I got so much excellent help from my last question that I have decided to take advantage as much as I can. Here is another question.(adsbygoogle = window.adsbygoogle || []).push({});

On page 202, I combine equations 8.18 and 8.19 and change bars for primes (to match eqn. 8.22)

[tex]g_{\alpha' \beta'} = \eta_{\alpha' \beta'} + \Lambda^{\mu}_{\alpha'}\Lambda^{\nu}_{\beta'}h_{\mu \nu}[/tex]

where [itex]\Lambda[/itex] is a boost. Here is eqn 8.21 with the non-linear terms deleted

[tex]\Lambda^{\alpha}_{\beta'} = \delta^{\alpha}_{\beta} - \xi^{\alpha}_{,\beta}[/tex]

Combining these I get:

[tex]g_{\alpha' \beta'} = \eta_{\alpha' \beta'} + (\delta^{\mu}_{\alpha} - \xi^{\mu}_{,\alpha})(\delta^{\nu}_{\beta} - \xi^{\nu}_{,\beta})h_{\mu \nu}[/tex]

using [itex]\eta_{\alpha' \beta'} = \eta_{\alpha \beta}[/itex], expanding the factors, and dropping the term quadratic in [itex]\xi[/itex] I get:

[tex]g_{\alpha' \beta'} = \eta_{\alpha \beta} + \delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}h_{\mu \nu} - \xi^{\mu}_{,\alpha}\delta^{\nu}_{\beta}h_{\mu \nu} - \delta^{\mu}_{\alpha}\xi^{\nu}_{,\beta}h_{\mu \nu}[/tex]

[tex]g_{\alpha' \beta'} = \eta_{\alpha \beta} + h_{\alpha \beta} - \xi^{\mu}_{,\alpha}h_{\mu \beta} - \xi^{\nu}_{,\beta}h_{\alpha \nu}[/tex]

Now, finally, comes my question:

How can I use eqn 8.23 to simplify this to 8.22? It looks like Schutz is using h as if it were [itex]\eta[/itex]

Here is 8.23

[tex]\xi_{\alpha} = \eta_{\alpha \beta}\xi^{\beta}[/tex]

Here is 8.22

[tex]g_{\alpha' \beta'} = \eta_{\alpha \beta} + h_{\alpha \beta} - \xi_{\alpha,\beta} - \xi_{\beta,\alpha}[/tex]

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# Another question from Schutz

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