Quadrivector

The theory of relativity (restricted, then general) postulated by pleasing Einstein to regard the three coordinates of space (for example height, width, depth) and time as formant an indissociable whole.

Description and calculations on this unit (baptized since Space time) led Einstein to prolong the concept of traditional Vecteur Newtonian (formed of three coordinates of space, the such vector Speed of a material point) to create a new mathematical object, the quadrivector .

A quadrivector thus corresponds to the whole from not three but four component, the additional component being associated with the coordinate of time T . Sometimes for dimensional reasons, one uses in his place the coordinate, possibly noted X 0 equalize with the product of the coordinate of time by the Speed of light, c T . According to the cases, time appears is in first (as the notation X suggests it 0 ), that is to say in the last. The raison d'être of the quadrivector is first of all to describe how changes the whole of the coordinates X , there , Z and T (or c T ) during a change of Reference frame. These changes are called Transformation of Lorentz into restricted relativity.

By extension, it appears that a great number of quantities can be seen like changing like the four coordinates of space time. Such objects are also quadrivecteurs. The space components of the quadrivecteurs always correspond to a three-dimensional vector in traditional Mécanique. The physical significance of the fourth component of the quadrivecteurs is on the other hand less immediate, though always in connection with the object considered.

Quadrivecteurs covariants and contravariants

A quadrivector can exist in two forms, known as covariante and contravariante . The distinction between these two forms is done using the position of the indices of the components of quadrivecteurs.

When the indices are noted while exposing, one speaks about indices covariants. When it are noted in bottom, one speaks about indices contravariants. For example, for a vector V :

the V has is the components covariantes vector,
the V are the components contravariantes vector.

The coordinates X , there , Z and T are covariantes quantities. One thus notes them in the form

(T, X, there, Z) = x^a

It is possible to pass from the form covariante to the shape contravariante of the components of one vector using an operation which can formally be assimilated to a matric multiplication, using a symmetrical matrix in general noted η . One has thus, for any vector

V_a = \ sum_b \ eta_ {ab} V^b

In this kind of summation, it is of habit to omit the sign summons, this one being implicit each time a inidie appears simultaneously in top and bottom in a formula. The preceding equation notes

V_a thus = \ eta_ {ab} V^b

The reverse transformation is done using the opposite matrix, noted with indices in top,

\ eta^ {ab} = \ left \ right^ {- 1}

There is thus

V^a = \ eta^ {ab} V_b

The matrix η is called Métrique of Minkowski. Its matric form is

\ eta_ {ab} = \ pm \ left (\ begin {array} {cccc} X^2 & 0 & 0 & 0 \ \ 0 & - 1 & 0 & 0 \ \ 0 & 0 & -1 & 0 \ \ 0 & 0 & 0 & -1 \ end {array} \ right)

The quantity X is worth C (speed of light) or 1 according to whether the temporal coordinate were taken equal to T or c T . The sign of η is unspecified, and depends on an arbitrary convention, called Convention of sign of metric the.

In general, a quadrivector is defined in a nonambiguous way by his form covariante or contravariante, the other form being deduced by the action from the matrix η.

The matrix η allows to calculate the standard of a quadrivector, or more generally the scalar Produit between two vector. There is thus

{\ mathbf {U}} \ cdot {\ mathbf {V}} = \ eta_ {ab} U^a V^b = \ pm \ left (X^2 U^0 V^0 - U^x V^x - U^y V^y - U^z V^z \ right)

Some examples of quadrivecteurs

Quadrivecteur position-time:

x^a = (T, {\ mathbf {R}}) = (T, X, there, Z)

x^a = (C T, {\ mathbf {R}}) = (C T, X, there, Z)

Quadrivecteur speed (Quadrivitesse):

u^a = \ frac

u^a = \ frac {C {\ rm D} \ tau}

where τ clean Temps is the , i.e. the time which would be indicated by a clock which would be attached to the object whose trajectory would have the corresponding Flight Path Vector. The quadrivector speed is by definition of set standard (by standard one understands the quantity

\ eta_ {ab} u^a u^b

, to see paragraph above), equalizes, according to the convention of coordinate and sign chosen with C 2 , - C 2 , 1, or -1. The temporal component of the quadrivector speed is determined by the condition which the standard is equal to the specified value.

Quadrivecteur impulse (Quadriimpulsion). For a particle of nonnull mass:

p^a = m \ frac

p^a = m \ frac {C {\ rm D} \ tau}

where τ clean Temps is the . The quadrivector impulse has a set standard, equalizes, according to the convention of coordinate and sign chosen with m 2 C 2 , - m 2 C 2 , m 2 , or - m 2 . The temporal component of the quadrivector speed is determined by the condition which the standard is equal to the specified value. One can show that it is identified (except for a constant) with the energy of the object such as it would be measured by a motionless observer compared to the coordinates X , there , Z .

Potentiel vector has . This quadrivector is defined with components contravariantes. Its space components are identified (with the sign near) with the potential vector of the electromagnetism, whose Rotationnel gives the Magnetic field. Its temporal component gives, except for a multiplicative factor, the electric Potentiel.

See too

Transformation of Lorentz

External bonds

Course on the theory of relativity of the university of Nantes

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