Null coordinates have been used to define plane gravitational waves which are an exact solution to Einstein’s equations. We reason here first about null coordinates.

Let t, x, y and z be vanilla Minkowski coordinates—i.e. gij = ηij = diag (−1, 1, 1, 1). We introduce null coordinates u and v replacing t and z with u and v. u = t + z and v = t − z. For these coordinates
ds2 = dx2 + dy2 + dz2 − dt2 = dx2 + dy2 + (du − dv)2/4 − (du + dv)2/4 =
dx2 + dy2 − du dv.

When we sort the coordinates u, v, x, y for matrix notation we get:
gij =
0−100
−1000
0010
0001
and ds2 = gijdxidxj where i and j range over {u, v, x, y}. Weird—and that’s for flat space-time.

For the plane wave moving in the z direction gij is a function only of v. We take gij =