# 2nd order correction to gauge transformation

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

## Answers and Replies

atyy
Science Advisor
Last edited by a moderator:
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
Science Advisor
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).