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the_doors
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Homework Statement
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})##
Homework 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}##.
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}##
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