The light cone 
in 
-dimensional Lorentzian space 
is the subset consisting
of all vectors
 |
(1)
|
whose Lorentzian inner product with itself, 
, is zero:
 |
(2)
|
Alternatively, 
is the collection of all lightlike vectors
in 
.
The identification of 
with Lorentzian space of signature 
leads to a natural decomposition
of such a vector 
into its 
component and its 
-subvector 
. Using this notation, the Lorentzian
inner product of 
with itself can be expressed as
 |
(3)
|
where 
is the l2-norm.
One can also define the light cone to be the collection of all vectors 
satisfying
 |
(4)
|
This particular perspective makes natural the distinction between positive lightlike and negative lightlike vectors.
The open set inside the light cone is defined by 
and consists of all timelike vectors. The open
set outside the light cone is defined by 
and consists of all spacelike vectors.
