where the quantities Wo" a o;r and to" determine properties of the background spacetime.4
Now the gauge transformation for hmm becomes
hmm 8 mo aP B + iP m a B
-I1 -I ~
+ p+p ( 8- roa o-A
where we have introduced (note that Bo is purely imaginary)
2 (4-27)
Bo" ={4xo o (~ ) + 5/uo a 2xToin00 oo a .
with c.c. indicating the complex conjugate. Integration of the backgrounds where xo" / 0
and to" / 0 using the Held technique has not made its way into the literature and is
beyond the scope of the present work. As a result, derivatives of xo" and to" appear in
Equations 4-27 but do no harm to our argument. ClIn..-!nig any gauge vector that satisfies
8 (mo p Eo Bo a g@ _.I~+~P C ,o ,)o a nO Ao = bo, (4-28)
will serve to impose the trace condition in the full IRG. Once again we have established
that 1 = 0 is both a necessary and sufficient condition for the existence of the full IRG.
Note that by setting xo" = cto = 0 (i.e., ignoring the C-metrics) in the background,
Ao = Bo = 0, and the result is virtually identical to Equations 4-18 and 4-19. There
is one simplification in that now p'o __ ,o [46]. The full extent of the remaining residual
gauge freedom in Equations 4-28 can be demonstrated along the same lines as used
in Section 4.3. As for the case of a type II background, it resides chiefly in the freely
specifiable (mo and (mo
4 FOT eXample, xo" / 0 leads to the accelerating C-metrics. The condition xo" = 0 implies
to" = 0 and so cto is also related to parameters in the C-metric.