Invariant classification of vacuum PP-waves

We solve the equivalence problem for vacuum PP-wave spacetimes by employing the Karlhede algorithm. Our main result is a suite of Cartan invariants that allows for the complete invariant classification of the vacuum pp-waves. In particular, we derive the invariant characterization of the G2 and G3 sub-classes in terms of these invariants. It is known [Collins91] that the invariant classification of vacuum pp-waves requires at most the fourth order covariant derivative of the curvature tensor, but no specific examples requiring the fourth order were known. Using our comprehensive classification, we prove that the q<=4 bound is sharp and explicitly describe all such maximal order solutions

Horizons that Gyre and Gimble: A Differential Characterization of Null Hypersurfaces

Motivated by the thermodynamics of black hole solutions conformal to stationary solutions, we study the geometric invariant theory of null hypersurfaces. It is well-known that a null hypersurface in a Lorentzian manifold can be treated as a Carrollian geometry. Additional structure can be added to this geometry by choosing a connection which yields a Carrollian manifold. In the literature various authors have introduced Koszul connections to study the study the physics on these hypersurfaces. In this paper we examine the various Carrollian geometries and their relationship to null hypersurface embeddings. We specify the geometric data required to construct a rigid Carrollian geometry, and we argue that a connection with torsion is the most natural object to study Carrollian manifolds. We then use this connection to develop a hypersurface calculus suitable for a study of intrinsic and extrinsic differential invariants on embedded null hypersurfaces; motivating examples are given, including geometric invariants preserved under conformal transformations.Comment: 18 pages, significant change

Invariant characterization of Szekeres models with positive cosmological constant

We present an invariant characterization of black holes in the Szekeres spacetime with positive cosmological constant. In the formation of the black holes, we locate geometric horizons, and show that they coincide with the more traditional apparent horizons in the Szekeres models. We also define an invariant approach for detecting shell crossings. It is shown that shell crossing regions in the Szekeres models can be contained within the geometric horizons for situations where no naked singularities form, allowing for the study of astrophysical models that are inhomogeneous and with a cosmological constant. A measure of inhomogeneity through the dipole functions in the Szekeres models is used to compute shell crossing surfaces along particular directions in the spacetime. An example of the method applied to the axially symmetric collapse of a quasispherical dust is given, motivated by previous work on primordial black hole formation. Future extensions and generalizations of this work are also discussed.acceptedVersio

Differential invariants of Kundt waves

Kundt waves belong to the class of spacetimes which are not distinguished by their scalar curvature invariants. We address the equivalence problem for the metrics in this class via scalar differential invariants with respect to the equivalence pseudo-group of the problem. We compute and finitely represent the algebra of those on the generic stratum and also specify the behavior for vacuum Kundt waves. The results are then compared to the invariants computed by the Cartan–Karlhede algorithm.publishedVersio