Lorentz transformation

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The Lorentz transformations , according to Hendrik Antoon Lorentz , are a class of coordinate transformations used to describe phenomena in different reference systems in physics . In a four-dimensional space - time they connect the time and place coordinates with which various observers indicate when and where events take place. The Lorentz transformations therefore form the basis of Albert Einstein's special theory of relativity .

The equivalent to the Lorentz transformations in three-dimensional Euclidean space are the Galileo transformations ; just as these get the distances and angles , the Lorentz transformations get the distances in non-Euclidean space-time ( Minkowski space ). Angles are not preserved in the Minkowski space, as the Minkowski space is not a standardized space .

The Lorentz transformations form a group in the mathematical sense, the Lorentz group :

• The sequential execution of Lorentz transformations can be described as a single Lorentz transformation.
• The trivial transformation from a frame of reference into the same is also a Lorentz transformation.
• For each Lorentz transformation there is an inverse transformation that transforms back into the original reference system.

Subclasses of the Lorentz transformations are the discrete transformations of spatial reflection , i.e. the inversion of all spatial coordinates, as well as time reversal , i.e. the reversal of the time arrow , and the continuous transformations of finite rotation as well as the special Lorentz transformations or Lorentz boosts. Continuous rotary movements of the coordinate systems do not belong to the Lorentz transformations. Sometimes only the special Lorentz transformations are referred to as Lorentz transformations for short.

definition

Components of the Lorentz Transformation

The Lorentz transformation includes all linear transformations of the coordinates between two observers. They are therefore transformations between two inertial systems whose coordinate origin, the reference point of the coordinate system at the point in time , coincides. A general Lorentz transformation therefore includes ${\displaystyle t=0}$

• Transformations between two observers who have different, constant speeds, called Lorentz Boost or special Lorentz transformation . They correspond to a rotation in the space-time sector of the non-Euclidean Minkowski space.
• Rotations of spatial coordinates
• Time and space reflections

Every general Lorentz transformation can be written as a series of these transformations. A Lorentz transformation, in which reflections are excluded and the orientation of time is preserved, is referred to as an actual, orthochronous Lorentz transformation.

Special Lorentz transformation for places and times

If the observer A is moving at constant speed in the direction opposite to another observer B , the coordinates that observer A ascribes to an event depend on the special Lorentz transformation ${\displaystyle v_{x}}$ ${\displaystyle x}$ ${\displaystyle \textstyle (t',x',y',z')}$

${\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {v_{x}}{c^{2}}}\,x\right)\\x'&=\gamma (x-v_{x}\,t)\\y'&=y\\z'&=z\\v'_{x}&=-v_{x}\end{aligned}}}$

with the coordinates of observer B for the same event, if the two reference systems have the same origin, i.e. coincide with one another at the time . Therein is the Lorentz factor . ${\displaystyle (t,x,y,z)}$ ${\displaystyle \textstyle t=t'=0}$ ${\displaystyle \textstyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

Inverse of the special Lorentz transformation

Since B moves at constant speed relative to A , if A does this at speed relative to B , one can exchange their roles according to the principle of relativity . In the transformation formulas, only the sign of the speed changes. In particular also applies ${\displaystyle -v}$ ${\displaystyle +v}$

${\displaystyle {\begin{aligned}t&=\gamma \left(t'+{\frac {v}{c^{2}}}\,x'\right)\\\end{aligned}}}$

While for A the time (clock) in B (with ) apparently runs more slowly than that in A , this also applies the other way around, i.e. H. for B the clock of A runs (with ) slower. ${\displaystyle x=0}$ ${\displaystyle x'=0}$

historical development

→ Main article : History of the Lorentz Transformation

The work of Woldemar Voigt (1887), Hendrik Antoon Lorentz (1895, 1899, 1904), Joseph Larmor (1897, 1900) and Henri Poincaré (1905) showed that the solutions of the equations of electrodynamics are mapped onto one another by Lorentz transformations or, in other words, that the Lorentz transformations are symmetries of Maxwell's equations .

At that time an attempt was made to explain the electromagnetic phenomena through a hypothetical ether , a transmission medium for electromagnetic waves. However, it turned out that there was no evidence of him. In 1887 Voigt presented transformation formulas which leave the wave equation invariant. However, the Voigt transformation is not reciprocal, so it does not form a group . Voigt assumed that the propagation speed of the waves in the rest system of the aether and in a reference system that moves relative to this with constant speed are the same, without giving an explanation. In his ether theory , Lorentz was able to explain this by the fact that the length scales shorten when moving in the direction of movement and that moving clocks show a slower running time, which he called local time. The transformations of lengths and times given by Lorentz formed a group and were therefore mathematically consistent. Even if in Lorentz's theory of aether a uniform movement towards the aether was not detectable, Lorentz stuck to the idea of ​​an aether.

Einstein's special theory of relativity replaced Newton's mechanics and the ether hypothesis. He derived his theory from the principle of relativity , that in a vacuum, neglecting gravitational effects, rest cannot be distinguished from uniform motion. In particular, light in a vacuum has the same speed for every observer . The time and location coordinates with which two uniformly moving observers designate events are then related to one another by a Lorentz transformation, instead of a Galileo transformation as in Newton's mechanics . ${\displaystyle c}$

properties

Speed ​​addition

→ Main article : Relativistic addition theorem for speeds

Two Lorentz boosts executed one after the other in the same direction with speed and again result in a Lorentz boost with the total speed ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$

${\displaystyle {\frac {v}{c}}={\frac {{\frac {v_{1}}{c}}+{\frac {v_{2}}{c}}}{1+{\frac {v_{1}}{c}}\cdot {\frac {v_{2}}{c}}}}.}$

The equation shows that the speed of light does not change with Lorentz transformations. If it is about the speed of light, that is , it is also the speed of light. ${\displaystyle v_{1}}$ ${\displaystyle {\tfrac {v_{1}}{c}}=1}$ ${\displaystyle v=c{\tfrac {1+v_{2}/c}{1+v_{2}/c}}=c}$

Lorentz boosts performed one after the other in different directions generally do not result in Lorentz boosts, but rather a general Lorentz transformation: The set of Lorentz boosts is not a subgroup of the Lorentz transformations.

Lorentz invariant

A quantity that does not change with Lorentz transformations is called the Lorentz invariant or Lorentz scalar . In a physical system or process, a Lorentz invariant describes a property that is observed with the same value from all inertial systems, e.g. B. the speed of light , the mass , the number of particles, the electrical charge etc. ${\displaystyle c}$ ${\displaystyle m}$

With a Lorentz boost towards, it can be shown that ${\displaystyle x}$

${\displaystyle c^{2}t'^{2}-x'^{2}=c^{2}t^{2}-x^{2}}$

must apply. The expression is therefore an invariant of the Lorentz transformation, i.e. H. constant in all coordinate systems associated with Lorentz transformations. ${\displaystyle \textstyle c^{2}t^{2}-x^{2}}$

In three spatial dimensions, the norm is the only way to form a Lorentz invariant. For example, the norm of the energy-momentum vector is the mass multiplied by , and the norm of the angular momentum vector is the Lorentz-invariant amount of the intrinsic angular momentum. The distance between two events, i.e. the norm of the difference between the four-vectors of the two world points, is also Lorentz invariant. With two four-vectors, their scalar product is also Lorentz invariant. A second order tensor has a trace of Lorentz, etc. ${\displaystyle \textstyle c^{2}t^{2}-(x^{2}+y^{2}+z^{2})}$ ${\displaystyle c}$ ${\displaystyle mc}$

Lorentz contraction and invariance of the transverse coordinates

→ Main article : Special Lorentz transformation  and Lorentz contraction

For a Lorentz boost with any directional speed , the coordinate vector of the event can be broken down into two components . The indices and denote the parallel or perpendicular direction to the speed . The transformed coordinates are then through ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {r}}=(x,y,z)}$ ${\displaystyle \textstyle {\vec {r}}={\vec {r_{\parallel }}}+{\vec {r_{\bot }}}}$ ${\displaystyle \parallel }$ ${\displaystyle \perp }$ ${\displaystyle {\vec {v}}}$

${\displaystyle t'=\gamma \left(t-{\frac {{\vec {v}}\cdot {\vec {r}}}{c^{2}}}\right),\qquad {\vec {r}}_{\parallel }'=\gamma \left({\vec {r}}_{\parallel }-{\vec {v}}t\right),\qquad {\vec {r}}_{\bot }'={\vec {r}}_{\bot }}$

given. A distance measured by the observers in the deleted system is only shortened in the direction of movement . This effect is called the Lorentz contraction. The relativity of simultaneity has no effect on scales perpendicular to the direction of movement . In summary, these equations in matrix notation with four-vectors (and the identity matrix ) are: ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {r_{\parallel }}}}$ ${\displaystyle {\vec {r_{\bot }}}}$ ${\displaystyle I_{3}}$

${\displaystyle {\begin{pmatrix}ct'\\{\vec {r}}'\end{pmatrix}}={\begin{pmatrix}\gamma &-\gamma {\vec {v}}^{T}/c\\-\gamma {\vec {v}}/c&\mathrm {I} _{3}+(\gamma -1){\vec {v}}{\vec {v}}^{T}/v^{2}\\\end{pmatrix}}{\begin{pmatrix}ct\\{\vec {r}}\end{pmatrix}}}$.

In the same way, electromagnetic fields can be broken down according to and into components. The (scalar) field coordinates are obtained ${\displaystyle {\vec {E}}'={\vec {E}}'_{\parallel }+{\vec {E}}'_{\perp }}$ ${\displaystyle {\vec {B}}'={\vec {B}}'_{\parallel }+{\vec {B}}'_{\perp }}$

${\displaystyle {\begin{aligned}E'_{\parallel }&=E_{\parallel }\\B'_{\parallel }&=B_{\parallel }\\E'_{\perp }&=\gamma \left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)_{\perp }\\B'_{\perp }&=\gamma \left({\vec {B}}-{\frac {{\vec {v}}\times {\vec {E}}}{c^{2}}}\right)_{\perp }.\end{aligned}}}$

In a non-relativistic approximation, i. H. for speeds , applies . In this case there is no need to differentiate between places and times in different reference systems and the following applies to the field sizes: ${\displaystyle v\ll c}$ ${\displaystyle \gamma \approx 1}$

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

Derivation

To keep the formulas simple, the distance that light travels in one second is chosen as the unit of length . Then time and length have the same unit of measurement and the dimensionless speed of light is . The speed is measured in units of the speed of light. ${\displaystyle c=1}$ ${\displaystyle v}$

The first derivation was based on the invariance of the wave equation in the context of elastic light theory. It was later shown that the Lorentz transformation formulas, which leave the expression and thus the shape of spherical light waves invariant, can be rigorously derived from the electromagnetic wave equation (and thus from the Maxwell equations ), provided that the requirement for linearity and reciprocity is taken into account. In the context of electrodynamics, the Lorentz transformation can also be derived taking into account the potential of a moving charge ( Liénard-Wiechert potential ). In addition, there is a larger group of spherical wave transformations which leave the expression invariant. However, only the Lorentz transformations with form all natural laws including mechanics symmetrically and go over to the Galileo transformation. ${\displaystyle \textstyle \delta x^{2}+\delta y^{2}+\delta z^{2}-c^{2}\delta t^{2}}$ ${\displaystyle \textstyle \lambda \left(\delta x^{2}+\delta y^{2}+\delta z^{2}-c^{2}\delta t^{2}\right)}$ ${\displaystyle \lambda =1}$ ${\displaystyle c\to \infty }$

Derivations in modern textbooks are mainly based on the interpretation of the transformations in the sense of the special theory of relativity, according to which they concern space and time themselves, and are independent of assumptions about electrodynamics. Einstein (1905) used two postulates: the principle of relativity and the principle of the constancy of the speed of light. More general derivations, which go back to Wladimir Ignatowski (1910), are based on considerations of group theory.

Derivation from linearity and the principle of relativity

The following considerations clarify how coordinates are related which inertial observers (observers who are firmly connected to an inertial system) use to name the time and place of events. The observers are supposed to be Anna and Bert, for example. Anna's coordinate system is given by and Berts by the deleted variables . They are right-angled coordinates. ${\displaystyle x,y,z,t}$ ${\displaystyle \textstyle x',y',z',t'}$

Linearity

For all uniformly moving observers, free particles run through straight world lines. Therefore, the transformation must map straight lines to straight lines. Mathematically, this means that the transformation is linear.

If both observers agree in the choice of the time zero point and the spatial origin, then the transformation sought is linear and homogeneous.

Bert moves with speed relative to Anna . The coordinate systems are oriented so that and lie on a straight line in one direction. Then you can limit yourself to the coordinates . ${\displaystyle v}$ ${\displaystyle x,x'}$ ${\displaystyle v}$ ${\displaystyle x,t}$

The Lorentz transformation we are looking for is then

${\displaystyle t'=at+bx,\quad x'=et+fx.}$

The unknowns are now to be determined. ${\displaystyle a,b,e,f}$

Cone of light

A light pulse that Anna is currently sending off on the spot is described by. Since the speed of light is absolute, it must apply to Bert . The equations with the plus sign require and the equations with the minus sign . It follows and or ${\displaystyle t=0}$ ${\displaystyle x=0}$ ${\displaystyle x=\pm t}$ ${\displaystyle x'=\pm t'}$ ${\displaystyle e+f=a+b}$ ${\displaystyle e-f=-a+b}$ ${\displaystyle e=b}$ ${\displaystyle f=a}$

${\displaystyle t'=at+bx,\quad x'=bt+ax.}$

This applies to all Lorentz transformations, regardless of the relative speed of the observer.

Relative speed

Anna describes Bert's movement through , Bert through his own . The Lorentz transformation from Anna's to Bert's coordinate system has to convert these two expressions into one another. From then follows , so ${\displaystyle x=vt}$ ${\displaystyle \textstyle x'=0}$ ${\displaystyle \textstyle x'=bt+avt=(b+av)t=0}$ ${\displaystyle b=-av}$

${\displaystyle t'=a(t-vx),\quad x'=a(x-vt).}$

The preliminary factor still remains to be determined. It cannot depend on the coordinates, otherwise the Lorentz transformation would be non-linear. So there remains a dependency on the relative speed. One writes . Since the Lorentz transformation should not depend on the direction of , the following applies . ${\displaystyle a}$ ${\displaystyle a=a(v)}$ ${\displaystyle v}$ ${\displaystyle a=a(|v|)}$

Factor

To determine the prefactor, another inertial observer, Clara, is introduced with the coordinates and the relative speed in relation to Bert. The Lorentz transformation from Bert's to Clara's coordinates must have the same form as the above because of the principle of relativity, i.e. ${\displaystyle \textstyle t'',x''}$ ${\displaystyle v'}$

${\displaystyle t''=a'(t'-v'x'),\quad x''=a'(x'-v't'),}$

it was abbreviated. ${\displaystyle a'=a(v')}$

You now combine the two transformations, so convert the coordinates of Anna into those of Clara. It is enough to calculate one of the two coordinates:

${\displaystyle t''=a'(t'-v'x')=a'(a(t-vx)-v'a(x-vt))=a'a(1+vv')\left(t-{\frac {v+v'}{1+vv'}}x\right).}$

If Clara is sitting next to Anna, is and the double-crossed coordinates are the same as the unlined ones. The factor disappears and the prefactor must be equal to 1. Because and then must ${\displaystyle v'=-v}$ ${\displaystyle \textstyle (v+v')/(1+vv')}$ ${\displaystyle \textstyle a'a(1+v'v)=a'a(1-v^{2})}$ ${\displaystyle \textstyle a(-v)a(v)\cdot (1-v^{2})=1}$ ${\displaystyle a(-v)=a(v)}$

${\displaystyle a(v)={\frac {1}{\sqrt {1-v^{2}}}}}$

be valid. With the abbreviation is ${\displaystyle \gamma =a(v)}$

${\displaystyle t'=\gamma (t-vx),\quad x'=\gamma (x-vt).}$

The Lorentz transformations are therefore

${\displaystyle t'=\gamma \left(t-\left({\frac {v}{c^{2}}}\right)x\right),\qquad x'=\gamma (x-vt),\qquad \gamma ={\frac {1}{\sqrt {1-({\frac {v}{c}})^{2}}}}.}$

Derivation from the time dilation

With an argument from Macdonald, the transformation formulas can be obtained from the time dilation . On a light front moving in the positive x-direction, the difference coordinate has the same value everywhere, as well . One looks at a front that goes through the event E and at some point (before or after) meets the moving coordinate origin O ', which must be slower than light. Because of the constant values, the difference coordinates at E are in the same relationship to one another as at point O '. It applies to this , as well as according to the dilation formula where is. The following therefore applies to the difference coordinates ${\displaystyle ct-x}$ ${\displaystyle \textstyle ct'-x'}$ ${\displaystyle \textstyle x'=0,\ x=vt}$ ${\displaystyle \textstyle t=\gamma t'}$ ${\displaystyle \textstyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}$

${\displaystyle ct-x=\left(1-{\frac {v}{c}}\right)\gamma (ct'-x')}$

Similarly, on a light front moving in the negative x-direction, the total coordinate has the same value everywhere, as well . Such a front also passes through E (with the same coordinates as above) and through O '(at a different point in time than above). In the equation analogous to the previous one, sums are now formed instead of differences, therefore it is ${\displaystyle ct+x}$ ${\displaystyle \textstyle ct'+x'}$

${\displaystyle ct+x=\left(1+{\frac {v}{c}}\right)\gamma (ct'+x')}$

Adding and subtracting the two equations gives as a function of . ${\displaystyle ct,x}$ ${\displaystyle ct',x'}$

Empirical derivation

→ Main article : Test theories of special relativity

Howard P. Robertson and others showed that the Lorentz transformation can also be derived empirically. For this it is necessary to provide general transformation formulas between different inertial systems with parameters that can be determined experimentally. It is believed that there is a single "preferred" inertial system in which the speed of light is constant, isotropic and independent of the speed of the source. Likewise, should Einstein Synchronization and synchronization be equivalent by slow Watches transport in this system. Let there be another system that is collinear with this system , whose spatial origin at the time coincides with the origin of the first system and in which the clocks and scales have the same internal constitution as in the first system. This second system moves relative to the first system at a constant speed along the common axis. The following sizes remain indefinite: ${\displaystyle X,Y,Z,T}$ ${\displaystyle x,y,z,t}$ ${\displaystyle T=t=0}$ ${\displaystyle X}$

• ${\displaystyle a(v)}$ Differences in timing,
• ${\displaystyle b(v)}$ Differences in the measurement of longitudinal lengths,
• ${\displaystyle d(v)}$ Differences in the measurement of transverse lengths,
• ${\displaystyle \varepsilon (v)}$ follows from the convention for clock synchronization.

This results in the following transformation formulas:

${\displaystyle {\begin{aligned}t&=a(v)T+\varepsilon (v)x\\x&=b(v)(X-vT)\\y&=d(v)Y\\z&=d(v)Z\end{aligned}}}$

${\displaystyle \varepsilon (v)}$is not measured directly, but follows from the clock synchronization convention. Here the Einstein synchronization is the simplest possibility, which results. The relationship between and is determined from the Michelson-Morley experiment , the relationship between and from the Kennedy-Thorndike experiment, and finally from the Ives-Stilwell experiment alone . The experiments resulted in and , which converts the above transformation into the Lorentz transformation. On the other hand, the Galileo transformation was excluded. ${\displaystyle \textstyle \varepsilon (v)=-v/c^{2}}$ ${\displaystyle b(v)}$ ${\displaystyle d(v)}$ ${\displaystyle a(v)}$ ${\displaystyle b(v)}$ ${\displaystyle a(v)}$ ${\displaystyle \textstyle 1/a(v)=b(v)=\gamma }$ ${\displaystyle d(v)=1}$ ${\displaystyle a(v)=b(v)=d(v)=1}$

Poincaré and Lorentz Group

→ Main articles : Lorentz Group  and Poincaré Group

The Poincaré group is the set of linear inhomogeneous transformations

${\displaystyle T_{\Lambda ,a}\colon x\mapsto T_{\Lambda ,a}x=x^{\prime },\quad x^{\prime \,m}=\Lambda ^{m}{}_{n}\,x^{n}+a^{m},\quad m,n\in \{0,1,2,3\},}$

which leave the distance between two four-vectors invariant. The subgroup of the homogeneous transformations forms the Lorentz group, that is the group of the linear transformations of on , which is the square of length ${\displaystyle \textstyle T_{\Lambda ,0}}$ ${\displaystyle \mathrm {O} (1,3)}$ ${\displaystyle \textstyle \mathbb {R} ^{4}}$ ${\displaystyle \textstyle \mathbb {R} ^{4}}$

${\displaystyle w^{2}=t^{2}-x^{2}-y^{2}-z^{2}}$

each vector from blank invariant. Let us write the square of length as a matrix product ${\displaystyle w=(t,x,y,z)}$ ${\displaystyle \textstyle \mathbb {R} ^{4}}$

${\displaystyle w^{\mathrm {T} }\,\eta \,w}$

of the column vector with the matrix ${\displaystyle w}$

${\displaystyle \eta ={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\\\end{pmatrix}}}$

and the transposed column of the line , it must transformed Lorentz for each vector are ${\displaystyle \textstyle w^{\mathrm {T} }}$ ${\displaystyle \Lambda w}$

${\displaystyle w^{\mathrm {T} }\,\Lambda ^{\mathrm {T} }\eta \,\Lambda \,w=w^{\mathrm {T} }\,\eta \,w.}$

This is the case if and only if the Lorentz transformation satisfies the equation

${\displaystyle \Lambda ^{\mathrm {T} }\eta \,\Lambda =\eta }$

Fulfills.

All solutions to this equation that do not reverse the direction of time and spatial orientation are of the form

${\displaystyle \Lambda =D_{1}\,\Lambda _{v}\,D_{2}.}$

There are and twists ${\displaystyle D_{1}}$ ${\displaystyle D_{2}}$

${\displaystyle D={\begin{pmatrix}1&\\&D_{3\times 3}\\\end{pmatrix}},\quad D_{3\times 3}^{\mathrm {T} }\,D_{3\times 3}=\mathbf {1} ,\quad \det D_{3\times 3}=1.}$

These rotations form the subgroup SO (3) of the Lorentz group. The matrix

${\displaystyle \Lambda _{v}={\begin{pmatrix}\gamma &-\gamma \,v&0&0\\-\gamma \,v&\gamma &0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}}$

causes the Lorentz transformation given above with one speed . The transformations ${\displaystyle |v|<1}$

${\displaystyle \Lambda =D\,\Lambda _{v}\,D^{-1}.}$

called Lorentz Boost . They transform to the coordinates of the moving observer, who moves with speed in the direction that results from the rotation from the direction. ${\displaystyle v}$ ${\displaystyle D}$ ${\displaystyle x}$

Lorentz transformations that do not change the sign of the time coordinate, the direction of time,

• ${\displaystyle \Lambda _{\ 0}^{0}\geq 1,}$

form the subgroup of the orthochronous Lorentz transformations. The Lorentz transformations with

• ${\displaystyle \det \Lambda =1}$

form the subgroup of the actual Lorentz transformations. The following applies to the orientation-true Lorentz transformations

• ${\displaystyle \Lambda _{\ 0}^{0}\cdot \det \Lambda \geq 1.}$

The time and orientation true Lorentz transformations

• ${\displaystyle \Lambda _{\ 0}^{0}\geq 1,\quad \det \Lambda =1,}$

form the actual orthochronous Lorentz group. It is coherent: every actual orthochronous Lorentz transformation can be converted into the identical mapping by continuously changing the six parameters, three for the axis of rotation and the angle of rotation and three for the relative speed of the two reference systems.

Time and space reflection

The Lorentz transformations that are not related to the related one get by the time mirroring or the space mirroring ${\displaystyle \mathbf {1} }$

${\displaystyle {\mathcal {T}}={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}},\quad {\mathcal {P}}={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\\\end{pmatrix}}}$

or both multiplied by the Lorentz transforms associated with the . The Lorentz group has four related components. ${\displaystyle \mathbf {1} }$ ${\displaystyle \mathrm {O} (1,3)}$

Overlay Group

The following considerations show that the group of linear transformations of the two-dimensional, complex vector space , the determinant of which has the special value , the so-called special linear group , is the simply connected superposition of the actual orthochronous Lorentz transformations. The subset of the special two-dimensional unitary transformation, overlaid SU (2) the group of rotations . ${\displaystyle \textstyle \mathbb {C} ^{2}}$ ${\displaystyle 1}$ ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ ${\displaystyle \mathrm {SO} (3)}$

Every Hermitian matrix is ​​of the form: ${\displaystyle 2\times 2}$

${\displaystyle {\hat {w}}={\begin{pmatrix}t+z&x-\mathrm {i} y\\x+\mathrm {i} y&t-z\end{pmatrix}}={\hat {w}}^{\mathrm {T} \,*}={\hat {w}}^{\dagger }.}$

Since it is reversibly uniquely designated by the four real parameters and since sums and real multiples of Hermitian matrices are again Hermitian and belong to the sums and multiples of four vectors , it is an element of a four-dimensional vector space. ${\displaystyle w=(t,x,y,z)}$ ${\displaystyle w}$

The determinant

${\displaystyle \det {\hat {w}}=t^{2}-x^{2}-y^{2}-z^{2}}$

is the square of length of the four-vector . ${\displaystyle w}$

If you multiply from the left with any complex matrix and from the right with its adjoint, the result is again Hermitian and can be written as , where linearly depends on. If from the special linear group of complex matrices, the determinants of which have the special value , then the square of the length of and agrees, so it is a Lorentz transformation. To everyone from belongs so fortune ${\displaystyle {\hat {w}}}$ ${\displaystyle 2\times 2}$ ${\displaystyle \textstyle M{\hat {w}}M^{\dagger }={\hat {u}}}$ ${\displaystyle {\hat {u}}}$ ${\displaystyle u=\Lambda w}$ ${\displaystyle w}$ ${\displaystyle M}$ ${\displaystyle 2\times 2}$ ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ ${\displaystyle 1}$ ${\displaystyle w}$ ${\displaystyle u=\Lambda w}$ ${\displaystyle \Lambda }$ ${\displaystyle M}$ ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$

${\displaystyle M{\hat {w}}M^{\dagger }={\widehat {\Lambda w}}}$

a Lorentz transformation from . More precisely, to every pair of complex matrices there belongs exactly one Lorentz transformation from the part of which is continuously related to. This part of the Lorentz group is a representation of the group . ${\displaystyle \Lambda }$ ${\displaystyle \mathrm {O} (1,3)}$ ${\displaystyle \pm M}$ ${\displaystyle 2\times 2}$ ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ ${\displaystyle \Lambda (M)=\Lambda (-M)}$ ${\displaystyle \mathrm {O} (1,3)}$ ${\displaystyle \mathbf {1} }$ ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$

The group is the product variety and simply coherent. The group of the actual orthochronous Lorentz transformations, however, is not simply connected: rotations around a fixed axis with angles that increase from to form a closed circle in the rotation group. One cannot continuously change these transformations into other rotations, so that this circle shrinks to one point. ${\displaystyle \mathrm {SL} (2,\mathbb {C} )}$ ${\displaystyle \mathbb {R} ^{3}\times S^{3}}$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha =2\pi }$

literature

• Charles Kittel , Walter D. Knight, Malvin A. Ruderman: Mechanics (= Berkeley Physics Course. Vol. 1). Vieweg, Braunschweig 1973, ISBN 3-528-08351-4 , p. 232: chap. 11.
• Norbert Dragon: Geometry of the theory of relativity. (PDF file; 2.37 MB)

Web links

• Interactive Java applet
• Video: Lorentz Transformations . Jörn Loviscach 2013, made available by the Technical Information Library (TIB), doi : 10.5446 / 19922 .
• Video: Lorentz transformation in detail . Jörn Loviscach 2013, made available by the Technical Information Library (TIB), doi : 10.5446 / 19921 .

Individual evidence

1. ↑ Harald Klingbeil: Electromagnetic Field Theory: A textbook and exercise book . Springer-Verlag, 2010, ISBN 3-8348-1403-2 , p. 497 ( limited preview in Google Book search).
2. ↑ arxiv : 1609.08647v1
3. ^ Christian Møller: The theory of relativity . 1952, § 18. The most general Lorentz transformation, p. 41 ( Internet Archive ).
4. ^ Klaus W. Kark: Antennas and radiation fields. Electromagnetic waves on lines, in free space and their radiation. 3rd, expanded edition. Vieweg + Teubner, Wiesbaden 2010, ISBN 978-3-8348-0553-9 , chap. 3.7.1, p. 46
5. ^ RP Feynman : Lectures On Physics, Vol. II, 26-3, Relativistic transformation of the fields
6. ↑ Max von Laue : Das Relativitätsprinzip , 2nd edition, Vieweg, Braunschweig 1913, pp. 38–41.
7. ^ Karl Stiegler: On the Deduction of the Lorentz-Einstein Transformation from Maxwell's Electromagnetic Field Equations . In: Proceedings of the Physical Society . 71, No. 3, 1958, pp. 512-513. doi : 10.1088 / 0370-1328 / 71/3/429 .
8. ↑ Feynman, RP: 21-6 The potentials for a charge moving with constant velocity; the Lorentz formula . In: The Feynman Lectures on Physics , Volume 2. Basic Books, New York 2013, ISBN 978-0-465-02416-2 .
9. ↑ Pal Palash B .: Nothing but relativity . In: European Journal of Physics . No. 3, 2003, p. 24. arxiv : physics / 0302045 . doi : 10.1088 / 0143-0807 / 24/3/312 .
10. ↑ Baccetti, Valentina; Tate, Kyle; Visser, Matt: Inertial frames without the relativity principle . In: Journal of High Energy Physics . 2012, p. 119. arxiv : 1112.1466 . bibcode : 2012JHEP ... 05..119B . doi : 10.1007 / JHEP05 (2012) 119 . ; See references 5 through 25.
11. ^ Alan Macdonald, Derivation of the Lorentz transformation. In: American Journal of Physics. Vol. 49, Issue 5, 1981, ISSN  0002-9505 , p. 493, updated version .