Abstract
We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.
Highlights
Metric-affine gravity is a set of theories in which the metric gμν and the affine connection Γμν ρ are taken to be independent variables
In metric-affine theories, the idea is that the affine connection Γμνρ should be determined by its own equation of motion, just as any other dynamical variable of the theory
For Lovelock gravities, the metric formalism is a consistent truncation of the Palatini formalism [24]
Summary
Metric-affine gravity (sometimes called the Palatini formalism) is a set of theories in which the metric gμν and the affine connection Γμν ρ are taken to be independent variables. In metric-affine theories, the idea is that the affine connection Γμνρ should be determined by its own equation of motion, just as any other dynamical variable of the theory It has been shown [1,2] (see [3]) that the physics described by the Einstein-Hilbert-Palatini action, S. For Lovelock gravities, the metric formalism is a consistent truncation of the Palatini formalism [24] It is by no means clear whether for these theories both formalisms are equivalent, as in the case of the Einstein-Hilbert action, in the sense that all allowed solutions of the connection equation of the metric-affine Lovelock lagrangians yield the same physics as the metric formalism.
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