arXiv subjects

Juan A. Valiente Kroon

Publications and source records attributed to Juan A. Valiente Kroon.

At least 19 recordsLinked to original sources

arXiv2026

Controlled regularity at future null infinity from past asymptotic initial data: the wave equation

We study the relationship between asymptotic characteristic initial data for the wave equation at past null infinity and the regularity of the solution at future null infinity on the Minkowski spacetime. By constructing estimates on a causal rectangle reaching the conformal boundary, we prove that the solution admits an asymptotic expansion near null and spatial infinity whose regularity is controlled quantitatively in terms of the regularity of the data at past null infinity. In particular, our method gives rise to solutions to the wave equation in a neighbourhood of spatial infinity satisfying the peeling behaviour, for data on past null infinity with non-compact support. Our approach makes use of Friedrich's conformal representation of spatial infinity in which we prove delicate non-degenerate Grönwall estimates. We describe the relationship between the solution and the data both in terms of Friedrich's conformal coordinates and the usual physical coordinates on Minkowski space.

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arXiv2026

Wellposedness of the initial boundary value problem for the conformal field equations

We provide a formulation of the initial boundary value problem for Friedrich's extended conformal Einstein field equations in which boundary data is prescribed on a timelike hypersurface located at a finite position in the spacetime. Our construction relies on a gauge based on the properties of conformal geodesics and requires the the boundary is ruled by timelike conformal geodesics. The consequences of this assumption on the timelike boundary are analysed and we identify a subset of maximally dissipative boundary conditions which are consistent with this assumption. For this class of consistent boundary conditions we establish the wellposedness of the initial boundary value problem and prove the propagation of the constraints.

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arXiv2025

The space spinor formalism and estimates for spinor fields

We show how the space spinor formalism for 2-component spinors can be used to construct estimates for spinor fields satisfying first order equations. We discuss the connection of the approach presented in this article with other strategies for the construction of estimates. In addition, we recast several concepts related to the notion of hyperbolicity in the context of spinor equations. The approach described in this article can be regarded as an adaptation to first order equations of the method of positive commutators for second order hyperbolic equations.

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arXiv2024

An asymptotic characterisation of the Kerr spacetime

We provide a characterisation of the Kerr spacetime close to future null infinity using the asymptotic characteristic initial value problem in a conformally compactified spacetime. Stewart's gauge is used to set up the past-oriented characteristic initial value problem. By a theorem of M. Mars characterising the Kerr spacetime, we provide conditions for the existence of an asymptotically timelike Killing vector on the development of the initial data by demanding that the spacetime is endowed with a Killing spinor. The conditions on the characteristic initial data ensuring the existence of a Killing spinor are, in turn, analysed. Finally, we write the conditions on the initial data in terms of the free data in the characteristic initial value problem. As a result, we characterise the Kerr spacetime using only a section of future null infinity and its intersection with an outgoing null hypersurface.

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arXiv2024

The ďAlembert solution in hyperboloidal foliations

We explicitly construct the analogue of the ďAlembert solution to the 1+1 wave equation in an hyperboloidal setting. This hyperboloidal ďAlembert solution is used, in turn, to gain intuition into the behaviour of solutions to the wave equation in a hyperboloidal foliation and to explain some apparently anomalous behaviour observed in numerically constructed solutions discussed in the literature.

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arXiv2024

New spinorial mass-quasilocal angular momentum inequality for initial data with marginally future trapped surface

We prove a new geometric inequality that relates the Arnowitt-Deser-Misner mass of initial data to a quasilocal angular momentum of a marginally future trapped surface inner boundary. The inequality is expressed in terms of a 1-spinor, which satisfies an intrinsic first-order Dirac-type equation. Furthermore, we show that if the initial data is axisymmetric, then the divergence-free vector used to define the quasilocal angular momentum cannot be a Killing field of the generic boundary.

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arXiv2024

BMS-supertranslation charges at the critical sets of null infinity

For asymptotically flat spacetimes, a conjecture by Strominger states that asymptotic BMS-supertranslations and their associated charges at past null infinity $\mathscr{I}^{-}$ can be related to those at future null infinity $\mathscr{I}^{+}$ via an antipodal map at spatial infinity $i^{0}$. We analyse the validity of this conjecture using Friedrich's formulation of spatial infinity, which gives rise to a regular initial value problem for the conformal field equations at spatial infinity. A central structure in this analysis is the cylinder at spatial infinity representing a blow-up of the standard spatial infinity point $i^{0}$ to a 2-sphere. The cylinder touches past and future null infinities $\mathscr{I}^{\pm}$ at the critical sets. We show that for a generic class of asymptotically Euclidean and regular initial data, BMS-supertranslation charges are not well-defined at the critical sets unless the initial data satisfies an extra regularity condition. We also show that given initial data that satisfy the regularity condition, BMS-supertranslation charges at the critical sets are fully determined by the initial data and that the relation between the charges at past null infinity and those at future null infinity directly follows from our regularity condition.

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arXiv2023

Hyperboloidal discontinuous time-symmetric numerical algorithm with higher order jumps for gravitational self-force computations in the time domain

Within the next decade the Laser Interferometer Space Antenna (LISA) is due to be launched, providing the opportunity to extract physics from stellar objects and systems, such as \textit{Extreme Mass Ratio Inspirals}, (EMRIs) otherwise undetectable to ground based interferometers and Pulsar Timing Arrays (PTA). Unlike previous sources detected by the currently available observational methods, these sources can \textit{only} be simulated using an accurate computation of the gravitational self-force. Whereas the field has seen outstanding progress in the frequency domain, metric reconstruction and self-force calculations are still an open challenge in the time domain. Such computations would not only further corroborate frequency domain calculations and models, but also allow for full self-consistent evolution of the orbit under the effect of the self-force. Given we have \textit{a priori} information about the local structure of the discontinuity at the particle, we will show how to construct discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae and hence recover higher order accuracy. In this work we demonstrate how this technique in conjunction with well-suited gauge choice (hyperboloidal slicing) and numerical (discontinuous collocation with time symmetric) methods can provide a relatively simple method of lines numerical algorithm to the problem. This is the first of a series of papers studying the behaviour of a point-particle prescribing circular geodesic motion in Schwarzschild in the \textit{time domain}. In this work we describe the numerical machinery necessary for these computations and show not only our work is capable of highly accurate flux radiation measurements but it also shows suitability for evaluation of the necessary field and it's derivatives at the particle limit.

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arXiv2023

Controlled regularity at future null infinity from past asymptotic initial data: massless fields

We study the relationship between asymptotic characteristic initial data at past null infinity and the regularity of solutions at future null infinity for the massless linear spin-s field equations on Minkowski space. By quantitatively controlling the solutions on a causal rectangle reaching the conformal boundary, we relate the (generically singular) behaviour of the solutions near past null infinity, future null infinity, and spatial infinity. Our analysis uses Friedrich's cylinder at spatial infinity together with a careful Grönwall-type estimate that does not degenerate at the intersection of null infinity and the cylinder (the so-called critical sets).

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arXiv2023

Asymptotic charges for spin-1 and spin-2 fields at the critical sets of null infinity

The asymptotic charges of spin-1 and spin-2 fields are studied near spatial infinity. We evaluate the charges at the critical sets where spatial infinity meets null infinity with the aim of finding the relation between the charges at future and past null infinity. To this end, we make use of Friedrich's framework of the cylinder at spatial infinity to obtain asymptotic expansions of the Maxwell and spin-2 fields near spatial infinity, which are fully determined in terms of initial data on a Cauchy hypersurface. Expanding the initial data in terms of spin-weighted spherical harmonics, it is shown that only a subset of the initial data, that satisfies certain regularity conditions, gives rise to well-defined charges at the point where future (past) infinity meets spatial infinity. Given such initial data, the charges are shown to be fully expressed in terms of the freely specifiable part of the data. Moreover, it is shown that there exists a natural correspondence between the charges defined at future and past null infinity.

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arXiv2023

A comparison of Ashtekar's and Friedrich's formalisms of spatial infinity

Penrose's idea of asymptotic flatness provides a framework for understanding the asymptotic structure of gravitational fields of isolated systems at null infinity. However, the studies of the asymptotic behaviour of fields near spatial infinity are more challenging due to the singular nature of spatial infinity in a regular point compactification for spacetimes with non-vanishing ADM mass. Two different frameworks that address this challenge are Friedrich's cylinder at spatial infinity and Ashtekar's definition of asymptotically Minkowskian spacetimes at spatial infinity that give rise to the 3-dimensional asymptote at spatial infinity $\mathcal{H}$. Both frameworks address the singularity at spatial infinity although the link between the two approaches had not been investigated in the literature. This article aims to show the relation between Friedrich's cylinder and the asymptote as spatial infinity. To do so, we initially consider this relation for Minkowski spacetime. It can be shown that the solution to the conformal geodesic equations provides a conformal factor that links the cylinder and the asymptote. For general spacetimes satisfying Ashtekar's definition, the conformal factor cannot be determined explicitly. However, proof of the existence of this conformal factor is provided in this article. Additionally, the conditions satisfied by physical fields on the asymptote $\mathcal{H}$ are derived systematically using the conformal constraint equations. Finally, it is shown that a solution to the conformal geodesic equations on the asymptote can be extended to a small neighbourhood of spatial infinity by making use of the stability theorem for ordinary differential equations. This solution can be used to construct a conformal Gaussian system in a neighbourhood of $\mathcal{H}$.

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arXiv2023

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner (ADM) mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the $1+1+2$ decomposition of spinorial equations.

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arXiv2023

Trapped surface formation for the Einstein-Scalar system

We consider the formation of trapped surfaces in the evolution of the Einstein-scalar field system without symmetries. To this end, we follow An's strategy to analyse the formation of trapped surfaces in vacuum and for the Einstein-Maxwell system. Accordingly, we set up a characteristic initial value problem (CIVP) for the Einstein-Scalar system with initial data given on two intersecting null hypersurfaces such that on the incoming slince the data is Minkowskian whereas on the outgoing side no symmetries are assumed. We obtain a scale-critical semi-global existence result by assigning a signature for decay rates to both the geometric quantities and the scalar field. The analysis makes use of a gauge due to J. Stewart and an adjustment of the Geroch-Held-Penrose (GHP) formalism, the T-weight formalism, which renders the connection between the Newman-Penrose (NP) quantities and the PDE analysis systematic and transparent.

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arXiv2022

Dain's invariant for black hole initial data

Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that approximate symmetries coincide with actual symmetries when they are present allows one to construct geometric invariants characterising the symmetry. In this paper, we extend Dain's construction of geometric invariants characterising stationarity to the case of initial data sets for the Einstein equations corresponding to black hole spacetimes. We prove the existence and uniqueness of solutions to a boundary value problem showing that one can always find approximate Killing vectors in black hole spacetimes and these coincide with actual Killing vectors when they are present. In the time-symmetric setting we make use of a 2+1 decomposition to construct a geometric invariant on a MOTS that vanishes if and only if the Killing initial data equations are locally satisfied.

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arXiv2022

Bach equation and the matching of spacetimes in conformal cyclic cosmology models

We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -- one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the use of the Bach equation. This equation is regular at the past conformal infinity for conformally flat and conformally Einstein spacetimes, so we will mostly focus on them here. An example of the electrovacuum spacetime which does not fall into this class and has regular conformal Bach tensor will be discussed in the appendix.

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arXiv2022

Conservative Evolution of Black Hole Perturbations with Time-Symmetric Numerical Methods

The scheduled launch of the LISA Mission in the next decade has called attention to the gravitational self-force problem. Despite an extensive body of theoretical work, long-time numerical computations of gravitational waves from extreme-mass-ratio-inspirals remain challenging. This work proposes a class of numerical evolution schemes suitable to this problem based on Hermite integration. Their most important feature is time-reversal symmetry and unconditional stability, which enables these methods to preserve symplectic structure, energy, momentum and other Noether charges over long time periods. We apply Noether's theorem to the master fields of black hole perturbation theory on a hyperboloidal slice of Schwarzschild spacetime to show that there exist constants of evolution that numerical simulations must preserve. We demonstrate that time-symmetric integration schemes based on a 2-point Taylor expansion (such as Hermite integration) numerically conserve these quantities, unlike schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This makes time-symmetric schemes ideal for long-time EMRI simulations.

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arXiv2021

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

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arXiv2021

The conformal Einstein field equations and the local extension of future null infinity

We make use of an improved existence result for the characteristic initial value problem for the conformal Einstein equations to show that given initial data on two null hypersurfaces $\mathcal{N}_\star$ and $\mathcal{N}'_\star$ such that the conformal factor (but not its gradient) vanishes on a section of $\mathcal{N}_\star$ one recovers a portion of null infinity. This result combined with the theory of the hyperboloidal initial value problem for the conformal Einstein field equations allows to show the semi-global stability of the Minkowski spacetime from characteristic initial data.

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