# 2nd order correction to gauge transformation

• matt91a
In summary, when making a coordinate transformation of the form xμ = x'μ + ξμ, the metric gμν changes to g'μν = gμν + ξα,μ hαν + ξβ,ν hμβ + ξα,μξβ,νhαβ at higher than linear order. A clear reference is http://arxiv.org/abs/gr-qc/9609040, which explains the use of Lie derivatives in expressing this 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

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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.

The exact transformation is xμ = x'μ + ξμ. Applied to the metric this is

gμν = ∂xα/∂x'μ ∂xβ/∂x'ν gαβ

= (δαμ + ξα)(δβν + ξβ)(ηαβ + hαβ)

= ημν + (hμν + ξα hαν + ξβ hμβ + ξαξβhαβ)

= ημν + h'μν

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).