Brinkmann coordinates (named for Hans Brinkmann) are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor can be written as
d
s
2
=
H
(
u
,
x
,
y
)
d
u
2
+
2
d
u
d
v
+
d
x
2
+
d
y
2
where
∂
v
, the coordinate vector field dual to the covector field
d
v
, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars. Physically speaking, it serves as the wave vector defining the direction of propagation for the pp-wave.
The coordinate vector field
∂
u
can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of
H
(
u
,
x
,
y
)
at that event. The coordinate vector fields
∂
x
,
∂
y
are both spacelike vector fields. Each surface
u
=
u
0
,
v
=
v
0
can be thought of as a wavefront.
In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended range of the coordinate variables
u
,
v
,
x
,
y
. Here we should take
−
∞
<
v
,
x
,
y
<
∞
,
u
0
<
u
<
u
1
to allow for the possibility that the pp-wave develops a null curvature singularity.
