This article looks at the process of deriving the so called** Palatini equation** and follows the demonstration found in D'Inverno *Introducing Einstein's relativity* , Chapter 11-1 (General Relativity from a variational principle, The Palatini equation).

Many tensor identities are derived most easily using the technique of geodesic coordinates in a Local Inertial Frame, where we choose an arbitrary point P at which the Christoffel symbols nullify, which in D'Inverno notation could be written as:

As we know from our article Riemann curvature tensor part III: Symmetries and independant components in this particular case, the Riemann tensor reduces to:

Looking now at a variation of the connection Γ^{a}_{bc} to a new connection Γ^{a}_{bc}(hat):

Then δΓ being the difference of two connections, is a tensor of type (1,2), and this variation results in a change in the Riemann tensor between two coordinate systems as:

since partial derivatives commute with variation and **is equivalent to covariant derivative in geodesic coordinates**.

We have now an equation with two tensor quantites on the left hand side (being the difference of two tensors) and on the right-hand side (as by defintion of a covariant derivative). So by definition, **if a tensor equatiion holds in one coordinate system it must hold in any coordinate system**, we can deduce the **Palatini equation - **in Inverno notation, it means we can remove the .dot over the equals sign:

And contraction on indice a and c gives the useful result:

**Remark**: this last result is used during the process of the Einstein's equation derivation from a variational approach Einstein-Hilbert action