The standard Lorentzian inner product on is given by
(1)
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i.e., for vectors and ,
(2)
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endowed with the metric tensor induced by the above Lorentzian inner product is known as Minkowski space and is denoted .
The Lorentzian inner product on is nothing more than a specific case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature : In this more general environment, the inner product of two vectors and has the form
(3)
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The Lorentzian inner product of two such vectors is sometimes denoted to avoid the possible confusion of the angled brackets with the standard Euclidean inner product (Ratcliffe 2006). Analogous presentations can be made if the equivalent metric signature (i.e., for Minkowski space) is used.
The four-dimensional Lorentzian inner product is used as a tool in special relativity, namely as a measurement which is independent of reference frame and which replaces the typical Euclidean notion of distance. For a four-vector in Minkowski space, the variables , , and can be thought of as space variables with as the time variable. In various literature, the time variable is sometimes labeled ; moreover, when used in general relativity, either of or may be used where denotes the speed of light and where denotes the imaginary unit (Misner et al. 1973). For simplicity, the formula (2) uses the conventions of real time coordinates and appropriately-chosen units so that the speed of light has the value .
For a vector , the sign of determines the type of : In particular, if , then is spacelike; if , then is lightlike; and if , then is called timelike. After a change of variables, it is possible to rewrite the Lorentzian inner product as above where is in the direction of a given timelike vector with . Such a change of variables corresponds to a change in reference frame. Collectively, the set of all reference frame changes form the Lorentz group, also called the orthogonal group (or when using the metric signature).