Lorentz transformation

Author: Leandro Alegsa Creation date: November 13, 2022

The Lorentz transformations is a set of equations that describe a linear transformation between a stationary reference frame and a reference frame in constant velocity. The equations are given by:

$x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ , $y'=y$ , $z'=z$ , $t'={\frac {t-{\frac {vx}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

where $x'$ represents the new x co-ordinate, $v$ represents the velocity of the other reference frame, $t$ representing time, and $c$ the speed of light.

On a Cartesian coordinate system, with the vertical axis being time (t), the horizontal axis being position in space along one axis (x), the gradients represent velocity (shallower gradient resulting in a greater velocity). If the speed of light is set as a 45° or 1:1 gradient, Lorentz transformations can rotate and squeeze other gradients while keeping certain gradients, like a 1:1 gradient constant. Points undergoing a Lorentz transformations on such a plane will be transformed along lines corresponding to $t^{2}-x^{2}=n^{2}$ where n is some number

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.