# Ricci tensor in locally Lorentz frame

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.

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

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. $$$$\Gamma^{\mu}_{\nu\sigma}(0)=0 \quad g_{\mu\nu}(x)=\eta_{\mu\nu}+\mathcal{O}(|x|^2) \tag{1}\label{1}$$$$ where we've used $$x$$ for the normal coordinates in a neighbourhood of $$p\mapsto(0,0,0,0)$$.
2. The general coordinate expression of the Riemann tensor is $$$$R^{\rho}{}_{\sigma\mu\nu} = \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} - \partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma} \tag{2a}\label{2a}$$$$ Switching to normal coordinates, using \eqref{1} and neglecting higher order contributions yields $$$$R^{\rho}{}_{\sigma\mu\nu} \overset{\ast}{=} \partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma} -\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}\tag{2b}\label{2b}$$$$
3. Contracting \eqref{2b} yields the Ricci tensor $$$$R_{\sigma\nu} \overset{\ast}{=} \partial_{\mu}\Gamma^{\mu}{}_{\nu\sigma} -\partial_{\nu}\Gamma^{\mu}{}_{\mu\sigma}\tag{3}\label{3}$$$$.