I was asked to do a problem about the gauge freedom one has in the weak field limit in general relativity. I am given a coordinate transformation [tex]x^{\mu '} = x^{\mu} - \zeta^{\mu}[/tex]. Now I have to show how the perturbation of the metric transforms under this coordinate transformation.(adsbygoogle = window.adsbygoogle || []).push({});

The result should be:

[tex] h_{\mu\nu}^{new} = h_{\mu\nu}^{old} - \zeta_{\mu ,\nu} - \zeta_{\nu ,\mu}[/tex]

The most simple way to do this, i thought, was to just use the transformational law for tensors.

so: [tex]h_{\mu ' \nu '} = \partial_{\mu '}x^{\mu} \partial_{\nu '}x^{\nu} h_{\mu \nu}[/tex]

Using the fact that we can neglect the product of the derivatives of the small perturbation, I find that:

[tex]h_{\mu ' \nu '} = h_{\mu\nu} - (\partial_{\mu '}\zeta^{\mu} + \partial_{\nu '}\zeta^{\nu})h_{\mu\nu}[/tex]

From this point I don't know how to get rid of the metric factor in the second term, because if I can lose it, i think i would be able to come to the correct form that is wanted.

Can someone give me some hints? :)

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# Homework Help: Weak field general relativity

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