arXiv subjects

Matthew Aadne

Publications and source records attributed to Matthew Aadne.

4 recordsLinked to original sources

arXiv2022

Progress on the Kundt conjecture

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.

math.DG↗

arXiv2020

Observer-based invariants for cosmological models

We consider the equivalence problem for cosmological models in four-dimensional gravity theories. A cosmological model is considered as a triple $(M, {\bf g},{\bf u})$ consisting of a spacetime $(M, {\bf g})$ and a preferred normalized time-like vector field ${\bf u}$ tangent to a congruence of fundamental observers. We introduce a modification of the Cartan-Karlhede algorithm by restricting to frames adapted to ${\bf u}$ and including the covariant derivatives of ${\bf u}$ along with the Riemann tensor and its covariant derivatives. To fix the frame we make use of quantities relative to the fundamental observers, such as the anisotropic pressure tensor, energy flux vector, electric and magnetic parts of the Weyl tensor and the kinematical quantities of ${\bf u}$. This provides a simpler way to construct a list of invariants relative to the fundamental observers that completely characterizes the model, independent of coordinates. As an illustration of the algorithm, we consider several well-known cosmological models from General Relativity.

gr-qc↗

arXiv2019

Kundt Structures

In this paper we consider a new approach to studying Kundt spacetimes through $G$-structures. We define a Lie-group $GN$ such that the $GN$-structures satisfying an integrability condition and an existence criterion, which we call Kundt structures, have the property that each metric belonging to the Kundt structure is automatically a Kundt spacetime. We find that the Lie algebra of infinitesimal automorphisms of such structures is given by a Lie algebra of nil-Killing vector fields. Lastly we characterize all left invariant Kundt structures on homogeneous manifolds.

math.DG↗

arXiv2019

Nil-Killing vector fields and type III deformations

This paper is concerned with deformations of Kundt metrics in the direction of type $III$ tensors and nil-Killing vector fields whose flows give rise to such deformations. We find various characterizations within the Kundt class in terms of nil-Killing vector fields and obtain a theorem classifying algebraic stability of tensors, which has an application in finding sufficient criteria for a type $III$ deformation of the metric to preserve spi's. This is used in order to specify Lie algebras of nil-Killing vector fields that preserve the spi's, for degenerate Kundt metrics. Using this we discuss the characterization of Kundt-CSI spacetimes in terms of nil-Killing vector fields.

math.DG↗