11.3 Gravitational Plane Waves
167
If we now use the transformation relation for the hat metric field in (11.14) we can
make all the elements with μ = 0 index equal to zero in some new primed system.
We do this beginning with the 0,1 element, and recall that f μ does not depend on x,
but only on z and ct, so that
h
01 = ¯
h 01 + f 0,1 + f 1,0 = ¯
h 01 + f 1,0 = ¯
h 01 + f 1,U .
(11.33)
This component can thus be made zero by choosing
f 1,U = − ¯
h 01 , f 1 (U ) = −
U
¯
h 01 dU
.
(11.34)
The same procedure makes the 0,2 element equal to zero in an obvious way. For the
0,0 element we obtain in similar fashion
h
00 = ¯
h 00 + 2 f 0,0 − f
β ,β = ¯
h 00 +
f 0,U − f 3,U
,
(11.35)
so the 0,0 element can be made zero by choosing
f 0,U − f 3,U = − ¯
h 00 , f 0 (U ) − f 3 (U ) = −
U
¯
h 00 dU
.
(11.36)
Thus, in the primed system the metric field has been reduced to an array with nonzero
elements in only the 1,2 block. Three equations for the four transformation functions
have been determined in the process.
Finally, to determine the 1,2 block in the primed frame we use the transformation
(11.14) again to find
h
11 = ¯
h 11 +
f 0,U + f 3,U
, h
22 = ¯
h 22 +
f 0,U + f 3,U
, h
12 = ¯
h 12 (11.37)
These two relations allow us to make the trace in the primed system zero. From
(11.37) the new trace is
h
11 + h
22 = ¯
h 11 + ¯
h 22 + 2
f 0,U + f 3,U
.
(11.38)
This will be zero if we choose
f 0,U + f 3,U = −
1
2
¯
h 11 + ¯
h 22
, f 0 + f 3 = −
1
2
U
¯
h 11 + ¯
h 22
dU
.
(11.39)