On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism

Understand

We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein-Hilbert action (with possible minimally coupled matter terms).

• We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field ${\cal A}_\mu$.
• We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparametrised geodesics and the Einstein equation are shared by all of them.
• Moreover, the Palatini connections are characterised precisely by these two properties, as well as by other properties of its parallel transport.