The 10 independent components in the Einstein's Field equations

I'm trying to understand the Einstein field equations, but there is something that is costing me to understand. From the Wikipedia: https://en.wikipedia.org/wiki/Einstein_field_equations, it says that in that the EFE (Einstein's field equation), "is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge fixing degrees of freedom". So what exactly are the 10 independent components? Are only values that you extract from data or there is something more? And how the Bianchi identities take four of them?

1 Answer

The $4\times 4$ tensor is symmetric, so the $6$ entries below the leading diagonal are equal to the $6$ above it. Therefore, there are only $4^2-6=4+6=10$ DOFs before we consider Bianchi; the sum $4+6$, another way of counting the DOFs, is one triangle plus a diagonal. Bianchi is a vector$=0$ constraint (explicitly $\nabla^\mu G_{\mu\nu}=0$), knocking out another $4$ DOFs. Wikipedia's explanation is perhaps poorly worded; I recommend Carroll's alternative here (Ctrl+F to the paragraph starting, "However, the Bianchi identity"). Carroll notes that the number of DOFs Bianchi removes is the number of unphysical DOFs in a transformation between two coordinate systems on $4$-dimensional spacetime. This supplements Wikipedia's explanation, in which we have $4$ gauge-fixing & $6$ physical DOFs.