Energy in Fourth Order Gravity

Abstract

In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.

Rigidity Theorems for Asymptotically Euclidean $Q$-singular Spaces

“…Proof. Using (27) we recognise in the right-hand side of (25) the energy density from the fourth order energy E(g) analysed in [4] and whose definition can be found in the introduction (10). Thus, from [4, Proposition 1], it follows that we can pass to limit r → ∞ under the conditions of this lemma, which establishes the claim.…”

“…Thus, 1 ∈ N GR 0 (δ) becomes naturally associated to time-like Killing fields (and their associated conservation laws) of AE initial data sets. 7 In parallel to the above comments, E(g) was introduced in [4] as a quantity defined on AE-slices of asymptotically Minkowskian space-time solutions to a set of fourth order equations, which is conserved through time translations. In particular, from the work of [4] and [3], the Q-curvature of the corresponding fourth order initial data set appears as a natural analogue to the scalar curvature for the corresponding relativistic initial data sets.…”

“…7 In parallel to the above comments, E(g) was introduced in [4] as a quantity defined on AE-slices of asymptotically Minkowskian space-time solutions to a set of fourth order equations, which is conserved through time translations. In particular, from the work of [4] and [3], the Q-curvature of the corresponding fourth order initial data set appears as a natural analogue to the scalar curvature for the corresponding relativistic initial data sets. Thus, E g (1) appears as the corresponding dynamical equivalent of the global Lagrangian approach defining E(g) in [4], with the same conserved energy on the space slices.…”

“…This lower bound for τ will be needed to appeal to the rigidity of the positive energy theorem associated to a fourth order invariant related to Q-curvature. This invariant was introduced as a canonically associated conserved quantity in a family of fourth order gravitational theories in [4], which, in the case of stationary space-time solutions reduces to…”

Einstein Type Systems on Complete Manifolds

“…Indeed the method employed is both robust (it considers a coupled system on a complete manifold in a general manner, and improves on pre-existing results in the way described above) and flexible (see for instance the L 2 results or remark 3.3 for an exploration of how it can adapted to other systems). This points to the possibility to transpose these iteration schemes to constraint systems related to (5), such as those of fourth order gravity where the fourth order phenomena suggest stepping out of the AE framework (see [9] for an exploration of these phenomena and [61] for the associated constraint equations).…”