<PERSON> equation for the geodesic deviation in the weak field limit
The geodesic deviation equation can be written in the following form $$ \nabla_U^2 \xi = R (U, \xi) U \tag{1} $$ where $R$ is the <PERSON> tensor. It can also be written component-wise using the <PERSON> tensor $$ (\nabla_U^2 \xi)^\alpha = R^\alpha_{\;\, \beta \mu \nu} \: U^\beta U^\mu \xi^\nu \tag{2} $$
I'm interested in this equation on a Riemannian manifold equipped with the following metric tensor $$ g = -(1+2\phi(x))\mathrm{d} t \otimes \mathrm{d} t + (1-2\phi (x))\left(\mathrm{d} x \otimes \mathrm{d} x + \mathrm{d} y \otimes \mathrm{d} y + \mathrm{d} z \otimes \mathrm{d} z \right) \tag{3} $$ with the standard torsion-free and metric-preserving connection $\nabla$. I'm only interested in the spatial part of the Jacobi equation, that starts from $U = e_0$ i.e. $$ (\nabla_t^2 \xi)^i = R^i_{\;\, 0 0 \nu} \,\xi^\nu \tag{4} $$
In textbooks it can be found that in the weak-field limit (keeping only the linear terms involving $\phi (x)$) this reduces to $$ \frac{\mathrm{d}^2 \xi^i}{\mathrm{d} t^2} = - \phi_{,ij} \, \xi^j \tag{5} $$ where comma indicates partial derivatives.
However, when I try this for $g$ above, I get a different result. 346