A distinguishing gravitational property for gravitational equation in higher dimensions

Abstract

It is well known that Einstein gravity is kinematic (meaning that there is no non-trivial vacuum solution; i.e. the Riemann tensor vanishes whenever the Ricci tensor does so) in 3 dimension because the Riemann tensor is entirely given in terms of the Ricci tensor. Could this property be universalized for all odd dimensions in a generalized theory? The answer is yes, and this property uniquely singles out pure Lovelock (it has only one N th order term in the action) gravity for which the N th order Lovelock-Riemann tensor is indeed given in terms of the corresponding Ricci tensor for all odd, d = 2N + 1, dimensions. This feature of gravity is realized only in higher dimensions and it uniquely picks out pure Lovelock gravity from all other generalizations of Einstein gravity. It serves as a good distinguishing and guiding criterion for the gravitational equation in higher dimensions.

Quasi-local energy and ADM mass in pure Lovelock gravity

“…Pure Lovelock gravities are a subset of Lovelock gravity theories labeled by a non-negative integer m that contain a single Lovelock invariant in the action. They are very similar to Einstein gravity m = 1 and extend many of its properties to higher dimensions [12][13][14].…”

“…For generic metrics, the limit differs from the ADM mass (2.16) as we will now show. 14 The regularization presented in [28] involves replacing all the extrinsic curvatures in the boundary term B (m) (3.11) with the vacuum subtracted ones ∆ K ij = K ij − K ij | (0) . Clearly it is a non-linear procedure, which is the reason why the form of the Hamilton's equations is not preserved.…”

A way of decoupling gravitational sources in pure Lovelock gravity

“…AB is a n order generalization of the Einstein tensor due to the topological density L n . As example G (1) AB is just the Einstein tensor associated with the Ricci scalar (Einstein Hilbert theory is a particular case of Lovelock theory), and G (2) AB is the Lanczos tensor H AB associated with the Gauss Bonnet Lagrangian.…”

“…Into the Lovelock gravities, we can find the Pure Lovelock theory. It is well known that the General Relativity has a no non-trivial vacuum solution (without cosmological constant) when d = 3 (i.e d = 2n + 1, where n = 1), one interesting feature is that Pure Lovelock keeps this property for d = 2n + 1 with n > 1, see reference [2]. On the other hand, including the cosmological constant, General Relativity has a unique (Anti) de Sitter ground state for Λ(< 0) > 0, in regarding this, other interesting feature of Pure Lovelock theory is that, it keeps this feature for n odd, however, for n even, this theory has a double Anti de Sitter or de Sitter ground state for Λ > 0 and does not have ground state for Λ < 0 [3,4].…”

High-energy collision of black holes in higher dimensions

“…A further extension of our work may consider boosted collisions of BHs in higherdimensional Lovelock gravity following the BH solutions and formalism of Refs. [77][78][79]. Such a program, however, might require more investigation to ensure availability of a well-posed initial-value formulation [80,81].…”