Progress on the Kundt conjecture

TL;DR

This paper advances the proof of the Kundt conjecture by establishing that certain curvature conditions imply the existence of a Kundt null congruence in higher dimensions, confirming the conjecture in new cases.

Contribution

It proves the Kundt conjecture in arbitrary dimensions under genericity or algebraic type conditions, extending previous results and removing some regularity assumptions.

Findings

Confirmed the Kundt conjecture for dimensions >4 under specific conditions

Showed only the third covariant derivative is needed in 3 and 4 dimensions

Introduced a new bilinear map for tensor analysis

Abstract

The Kundt conjecture states that a Lorentzian manifold of arbitrary dimension which is not characterized by its scalar polynomial curvature invariants (SPIs) allows for a non-twisting, non-shearing and non-expanding (in short, Kundt) null congruence of geodesics. The conjecture has been proven for dimensions 3 and 4. A necessary condition for a spacetime not to be characterized by SPIs is that all covariant derivatives of the Riemann tensor are of aligned type II or more special in the null alignment classification. In arbitrary dimensions, we prove that this property indeed requires the presence of a Kundt null congruence when a certain genericity condition holds, or when the trac-free Ricci orWeyl tensor is of genuine type III or N, thus confirming the validity of the Kundt conjecture in these cases. We also strenghten the results for dimensions 3 and 4 by removing regularity assumptions and showing that only the third covariant derivative is needed to obtain the results. A key tool in our proofs is a new bilinear map acting on tensors of related boost orders relative to a null direction.