A discerning gravitational property for gravitational equation in higher dimensions

TL;DR

This paper demonstrates that pure Lovelock gravity uniquely exhibits a kinematic property in all odd higher dimensions, serving as a criterion to distinguish it from other gravitational theories.

Contribution

It generalizes the kinematic property of Einstein gravity in 3D to all odd dimensions for pure Lovelock gravity, providing a new criterion for identifying higher-dimensional gravitational equations.

Findings

Pure Lovelock Riemann tensor is expressed in terms of Ricci in odd dimensions.

This property is unique to pure Lovelock gravity among higher-dimensional theories.

Serves as a guiding criterion for gravitational equations in higher dimensions.

Abstract

It is well-known that Einstein gravity is kinematic (no non-trivial vacuum solution;i.e. Riemann vanishes whenever Ricci does so) in $3$ dimension because Riemann is entirely given in terms of Ricci. 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 action) gravity for which $N$th order Lovelock Riemann tensor is indeed given in terms of corresponding Ricci 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 discerning and guiding criterion for gravitational equation in higher dimensions.