2nd order correction to gauge transformation

[matt91a]

In the weak field approximation,

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$

If we make a coordinate transformation of the form

[itex]x^{\mu'}=x^{\mu}+\xi^{\mu}(x)[\itex]

it changes [itex]h_{\mu\nu}[\itex] to

[itex]h'_{\mu\nu}=h_{\mu\nu}+\xi_{\mu,\nu}+\xi_{\nu,\mu}+O(\xi^{2})[\itex]

I was wondering if anyone could shed some light on what form the higher order terms take. I have an inkling it's terms from a taylor series expansion but I'm not sure.

Thanks

 

[atyy]

Maybe try section 4.3 of http://relativity.livingreviews.org/Articles/lrr-2006-3/

Or probably more relevant, section 2.2 of http://arxiv.org/abs/astro-ph/0006423v1

 

[matt91a]

Thanks for your reply but I'm afraid it doesn't shed any more light on it. I'm not sure if [itex]\xi^{\mu}(x)[\itex] being a Killing vector has anything to do with it.

 

[Bill_K]

The exact transformation is x^μ = x'^μ + ξ^μ. Applied to the metric this is

g_μν = ∂x^α/∂x'^μ ∂x^β/∂x'^ν g_αβ

= (δ^α_μ + ξ^α_,μ)(δ^β_ν + ξ^β_,ν)(η_αβ + h_αβ)

= η_μν + (h_μν + ξ^α_,μ h_αν + ξ^β_,ν h_μβ + ξ^α_,μξ^β_,νh_αβ)

= η_μν + h'_μν

 

[Sam Gralla]

A clear reference is http://arxiv.org/abs/gr-qc/9609040, although this is not that accessible to the beginner. One subtlety is that if you take the transformation x = x' + xi to be exact, then xi is no longer the generator of the diffeomorphism at higher than linear order. So most people use equation (1.1) of the above reference for the coordinate transformation. With that form you can express everything in terms of Lie derivatives, equation (1.3).