# Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.

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 More on Killing vector fields: physics.stackexchange.com/q/39124/2451 – Qmechanic♦ Oct 15 '12 at 8:15

In terms of classical general relativity: Einstein's equations $$G_{ab} = 8\pi T_{ab}$$ can be formulated, in local coordinates, as a system of second order partial differential equations for the metric unknown $g_{ab}$. The matter field equations further generate some family of partial differential equations.
• Noether's theorem tells us that for Einstein's equation (which admits a Lagrangian formulation), associated to each Killing vector field $X^a$ is a conservation law. One can simply see this by considering the current $$J^{(X)}_a = T_{ab} X^b$$ Its divergence is $$\nabla^a J^{(X)}_a = \nabla^a T_{ab} X^b + T_{ab} \nabla^a X^b$$ The first term vanishes since the energy momentum tensor is divergence free. Using that the energy-momentum tensor is symmetric, we write $$\nabla^a J^{(X)}_a = \frac12 T_{ab} \left( \nabla^a X^b + \nabla^b X^a\right)$$ As a consequence of Killing's equations, if $X^a$ is a Killing field, the term inside the parenthesis evaluates to 0. So $J^{(X)}$ is divergence free. Applying Stokes' theorem we then see a conservation law.