Wave Metric

using $e x p (ϕ (x))$

g_{a b} = e_{a b} e x p (ϕ)

g^{a b} = (e^{- 1})^{a b} e x p (- ϕ)

g_{a b, c} = e_{a b} e^{ϕ} ϕ_{, c} = g_{a b} ϕ_{, c}

Γ_{a b c} = \frac{1}{2} (g_{a b} ϕ_{, c} + g_{a c} ϕ_{, b} - g_{b c} ϕ_{, a})

{Γ^{a}}_{b c} = \frac{1}{2} (δ_{b}^{a} ϕ_{, c} + δ_{c}^{a} ϕ_{, b} - g_{b c} g^{a d} ϕ_{, d})

{Γ^{a}}_{b c, d} = \frac{1}{2} (δ_{b}^{a} ϕ_{, c d} + δ_{c}^{a} ϕ_{, b d} - g_{b c} ϕ_{, d} g^{a e} ϕ_{, e} - g_{b c} {g^{a e}}_{, d} ϕ_{, e} - g_{b c} g^{a e} ϕ_{, e d})

= \frac{1}{2} (δ_{b}^{a} ϕ_{, c d} + δ_{c}^{a} ϕ_{, b d} - g_{b c} g^{a e} ϕ_{, e d})

{Γ^{a}}_{u c} {Γ^{u}}_{b d} = \frac{1}{4} (δ_{u}^{a} ϕ_{, c} + δ_{c}^{a} ϕ_{, u} - g_{u c} g^{a e} ϕ_{, e}) (δ_{b}^{u} ϕ_{, d} + δ_{d}^{u} ϕ_{, b} - g_{b d} g^{u f} ϕ_{, f})

= \frac{1}{4} (+ δ_{u}^{a} ϕ_{, c} δ_{b}^{u} ϕ_{, d} + δ_{u}^{a} ϕ_{, c} δ_{d}^{u} ϕ_{, b} - δ_{u}^{a} ϕ_{, c} g_{b d} g^{u f} ϕ_{, f} + δ_{c}^{a} ϕ_{, u} δ_{b}^{u} ϕ_{, d} + δ_{c}^{a} ϕ_{, u} δ_{d}^{u} ϕ_{, b} - δ_{c}^{a} ϕ_{, u} g_{b d} g^{u f} ϕ_{, f} - g_{u c} g^{a e} ϕ_{, e} δ_{b}^{u} ϕ_{, d} - g_{u c} g^{a e} ϕ_{, e} δ_{d}^{u} ϕ_{, b} + g_{u c} g^{a e} ϕ_{, e} g_{b d} g^{u f} ϕ_{, f})

= \frac{1}{4} (+ δ_{b}^{a} ϕ_{, c} ϕ_{, d} + δ_{d}^{a} ϕ_{, c} ϕ_{, b} - ϕ_{, c} g_{b d} g^{a e} ϕ_{, e} + δ_{c}^{a} ϕ_{, b} ϕ_{, d} + δ_{c}^{a} ϕ_{, d} ϕ_{, b} - δ_{c}^{a} g_{b d} g^{e f} ϕ_{, e} ϕ_{, f} - g_{c d} g^{a e} ϕ_{, e} ϕ_{, b} + g_{b d} g^{a e} ϕ_{, e} ϕ_{, c} - g_{b c} g^{a e} ϕ_{, e} ϕ_{, d})

= \frac{1}{4} (δ_{b}^{a} ϕ_{, c} ϕ_{, d} + 2 δ_{c}^{a} ϕ_{, b} ϕ_{, d} - δ_{c}^{a} g_{b d} g^{e f} ϕ_{, e} ϕ_{, f} + δ_{d}^{a} ϕ_{, c} ϕ_{, b} - g_{b c} g^{a e} ϕ_{, e} ϕ_{, d} - g_{c d} g^{a e} ϕ_{, e} ϕ_{, b})

R_{a b} = \frac{1}{4} (+ δ_{a}^{c} ϕ_{, c} ϕ_{, b} + 2 δ_{c}^{c} ϕ_{, a} ϕ_{, b} - δ_{c}^{c} g_{a b} g^{e f} ϕ_{, e} ϕ_{, f} + δ_{b}^{c} ϕ_{, c} ϕ_{, a} - g_{a c} g^{c e} ϕ_{, e} ϕ_{, b} - g_{c b} g^{c e} ϕ_{, e} ϕ_{, a})

= 2 ϕ_{, a} ϕ_{, b} - g_{a b} g^{c d} ϕ_{, c} ϕ_{, d}

R = 2 g^{a b} ϕ_{, a} ϕ_{, b} - 4 g^{a b} ϕ_{, a} ϕ_{, b}

= - 2 g^{a b} ϕ_{, a} ϕ_{, b}

- \frac{R}{2} = g^{a b} ϕ_{, a} ϕ_{, b}

R_{a b} = 2 ϕ_{, a} ϕ_{, b} + \frac{1}{2} g_{a b} R

for

R = 0

this gives

R_{a b} = 2 ϕ_{, a} ϕ_{, b}