arXiv subjects

Marc Mars

Publications and source records attributed to Marc Mars.

At least 19 recordsLinked to original sources

arXiv2026

Spacetime Bartnik Mass Positivity and Temporal Monotonicity for Black Holes

We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are: compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.

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arXiv2026

Cosmological Spacetimes with Sign-Changing Spatial Curvature and Topological Transitions

Observational evidence, together with practical computations and modeling, supports a Euclidean spatial sector in the current cosmological model based on the FLRW metric. This, however, would imply that the total amount of matter and energy immediately after the Big Bang must have been infinite, an implication that could only be avoided through a transition from a closed to an open universe, a process forbidden in standard FLRW models. In this article, we investigate the spacetimes resulting from promoting the spatial curvature $k$ in FLRW spacetimes to a time-dependent function, $k \to k(t)$, allowing it to change sign and thereby allowing changes in the topology of the constant-$t$ slices. Although previously dismissed due to a classical theorem by Geroch, such transitions are shown to be consistent with global hyperbolicity when the comoving time is distinct from a Cauchy time, as recent work by one of the authors demonstrates. We construct three distinct geometries exhibiting this behavior using different representations of constant-curvature spaces. We analyze their global properties and identify mild conditions under which they remain globally hyperbolic. Furthermore, we characterize their Killing vectors, proving a general result for spherically symmetric spacetimes and compare them with known geometries in the literature.

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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 $\Omega$ 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.

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arXiv2025

Horizon Data: Existence Results and a Near-Horizon Equation on General Null Hypersurfaces

In a spacetime $(\mathcal{M},g)$, a horizon is a null hypersurface where the deformation tensor $\mathcal{K}:=\pounds_{\eta}g$ of a null and tangent vector $\eta$ satisfies certain restrictions. In this work, we develop a formalism to study the geometry of \textit{general} horizons (i.e. characterized by any $\mathcal{K}$), based on encoding the zeroth and first transverse derivatives of $\mathcal{K}$ on null hypersurfaces detached from any ambient spacetime. We introduce the notions of \textit{$\mathcal{K}$-tuple} and \textit{non-isolation tensor}. The former encodes the order zero of $\mathcal{K}$, while the latter is a symmetric $2$-covariant tensor that codifies the ``degree of isolation" of a horizon. In particular, the non-isolation tensor vanishes for homothetic, Killing and isolated horizons. As an application we derive a \textit{generalized near-horizon equation}, i.e., an identity that holds on any horizon (regardless of its topology or whether it contains fixed points), which relates the non-isolation tensor, a certain torsion one-form, and curvature terms. By restricting this equation to a cross-section one can recover the near-horizon equation of isolated horizons and the master equation of multiple Killing horizons. Our formalism allows us to prove two existence theorems for horizons. Specifically, we establish the necessary and sufficient conditions for a non-degenerate totally geodesic horizon with any prescribed non-isolation tensor to be embeddable in a spacetime satisfying any (non-necessarily $\Lambda$-vacuum) field equations. We treat first the case of arbitrary topology, and then show how the result can be strengthened when the horizon admits a cross-section.

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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.

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arXiv2025

Classification of $\Lambda \neq 0$-vacuum algebraically special spacetimes with conformally flat $\mathscr I$ from Weyl tensor expansion

We introduce a general algebraic decomposition of Riemann-like and Weyl-like tensors with respect to a non-null vector $u$. We derive Gauss, Codazzi and Ricci-type identities for the Weyl tensor, that allow to relate the components of the spacetime Weyl tensor with intrinsic quantities of the hypersurfaces orthogonal to $u$. Restricting to the case of $\Lambda$-vacuum spacetimes (with $\Lambda \neq 0$ and any dimension) admiting a conformal compactification, we then study the behaviour of the Weyl tensor near $\mathscr I$ by means of an asymptotic expansion {\it \`a la} Fefferman-Graham, where the first terms are explicitly computed. We use these tools to characterize four dimensional algebraically special spacetimes with locally conformally flat $\mathscr{I}$, showing they match exactly the so-called {\it Kerr-de Sitter-like class with conformally flat $\scri$}, thus providing a geometric characterization of this class of spacetimes.

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arXiv2025

Killing and homothetic initial data for general hypersurfaces

In this paper we present a collection of general identities relating the deformation tensor $\mathcal{K}=\mathcal{L}_{\eta}g$ of an arbitrary vector field $\eta$ with the tensor $\Sigma=\mathcal{L}_{\eta}\nabla$ 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.

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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 {\Lambda}-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.

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arXiv2024

Marginally outer trapped tubes in de Sitter spacetime

We prove two results which are relevant for constructing marginally outer trapped tubes (MOTTs) in de Sitter spacetime. The first one holds more generally, namely for spacetimes satisfying the null convergence condition and containing a timelike conformal Killing vector with a "temporal function". We show that all marginally outer trapped surfaces (MOTSs) in such a spacetime are unstable. This prevents application of standard results on the propagation of stable MOTSs to MOTTs. On the other hand, it was shown recently that, for every sufficiently high genus, there exists a smooth, complete family of CMC surfaces embedded in the round 3-sphere S3. This family connects a Lawson minimal surface with a doubly covered geodesic 2-sphere. We show by a simple scaling argument that this result translates to an existence proof for complete MOTTs with CMC sections in de Sitter spacetime. Moreover, the area of these sections increases strictly monotonically. We compare this result with an area law obtained before for holographic screens.

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arXiv2024

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 $\Lambda$-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.

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arXiv2024

Null hypersurface data and ambient vector fields: Killing horizons of order zero/one

In this work, we study null hypersurfaces admitting a privileged vector field $\eta$ which is null and tangent at the hypersurface. We derive an identity that relates the deformation tensor of $\eta$ with tensor fields codifying the intrinsic and extrinsic geometry of the hypersurface. This is done without imposing restrictions either on the topology of the hypersurface or on the (possibly empty) subset of points where $\eta$ vanishes. We introduce a generalized notion of surface gravity $\kappa$ that extends smoothly the usual one to the fixed points. We also analyze the properties of the Lie derivative of the Levi-Civita connection along $\eta$. This analysis allows us to introduce three new notions of abstractly defined horizons (i.e. not necessarily viewed as embedded submanifolds) that we then compare with the standard concepts of non-expanding, weakly isolated and isolated horizons. The former generalize the latter to completely general topologies and to horizons admitting fixed points. Finally, we study the structure of the fixed points on these new horizons.

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arXiv2024

Abstract null geometry, energy-momentum map and applications to the constraint tensor

We introduce and study the notion of null manifold. This is a smooth manifold ${\mathcal N}$ endowed with a degenerate metric $\gamma$ with one-dimensional radical at every point. We also define the notion of ruled null manifold, which is a special case of null manifolds. We prove that ruled null manifolds are in one-to-one correspondence with equivalence classes of null metric hypersurface data. This correspondence is used to endow any null manifold $({\mathcal N},\gamma)$ with a family of torsion-free connections related to each other by a well-defined gauge group. The whole construction allows one to define and use geometric notions on arbitrary null manifolds. The paper has a second part where we introduce a canonical map on any null metric hypersurface data and use its algebraic properties to define a canonical decomposition of any symmetric (0,2)-covariant tensor. This decomposition, together with two new differential operators compatible with this splitting, are used to decompose the constraint tensor in full generality and at the purely abstract level. This leads to a hierarchical structure of the (detached) Einstein vacuum null constraint equations without the need of introducing special coordinates or special foliations. The results are applied to study null shells arising from the matching of two spacetimes across null boundaries. The equations governing such objects are obtained in hierarchical form without imposing any topological, gauge or coordinate conditions on the shell.

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arXiv2024

Hypersurface data: General properties and Birkhoff theorem in spherical symmetry

The notions of (metric) hypersurface data were introduced in [Mars,2013] as a tool to analyze, from an abstract viewpoint, hypersurfaces of arbitrary signature in pseudo-riemannian manifolds. In this paper, general geometric properties of these notions are studied. In particular, the properties of the gauge group inherent to the geometric construction are analyzed and the metric hypersurface connection and its corresponding curvature tensor are studied. The results set up the stage for various potential applications. The particular but relevant case of spherical symmetry is considered in detail. In particular, a collection of gauge invariant quantities and a radial covariant derivative is introduced, such that the constraint equations of the Einstein field equations with matter can be written in a very compact form. The general solution of these equations in the vacuum case and Lorentzian ambient signature is obtained, and a generalization of the Birkhoff theorem to this abstract hypersurface setting is derived.

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arXiv2023

On staticity of bifurcate Killing horizons

We show that bifurcate Killing horizons with closed torsion form, in spacetimes of arbitrary dimension satisfying a Ricci-structure condition, arise from static Killing vectors. The result applies in particular to $\Lambda$-vacuum spacetimes.

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arXiv2023

The constraint tensor for null hypersurfaces

In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold $S$ of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to $S$ takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.

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arXiv2023

Abstract Formulation of the Spacetime Matching Problem and Null Thin Shells

The formalism of hypersurface data is a framework to study hypersurfaces of any causal character abstractly (i.e. without the need of viewing them as embedded in an ambient space). In this paper we exploit this formalism to study the general problem of matching two spacetimes in a fully abstract manner, as this turns out to be advantageous over other approaches in several respects. We then concentrate on the case when the boundaries are null and prove that the whole matching is determined by a diffeomorphism $\varphi$ on the abstract data set. By exploiting the gauge structure of the formalism we find explicit expressions for the gravitational/matter-energy content of any null thin shell. The results hold for arbitrary topology. A particular case of interest is when more than one matching is allowed. Assuming that one such matchings has already been solved, we provide explicit expressions for the gravitational/matter-energy content of any other shell in terms of the known one. This situation covers, in particular, all cut-and-paste constructions, where one can simply take as known matching the trivial re-attachment of the two regions. We include, as an example, the most general matching of two regions of the (anti-)de Sitter or Minkowski spacetime across a totally geodesic null hypersurface.

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arXiv2023

New characterization of Robertson-Walker geometries involving a single timelike curve

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Roberson-Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow.

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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.

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