arXiv subjects

Edgar Gasperin

Publications and source records attributed to Edgar Gasperin.

At least 19 recordsLinked to original sources

arXiv2026

The linearised conformal Einstein field equations around a Petrov-type~D spacetime: the conformal Teukolsky equation

While the Teukolsky equation plays a central role in traditional treatments of perturbations of algebraically special spacetimes, its relation to Friedrich's conformal Einstein field equations (CEFEs) remains largely unexplored. Here we develop a conformal formulation of black-hole perturbation theory based on the CEFEs and derive the conformal Teukolsky equation. Starting from a transparent review of Friedrich's regularisation strategy, this work establishes a direct connection between mainstream curvature-based linear perturbation theory and conformal formulations of general relativity. This perspective is timely given the growing relevance of hyperboloidal frameworks in black-hole perturbation theory, where conformal compactification is introduced at the level of an already linearised effective wave equation. Here instead, the conformal factor is a dynamical variable within the field equations. In the non-linear equations there is a coupling between conformal and curvature perturbations; however, when linearised around a Petrov-type D background, the conformal factor decouples from the equations governing the Newman-Penrose components $\phi_0$ and $\phi_4$ of the rescaled Weyl tensor. The resulting equation preserves the structural form of the classical Teukolsky equation while remaining regular at the conformal boundary. This provides a geometric interpretation of the hyperboloidal master variable and an entry point into the CEFE framework. We further derive the conformal Teukolsky equation for a conformal representation of Kerr spacetime where spatial infinity is realised as a blown-up cylinder. By bridging conformal and traditional approaches to black-hole perturbation theory, the framework highlights a geometrically regular representation of perturbative dynamics that may inform extensions beyond the linear regime.

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arXiv2025

Asymptotic charges of a quadrupolar naked singularity

The purpose of this article is to compute the asymptotic charges of a vacuum solution to the Einstein field equations describing a naked singularity with a non-vanishing quadrupole moment, known in the literature as the Zipoy-Voorhees spacetime (q-metric). In addition to the well-known asymptotic quantities such as the Bondi-Sachs energy-momentum, the BMS charges and NP constants of this spacetime are computed. Explicit calculations of the latter are relatively scarce in the literature. Moreover, it has been proven that the NP constants of asymptotically flat, stationary, vacuum, and algebraically special spacetimes vanish (for instance, those of the Kerr spacetime). A by-product of the present analysis is to show that the algebraically special condition in the aforementioned result appears to be crucial, since the q-metric provides a counterexample to the conjecture that all asymptotically flat, stationary, vacuum, and asymptotically algebraically special spacetimes (a weaker version of the algebraically special condition) have vanishing NP constants.

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arXiv2025

An invariant measure of deviation from Petrov type D at the level of initial data

In this article, we describe a simple covariant characterisation of initial data sets which give rise to Petrov type D vacuum spacetime developments. As an application, we derive an integral invariant which, when restricted to the appropriate class of asymptotically Euclidean initial data sets, vanishes if and only if the initial dataset is isometric to initial data for the Kerr spacetime. As such, the invariant can be considered a measure of non-Kerrness on such initial data sets. In contrast with other similar invariants constructed through the notion of 'approximate Killing spinors', the present invariant is algebraic in the sense that it is algorithmically computable directly from initial data without having to solve any PDEs on the initial data hypersurface.

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arXiv2024

Twistor initial data characterisation of pp-waves

This note gives a concise derivation of a twistor-initial-data characterisation of pp-wave spacetimes in vacuum. The construction is based on a similar calculation for the Minkowski spacetime in [Class. Quantum Grav. 28 075010]. The key difference is that for the Minkowski spacetime a necessary condition is that $\nabla_{A}{}^{A'}\bar{\kappa}_{A'} \neq 0$. In this note it is shown that if $\nabla_{A}{}^{A'}\bar{\kappa}_{A'}=0$ then the development is a pp-wave spacetime. Furthermore, it is shown that such condition propagates off the initial hypersurface, which, in turn, gives a twistor initial data characterisation of pp-waves.

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arXiv2023

Spin-0 fields and the NP-constants close to spatial infinity in Minkowski spacetime

The NP constants of massless spin-0 fields propagating in Minkowski spacetime are computed close to spatial and null infinity by means of Friedrich's \emph{$i^0$-cylinder}. Assuming certain regularity condition on the initial data ensuring that the field extends analytically to the critical sets, it is shown that the NP constants at future $\mathscr{I}^{+}$ and past null infinity $\mathscr{I}^{-}$ are independent of each other. In other words, the classical NP constants at $\mathscr{I}^{\pm}$ stem from different parts of the initial data given on a Cauchy hypersurface. In contrast, it is shown that, using a slight generalisation of the classical NP constants, the associated quantities ($i^0$-cylinder NP constants) do not require the regularity condition being satisfied and give rise to conserved quantities at $\mathscr{I}^{\pm}$ that are determined by the same piece of initial data which, in turn, correspond to the terms controlling the regularity of the field. Additionally, it is shown how the conservation laws associated to the NP constants can be exploited to construct, in flat space, heuristic asymptotic-system expansions which are sensitive to the logarithmic terms at the critical sets.

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arXiv2023

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in Generalised Harmonic Gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background -- the inversion (conformal) representation of the Minkowski spacetime -- and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

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arXiv2023

Polyhomogeneous spin-0 fields in Minkowski spacetime

The asymptotic behaviour of massless spin-0 fields close to spatial and null infinity in Minkowski spacetime is studied by means of Friedrich's cylinder at spatial infinity. The results are applied to a system of equations called the good-bad-ugly which serves as a model for the Einstein field equations in generalised harmonic gauge. The relation between the logarithmic terms (polyhomogeneity) appearing in the solution obtained using conformal methods and those obtained by means of a heuristic method based on Hörmander's asymptotic system is discussed. This review article is based on Class. Quantum Grav. 40 055002 and arXiv:2304.11950.

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arXiv2022

The good-bad-ugly system near spatial infinity on flat spacetime

A system of equations that serves as a model for the Einstein field equation in generalised harmonic gauge called the good-bad-ugly system is studied in the region close to null and spatial infinity in Minkowski spacetime. This analysis is performed using H. Friedrich's cylinder construction at spatial infinity and defining suitable conformally rescaled fields. The results are translated to the physical set up to investigate the relation between the polyhomogeneous expansions arising from the analysis of linear fields using the $i^0$-cylinder framework and those obtained through a heuristic method based on Hörmander's asymptotic system.

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arXiv2022

Peeling in Generalized Harmonic Gauge

It is shown that a large class of systems of non-linear wave equations, based on the good-bad-ugly model, admit formal solutions with polyhomogeneous expansions near null infinity. A particular set of variables is introduced which allows us to write the Einstein field equations in generalized harmonic gauge as a good-bad-ugly system and the functional form of the first few orders in such an expansion is found by applying the aforementioned result. Exploiting these formal expansions of the metric components, the peeling property of the Weyl tensor is revisited. The question addressed is whether or not the use of generalized harmonic gauge, by itself, causes a violation of peeling. Working in harmonic gauge, it is found that log-terms that prevent the Weyl tensor from peeling do appear. The impact of gauge source functions and constraint additions on the peeling property is then considered. Finally, the special interplay between gauge and constraint addition, as well as its influence on the asymptotic system and the decay of each of the metric components, is exploited to find a particular gauge which suppresses this specific type of log-term to arbitrarily high order.

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arXiv2022

The conformal Killing spinor initial data equations

We obtain necessary and sufficient conditions for an initial data set for the vacuum conformal Einstein field equations to give rise to a spacetime development in possession of a Killing spinor. The fact that the conformal Einstein field equations are used in our derivation allows for the possibility of the initial hypersurface $\mathcal{S}$ intersecting non-trivially with (or even being a subset of) null infinity $\mathscr{I}$. For conciseness, these conditions are derived assuming that the initial hypersurface is spacelike. Hence, in particular, these conformal Killing spinor initial data equations encode necessary and sufficient conditions for the existence of a Killing spinor in the development of asymptotic initial data on spacelike components of $\mathscr{I}$.

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arXiv2021

Energy scales and black hole pseudospectra: the structural role of the scalar product

A pseudospectrum analysis has recently provided evidence of a potential generic instability of black hole (BH) quasinormal mode (QNM) overtones under high-frequency perturbations. Such instability analysis depends on the assessment of the size of perturbations. This is encoded in the scalar product and its choice is not unique. We address the impact of the scalar product choice, founding it on the physical energy scales of the problem. The article is organized in three parts: basics, applications and heuristic proposals. In the first part, we revisit the energy scalar product used in the hyperboloidal approach to QNMs, extending previous effective analyses and placing them on solid spacetime basis. The second part focuses on applications of the scalar product in the QNM problem: i) we demonstrate that the QNM instability is not an artifact of previous spectral numerical schemes, by implementing a finite elements calculation from a weak formulation; ii) using Keldysh's theorem, we provide QNM resonant expansions for the gravitational waveform, with explicit expressions of the expansion coefficients; iii) we propose the notion of "epsilon-dual QNM expansions" to exploit BH QNM instability in BH spectroscopy, exploiting both non-perturbed and perturbed QNMs, the former informing on large scales and the latter probing small scales. The third part enlarges the conceptual scope of BH QNM instability proposing: a) spiked perturbations are more efficient in triggering BH QNM instabilities than smooth ones, b) a general picture of the BH QNM instability problem is given, supporting the conjecture (built on Burnett's conjecture) that Nollert-Price branches converge universally to logarithmic Regge branches in the high-frequency limit and c) aiming at a fully geometric description of QNMs, BMS states are hinted as possible asymptotic/boundary degrees of freedom for an inverse scattering problem.

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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

High Order Asymptotic Expansions of a Good-Bad-Ugly Wave Equation

A heuristic method to find asymptotic solutions to a system of non-linear wave equations near null infinity is proposed. The non-linearities in this model, dubbed good-bad-ugly, are known to mimic the ones present in the Einstein field equations (EFE) and we expect to be able to exploit this method to derive an asymptotic expansion for the metric in General Relativity (GR) close to null infinity that goes beyond first order as performed by Lindblad and Rodnianski for the leading asymptotics. For the good-bad-ugly model, we derive formal expansions in which terms proportional to the logarithm of the radial coordinate appear at every order in the bad field, from the second order onward in the ugly field but never in the good field. The model is generalized to wave operators built from an asymptotically flat metric and it is shown that it admits polyhomogeneous asymptotic solutions. Finally we define stratified null forms, a generalization of standard null forms, which capture the behavior of different types of field, and demonstrate that the addition of such terms to the original system bears no qualitative influence on the type of asymptotic solutions found.

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arXiv2020

Zero rest-mass fields and the Newman-Penrose constants on flat space

Zero rest-mass fields of spin 1 (the electromagnetic field) and spin 2 propagating on flat space and their corresponding Newman-Penrose (NP) constants are studied near spatial infinity. The aim of this analysis is to clarify the correspondence between data for these fields on a spacelike hypersurface and the value of their corresponding NP constants at future and past null infinity. To do so, Friedrich's framework of the cylinder at spatial infinity is employed to show that, expanding the initial data in terms spherical harmonics and powers of the geodesic spatial distance $ρ$ to spatial infinity, the NP constants correspond to the data for the second highest possible spherical harmonic at fixed order in $ρ$. In addition, it is shown that for generic initial data within the class considered in this article, there is no natural correspondence between the NP constants at future and past null infinity ---for both the Maxwell and spin-2 field. However, if the initial data is time-symmetric then the NP constants at future and past null infinity have the same information.

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arXiv2019

The Hyperboloidal Numerical Evolution of a Good-Bad-Ugly Wave Equation

One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a naive choice of evolved variables. Here we consider a system of wave equations that satisfy the weak-null condition and serve as a model system with similar nonlinearities to those present in the Einstein field equations in generalized harmonic gauge. We show that these equations can be explicitly regularized by a nonlinear change of variables. Working in spherical symmetry, a numerical implementation of this model using compactified hyperboloidal slices is then presented. Clean convergence is found for the regularized system. Although more complicated, it is expected that general relativity can be treated similarly.

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arXiv2018

The Weak Null Condition in Free-evolution Schemes for Numerical Relativity: Dual Foliation GHG with Constraint Damping

All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expected rates of decay of the metric near null-infinity in a free-evolution setting. We show, within the asymptotic expansion, that by carefully modifying the non-principal part of the field equations by the addition of constraints, we are able to obtain optimal decay rates even when the constraints are violated. The light-speed condition can hence be satisfied, which paves the way for the explicit numerical treatment of future null-infinity. We then study the behavior of the Trautman-Bondi mass under the decay results predicted by the asymptotic expansion. Naively the mass seems to be unbounded, but we see first that the divergent terms can be replaced with a combination of the constraints and the Einstein field equations, and second that the Bondi mass loss formula is recovered within the framework. Both of the latter results hold in the presence of small constraint violations.

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arXiv2017

Conformal geodesics in spherically symmetric vacuum spacetimes with Cosmological constant

An analysis of conformal geodesics in the Schwarzschild-de Sitter and Schwarzschild-anti de Sitter families of spacetimes is given. For both families of spacetimes we show that initial data on a spacelike hypersurface can be given such that the congruence of conformal geodesics arising from this data cover the whole maximal extension of canonical conformal representations of the spacetimes without forming caustic points. For the Schwarzschild-de Sitter family, the resulting congruence can be used to obtain global conformal Gaussian systems of coordinates of the conformal representation. In the case of the Schwarzschild-anti de Sitter family, the natural parameter of the curves only covers a restricted time span so that these global conformal Gaussian systems do not exist.

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arXiv2017

Polyhomogeneous expansions from time symmetric initial data

We make use of Friedrich's construction of the cylinder at spatial infinity to relate the logarithmic terms appearing in asymptotic expansions of components of the Weyl tensor to the freely specifiable parts of time symmetric initial data sets for the Einstein field equations. Our analysis is based on the assumption that a particular type of formal expansions near the cylinder at spatial infinity corresponds to the leading terms of actual solutions to the Einstein field equations. In particular, we show that if the Bach tensor of the initial conformal metric does not vanish at the point at infinity then the most singular component of the Weyl tensor decays near null infinity as $O(\tilde{r}^{-3}\ln \tilde{r})$ so that spacetime will not peel. We also provide necessary conditions on the initial data which should lead to a peeling spacetime. Finally, we show how to construct global spacetimes which are candidates for non-peeling polyhomogeneous) asymptotics.

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