Pure Lovelock Kasner metrics

TL;DR

This paper classifies Kasner solutions in pure Lovelock gravity, revealing unique geometric structures and symmetries, and explores their implications for singularity behavior and exotic solutions like complex exponents.

Contribution

It provides a complete classification of Kasner metrics in pure Lovelock theories and analyzes their geometric properties and solution space in various dimensions.

Findings

Vacuum in critical odd dimensions is kinematic, with vanishing Lovelock-Riemann tensor.

Complete classification of isotropic and vacuum Kasner metrics in these theories.

Existence of solutions with complex exponents leading to closed timelike curves.

Abstract

We study pure Lovelock vacuum and perfect fluid equations for Kasner-type metrics. These equations correspond to a single $N$th order Lovelock term in the action in $d=2N+1,\,2N+2$ dimensions, and they capture the relevant gravitational dynamics when aproaching the big-bang singularity within the Lovelock family of theories. Pure Lovelock gravity also bears out the general feature that vacuum in the critical odd dimension, $d=2N+1$, is kinematic; i.e. we may define an analogue Lovelock-Riemann tensor that vanishes in vacuum for $d=2N+1$, yet the Riemann curvature is non-zero. We completely classify isotropic and vacuum Kasner metrics for this class of theories in several isotropy types. The different families can be characterized by means of certain higher order 4th rank tensors. We also analyze in detail the space of vacuum solutions for five and six dimensional pure Gauss-Bonnet theory. It possesses an interesting and illuminating geometric structure and symmetries that carry over to the general case. We also comment on a closely related family of exponential solutions and on the possibility of solutions with complex Kasner exponents. We show that the latter imply the existence of closed timelike curves in the geometry.