arXiv subjects

Gabriel Sánchez-Pérez

Publications and source records attributed to Gabriel Sánchez-Pérez.

8 recordsLinked to original sources

arXiv2026

Abstract null hypersurfaces and characteristic initial value problems in General Relativity

This thesis is framed within the field of Mathematical Relativity and is organized into six chapters. After an introduction to the topic in Chapter 1, Chapter 2 reviews and further develops the formalism of hypersurface data, which provides the unifying framework for the entire thesis. In Chapter 3 we study the characteristic Cauchy problem from a fully detached perspective. Chapter 4 is devoted to the Killing initial data problem, also analyzed within this detached framework. In Chapter 5 we investigate the transverse (or asymptotic) expansion of the metric at a general null hypersurface. Finally, Chapter 6 addresses the geometry of conformal null infinity.

gr-qc↗

arXiv2026

Transverse expansion of the metric at null infinity

In this paper we analyze the conformal Einstein equations to all orders at null infinity without imposing any restriction on the spacetime dimension, the topology of $\mathscr{I}$, or fall-off conditions for the Weyl tensor. In particular, we study how the equations constrain the geometry of null infinity when it is assumed to be foliated by cross-sections, not necessarily spheres. Our approach is coordinate-free and treats the conformal factor $Ω$ as a dynamical variable. After identifying the free data at $\mathscr{I}$, we show that any two asymptotically flat spacetimes sharing the same free data at null infinity are necessarily isometric to infinite order. In addition, we provide a detached definition of null infinity and prove an existence theorem for asymptotically flat spacetimes solving the field equations to infinite order at $\mathscr{I}$ realizing the prescribed initial data.

gr-qc↗

arXiv2025

Conformal characterization of the Fefferman-Graham ambient metric

In this paper, we study the asymptotic structure of the Fefferman-Graham ambient metric. We prove that every straight ambient metric admits a conformal completion with a well-defined null infinity, and that the asymptotic expansion of the metric at infinity can be related to that at the homothetic horizon. Furthermore, in even dimensions, we show that the Fefferman-Graham obstruction tensor naturally arises in the geometry at infinity. By identifying the fundamental properties that this particular conformal extension exhibits, and analyzing their sufficiency, we arrive at the main result of the paper, namely the identification of a set of conformally covariant conditions that completely characterize the ambient metric from a conformal perspective. In particular, our result relaxes the requirement of the homothety one-form being exact.

gr-qc↗

arXiv2025

Transverse expansion of the metric at null hypersurfaces II. Existence results and application to Killing horizons

This paper finishes the series of two papers that we started with [arXiv:2405.05377], where we analyzed the transverse expansion of the metric at a general null hypersurface. While [arXiv:2405.05377] focused on uniqueness results, here we show existence of ambient manifolds given the full asymptotic expansion at the null hypersurface. When such expansion fulfills a set of "constraint equations" we prove that the ambient manifold solves the Einstein equations to infinite order at the hypersurface. Our approach does not make any assumptions regarding the dimension or topology of the null hypersurface and is entirely covariant. Furthermore, when the hypersurface exhibits a product topology we find the minimum amount of data on a cross-section that ensures the existence of an ambient space solving the Einstein equations to infinite order on the hypersurface. As an application we recall the notion of abstract Killing horizon data (AKH) introduced in [arXiv:2405.05377], namely the minimal data needed to define a non-degenerate Killing horizon from a detached viewpoint, and we prove that every AKH of arbitrary dimension and topology gives rise to an ambient space solving the Λ-vacuum equations to infinite order and with the given data as Killing horizon. Our result also includes the possibility of the Killing vector having zeroes at the horizon.

gr-qc↗

arXiv2025

Killing and homothetic initial data for general hypersurfaces

In this paper we present a collection of general identities relating the deformation tensor $\mathcal{K}=\mathcal{L}_ηg$ of an arbitrary vector field $η$ with the tensor $Σ=\mathcal{L}_η\nabla$ on an abstract hypersurface $\mathcal{H}$ of any causal character. As an application we establish necessary conditions on $\mathcal{H}$ for the existence of a homothetic Killing vector on the spacetime where $\mathcal{H}$ is embedded. The sufficiency of these conditions is then analysed in three specific settings. For spacelike hypersurfaces, we recover the well-known homothetic KID equations [10, 13] in the language of hypersurface data. For two intersecting null hypersurfaces, we generalize a previous result [7], valid for Killings, to the homothetic case and, moreover, demonstrate that the equations can be formulated solely in terms of the initial data for the characteristic Cauchy problem, i.e., without involving a priori spacetime quantities. This puts the characteristic KID problem on equal footing with the spacelike KID problem. Furthermore, we highlight the versatility of the formalism by addressing the homothetic KID problem for smooth spacelike-characteristic initial data. Other initial value problems, such as the spacelike-characteristic with corners, can be approached similarly.

gr-qc↗

arXiv2025

Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons

This is the first in a series of two papers with sequel [arXiv:2501.03983] where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a non-degenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in $Λ$-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.

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arXiv2023

Covariant definition of Double Null Data and geometric uniqueness of the characteristic initial value problem

The characteristic Cauchy problem of the Einstein field equations has been recently addressed from a completely abstract viewpoint by means of hypersurface data and, in particular, via the notion of double null data. However, this definition was given in a partially gauge-fixed form. In this paper we generalize the notion of double null data in a fully diffeomorphism and gauge covariant way, and show that the definition is complete by proving that no extra conditions are needed to embed the double null data in some spacetime. The second aim of the paper is to show that the characteristic Cauchy problem satisfies a geometric uniqueness property. Specifically, we introduce a natural notion of isometry at the abstract level such that two double null data that are isometric in this sense give rise to isometric spacetimes.

gr-qc↗

arXiv2023

Double Null Data and the Characteristic Problem in General Relativity

General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in General Relativity.

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