Contravariant metric components

[michael879]

I realize this is a "simple" mathematical exercise, in theory, but I'm having a lot of trouble finding some algorithmic way to do it. The problem is this: I want to expand the contravariant metric tensor components $$g^{\mu\nu}$$ in terms of the covariant metric tensor $$g_{\mu\nu}$$. The first order calculation is very simple, but the second order one seems incredibly tedious and error prone. Could someone either show me this expansion to at least 2nd order or direct me to a source that does? I'm trying to do some stuff with quantum gravity and I don't want to spend all this time on such a trivial problem if the answer is already out there (I have looked).

Also, here is the method I have been using:
$$g^{\lambda\mu}g_{\mu\sigma} = \delta^\lambda_\sigma$$
$$g^{\mu\nu} \equiv \eta^{\mu\nu} + sqrt(G)*h^{\mu\nu}$$
$$g_{\mu\nu} \equiv \eta_{\mu\nu} + sqrt(G)*h_{\mu\nu}$$

and then just use the resulting formula to calculate the contravariant h in terms of the covariant h order by order (use all possible combinations of h, $$\eta$$, and $$\partial$$ for each order of h and assign some constant that will be solved for). The problem is that the number of terms at each order grows very fast.

 

[Bill_K]

Let's do it in matrix form first and then put the indices in. Let N = η_μν, H = h_μν, and G = g^μν.

G = (N + H)^-1 = ((I + HN^-1)N)^-1 = N^-1(I + HN^-1)^-1 = N^-1 - N^-1HN^-1 + N^-1HN^-1HN^-1 + ...

Therefore

g^μν = η^μν - η^μαh_αβη^βν + η^μαh_αβη^βγh_γεη^εν + ...