$$\require{cancel}$$
2. Show that the metric ds2 = dt2 − A dx2 − B dy2 − dz2 with $$\begin{split} A &= 1 - f + \frac{3}{8} f^{2} - \frac{25}{416} f^{3} + \frac{15211}{10729472} f^{5} \\ B &= 1 + f + \frac{3}{8} f^{2} + \frac{25}{416} f^{3} - \frac{15211}{10729472} f^{5} \\ f &= Ae^{k(t-z)} \end{split}$$is an approximate solution to the vacuum field equations, provided that k is real — which prevents this from being a physically realistic, oscillating wave. Find the next nonvanishing term in each series.