Abstract
We define the concept of Levi-Civita truncation for a Lagrangian in the Palatini formulation with an arbitrary connection, and show that its consistency uniquely identifies the Lovelock Lagrangians.
Highlights
In the Palatini formulation of gravity, the connection is treated in line with gauge field theory as an independent variable
Since in the truncated theory the Riemann tensor is built with the Levi Civita (LC) connection, keeping all its indices down will allow us to take full advantage of the symmetry properties of Rμνρσ which will be inherited as symmetry properties by the density tensor
It may be noted that all previous considerations of this problem referred to a symmetric connection in the metric affine formulation while here we deal with an arbitrary connection without any restrictions
Summary
In the Palatini formulation of gravity, the connection is treated in line with gauge field theory as an independent variable. In the usual derivations of the equivalence, the symmetry of the connection, which means vanishing torsion, is always assumed in the metric-affine formulation, whereas for the vielbein-affine formulation is assumed the antisymmetry of the spin-connection, which means metricity This was the situation until very recently when in the works [6, 7], the equivalence is established for an arbitrary connection without any condition on it. We believe that the result is true, it appears that the arguments are not completely satisfactory This is an interesting characterization of the Lovelock Lagrangians. Our main result is that the consistency of such a truncation uniquely identifies the Lovelock polynomial Lagrangians.
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