

$M\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; (1/c^2)\; \backslash \; Sigma\_jE\_j\; \backslash .$

By taking account of the fact that energy E of each particle J (in the reference mark considered) is the sum of energy m  C 2 correspondent with his rest mass m and of his kinetic energy K (always in the reference mark considered), one still can écrire :

$M\_\; \backslash \; mathrm\; \{syst\; \backslash \; serious\; eme\}\; =\; \backslash \; Sigma\_jm\_j\; +\; (\backslash \; Sigma\_jK\_j/c^2)\; \backslash .$

This last formula shows the annoncé  result;: the total mass of a system of particles is higher than the sum of the individual masses of the particles . The value of the total mass of the system thus obtained is independent of the reference mark in which one evaluates it (but to find his value we chose the simple reference mark in whom the total momentum is null).

One can check this result on the numerical example of the collision to a treated dimension above (one measures energy, impulse and mass in a unit of arbitrary mass). There were two particles has and B respectively mass 8 and 12. The total mass M of the system is calculated by the formula M   2   =  E   2   -   p 2 , with E   =  30 and p   =  10. One finds consequently M   =√ 800  =  28,28, which is indeed larger than the sum 8+12=20 of the masses of has and B.

Conversion of mass into energy in the Sun

The conservation of the quadrivector energy-impulse explains why in a reaction the mass of a system can not be preserved and be transformed into energy, partly or entirely. It is what occurs in the reactions from fission, of fusion and annihilation of particles. We will describe here the helium hydrogen fusion in the Sun, a reaction allowing to include/understand how our beneficial star has an energy source having made it able to shine during some five billion years and ensuring him to continue to do it for one comparable length of time.

In the Années 1900 the physicists were confronted with a mystery of size, that of the mechanism allowing the Sun to emit its light. While basing itself on the thickness of layers of sedimentary rocks terrestrial the geologists of the XIX {{E}} century estimated the age of the Earth at a few hundreds of million years. By proposing its theory of the evolution species, Charles Darwin led to concordant figures. However physicists theorists as William Thomson (Lord Kelvin) affirmed that the Sun had not been able to shine since much more than ten million years. Their arrogant attitude was stigmatized besides by other scientists, in particular by the geologist Thomas Chamberlin, this last suggesting at the time of a congress in 1899 qu ' an energy of still unknown origin could reside within the matter located at the center of the Sun. Chamberlin advanced indeed that the exceptional physical conditions reigning in these areas of strong temperature and strong pressure were likely to release energy of an atomic type. The history confirmed its intuitions (in addition founded). Thanks to work of George Gamow, Hans Bethe and Carl Friedrich von Weizsäcker one will end up reaching in the last Années 1930 a detailed knowledge of the reactions having allowed the Sun to shine since some five billion years. These physicists showed that the Sun functioned the made-to-order of a gigantic nuclear reactor confined by the gravitation releasing a considerable quantity of energy in accordance with the equations of Einstein.

Here figures concerned. The solar Luminosity, or quantity of energy which the star emits a second in all space, is

$L\_\; \{\backslash \; odot\}\; \backslash \; simeq\; 4\; \backslash \; times\; 10^\; \{33\}\; \backslash ;\; \{\backslash \; rm\; erg\}\; \backslash \; cdot\; \{\backslash \; rm\; S\}\; ^\; \{-\; 1\}$.

To express this energy in units of mass (for example in grams, according to the practice of the astronomers), it is enough to divide by C 2 , which gives an equivalent mass M   =  4,4× 10 12 grams.

By a series of reactions constituting the Chain proton-proton, possible at a sufficiently high temperature, four protons (or hydrogen cores) are transformed in the internal areas of the Sun into a helium core, composed of two neutrons and two protons. A Proton having a mass of 1,67262× 10 -24   G the four protons of origin have a mass 4× 1,67262× 10 -24   =  6,69048× 10 -24   G. The core of Helium, or particle α, has a mass of 6,64466× 10 -24   G. The difference, 0,04582× 10 -24   G, is released in the form of light.

It is noticed that only a weak part of the mass used is converted into energy. The report/ratio of the hydrogen mass consumed with the really transformed mass is 6,69048/0,04582~150. Thus for each gram converted into energy these are 150 grams of hydrogen which are used. (Conversely that means that approximately 0,7 pourcent initial hydrogen mass is converted into energy.) Consequently to transform 4,4× 10 12 grams a second 150 times will have “to be burned” more, that is to say 6,6× 10 14 grams of hydrogen a second. As the mass of the Sun is of 2× 10 33 grams, if all this quantity of matter were consumed, the Sun could provide its current energy, at the current rate, during 2× 10 33 /  6,6× 10 14   =  3× 10 18 seconds. As a Année is worth approximately 3× 10 7 seconds, we end to one total lifespan of 100 billion (10 11 ) of years.

Evolution of a star in general, and Sun in particular, is more complicated than this elementary diagram of conversion of helium hydrogen. All hydrogen is not burned out of helium and other reactions of burn-up intervene. With final, when more elaborate stellar models are considered, the current theories predict one total lifespan of some 10 billion years for star of a solar type. The Sun having already used approximately 5 billion, it him would remain about it about as much “to be lived”. (But nobody any more will be there to be pilot of a future who so largely exceeds the temporal scale of the humanité !)

Electromagnetism and restricted relativity

In Newtonian space with three dimensions, a particle of load Q placed in an electric field $\backslash \; vec\; \{E\}$ and a magnetic field $\backslash \; vec\; \{B\}$ is subjected to the Force of Lorentz and the equation which governs its movement is

$D\; \backslash \; vec\; \{p\}\; /dt\; =\; \backslash ,\; Q\; \backslash ,\; (\backslash \; vec\; E\; \backslash \; +\; \backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; \backslash .$

To transpose this formula in relativistic mechanics, one will have to consider the quadrivector energy-impulse $\backslash \; mathbf\; \{p\}$ in the place of the vector $\backslash \; vec\; \{p\}$ and to evaluate the rate of variation of this quadrivector not in the reference mark of an unspecified observer galiléen but in the clean reference mark of the particle. The member of left will be thus of the form $d\; \backslash \; mathbf\; \{p\}\; /d\; \backslash \; tau$, where $\backslash \; tau$ is the clean time of the particle charged. On the right one will find an object independent of the selected reference mark and which moreover will be inevitably a linear function the speed $\backslash \; vec\; \{v\}$ of the particle. Indeed the space part of the equation of dynamics is linear in $\backslash ,\; \backslash \; vec\; \{v\}\; \backslash ,$ since she is written

$d\; \backslash \; vec\; \{p\}\; /d\; \backslash \; tau\; =\; \backslash \; gamma\; D\; \backslash \; vec\; \{p\}\; /dt=\; \backslash \; gamma\; Q\; (\backslash \; vec\; E\; \backslash \; +\; \backslash \; vec\; \{v\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; =\; Q\; (u\_0\; \backslash \; vec\; \{E\}\; /c\; +\; \backslash \; vec\; \{U\}\; \backslash \; wedge\; \backslash \; vec\; \{B\})\; \backslash .$

In this expression $\backslash ,\; u\_0\; \backslash ,$ and $\backslash \; vec\; \{U\}$ are the components in a Lorentzian reference mark of the quadrivector speed $\backslash \; mathbf\; \{U\}\; \backslash ,$, which thus can écrire :

$\backslash \; mathbf\; \{U\}\; =\; (u\_0,\; \backslash \; vec\; \{U\})\; =\; \backslash \; left\; (\backslash \; frac\; \{C\}\; \{\backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\},\; \backslash \; frac\; \{\backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; -\; (v^2/c^2)\}\}\; \backslash \; right)\; \backslash \; equiv\; (\backslash \; gamma\; C,\; \backslash \; gamma\; \backslash \; vec\; \{v\})\; \backslash .$

Explicitly the equation above breaks up on the three axes in the following way  :

$$

\ begin {boxes} dp_x/d \ tau = Q (u_0 E_x/c +v_y B_z - v_zB_y) \ \ dp_y/d \ tau = Q (u_0 E_y/c +v_z B_x - v_xB_z) \ \ dp_z/d \ tau = Q (u_0 E_z/c +v_x B_y - v_yB_x) \ end {boxes}

On its side the temporal component of the equation of the dynamics (which corresponds to the law giving the variation of energy) is written

$dp\_0/d\; \backslash \; tau\; =\; \backslash \; gamma\; D\; (W/c)/dt\; =\; \backslash \; gamma\; Q\; (\backslash \; vec\; \{E\}\; /c)\; \backslash \; cdot\; \backslash \; vec\; \{v\}\; \backslash \; equiv\; Q\; (\backslash \; vec\; \{E\}\; /c)\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; \backslash ,$

where W is the work of the force $q\; \backslash \; vec\; \{E\}\; \backslash .$

By gathering the equations written above within the framework of a space time with four dimensions, the rate of variation of the quadrivector energy-impulse is given by

$$

\begin{pmatrix} dp_0/d \ tau \ \ dp_x/d \ tau \ \ dp_y/d \ tau \ \ dp_z/d \ tau \end{pmatrix}

Q

\begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \ \ E_x/c & 0 & B_z & - B_y \ \ E_y/c & - B_z & 0 & B_x \ \ E_z/c & B_y & - B_x & 0 \end{pmatrix} \begin{pmatrix} u_0\\u_x\\u_y\\u_z \end{pmatrix}

The matric equation which we have just written watch that in restricted relativity the magnetic field and the electric field constitute a single entity. Actually the preceding presentation is somewhat incorrect insofar as to benefit from all the power of the relativistic theory it is necessary to call upon the tensors. The matric equation above is the translation in terms of components of the equation tensorial, independent, it, of any frame of reference

$d\; \backslash \; mathbf\; \{p\}\; /d\; \backslash \; tau\; =\; Q\; \backslash \; mathbf\; \{F\}\; (\backslash \; mathbf\; \{U\})\; \backslash .$

$\backslash \; mathbf\; \{F\}$ is the tensor electromagnetic field (or tensor of Maxwell or tensor of Faraday). It is this object which represents the electromagnetic field physically. Its components in a certain frame of reference are given by the matrix written above.

Vocabulary

1. Observant: human or detector receiving set having a clock allowing him to read the hour and possibly, if it belongs to a group made up, carrying a mark indicating its position.
2. Reference frame galiléen, also called “reference frame of Lorentz” or “inertial reference frame”: together of observers moving freely in space far from any mass, whose mutual distances do not change during time (they are at rest the ones compared to the others) and who synchronized their clocks.
3. Event: an event is an event (!), such as for example the birth of an individual, the departure of a rocket or the shooting of a detonator. It is independent of the coordinates of times and space which possibly make it possible to locate it. However it is convenient to locate the events by their coordinates in practice, namely the point where the event occurs and dates it to which it occurs.

In restricted relativity a length and a time should be measured with the same unit (what we did not do here in a systematic way). In astronomy one chooses a unit of time and one measures a distance by time that it is necessary for the light to cover it. For example that a galaxy is located at five million light-years of ours means that the light puts five million years to traverse the distance which separates us. Let us notice that in the everyday life one will say easily that Paris, for example, is at three hours of train of Montpellier, which amounts exactly measuring a distance in time. ; to also see: question of units

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