Lovelock gravity at the crossroads of Palatini and metric formulations

Abstract

We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor, L(gµν, R µαβν ). A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove that within the class of modified gravity theories we consider, only the Lovelock gravity satisfies this requirement.

AdS7/CFT6, Gauss-Bonnet gravity, and viscosity bound

“…It is usually referred to as Lovelock gravity [25][26][27] and is the most general theory of gravitation whose equations of motion contain at most second order derivatives of the metric. Recently, the Palatini and metric formulations of Lovelock gravity have been shown to be equivalent [28].…”

A Bird's Eye View of f (R)-Gravity

“…3 Palatini ) (R f gravity was proposed as an alternative to dark energy, on the same footing as metric ) (R f models. 3 By imposing that the metric and Palatini variations generate the same field equations, Lovelock gravity is selected [74]. GR is a special case of Lovelock theory.…”

“…In fact, exit from the radiation or any era can be obtained as follows. In the approach dubbed "designer ) (R f gravity" in [74], the desired expansion history of the universe can be obtained by specifying the desired scale factor ) (t a and integrating an ordinary differential equation for the function ) (R f that produces the chosen ) (t a [79]. In general, the solution to this ODE is not unique and can assume a form that appears rather contrived in comparison with simple forms adopted in most popular models.…”

Spectra, vacua, and the unitarity of Lovelock gravity inD-dimensional AdS spacetimes

“…One can have a further reduction in representing the maximally symmetric vacua and the free part of the Lovelock theory with an equivalent action by using the fact discussed above: the vacua and the propagator of the Einstein-GB theory can be represented with cosmological Einstein's gravity having the modified parameters given in (6). Then, we can write…”

“…Moreover, Lovelock gravity is the only higher derivative theory that does not suffer the Buchdahl's inequality [5] that exists between the metric formulation and the Palatini formulation-which assumes a generic connection a prioriof higher derivative gravity theories [6]. This result is quite interesting, since it yields a dynamical derivation of equivalence principle for a class of torsion-free theories [6,7]. Another property of (1) is that for even D, the highest-order term does not contribute to the field equations, since its variation is a total derivative (i.e.…”