Editing Lorentz transformation (section)

{{Short description|Family of linear transformations}}
[[File:Hendrik Antoon Lorentz.jpg|thumb|[[Hendrik Lorentz]] in 1916.]]
{{spacetime|cTopic=Mathematics}}

In [[physics]], the '''Lorentz transformations''' are a six-parameter family of [[Linear transformation|linear]] [[coordinate transformation|transformations]] from a [[Frame of Reference|coordinate frame]] in [[spacetime]] to another frame that moves at a constant [[velocity]] relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch [[physicist]] [[Hendrik Lorentz]].

The most common form of the transformation, parametrized by the real constant <math>v,</math> representing a velocity confined to the {{mvar|x}}-direction, is expressed as<ref>{{cite book |title=The Rotation and Lorentz Groups and Their Representations for Physicists |edition=illustrated |last1= Rao|first1= K. N. Srinivasa |publisher=John Wiley & Sons |year=1988 |isbn=978-0-470-21044-4 |page=213 |url=https://books.google.com/books?id=XRJIsf5zoM0C}} [https://books.google.com/books?id=XRJIsf5zoM0C&pg=PA210 Equation 6-3.24, page 210]</ref><ref>{{harvnb|Forshaw|Smith|2009}}</ref>
<math display="block">\begin{align}
t' &= \gamma \left( t  - \frac{vx}{c^2} \right)  \\
x' &= \gamma \left( x - v t \right)\\
y' &= y \\
z' &= z
\end{align}</math>
where {{math|(''t'', ''x'', ''y'', ''z'')}} and {{math|(''t′'', ''x′'', ''y′'', ''z′'')}} are the coordinates of an event in two frames with the spatial origins coinciding at {{math|1=''t'' = ''t′'' = 0}}, where the primed frame is seen from the unprimed frame as moving with speed {{mvar|v}} along the {{mvar|x}}-axis, where {{mvar|c}} is the [[speed of light]], and 
<math display="block"> \gamma = \frac{1}{\sqrt{1 - v^2 / c^2 }}</math>
is the [[Lorentz factor]]. When speed {{mvar|v}} is much smaller than {{mvar|c}}, the Lorentz factor is negligibly different from 1, but as {{mvar|v}} approaches {{mvar|c}}, <math>\gamma</math> grows without bound. The value of {{mvar|v}} must be smaller than {{mvar|c}} for the transformation to make sense.

Expressing the speed as a fraction of the speed of light, <math display="inline"> \beta = v/c,</math> an equivalent form of the transformation is<ref>{{harvnb|Cottingham|Greenwood|2007|p=[https://books.google.com/books?id=Dm36BYq9iu0C&pg=PA21 21]}}</ref>
<math display="block">\begin{align}
ct' &= \gamma \left( c t - \beta x \right)  \\
x' &= \gamma \left( x - \beta ct \right)  \\
y' &= y \\
z' &= z.
\end{align}</math>

Frames of reference can be divided into two groups: [[Inertial frame of reference|inertial]] (relative motion with constant velocity) and [[Non-inertial reference frame|non-inertial]] (accelerating, moving in curved paths, rotational motion with constant [[angular velocity]], etc.). The term "Lorentz transformations" only refers to transformations between ''inertial'' frames, usually in the context of special relativity.

In each [[frame of reference|reference frame]], an observer can use a local coordinate system (usually [[Cartesian coordinates]] in this context) to measure lengths, and a clock to measure time intervals. An [[Event (relativity)|event]] is something that happens at a point in space at an instant of time, or more formally a point in [[spacetime]]. The transformations connect the space and time coordinates of an [[Event (relativity)|event]] as measured by an observer in each frame.<ref group=nb>One can imagine that in each inertial frame there are observers positioned throughout space, each with a synchronized clock and at rest in the particular inertial frame. These observers then report to a central office, where all reports are collected. When one speaks of a ''particular'' observer, one refers to someone having, at least in principle, a copy of this report. See, e.g., {{harvtxt|Sard|1970}}.</ref>

They supersede the [[Galilean transformation]] of [[Newtonian physics]], which assumes an [[absolute space and time]] (see [[Galilean relativity]]). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different [[velocity|velocities]] may measure different [[Length contraction|distances]], [[time dilation|elapsed times]], and even different [[Relativity of simultaneity|orderings of events]], but always such that the [[speed of light]] is the same in all inertial reference frames. The invariance of light speed is one of the [[postulates of special relativity]].

Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of [[light]] was observed to be independent of the [[frame of reference|reference frame]], and to understand the symmetries of the laws of [[electromagnetism]]. The transformations later became a cornerstone for [[special relativity]].

The Lorentz transformation is a [[linear transformation]]. It may include a rotation of space; a rotation-free Lorentz transformation is called a '''Lorentz boost'''. In [[Minkowski space]]—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the [[spacetime interval]] between any two events. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a [[hyperbolic rotation]] of Minkowski space. The more general set of transformations that also includes translations is known as the [[Poincaré group]].