Special Lorentz transformation

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The special Lorentz transformations (also Lorentz boosts or just boosts ), according to Hendrik Antoon Lorentz , are a subclass of the Lorentz transformations . They are needed in the special theory of relativity to convert quantities in two reference systems into one another, whose coordinate axes are parallel and which move at a constant speed relative to one another. Formally, they are those Lorentz transformations that contain no space reflection , no time reversal and no rotation . While these three classes result from their classical analogues trivially as block diagonal matrices , the special Lorentz transformations link the time-like and spatial components of a physical quantity.

For low speeds, the special Lorentz transformations change over to the Galilei transformations .

Physical background

→ Main article : Spacetime

In the special theory of relativity, the classical quantities of Euclidean three-dimensional space and (one-dimensional) time are linked to form a four-dimensional space - time . Physical objects such as three-dimensional vectors and higher-level tensors must therefore be inserted into four-dimensional objects ( four-vector , four-tensor ). This happens, for example, through the standardization of place and time in space-time or through the combination of the scalar quantity energy and the vector quantity impulse in the four-pulse . While in classical physics such scalar quantities as the energy assume the same value in each reference system, this changes in special relativity by linking the time-like ("zeroth") component of a four-vector with its spatial components through the special Lorentz transformations. ${\displaystyle x^{\mu }={\begin{pmatrix}ct&{\vec {x}}\end{pmatrix}}^{T}}$ ${\displaystyle p^{\mu }={\begin{pmatrix}E/c&{\vec {p}}\end{pmatrix}}^{T}}$

presentation

The special Lorentz transformations can be represented as four-dimensional matrices that operate on space-time. In its most general form, a Lorentz matrix is ​​the following , which describes a special Lorentz transformation between two reference systems that move with one another at one speed : ${\displaystyle \Lambda }$ ${\displaystyle {\vec {v}}}$

${\displaystyle \Lambda _{\;\;\nu }^{\mu }({\vec {\beta }})={\begin{pmatrix}\gamma &-\gamma \beta _{j}\\-\gamma \beta _{i}&\delta _{ij}+(\gamma -1){\frac {\beta _{i}\beta _{j}}{\beta ^{2}}}\end{pmatrix}}\equiv (\Lambda _{\;\;\nu }'^{\mu })^{-1}}$

${\displaystyle \Lambda _{\;\;\nu }'^{\mu }({\vec {\beta }})={\begin{pmatrix}\gamma &\gamma \beta _{j}\\\gamma \beta _{i}&\delta _{ij}+(\gamma -1){\frac {\beta _{i}\beta _{j}}{\beta ^{2}}}\end{pmatrix}}\equiv (\Lambda _{\;\;\nu }^{\mu })^{-1}}$

It is

• ${\displaystyle {\vec {\beta }}}$the speed related to the speed of light , ${\displaystyle c}$ ${\displaystyle {\vec {\beta }}={\vec {v}}/c}$
• ${\displaystyle \gamma }$the Lorentz factor , ${\displaystyle \gamma =(1-\beta ^{2})^{-1/2}}$
• ${\displaystyle \delta _{ij}}$the Kronecker Delta

The Lorentz matrix is ​​a four-dimensional matrix. To perform a Lorentz transformation, every space-time index of a tensor has to be contracted with a Lorentz matrix. For example, in a new reference system in which all quantities are marked with a dash, the following applies to the new location and time variables

${\displaystyle x'^{\mu }=\Lambda _{\;\;\nu }^{\mu }x^{\nu }\Leftrightarrow {\begin{pmatrix}ct'\\x'_{i}\end{pmatrix}}={\begin{pmatrix}\gamma &-\gamma \beta _{j}\\-\gamma \beta _{i}&\delta _{ij}+(\gamma -1){\frac {\beta _{i}\beta _{j}}{\beta ^{2}}}\end{pmatrix}}{\begin{pmatrix}ct\\x_{j}\end{pmatrix}}={\begin{pmatrix}\gamma ct-\gamma \beta _{j}x_{j}\\x_{i}-\gamma \beta _{i}ct+(\gamma -1)\beta _{i}{\frac {\beta _{j}x_{j}}{\beta ^{2}}}\end{pmatrix}}}$

or for higher level tensors such as the field strength tensor

${\displaystyle F'^{\mu \nu }=\Lambda _{\;\;\alpha }^{\mu }\Lambda _{\;\;\beta }^{\nu }F^{\alpha \beta }}$.

The inverse of the Lorentz matrix is ​​the Lorentz matrix in which the sign of the speed is reversed: This is understandable insofar as the Lorentz transformation transforms back into the original reference system with the opposite speed. The Lorentz matrix has the determinant and a positive component . This gives it the spatial and temporal orientation as well as the norm . The maintenance of the norm is definitional for the Lorentz group, the other two properties have special Lorentz transformations with rotations in common. ${\displaystyle \Lambda ^{-1}(\beta )=\Lambda (-\beta )}$ ${\displaystyle \det \Lambda =1}$ ${\displaystyle \Lambda _{0}^{0}>0}$

Non-relativistic borderline case

In the non-relativistic borderline case , the special Lorentz transformations pass into Galileo transformations. This can be seen in that in this case is and the Lorentz matrix as ${\displaystyle v\ll c}$ ${\displaystyle \gamma \approx 1}$

${\displaystyle \Lambda _{\;\;\nu }^{\mu }(\beta )\approx {\begin{pmatrix}1&-\beta _{j}\\-\beta _{i}&\delta _{ij}\end{pmatrix}}}$

can be written. So is

${\displaystyle {\begin{pmatrix}ct'\\x'_{i}\end{pmatrix}}\approx {\begin{pmatrix}ct-\beta _{j}x_{j}\\x_{i}-\beta _{j}ct\end{pmatrix}}\approx {\begin{pmatrix}ct\\x_{i}-v_{i}t\end{pmatrix}}}$.

In the non-relativistic limit case, it should be noted that the limit values ​​of the Lorentz matrix and its inverses are no longer inverse to one another. So if you transform back and forth, this should be done with the relativistically correct matrices and the limit value should only be formed in the end result. For example, the inverse of the matrix given here has the following form: ${\displaystyle \Lambda _{\;\;\nu }^{\mu }}$ ${\displaystyle \Lambda _{\;\;\nu }'^{\mu }}$

${\displaystyle {\begin{pmatrix}1&-\beta &0&0\\-\beta &1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}^{-1}={\begin{pmatrix}{\frac {1}{1-\beta ^{2}}}&{\frac {\beta }{1-\beta ^{2}}}&0&0\\{\frac {\beta }{1-\beta ^{2}}}&{\frac {1}{1-\beta ^{2}}}&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}\neq \lim _{\gamma \rightarrow 1}\Lambda _{\;\;\nu }'^{\mu }}$

This does not correspond to the limit of the inverse given above!

One-dimensional case

Every special Lorentz transformation along a direction can be described by the rotation of the frame of reference, a special Lorentz transformation along the direction and a rotation back. Since the coordinate axes can be freely chosen a priori, the reduction to the one-dimensional case is sufficient for many physical problems. For a boost in the direction of a coordinate axis , the transformation matrix is ​​simplified to: ${\displaystyle {\vec {\beta }}}$ ${\displaystyle x_{1}}$ ${\displaystyle {\vec {e}}_{1}}$

${\displaystyle \Lambda ({\vec {\beta }}=\beta {\vec {e}}_{1})={\begin{pmatrix}\gamma &-\beta \gamma &&\\-\beta \gamma &\gamma &&\\&&1&\\&&&1\end{pmatrix}}}$

From this special case it becomes clear that a Lorentz boost only changes the time-like (zeroth) and the spatial components of a four-vector along the direction of velocity, while the components orthogonal to it remain unchanged.

Analogies to rotation

With the introduction of rapidity , the boost along an axis can be as ${\displaystyle \theta =\operatorname {artanh} \beta }$

${\displaystyle \Lambda (\theta ,{\vec {e}}_{1})={\begin{pmatrix}\cosh \theta &\sinh \theta &&\\\sinh \theta &\cosh \theta &&\\&&1&\\&&&1\end{pmatrix}}}$

to be written. In this form, the Lorentz matrix is ​​analogous to a rotation matrix in Euclidean space, which describes the rotation by an angle around the -axis: ${\displaystyle R(\alpha )}$ ${\displaystyle \alpha }$ ${\displaystyle {\vec {e}}_{1}}$

${\displaystyle R(\alpha ,{\vec {e}}_{1})={\begin{pmatrix}1&&&\\&1&&\\&&\cos \alpha &-\sin \alpha \\&&\sin \alpha &\cos \alpha \end{pmatrix}}}$.

In this notation, a boost can be understood as a kind of rotation in a non-Euclidean geometry, in which the angle is replaced by the rapidity and the angle functions by the hyperbolic functions . The difference in one of the signs is created by the Lorentz metric with the signature (1, -1, -1, -1).

Likewise analogous to a rotation in the description by rapidities is that the successive execution of special Lorentz transformations with rapidities and can be written as a single Lorentz transformation with the sum of the rapidities, so it applies ${\displaystyle \theta _{1}}$ ${\displaystyle \theta _{2}}$

${\displaystyle \Lambda (\theta _{1},{\vec {e}}_{1})\Lambda (\theta _{2},{\vec {e}}_{1})=\Lambda (\theta _{1}+\theta _{2},{\vec {e}}_{1})}$,

which is a consequence of the relativistic addition theorem for speeds . As a result, the set of special Lorentz transformations along a coordinate axis forms a group .

In contrast to the rotations, however, the totality of the special Lorentz transformations does not form a subgroup of the Lorentz group. A composition of two boosts along two different axes cannot be written as a boost along a single axis. This can be illustrated by a simple example:

${\displaystyle \Lambda (\theta _{1},{\vec {e}}_{1})\Lambda (\theta _{2},{\vec {e}}_{2})={\begin{pmatrix}\cosh \theta _{1}\cosh \theta _{2}&-\sinh \theta _{1}&-\cosh \theta _{1}\sinh \theta _{2}&\\-\sinh \theta _{1}\cosh \theta _{2}&\cosh \theta _{1}&\sinh \theta _{1}\sinh \theta _{2}&\\-\sinh \theta _{2}&&\cosh \theta _{2}&\\&&&1\end{pmatrix}}}$

This matrix is ​​no longer symmetrical and therefore cannot be represented as a single special Lorentz matrix. The smallest group that contains all special Lorentz transformations is the actual orthochronous Lorentz group. In addition to the Lorentz boosts, it contains the rotations.

Lorentz transformation of the electric and magnetic field

The three-dimensional vectors of the electric field and the magnetic flux density cannot be converted into four-vectors, but are components of the antisymmetric electromagnetic field strength tensor ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$

${\displaystyle F^{\mu \nu }={\begin{pmatrix}0&-E_{1}/c&-E_{2}/c&-E_{3}/c\\E_{1}/c&0&-B_{3}&B_{2}\\E_{2}/c&B_{3}&0&-B_{1}\\E_{3}/c&-B_{2}&B_{1}&0\end{pmatrix}}}$

and therefore transform nontrivially as follows:

${\displaystyle {\begin{aligned}&{\vec {E}}'=\gamma \left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)+(1-\gamma ){\frac {{\vec {E}}\cdot {\vec {v}}}{v^{2}}}{\vec {v}}\\&{\vec {B}}'=\gamma \left({\vec {B}}-{\frac {1}{c^{2}}}{\vec {v}}\times {\vec {E}}\right)+(1-\gamma ){\frac {{\vec {B}}\cdot {\vec {v}}}{v^{2}}}{\vec {v}}\end{aligned}}}$

The inverse transformation is found by applying the Lorentz matrix in reverse (matrix inversion), which results in two sign changes in the structure of the previous equation. This change in sign is accompanied by the fact that . So the back and forth transformations have the same form: ${\displaystyle {\vec {v}}'=-{\vec {v}}}$

${\displaystyle {\begin{aligned}&{\vec {E}}=\gamma \left({\vec {E}}'-{\vec {v}}\times {\vec {B}}'\right)+(1-\gamma ){\frac {{\vec {E}}'\cdot {\vec {v}}}{v^{2}}}{\vec {v}}\\&{\vec {B}}=\gamma \left({\vec {B}}'+{\frac {1}{c^{2}}}{\vec {v}}\times {\vec {E}}'\right)+(1-\gamma ){\frac {{\vec {B}}'\cdot {\vec {v}}}{v^{2}}}{\vec {v}}\end{aligned}}}$

Even at low speeds in the Galileo approximation, effects occur that can only be explained by the Lorentz transformation, because then the following applies:

${\displaystyle {\begin{aligned}&{\vec {E}}'\approx {\vec {E}}+{\vec {v}}\times {\vec {B}}\\&{\vec {B}}'\approx {\vec {B}}-{\frac {1}{c^{2}}}{\vec {v}}\times {\vec {E}}\end{aligned}}}$

This nontrivial transformation behavior can be explained by the fact that observers make completely different observations in different systems: In a system with a static electrical charge, no current flows, so that the observer only perceives an electrical field. A current flows in a reference system that moves relative to it, so that an observer perceives a magnetic field in addition to an electric field.

The transformation property of the electric field leads to the appearance of the Lorentz force : In a system with a static charge, the force acts on a test charge in the electric field ${\displaystyle q}$

${\displaystyle {\vec {F}}=q{\vec {E}}}$

The force on a moving charge can alternatively be written as a force on a static charge in a moving frame of reference. Then applies

${\displaystyle {\vec {F}}'=q{\vec {E}}+q{\vec {v}}\times {\vec {B}}}$

with the Lorentz force . ${\displaystyle {\vec {F}}_{\text{L}}=q{\vec {v}}\times {\vec {B}}}$