Physics 523, Introduction to Relativity
Homework 7
Due Tuesday, 13
th
December 2011
Hans Bantilan
Gravitational Plane Wave
Consider an exact gravitational plane wave solution to the Einstein equations, expressed in doublenull
coordinates (
u, v, x, y
) as
g
ij
dx
i
dx
j
=

(
dudv
+
dvdu
) +
a
2
(
u
)
dx
2
+
b
2
(
u
)
dy
2
.
a.
The calculation of Christoffel coefficients and Riemann tensor components associated with
g
ij
is
standard. The Ricci tensor associated with
g
ij
can be expressed in terms of the functions
a
(
u
) and
b
(
u
)
R
ij
=
diag

a
(
u
)
a
(
u
)

b
(
u
)
b
(
u
)
,
0
,
0
,
0
.
b.
The Einstein equations in vacuum can be written as
R
ij
= 0
.
By part a, we see that this is equivalent to the statement that
a
(
u
)
a
(
u
)
+
b
(
u
)
b
(
u
)
= 0
.
c.
From the conclusions of part b, we observe that the functions
a
(
u
) and
b
(
u
) can be determined in
terms of a single arbitrary function
f
(
u
), such that
a
(
u
) =
f
(
u
)
a
(
u
)
b
(
u
) =

f
(
u
)
b
(
u
)
.
We also observe that the secondorder ordinary differential equations for
a
(
u
) and
b
(
u
) are of the most
general form (ie: given some arbitrary
f
(
u
), any secondorder ODE can be brought into this form). To
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 '09
 FRANSPRETORIUS
 Work, General Relativity, Diag, rij, gravitational plane wave

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