Post-Minkowskian expansion of some quantities in Post-Newtonian theory

Question

I'm studying Post-Newtonian theory on the book "Gravity" by Poisson and Will and I found a few formulas that I can't obtain by myself. I'm pretty sure it must be quite simple, but can't find the right method to get the result.

First of all, a really small introduction to the topic: instead of using the usual metric $g_{\alpha \beta}$, the inverse gothic metric $\mathfrak{g}^{\alpha \beta} = \sqrt{-g}\, g^{\alpha \beta}$ is used, where $g = \det (g_{\alpha \beta}) = \det (\mathfrak{g}^{\alpha \beta})$ (so that the relation can be easily reversed to obtain the usual metric). Then this inverse gothic metric is split in $\mathfrak{g}^{\alpha \beta} = \eta^{\alpha \beta} - h^{\alpha \beta}$, where $\eta^{\alpha \beta}$ is the usual Lorentzian metric $\operatorname{diag}(1, -1, -1, -1)$ (or with the opposite signs, if you prefer) and $h^{\alpha \beta}$ is a small correction of order $G$ (the gravitational constant, a placeholder for something a little bit more complicated and much more meaningful containing $G$).

It's easy to find that $g_{\alpha \beta} = \sqrt{-g}\, \mathfrak{g}_{\alpha \beta}$, but now it gets tricky. On pages 335-336 it is stated that we can perform a Post-Minkowskian expansion of a few quantities, i.e. they can be expressed as $\sum \limits_{n = 0}^{\infty} G^n k_{(n)}$, where $k_{(n)}$ are functions. The formulas are these (7.20a - 7.20d): $$g_{\alpha \beta} = \eta_{\alpha \beta} + h_{\alpha \beta} - \frac{1}{2} h \eta_{\alpha \beta} + h_{\alpha \mu} h^{\mu}{}_{\beta} - \frac{1}{2} h h_{\alpha \beta} + \left( \frac{1}{8} h^2 + \frac{1}{4} h^{\mu \nu} h_{\mu \nu} \right) \eta_{\alpha \beta} + O(G^3)$$ $$g^{\alpha \beta} = \eta^{\alpha \beta} - h^{\alpha \beta} + \frac{1}{2} h \eta^{\alpha \beta} - \frac{1}{2} h h^{\alpha \beta} + \left( \frac{1}{8} h^2 + \frac{1}{4} h^{\mu \nu} h_{\mu \nu} \right) \eta^{\alpha \beta} + O(G^3)$$ $$(-g) = 1 - h + \frac{1}{2} h^2 - \frac{1}{2} h^{\mu \nu} h_{\mu \nu} + O(G^3)$$ $$\sqrt{-g} = 1 - \frac{1}{2}h + \frac{1}{8} h^2 - \frac{1}{4} h^{\mu \nu} h_{\mu \nu} + O(G^3)$$

"Post-Minkowskian expansion" seems to have a vague meaning, as can be deduced from the definition, but I suppose here it's just a synonym of "Taylor expansion around $h^{\alpha \beta} = 0$".

If that's the case, I'm not sure what to expand and how, since it's not obvious or possible to isolate the quantity I'm looking for because of paired indices and products between tensors (and by tensor I mean - wrongly - everything that has indices). If that's not the case, then how can I calculate the expansion?

I tried to find the solution elsewhere, but it seems that the literature regarding Post-Newtonian theory is not so abundant and this specific topic isn't covered by anyone else, at least in the articles I found.

Edit: I forgot to specify that every index rasing/lowering and every contraction is made with respect to $\eta$. So for example $h = \eta^{\mu \nu} h_{\mu \nu}$ and $h^{\mu}{}_{\nu} = \eta^{\mu \sigma} h_{\sigma \nu}$.

Answer 1

For eq3 and eq4 it is a simple exercise to show that in general $$ \text{det }(\mathbb{1}- \mathbb{M}) = 1 - \text{tr }\mathbb{M} + \frac{1}{2} \text{tr }^2\mathbb{M} - \frac{1}{2}\text{tr }\mathbb{M}^2 + \mathcal{O}\left(\mathbb{M}^3\right) $$ hint: use the Levi-civita identity for the determinant Applying this formula immediately gives you det$(-g)=$det$(-\mathfrak{g})=\ldots$, and the square root by taylor expanding again.

Then for eq2 just use $$ g^{\alpha\beta}=\frac{1}{\sqrt{\text{det}(-g)}}\mathfrak{g}^{\alpha\beta}=\frac{1}{\sqrt{\text{det}(-\mathfrak{g})}}\mathfrak{g}^{\alpha\beta} = \frac{\eta^{\alpha\beta}-h^{\alpha\beta}}{1-\frac{1}{2}h + \frac{1}{8}h^2-\frac{1}{4}h^{\alpha\beta}h_{\alpha\beta}+\ldots} $$ and just do a taylor expansion (denomenator is just scalars).

Finally you can deduce eq1 from eq2 by inverting the latter perturbatively