# Varying the action with respect to metric

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1. Apr 19, 2015

### the_doors

1. The problem statement, all variables and given/known data
i want to find the variation of this action with respect to $g^{\mu\nu}$ , where $N_\mu(x^\nu)$ is unit time like four velocity and $\phi$ is scalar field.
$I_{total}=I_{BD}+I_{N}$
$I_{BD}=\frac{1}{16\pi}\int dx^4\sqrt{g}\left\{\phi R-\frac{\omega}{\phi}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right\}$
$I_N=\frac{1}{16\pi}\int dx^4\sqrt{g}\{\zeta(x^{\nu})(g^{\mu\nu}N_{\mu}N_{\nu}+1)+2\phi F_{\mu\nu}F^{\mu\nu}-\phi N_\mu N^{\nu}(2F^{\mu\lambda}\Omega_{\nu\lambda}+ F^{\mu\lambda}F_{\nu\lambda}+\Omega^{\mu\lambda}\Omega_{\nu\lambda}-2R_{\mu}^{\nu}+\frac{2\omega}{\phi^2}\nabla_{\mu}\phi\nabla^{\nu}\phi)\}$
where
$F_{\mu\nu}=2(\nabla_{\mu}N_{\nu}-\nabla_{\nu}N_{\mu})$
$\Omega_{\mu\nu}=2(\nabla_{\mu}N_{\nu}+\nabla_{\nu}N_{\mu})$

2. Relevant equations
i know the variation of Brans-Dicke and its field equations , but i have no idea about $\delta I_N$ with respect to $\delta g^{\mu\nu}$.

3. The attempt at a solution

$\delta \sqrt{-g}=\frac{-1}{2}\sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}$
and $\delta R=(R_{\mu\nu}+g_{\mu\nu}\Box-\nabla_\mu \nabla_\nu)\delta g^{\mu\nu}$

Last edited: Apr 19, 2015
2. Apr 19, 2015

### Orodruin

Staff Emeritus
Did you enter the problem exactly as stated? Your expression is adding terms with different free indices.

3. Apr 19, 2015

### the_doors

No, this is part of action which i have problem with that

4. Apr 19, 2015

### Orodruin

Staff Emeritus
The point is that your expression, as it stands, makes absolutely no sense. You cannot add tensors of different type ...

5. Apr 19, 2015

### the_doors

now i will edit post with the entire action.