- #1
michael879
- 696
- 7
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.
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.
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