Question

How to invert this metric?

Answer 1

Reading this article i find a result that i am not sure how to obtain (page 3 eq 3). It is about the inversion of a metric of the type
In order to invert this metric i thought to write the inverse metric as
and then imposing the relation
i was hoping to obtain the desired result but this is not the case. I am asking for suggestions, hints or references.

Answer 2

 Inversion of metric.........below⬇️⬇️........

gμν=Alμν+BHμν.gμν=Alμν+BHμν.

In order to invert this metric i thought to write the inverse metric as

gμν=alμν+bHμνgμν=alμν+bHμν

and then imposing the relation

gμαgαν=δνμ