Lorentz transformation
Linear coordinate transformations linking inertial frames in special relativity; they preserve the spacetime interval and produce time dilation, length contraction, and relativity of simultaneity.
The Lorentz transformation is the set of linear relations between space and time coordinates measured in two inertial reference frames that move at constant velocity relative to one another. These formulas replace the Galilean transform when observations involve speeds comparable to the speed of light, c. In their simplest form (boost along the x axis) they relate coordinates (x,y,z,t) in one frame to (x',y',z',t') in another moving at velocity v:
x' = (x - v t) / sqrt(1 - v^2/c^2)
y' = y
z' = z
t' = (t - v x/c^2) / sqrt(1 - v^2/c^2)
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Basic features and physical consequences
The most important invariant under Lorentz transformations is the spacetime interval: s^2 = c^2 t^2 - x^2 - y^2 - z^2. This quantity has the same numeric value in every inertial frame and replaces absolute time as an invariant. From the transform follow the familiar effects of special relativity: time dilation (moving clocks run slower), length contraction (moving rods are measured shorter along the direction of motion), and the relativity of simultaneity (events that are simultaneous in one frame need not be simultaneous in another). These consequences can be derived algebraically from the transformation equations or visualized geometrically.
Geometric interpretation
In Minkowski spacetime a Lorentz boost acts like a hyperbolic rotation mixing time and one spatial coordinate. Lines of constant velocity correspond to slopes on a t–x diagram; the transformations preserve the light cone defined by paths at speed c, often drawn at 45° when c is set to unity. Points transform along hyperbolae of constant interval, t^2 - x^2 = constant, which play the role of circles in ordinary rotations. This hyperbolic viewpoint introduces the parameter rapidity, an additive measure of boost analogous to angle in Euclidean rotations.
Formulas and related relations
Beyond the primary coordinate formulas, several useful derived relations appear repeatedly in applications: the relativistic velocity-addition formula (u' = (u - v)/(1 - uv/c^2)), the Lorentz factor gamma = 1/sqrt(1 - v^2/c^2), and transformation rules for energy, momentum, and electromagnetic fields. These rule how four-vectors and tensors change between frames, ensuring that physical laws retain the same form in all inertial frames.
History and significance
The transformation bears the name of Hendrik Lorentz, who developed related equations in the late 19th and early 20th centuries while studying electrodynamics. Henri Poincaré clarified their group properties, and Albert Einstein showed in 1905 that they follow from two simple postulates: the equivalence of physical laws in all inertial frames and the constancy of the speed of light. For historical context and derivations see Lorentz's original work and modern expositions of special relativity.
Uses, examples and distinctions
- Practical: necessary corrections for GPS satellite timing and particle accelerator design.
- Conceptual: distinguishes special relativity from Galilean relativity by enforcing a finite invariant speed and nonabsolute time.
- Mathematical: Lorentz transformations form a subgroup of the Poincaré group (which also includes translations), and they are represented by 4×4 matrices acting on four-vectors.
For a compact geometric introduction to boosts and rapidity see further reading on Minkowski geometry. The family of Lorentz transformations underlies much of modern physics and provides the minimal kinematic framework compatible with the measured constancy of the speed of light.
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 
, 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 
in the 
direction with respect to another observer Bthen the coordinates 
which observer A attributes to an event, depend on the special Lorentz transformation
with the coordinates 
of the observer B for the same event, if the two reference frames have the same origin, i.e. coincide at time . 
In this, γ 
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
While for A the time (clock) in B (with 
) 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 
) 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.
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AlegsaOnline.com Lorentz transformation Leandro Alegsa
URL: https://en.alegsaonline.com/art/59234