arXiv subjects

Leonardo García-Heveling

Publications and source records attributed to Leonardo García-Heveling.

13 recordsLinked to original sources

arXiv2026

On the affine parametrization of null geodesics in low regularity

In low-regularity spacetimes and Lorentzian length spaces, achronal causal curves play the role of null (pre-)geodesics. Because of the lack of a geodesic equation, they do not come with a canonical parametrization. In this context, we discuss a notion of affine parametrization via limits of affinely parametrized timelike geodesics. However, we point out a major drawback: an example where this approximation procedures gives a non-unique result, in a way that even completeness or incompleteness of the limit null geodesic is not well-defined. This example involves discontinuous gluing of two Lorentzian metrics across a null hypersurface to give a well-behaved Lorentzian length space and as such is, just like the parametrization problem itself, manifestly Lorentzian. In view of this, we explore possibilities for a Penrose-type singularity theorem using timelike geodesics.

math.DG↗

arXiv2026

Differentiability and Other Properties of the Cosmological Volume Function

In a previous work, the regular cosmological volume function $τ_V$ was introduced as an alternative to the regular cosmological time function of Andersson, Galloway, and Howard. Building on work by Chruściel, Grant and Minguzzi, in this paper we show that in many cases of interest, $τ_V$ is a continuously differentiable temporal function. This leads to a canonical splitting of the metric tensor, and induces a canonical ``Wick-rotated" Riemannian metric. We also provide some further results and examples related to the cosmological time and volume functions.

gr-qc↗

arXiv2026

Isometries of spacetimes without observer horizons

We study the isometry groups of (non-compact) Lorentzian manifolds with well-behaved causal structure, aka causal spacetimes satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving isometries acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy temporal function, and a splitting of the isometry group into a compact subgroup and a subgroup roughly corresponding to time translations. The latter can only be the trivial group, $\mathbb{Z}$, or $\mathbb{R}$.

math.DG↗

arXiv2025

Conformal transformations of spacetimes without observer horizons

We prove that for a certain class of Lorentzian manifolds, namely causal spacetimes without observer horizons, conformal transformations can be classified into two types: escaping and non-escaping. This means that successive powers of a given conformal transformation will either send all points to infinity, or none. As an application, we classify the conformal transformations of Einstein's static universe. We also study the question of essentiality in this context, i.e. which conformal transformations are isometric for some metric in the conformal class.

math.DG↗

arXiv2025

Minimal Elements of the Causal Boundary with Applications to Spacetime Splitting

In 1972, Geroch, Kronheimer, and Penrose introduced what is now called the causal boundary of a spacetime. This boundary is constructed out of Terminal Indecomposable Past sets (TIPs) and their future analogues (TIFs), which are the pasts and futures of inextendible causal curves. The causal boundary is a key tool to understand the global structure of a spacetime. In this paper, we show that in a spacetime with compact Cauchy surfaces, there is always at least one minimal TIP and one minimal TIF, minimal meaning that it does not contain another TIP (resp.\ TIF) as a proper subset. We then study the implications of the minimal TIP and TIF meeting each other. This condition generalizes some of the "no observer horizon" conditions that have been used in the literature to obtain partial solutions of the Bartnik splitting conjecture. We also show that such a no observer horizons condition is satisfied when the spacetime has a (possibly discrete) timelike conformal symmetry, generalizing a result of Costa e Silva, Flores, and Herrera about conformal Killing vector fields.

gr-qc↗

arXiv2025

Mean Curvature, Singularities and Time Functions in Cosmology

In this contribution, we study spacetimes of cosmological interest, without making any symmetry assumptions. We prove a rigid Hawking singularity theorem for positive cosmological constant, which sharpens known results. In particular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ in timelike directions and containing a compact Cauchy hypersurface with mean curvature $H \geq n$ is timelike incomplete. We also study the properties of cosmological time and volume functions, addressing questions such as: When do they satisfy the regularity condition? When are the level sets Cauchy hypersurfaces? What can one say about the mean curvature of the level sets? This naturally leads to consideration of Hawking type singularity theorems for Cauchy surfaces satisfying mean curvature inequalities in a certain weak sense.

gr-qc↗

arXiv2024

Time functions on Lorentzian length spaces

In general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by $K$-causality, show that a modified notion of Geroch's volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

gr-qc↗

arXiv2024

Volume Singularities in General Relativity

We propose a new notion of singularity in General Relativity which complements the usual notions of geodesic incompleteness and curvature singularities. Concretely, we say that a spacetime has a volume singularity if there exist points whose future or past has arbitrarily small spacetime volume: in particular, smaller than a Planck volume. From a cosmological perspective, we show that the (geodesic) singularities predicted by Hawking's theorem are also volume singularities. In the black hole setting, we show that volume singularities are always hidden by an event horizon, prompting a discussion of Penrose's cosmic censorship conjecture.

gr-qc↗

arXiv2024

Radial Gravitational Collapse Causes Timelike Incompleteness

We show that a globally hyperbolic spacetime containing a trapped surface and satisfying the strong energy condition and a condition on certain radial tidal forces must be timelike geodesically incomplete. This constitutes a "timelike" version of Penrose's celebrated singularity theorem. Recall that the latter concludes that certain spacetimes are null incomplete, providing the first theoretical evidence that black holes actually exist in our Universe. By concluding timelike instead of null incompleteness, we obtain, at the expense of stronger assumptions, a clearer physical interpretation and the existence of an event horizon.

gr-qc↗

arXiv2024

Global Hyperbolicity through the Eyes of the Null Distance

No Hopf-Rinow Theorem is possible in Lorentzian Geometry. Nonetheless, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of a time function. Our approach is based on the observation that null distances behave particularly well for weak temporal functions in terms of regularity and causality. Specifically, we also show that the null distances of Cauchy temporal functions and regular cosmological time functions encode causality globally.

math.DG↗

arXiv2024

Topology change with Morse functions: progress on the Borde-Sorkin conjecture

Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde and Sorkin proposed a way to avoid this dilemma by considering topology changing spacetimes constructed from Morse functions, where the metric is allowed to vanish at isolated points. They conjectured that these Morse spacetimes are causally continuous (hence quite well behaved), as long as the index of the Morse points is different from $1$ and $n-1$. In this paper, we prove a special case of this conjecture. We also argue, heuristically, that the original conjecture is actually false, and formulate a refined version of it.

gr-qc↗

arXiv2022

Causal bubbles in globally hyperbolic spacetimes

We give an example of a spacetime with a continuous metric which is globally hyperbolic and exhibits causal bubbling. The metric moreover splits orthogonally into a timelike and a spacelike part. We discuss our example in the context of energy conditions and the recently introduced synthetic timelike curvature-dimension (TCD) condition. In particular we observe that the TCD-condition does not, by itself, prevent causal bubbling.

gr-qc↗

arXiv2021

Causality theory of spacetimes with continuous Lorentzian metrics revisited

We consider the usual causal structure $(I^+,J^+)$ on a spacetime, and a number of alternatives based on Minguzzi's $D^+$ and Sorkin and Woolgar's $K^+$, in the case where the spacetime metric is continuous, but not necessarily smooth. We compare the different causal structures based on three key properties, namely the validity of the push-up lemma, the openness of chronological futures, and the existence of limit causal curves. Recall that if the spacetime metric is smooth, $(I^+,J^+)$ satisfies all three properties, but that in the continuous case, the push-up lemma fails. Among the proposed alternative causal structures, there is one that satisfies push-up and open futures, and one that has open futures and limit curves. Furthermore, we show that spacetimes with continuous metrics do not, in general, admit a causal structure satisfying all three properties at once.

gr-qc↗