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The pp-wave metrics describe various types of gravitational waves and are given by:
d
s
2
=
H
(
u
,
x
,
y
)
d
u
2
+
2
d
u
d
v
+
d
x
2
+
d
y
2
{\displaystyle ds^{2}=H(u,x,y)du^{2}+2dudv+dx^{2}+dy^{2}}
where
H
{\displaystyle H}
is a function of at most 3 variables. An important class of pp-waves are the plane gravitational waves and are obtained by choosing
H
(
u
,
x
,
y
)
=
a
(
u
)
x
2
+
b
(
u
)
y
2
+
c
(
u
)
x
y
{\displaystyle H(u,x,y)=a(u)x^{2}+b(u)y^{2}+c(u)xy}
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