Question

How many degrees of freedom to diagonalize the metric?

Answer 1

In A. Zee's Einstein Gravity in a Nutshell , he starts with the following expansion of the metric at some point of a Riemannian manifold, with coordinates that have the origin at P : ⬆⬆

Answer 2

the discover went under this formula

gμν(x)=gμν(0)+Aμν,λxλ+Bμν,λσxλxσ+...gμν(x)=gμν(0)+Aμν,λxλ+Bμν,λσxλxσ+...