 Solution

ID: past-exam-of-the-general-relativity-ii-course-of-the-university-of-oxford/2017/1/a/solution

K = \partial_{v}

has constant components

K^{a} = δ_{v}^{a}

. Its norm is

g_{ab} K^{a} K^{b} = g_{vv} = 0,

so it is a nonzero null vector everywhere. The Lie derivative of the metric tensor is

(L_{K} g)_{ab} = K^{c} \partial_{c} g_{ab} + g_{c b} \partial_{a} K^{c} + g_{a c} \partial_{b} K^{c} = \partial_{v} g_{ab} = 0.

Thus

K = \partial_{v} is a null Killing vector field .

K_{b} = g_{b v} = δ_{b}^{u}

and the Christoffel symbols found below give

\nabla_{a} K_{b} = - Γ^{u}_{ab} = 0

. This stronger property is characteristic of this plane-fronted gravitational wave.

 New to topics? Read the docs here!