Lorentz transformation

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The Lorentz transformations, after Hendrik Antoon Lorentz, are a class of coordinate transformations that in physics transform descriptions of phenomena in different reference frames into one another. In a four-dimensional spacetime, they connect the time and location coordinates used by different observers to indicate when and where events occur. The Lorentz transformations therefore form the basis of Albert Einstein's special theory of relativity.

The equivalent of the Lorentz transformations in three-dimensional Euclidean space are the Galilean transformations; just as these preserve distances and angles, the Lorentz transformations preserve distances in non-Euclidean spacetime (Minkowski space). Angles are not preserved in Minkowski space, since Minkowski space is not a normalized space.

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

The successive execution of Lorentz transformations can be described as a single Lorentz transformation.
The trivial transformation from one reference frame to the same is also a Lorentz transformation.
For every Lorentz transformation there is an inverse transformation, which transforms back to the original reference frame.

Subclasses of Lorentz transformations are the discrete transformations of space mirroring, i.e. the inversion of all spatial coordinates, as well as the time inversion, i.e. the inversion of the arrow of time, and the continuous transformations of finite rotation as well as the special Lorentz transformations or Lorentz boosts. Continuous rotations of the coordinate systems do not belong to the Lorentz transformations. Sometimes also only the special Lorentz-transformations are called Lorentz-transformations shortening.

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 time t=0 , coincides. A general Lorentz transformation therefore comprises

Transformations between two observers having a different constant velocity, called Lorentz boost or special Lorentz transformation. They correspond to a rotation in the space-time sector of the non-Euclidean Minkowskian space.
Rotations of the spatial coordinates
Time and space reflections

Every general Lorentz transformation can be written as a succession of these transformations. A Lorentz transformation in which reflections are excluded and the orientation of time is preserved is called a proper, orthochronous Lorentz transformation.

Special Lorentz transformation for places and times

If observer A is moving with constant velocity $v_{x}$ in the direction with respect to another observer Bthen the coordinates $\textstyle (t',x',y',z')$ which observer A attributes to an event, depend on the special Lorentz transformation

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

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

Inverse of the Special Lorentz Transformation

Since B moves relative to A with constant velocity , if A does so relative to B with velocity , one can swap their roles according to the principle of relativity. In the transformation formulas, only the sign of the velocity changes. In particular, the following also holds

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

While for A the time (clock) in B (with x=0 ) appears to run slower than that in A, this is also true the other way round, i.e. for B the clock of A (with x'=0 ) runs slower.

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 each other by Lorentz transformations, or in other words, that the Lorentz transformations are symmetries of Maxwell's equations.

At that time, attempts were made to explain electromagnetic phenomena by a hypothetical ether, a transmission medium for electromagnetic waves. However, it turned out that no trace of it could be detected. 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 velocity of propagation of waves in the rest system of the ether and in a reference system moving relative to it with constant velocity is the same, without giving an explanation for this. In his aether theory, Lorentz was able to explain this by the fact that length scales shorten when moving in the direction of motion and that moving clocks indicate a slower moving time, which he called local time. The transformations of lengths and times given by Lorentz formed a group and were thus mathematically consistent. Even if in Lorentz' aether-theory a uniform motion towards the aether could not be proved, Lorentz held on to the idea of an aether.

Einstein's special theory of relativity superseded 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 velocity for any observer. The time and location coordinates by which two uniformly moving observers denote events are then related by a Lorentz transformation, rather than by a Galileo transformation as in Newton's mechanics.