Vanishing Scalar Invariant Spacetimes in Higher Dimensions

TL;DR

This paper characterizes higher-dimensional Lorentzian spacetimes with all scalar curvature invariants vanishing, linking this property to the existence of a special null direction with specific Riemann tensor properties.

Contribution

It provides a necessary and sufficient condition for vanishing scalar invariants in higher-dimensional Lorentzian manifolds based on null direction properties.

Findings

All scalar curvature invariants vanish iff a specific null direction exists.

The null direction is non-expanding, non-twisting, and geodesic.

The Riemann tensor has negative boost order along this null direction.

Abstract

We study manifolds with Lorentzian signature and prove that all scalar curvature invariants of all orders vanish in a higher-dimensional Lorentzian spacetime if and only if there exists an aligned non-expanding, non-twisting, geodesic null direction along which the Riemann tensor has negative boost order.