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In the book A Relativist's Toolkit by Eric Poisson, section 4.1.4, page $123$, it is written that in a local Lorentz frame at a point $P$:

$$\delta R_{\alpha \beta} \stackrel{*}{=} \delta\left(\Gamma^\mu{}_{\alpha \beta, \mu}-\Gamma^\mu{}_{\alpha \mu, \beta}\right)$$

I am not able to get it from the expression:

$$R_{ij} = \partial_k \Gamma^k{}_{ij} - \partial_j \Gamma^k{}_{ik} + \Gamma^k{}_{ij} \Gamma^m{}_{km} - \Gamma^k{}_{im} \Gamma^m{}_{jk} $$

I will be glad if someone can help me in this.

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    $\begingroup$ In a local Lorentz frame all the Christoffel symbols are zero, though their derivatives are not, so your second equation simplifies considerably. $\endgroup$ Commented Apr 8 at 15:25

1 Answer 1

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  1. In normal coordinates centered at $p\in M$, the Christoffel symbols of the Levi-Civita connection are equal to zero at $p$ and the metric is Minkowskian up to second order corrections i.e. \begin{equation} \Gamma^{\mu}_{\nu\sigma}(0)=0 \quad g_{\mu\nu}(x)=\eta_{\mu\nu}+\mathcal{O}(|x|^2) \tag{1}\label{1} \end{equation} where we've used $x$ for the normal coordinates in a neighbourhood of $p\mapsto(0,0,0,0)$.
  2. The general coordinate expression of the Riemann tensor is \begin{equation} R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma} \tag{2a}\label{2a} \end{equation} Switching to normal coordinates, using \eqref{1} and neglecting higher order contributions yields \begin{equation} R^{\rho}{}_{\sigma\mu\nu} \overset{\ast}{=} \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} -\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}\tag{2b}\label{2b} \end{equation}
  3. Contracting \eqref{2b} yields the Ricci tensor \begin{equation} R_{\sigma\nu} \overset{\ast}{=} \partial_{\mu}\Gamma^{\mu}{}_{\nu\sigma} -\partial_{\nu}\Gamma^{\mu}{}_{\mu\sigma}\tag{3}\label{3} \end{equation}.
  4. OP's equation is the variation of the Ricci tensor using the coordinate expression \eqref{3}.

References

  1. A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Eric Poisson. CUP, 2007. Page 123.
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