- #1

gouranja

- 11

- 0

I don't think this belongs in the homework section since this is a graduate course.

My question is regarding making a field theory generally covariant by including a metric tensor [tex]g_{\mu\nu}(x)[/tex]in the Lagransian density, and it's transformation under infinitesimal coordinate change.

If I transform the coordinates according to:

[tex]x^{\mu}\rightarrow x^{\mu}'=x^{\mu}+\epsilon^{\mu}(x)[/tex]

The metric must be transformed according to:

[tex]g_{\mu\nu}(x)\rightarrow g'_{\mu'\nu'}(x')=\frac{\partial x^{\alpha}}{\partial x^{\mu}'}\frac{\partial x^{\beta}}{\partial x^{\nu}'}g_{\alpha\beta}(x)[/tex]

which I understand well. But according to the textbook I'm reading the infitesimal result is:

[tex]\delta g_{\mu\nu}(x)=\epsilon_{\mu;\nu}+\epsilon_{\nu;\mu}[/tex]

(with covariant derivatives).

I have tried using:

[tex]\frac{\partial x^{\alpha}}{\partial x^{\mu}'}\simeq\delta_{\mu}^{\alpha}-\frac{\partial\epsilon^{\alpha}}{\partial x^{\mu}}+O(\epsilon^{2})[/tex]

to obtain:

[tex]\delta g_{\mu\nu}(x)=-g_{\mu\alpha}\partial_{\nu}\epsilon^{\alpha}-g_{\nu\alpha}\partial_{\mu}\epsilon^{\alpha}[/tex]

but I don't know how to obtain the textbook result.

Can someone clue me in on how to do it?

Thanks