A Lorentz transformation 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.
An element 
in four-space which is invariant under a Lorentz transformation is said to be a Lorentz invariant; examples include scalars, elements
of the form 
,
and the interval 
between two events (Thorn 2012).
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 set of all Lorentz transformations is known as the inhomogeneous Lorentz group or the Poincaré group. Similarly, the set of Lorentz transformations with
is known as the homogeneous Lorentz group. Restricting the transformations by the
additional requirements
 |
(10)
|
and
 |
(11)
|
where 
denotes the tensor determinant, give the proper inhomogeneous and proper homogeneous
Lorentz groups.
Any proper homogeneous Lorentz transformation can be expressed as a product of a so-called boost and a rotation.
