A Lorentz transformation, often shortened to a Lorentz transform, is a four-dimensional transformation
 |
(1)
|
satisfied by all four-vectors 
, where 
is a so-called Lorentz
tensor. Lorentz tensors are restricted by the
conditions
 |
(2)
|
with 
the Minkowski metric (Weinberg 1972, p. 26;
Misner et al. 1973, p. 68).
Here, the tensor indices run over 0, 1, 2, 3, with 
being the time coordinate
and 
being space coordinates, and Einstein
summation is used to sum over repeated indices.
There are a number of conventions, but a common one used by Weinberg (1972) is to
take the speed of light 
to simplify computations and allow 
to be written simply as 
for 
. The group of Lorentz transformations
in Minkowski space 
is known as the Lorentz
group.
A Lorentz invariant is a scalar quantity unchanged by every Lorentz transformation. Examples include the quadratic
form defined by the Minkowski metric, such
as 
,
and the squared interval 
between two events (Thorn 2012). The four-vector
itself generally changes under the transformation.
Note that while some authors (e.g., Weinberg 1972, p. 26) use the term "Lorentz transformation" to refer to the inhomogeneous transformation
 |
(3)
|
where 
is a constant tensor, the
preferred term for transformations of this form is Poincaré
transformation (Misner et al. 1973, p. 68). The corresponding group of Poincaré
transformations is known as the Poincaré
group.
In the theory of special relativity, the Lorentz transformation replaces the Galilean transformation as the valid transformation law between reference frames moving with respect to one another at constant velocity. The Lorentz transformation serves this important role by virtue of the fact that it leaves the so-called proper time
 |  |  |
(4)
|
 |  |  |
(5)
|
invariant. (Here, the convention 
is used.) To see this, note that
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
(Weinberg 1972, p. 27).
The transformations with arbitrary translations 
form the inhomogeneous Lorentz group or Poincaré
group. The transformations with 
form the homogeneous Lorentz
group. Restricting the transformations by
the additional requirements
 |
(10)
|
and
 |
(11)
|
where 
denotes the determinant, gives the proper orthochronous
inhomogeneous and homogeneous Lorentz groups. The
condition 
alone defines the proper subgroup, whereas 
imposes preservation of time orientation.
Any proper orthochronous homogeneous Lorentz transformation can be expressed as a product of a so-called boost and a rotation.
