This article presents the transformations of Lorentz under a technical side. The reader eager to obtain more general information on this subject will be able to refer to the article restricted Relativité .

Within the framework of the restricted Relativity, the transformation of Lorentz corresponds to the law of change of Référentiel galiléen under which the equations of the Physique must be preserved. It is in particular to make so that the Maxwell's equations are written with identical in any reference frame galiléen that Hendrik Antoon Lorentz introduced this law mathematically before Albert Einstein carries out all the physical range of it. The index property of this transformation is that the Speed of light is the same one in any reference frame galiléen. One can also show that it is the single law of transformation Linéaire, Homogène and Isotrope of the Espace-temps having this property. This article offers a mathematically detailed presentation of the transformation.

Introduction

It is about a whole of mathematical relations which make it possible to change the point of view by which one describes the events without speed of light being changed.

The transformations of Lorentz intervene in restricted Relativité and in the relativistic Calculs as a " transformations" linear mathematics such as the Speed of light is the same one in all the inertial reference frames.

The most astonishing consequence is that the Temps is not any more " universel" but depend on the référentiel.
These equations enable us to find the coordinates of space and of time of a reference frame of inertia has, once one knows the coordinates of a reference frame of inertia B and the relative Speed between the reference frames has and B.

Elementary presentation

See first of all the matrices in mathematics.

Are two reference frames $\backslash \; mathbb\; R$ and $\backslash \; mathbb\; R\text{'}$ in rectilinear translation one compared to the other on parallel axes, with a relative speed v according to axis OX. Are $(X,\; T)\; \backslash \; quad$ them space-time coordinates of an event in the reference frame $\backslash \; mathbb\; R$, and $(x\text{'},\; you)\; \backslash \; quad$ its coordinates in the reference frame $\backslash \; mathbb\; R\text{'}\; \backslash \; quad$. (For to simplify the notations, one will not hold account in this paragraph of both others space components there and Z ).

It is supposed that the transformation is carried out by means of a linear operator:

S \ end {pmatrix} \ begin {pmatrix} X \ \ T \ end {pmatrix} There are four unknown factors, one thus needs four equations from where:

• 1) In the reference frame $\backslash \; mathbb\; R$, the reference frame $\backslash \; mathbb\; R\text{'}$ moves at the speed v , there is $x\; =\; vt\; \backslash \; quad$ if and only if $x\text{'}\; =\; 0\; \backslash \; quad$:

$x\text{'}\; =\; px\; +\; qt\; \backslash \; quad$ thus $0\; =\; pvt\; +\; qt\; \backslash \; quad$

what gives like first equation $0\; =\; statement\; +\; Q\; \backslash \; quad\; (1)$

• 2) Réciproquement, in the reference frame $\backslash \; mathbb\; R\text{'}$, the reference frame $\backslash \; mathbb\; R$ moves at the speed - v , so that $x\text{'}\; =\; -\; vt\text{'}\; \backslash \; quad$ if and only if $x=0\; \backslash \; quad$:

$x\text{'}\; =\; qt\; \backslash \; quad$ and $t\text{'}\; =\; St\; \backslash \; quad$

that is to say: $-vt\text{'}\; =\; \{qt\text{'}\; \backslash \; over\; S\}\; \backslash \; quad$

giving a second equation: $\{Q\; \backslash \; over\; S\}\; =\; -\; v$ or $q\; +\; sv\; =\; 0\; \backslash \; quad\; (2)$

• 3) So that speed of light C is the same one in the two reference marks, it is necessary that $x\; =\; ct\; \backslash \; quad$ if and only if $x\text{'}\; =\; ct\text{'}\; \backslash \; quad$:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; x\text{'}\; =\; pct\; +\; qt\; \backslash \; \backslash \; t\text{'}=\; rct\; +\; St\; \backslash \; end\; \{matrix\}\; \backslash \; right.$ is: $pct\; +\; qt\; =\; rc^2t\; +\; sct\; \backslash \; quad$

what gives: $pc+q\; =\; C\; (rc+s)\; \backslash \; quad\; (3)$

Of these three equations, one from of deduced that:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; Q\; =\; -\; statement\; \backslash \; \backslash \; S\; =\; p\; \backslash \; \backslash \; R\; =\; -\; pv/c^2\; \backslash \; end\; \{matrix\}\; \backslash \; right.$

so that:

p \ end {pmatrix} \ begin {pmatrix} X \ \ T \ end {pmatrix} \ quad (4)

• 4) If the matrix of transformation is reversed, one finds reciprocally that:

But this last transformation (5) must also result from the transformation (4) into exchanging the roles of the two reference frames and thus by changing the sign speed v . If it is supposed that the quantity p depends only on the module of v and thus that p is the same one for v and - v , one must thus have:

\ begin {pmatrix} p & statement \ \ pv/c^2 & p \ end {pmatrix} what imposes that $p\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{1\; -\; v^2/c^2\}\}$.

The transformation obtained is thus:

- v \ \ - v/c^2 & 1 \ end {pmatrix} \ begin {pmatrix} X \ \ T \ end {pmatrix} that one will also write in the form:

$\backslash \; begin\; \{pmatrix\}\; x\text{'}\; \backslash \; \backslash \; ct\text{'}\; \backslash \; end\; \{pmatrix\}\; =\; \{1\; \backslash \; over\; \backslash \; sqrt\; \{1\; -\; v^2/c^2\}\}\; \backslash \; begin\; \{pmatrix\}\; 1$

& - v/c \ \ - v/c & 1 \ end {pmatrix} \ begin {pmatrix} X \ \ ct \ end {pmatrix}

More complete presentation

The transformations of Galileo preserve the scalar Produit: : $\backslash \; vec\; \{has\}\; \backslash \; cdot\; \backslash \; vec\; \{B\}\; =\; \backslash \; vec\; \{has\}\; “\backslash \; cdot\; \backslash \; vec\; \{B\}\; “$

In the Espace-temps of Minkowski, the metric tensor is:

$\backslash \; eta\_\; \{\backslash \; alpha\; \backslash \; beta\}\; =\; \backslash \; left$

What wants to say that one must differentiate the coordinates covariantes, of the coordinates contravariantes. The pseudo-standard is defined: : $ds^2=\; \backslash \; eta\_\; \{\backslash \; alpha\; \backslash \; beta\}\; dx^\; \{\backslash \; alpha\}\; dx^\; \{\backslash \; beta\}\; =dx\_\; \{\backslash \; alpha\}\; dx^\; \{\backslash \; alpha\}\; =c^2dt^2-dx^2-dy^2-dz^2$

the transformations of Lorentz must preserve the pseudo-standard: : $dx\_\; \{\backslash \; alpha\}\; dx^\; \{\backslash \; alpha\}\; =dx”\; \_\; \{\backslash \; alpha\}\; dx\text{'}^\; \{\backslash \; alpha\}$

The transformations of Lorentz must be linear with constant coefficients. Subsequently, the preceded indices correspond to the coordinates in the reference frame $\backslash \; mathbb\; \{R\text{'}\}$, moreover the repetitions of Greek letters will want to say summation from 0 to 3, and the repetitions of Latin letters from 1 to 3.

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}\; \backslash \; eta\_\; \{\backslash \; alpha\; \backslash \; beta\}\; x^\; \{\backslash \; alpha\}\; x^\; \{\backslash \; beta\}\; =\; \backslash \; sum\_\; \{\backslash \; alpha=0\}\; ^\; \{3\}\; \backslash \; sum\_\; \{\backslash \; beta=0\}\; ^\; \{3\}\; \backslash \; eta\_\; \{\backslash \; alpha\; \backslash \; beta\}\; x^\; \{\backslash \; alpha\}\; x^\; \{\backslash \; beta\}\; \backslash \; \backslash \; \backslash \; delta\_\; \{ij\}\; x^\; \{I\}\; x^\; \{J\}\; =\; \backslash \; sum\_\; \{i=1\}\; ^\; \{3\}\; \backslash \; sum\_\; \{j=1\}\; ^\; \{3\}\; \backslash \; eta\_\; \{ij\}\; x^\; \{I\}\; x^\; \{J\}\; \backslash end\{matrix\}\backslash right.$

the transformations are written in the matric form:

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

x^ {\ mu'} \ rightarrow x^ \ mu=L_ {\ nu'} ^ {\ driven} x^ {\ nu'} \ \ y_ {\ mu'} \ rightarrow y_ \ mu=L_ {\ driven} ^ {\ nu'} y_ {\ nu'} \end{matrix}\right. The scalar pseudo-products are invariants not transformations of Lorentz: $x^\; \{\backslash \; driven\}\; y\_\; \{\backslash \; driven\}\; =x^\; \{\backslash \; mu\text{'}\}\; y\_\; \{\backslash \; mu\text{'}\}\; =y\_\; \{\backslash \; lambda\text{'}\}\; L\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; driven\}\; L\_\; \{\backslash \; driven\}\; ^\; \{\backslash \; lambda\text{'}\}\; x^\; \{\backslash \; rho\text{'}\}$ that is to say thus: $L\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; driven\}\; L\_\; \{\backslash \; driven\}\; ^\; \{\backslash \; lambda\text{'}\}\; =\; \backslash \; delta\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; lambda\text{'}\}$ where $\backslash \; delta\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; lambda\text{'}\}$ is the Symbole of Kronecker. The reverse of the matrix $L\_\; \{\backslash \; driven\}\; ^\; \{\backslash \; nu\text{'}\}$ its is transposed: $L\_\; \{\backslash \; nu\text{'}\}\; ^\; \{\backslash \; driven\}$ The transformation of the metric tensor is found by having with the spirit the invariance of the scalar pseudo-product:

$x\_\; \{\backslash \; driven\}\; x^\; \{\backslash \; driven\}\; =x\_\; \{\backslash \; lambda\text{'}\}\; x^\; \{\backslash \; lambda\text{'}\}$

$\backslash \; eta\_\; \{\backslash \; mu\text{'}\; \backslash \; nu\text{'}\}\; =\; \backslash \; eta\_\; \{\backslash \; lambda\; \backslash \; rho\}\; L\_\; \{\backslash \; mu\text{'}\}\; ^\; \{\backslash \; lambda\}\; L\_\; \{\backslash \; nu\text{'}\}\; ^\; \{\backslash \; rho\}$

One from of deduced that $(det\; L)\; ^2=1$ thus $det\; L=1$ or $det\; L=-1$ in the continuation, one will place oneself if the determinant is positive, and $L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; >0$ called clean group orthochrone of Lorentz. The transformations are written then:

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

dx^ {0} =L_ {0 '} ^ {0} dx^ {0 '} +L_ {k'} ^ {0} dx^ {k'} \ \ dx^ {I} =L_ {0 '} ^ {I} dx^ {0 '} +L_ {k'} ^ {I} dx^ {k'} \end{matrix}\right.

One considers a body at rest in the reference mark $\backslash \; mathbb\; \{R\text{'}\}$, then $dx\text{'}^\; \{K\}\; =0$, from where:

$\backslash \; frac\; \{dx^i\}\; \{dx^0\}\; =\; \backslash \; frac\; \{L\_\; \{0\; \text{'}\}\; ^\; \{I\}\}\; \{L\_\; \{0\; \text{'}\}\; ^\; \{0\}\}$

that is to say:

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

L_ {0 '} ^ {I} = \ beta^ {I} L_ {0 '} ^ {0} \ \ L_ {0} ^ {i'} =L_0^ {0 '} \ beta^ {i'} \ end {matrix} \ right.& (1) \ end {matrix}

Then there are these relations to show:

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

L_ {0 '} ^ {I} =-L_ {I} ^ {0 '} &L_ {i'} ^ {0} =-L_ {0} ^ {i'} &L_ {i'} ^ {K} =-L_ {K} ^ {i'} & (2) \ \ L_ {I} ^ {0 '} =L_0^ {0 '} \ beta_ {I} &L_ {0} ^ {i'} =L_ {0 '} ^ {0} \ beta_ {i'} && (3) \ \ L_ {0 '} ^ {0} \ beta^i=-L_ {k'} ^ {I} \ beta^ {k'} &L_ {0} ^ {0 '} \ beta^ {i'} =-L_ {i'} ^ {K} \ beta^ {K} && (4) \ \ L_ {0 '} ^ {0} = \ pm \ frac {1} {\ sqrt {1 \ beta^2}} &L_ {0} ^ {0 '} = \ pm \ frac {1} {\ sqrt {1 \ beta'^2}} && (5) \ \ detL_ {k'} ^ {I} =L_ {0 '} ^ {0} &detL_ {K} ^ {i'} =L_ {0} ^ {0 '} && (6) \ \ \ beta^2= \ beta'^2 \ leftrightarrow L_ {0 '} ^ {0} =L_ {0} ^ {0 '} = \ gamma= \ frac {1} {\ sqrt {1 \ beta^2}} &&& (7) \end{matrix}\right. For the expressions (2), it is enough to use the relation: $L\_\; \{\backslash \; mu\text{'}\}\; ^\; \{\backslash \; naked\}\; =\; \backslash \; eta^\; \{\backslash \; naked\; alpha\; \backslash \}\; \backslash \; eta\_\; \{\backslash \; beta\text{'}\; \backslash \; mu\text{'}\}\; L\_\; \{\backslash \; alpha\}\; ^\; \{\backslash \; beta\text{'}\}$ with $\backslash \; nu=i$, $\backslash \; mu=0$ and $\backslash \; mu\text{'}=\; \backslash \; nu\text{'}=0\text{'}$ that is to say:

$L\_\; \{0\; \text{'}\}\; ^\; \{I\}\; =\; \backslash \; eta^\; \{\backslash \; alpha\; I\}\; \backslash \; eta\_\; \{\backslash \; beta\text{'}\; 0\; \text{'}\}\; L\_\; \{\backslash \; alpha\}\; ^\; \{\backslash \; beta\text{'}\}\; =\; \backslash \; eta^\; \{II\}\; \backslash \; eta\_\; \{0\; \text{'}\; 0\; \text{'}\}\; L\_\; \{I\}\; ^\; \{0\; \text{'}\}\; =-L\_\; \{I\}\; ^\; \{0\; \text{'}\}$

For the expressions (3):

$L\_\; \{I\}\; ^\; \{0\; \text{'}\}\; =-L\_\; \{0\; \text{'}\}\; ^\; \{I\}\; =\; \backslash \; beta^\; \{I\}\; L\_\; \{0\}\; ^\; \{0\; \text{'}\}\; =L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; \backslash \; beta\_\; \{I\}$

For the expressions (4), we start from $L\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; driven\}\; L\_\; \{\backslash \; driven\}\; ^\; \{\backslash \; sigma\text{'}\}\; =\; \backslash \; delta\_\; \{\backslash \; rho\text{'}\}\; ^\; \{\backslash \; sigma\text{'}\}$, with $\backslash \; rho\text{'}=0\text{'}$ and $\backslash \; sigma\text{'}=i\text{'}$

$L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; L\_\; \{0\}\; ^\; \{i\text{'}\}\; +L\_\; \{0\; \text{'}\}\; ^\; \{K\}\; L\_\; \{K\}\; ^\; \{i\text{'}\}\; =\; \backslash \; delta\_\; \{0\; \text{'}\}\; ^\; \{i\text{'}\}\; =0$

$L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; L\_\; \{0\}\; ^\; \{i\text{'}\}\; -\; L\_\{0\; \text{'}\}\; ^\; \{0\}\; \backslash \; beta\_\; \{K\}\; L\_\; \{K\}\; ^\; \{i\text{'}\}\; =0$

$L\_\; \{0\}\; ^\; \{0\; \text{'}\}\; \backslash \; beta^\; \{i\text{'}\}\; =\; \backslash \; beta\_\; \{K\}\; L\_\; \{K\}\; ^\; \{i\text{'}\}\; =-L\_\; \{K\}\; ^\; \{i\text{'}\}\; \backslash \; beta^\; \{K\}$

$L\_\; \{0\}\; ^\; \{0\; \text{'}\}\; \backslash \; beta^\; \{i\text{'}\}\; =-L\_\; \{i\text{'}\}\; ^\; \{K\}\; \backslash \; beta^\; \{K\}$

For the expressions (5) the relations of transformations of the metric tensor give:

$\backslash \; eta\_\; \{\backslash \; mu\text{'}\; \backslash \; nu\text{'}\}\; =L\_\; \{\backslash \; mu\text{'}\}\; ^\; \{\backslash \; rho\}\; L\_\; \{\backslash \; nu\text{'}\}\; ^\; \{\backslash \; sigma\}\; \backslash \; eta\_\; \{\backslash \; rho\; \backslash \; sigma\}$, by taking $\backslash \; mu\text{'}=\; \backslash \; nu\text{'}=0\text{'}$

$1=L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; \backslash \; eta\_\; \{00\}\; +L\_\; \{0\; \text{'}\}\; ^\; \{I\}\; L\_\; \{0\; \text{'}\}\; ^\; \{J\}\; \backslash \; eta\_\; \{ij\}\; =\; (L\_\; \{0\; \text{'}\}\; ^\; \{0\})\; ^2\; (1+\; \backslash \; eta\_\; \{ij\}\; \backslash \; beta^\; \{I\}\backslash \; beta^\; \{J\})$

$L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; =\; \backslash \; pm\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1\; \backslash \; beta^2\}\}$

For the expressions (6): with $L\_\; \{0\; \text{'}\}\; ^\; \{I\}\; =L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; \backslash \; beta^\; \{I\}$ and $L\_\; \{k\text{'}\}\; ^\; \{0\}\; =L\_\; \{k\text{'}\}\; ^\; \{I\}\; \backslash \; beta\_\; \{I\}$ while noticing: $\backslash \; eta\_\; \{\backslash \; mu\text{'}\; \backslash \; nu\text{'}\}\; =L\_\; \{\backslash \; mu\text{'}\}\; ^\; \{\backslash \; lambda\}\; L\_\; \{\backslash \; nu\text{'}\}\; ^\; \{\backslash \; rho\}\; \backslash \; eta\_\; \{\backslash \; lambda\; \backslash \; rho\}$ for $\backslash \; mu\text{'}=i\text{'}$ and $\backslash \; nu\text{'}=j\text{'}$ one obtains:

$\backslash \; eta\_\; \{I\; k\text{'}\}\; =L\_\; \{I\}\; ^\; \{0\}\; L\_\; \{k\text{'}\}\; ^\; \{0\}\; \backslash \; eta\_\; \{00\}\; +L\_\; \{I\}\; ^\; \{m\}\; L\_\; \{k\text{'}\}\; ^\; \{I\}\; \backslash \; eta\_\; \{semi\}$

gold: $L\_\; \{I\}\; ^\; \{0\}\; =L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; \backslash \; beta\_\; \{I\}\; =-L\_\; \{0\}\; ^\; \{0\; \text{'}\}\; \backslash \; beta^\; \{I\}\; =L\_\; \{m\}\; ^\; \{I\}\; \backslash \; beta^\; \{m\}\; =-L\_\; \{I\}\; ^\; \{m\}\; \backslash \; beta\_\; \{m\}$

from where:

$-\; \backslash \; eta\_\; \{I\; k\text{'}\}\; =L\_\; \{I\}\; ^\; \{m\}\; L\_\; \{k\text{'}\}\; ^\; \{I\}\; \backslash \; delta\_\; \{semi\}\; -\; L\_\; \{I\}\; ^\; \{m\}\; L\_\; \{k\text{'}\}\; ^\; \{I\}\; \backslash \; beta\_\; \{m\}\; \backslash \; beta\_\; \{I\}\; =\; L\_\; \{I\}\; ^\; \{m\}\; L\_\; \{k\text{'}\}\; ^\; \{I\}\; (\backslash \; delta\_\; \{semi\}\; -\; \backslash \; beta\_\; \{m\}\; \backslash \; beta\_\; \{I\})$

One takes the determinant:

$1=\; (1\; \backslash \; beta^2)\; \{\backslash \; cdot\}\; (detL\_\; \{k\text{'}\}\; ^\; \{I\})\; ^2$

$detL\_\; \{k\text{'}\}\; ^\; \{I\}\; =L\_\; \{0\; \text{'}\}\; ^\; \{0\}$

For the expressions (7): We consider the clean group orthochrone of Lorentz, therefore $L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; >0$ moreover $L\text{'}=L^\; \{-\; 1\}$ (orthogonal matrices), one thus has: $L\_\; \{0\; \text{'}\}\; ^\; \{0\}\; =L\_\; \{0\}\; ^\; \{0\; \text{'}\}$, one thus has $\backslash \; beta^2=\; \backslash \; beta\text{'}^2$

See too

• Hendrik Antoon Lorentz
• restricted Relativity
• General relativity
• relativistic Calculations
• Covariance
• Invariance of Lorentz

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