A Ricci tensor for Fermi normal coordinates

[TimWilliams87]

TL;DR

Can the Ricci tensor be written simply in terms of second derivatives of the metric by the relation with the Riemann tensor?

I am learning about Fermi normal coordinates for an inertial observer on a reference curve from the textbook ''Advanced general relativity'' by Eric Poisson. The metric is written as g = eta + h, where eta is the Minkowski metric and h is the spacetime curvature perturbation close to the geodesic up to order x^2. t is proper time along the geodesic.

In these coordinates, the metric can be expressed as

$$ g_{tt} = -1 -R_{tatb}(t)x^ax^b + O(x^3), $$

$$ g_{ta} = \frac{2}{3}R_{tbac}(t) + O(x^3), $$

$$ g_{ab} = \delta_{ab} - \frac{1}{3}R_{acbd}(t)x^cx^d + O(x^3), $$

where $R_{abcd}$ is the Riemann tensor. It is stated in Poisson that these are related to statements regarding second derivatives of the metric (which I assume are just spatial derivatives only of the perturbing part of the metric h).

Since the components of the Riemann tensor can be written in terms of second derivatives of the metric, can one write components of the Ricci tensor simply in terms of second derivatives of the metric? So, for example, what is the component of the Ricci tensor R_{00}?

We do have as usual the relation

$$ R_{00} = g^{ii}R_{0i0i} , $$

but this seems like it would become complicated.

 

[PeterDonis]

@TimWilliams87 please review the PF LaTeX Guide. LaTeX formulas here are delimited by double dollar signs, for equations standing alone, or double pound signs for inline LaTeX. I have used magic moderator powers to edit your OP accordingly.