# Weak field approx raising index using minkowski metric

1. Mar 26, 2015

### binbagsss

I have the expression $g_{ab}=\eta_{ab}+\epsilon h_{ab}$,

The indices on $h^{ab}$ are raised with $\eta^{ab}$ to give $g^{ab}=\eta^{ab}-\epsilon h^{ab}$

I am not seeing where the minus sign comes from.

So I know $\eta^{ab}\eta_{bc}=\delta^{a}_{c}$ and $h^{ab}=\eta^{ac}\eta^{bd}h_{cd}$ ,
And that $\eta_{ab}$ has signature $(-,+,+,+)$,

Any help greatly appreciated.

2. Mar 26, 2015

### Mentz114

If $\epsilon$ is an infinitessimal then the sign changes between the covariant and contravariant delta so that the contraction is unchanged to order $\epsilon$

3. Mar 27, 2015

Thanks .