Variations

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Variations of the Christoffel symbols, Riemann tensor, Ricci tensor, curvature scalar, and inverse metric:

\begin{array}{rcl} \delta \Gamma^a_{bc} & = & \frac{1}{2}g^{ad} \Bigl( D_b \delta g_{dc} + D_c \delta g_{bd} - D_d \delta g_{bc} \Bigr) \\ \delta R^a{}_{bcd} & = & D_c \delta\Gamma^a_{bd} - D_d \delta\Gamma^a_{bc} \\ \delta R_{bd} & = & D_a \delta\Gamma^a_{bd} - D_d \delta\Gamma^a_{ba} \\ \delta R & = & D^a D^b \delta g_{ab} - D^a D_a \delta (\ln |g|) - R^{ab}\delta g_{ab} \\ \delta g^{ab} & = & -g^{ac} \delta g_{cd}\, g^{db} \end{array}

where D_a is the covariant derivative compatible with the metric g_{ab}. The variations \delta are general. They can be replaced by, say, coordinate derivatives or time derivatives. Bryce DeWitt's favorite identity is

\delta(\ln(\det M)) = Tr(M^{-1} \delta M)

where M is a matrix and Tr denotes the trace. Other useful relations, where S denotes a scalar, include

\begin{array}{rcl} \delta D_a D_b S & = & D_a D_b \delta S - \frac{1}{2} D^c S \Bigl( D_a\delta g_{bc} + D_b\delta g_{ac} - D_c\delta g_{ab} \Bigr) \\ \delta D^2 S & = & D^2 \delta S - (D^a D^b S) \delta g_{ab} - (D^a S) D^b\delta g_{ab} + (D^a S) D_a \delta (\ln\sqrt{|g|}) \end{array}
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