arXiv subjects

Marica Minucci

Publications and source records attributed to Marica Minucci.

8 recordsLinked to original sources

arXiv2026

Bilinear products and the orthogonality of quasinormal modes on hyperboloidal foliations

We explore the properties of bilinear products for black-hole quasinormal modes (QNMs) formulated on hyperboloidal foliations. We find that, although QNM solutions are smooth and finite on future-directed hyperboloids, the integrand of the bilinear form with respect to which the modes are orthogonal is still divergent. This is a result of the reflection (equivalently, CPT) transformation required in the definition of the products, which modifies the behaviour of the integrand at the boundaries. We present several regularisation procedures that yield a finite and well-defined bilinear form. In addition, we examine an alternative definition of the bilinear products that incorporates flux contributions, discussing its advantages and limitations. Finally, we define the QNM excitation factors and coefficients within the hyperboloidal framework in terms of the bilinear products, and compute them explicitly for a choice of mode numbers and constant initial data. For concreteness, we work with the QNMs associated to scalar perturbations of the Schwarzschild family of spacetimes.

gr-qc↗

arXiv2025

The Confluent Heun functions in Black Hole Perturbation Theory: a spacetime interpretation

This work provides a spacetime interpretation of the confluent Heun functions within black hole perturbation theory (BHPT) and explores their relationship to the hyperboloidal framework. In BHPT, the confluent Heun functions are solutions to the radial Teukolsky equation, but they are traditionally studied without an explicit reference to the underlying spacetime geometry. Here, we show that the distinct behaviour of these functions near their singular points reflects the structure of key surfaces in black hole spacetimes. By interpreting homotopic transformations of the confluent Heun functions as changes in the spacetime foliation, we connect these solutions to different regions of the black hole's global structure, such as the past and future event horizons, past and future null infinity, spatial infinity, and even past and future timelike infinity. We also discuss the relationship between the confluent Heun functions and the hyperboloidal formulation of the Teukolsky equation. Although neither confluent Heun form of the radial Teukolsky equation can be interpreted as hyperboloidal slices, this approach offers new insights into wave propagation and scattering from a global black hole spacetime perspective.

gr-qc↗

arXiv2024

A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature

In this article, it is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of de Sitter-like spacetimes with spatial sections of negative scalar curvature. This class of spacetimes admits a smooth conformal extension with a space-like conformal boundary. Central to the analysis is the use of conformal Gaussian systems to obtain a hyperbolic reduction of the conformal Einstein field equations for which standard Cauchy stability results for symmetric hyperbolic systems can be employed. The use of conformal methods allows us to rephrase the question of the global existence of solutions to the Einstein field equations into considerations of finite existence time for the conformal evolution system.

gr-qc↗

arXiv2023

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows us to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.

gr-qc↗

arXiv2023

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Class. Quantum Grav. 38 145026 and Class. Quantum Grav. 40 145005.

gr-qc↗

arXiv2022

The Maxwell-scalar field system near spatial infinity

We make use of Friedrich's representation of spatial infinity to study asymptotic expansions of the Maxwell-scalar field system near spatial infinity. The main objective of this analysis is to understand the effects of the non-linearities of this system on the regularity of solutions and polyhomogeneous expansions at null infinity and, in particular, at the critical sets where null infinity touches spatial infinity. The main outcome from our analysis is that the nonlinear interaction makes both fields more singular at the conformal boundary than what is seen when the fields are non-interacting. In particular, we find a whole new class of logarithmic terms in the asymptotic expansions which depend on the coupling constant between the Maxwell and scalar fields. We analyse the implications of these results on the peeling (or rather lack thereof) of the fields at null infinity.

gr-qc↗

arXiv2019

A new perspective on the Ermakov-Pinney and scalar wave equations

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations can be obtained by differentiation of a first-order non-linear differential equation. The second part of the paper proves that, similarly, the equation for the amplitude function for the parametrix of the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary 1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi)=0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2=0. The auxiliary 1-forms (psi,rho) are evaluated explicitly in Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained. Thus, the novel method developed in this paper can be used with profit in physical applications.

math.GM↗

arXiv2019

Hyperbolic Equations and General Relativity

This is the author Master's Thesis and its main purpose is to demonstrate that it is possible to formulate Einstein's field equations as an initial value problem. The first chapter concerns the hyperbolic equations theory. The definition of hyperbolic equation is given and the concept of wavelike propagation is presented. Hence, once Riemann kernel definition is given, Riemann method to solve an hyperbolic equation in two variables is showed. The second chapter is about the fundamental solution and its relation to Riemann kernel. Therefore, the study of the fundamental solutions proceeds by showing how to build them and by providing some examples of solution with odd and even number of variables. In chapter four, by following Foures-Bruhat, the problem of finding a solution to the Cauchy problem for Einstein field equations in vacuum with non analytic initial data is presented by studying under which assumptions second-order systems of partial differential equations, linear and hyperbolic, with n functions and four variables admit a solution. Thus, in chapter five, it is showed how to turn non-linear systems of partial differential equations into linear systems of the same type for which the previous results hold. Eventually, in chapter six, the existence and uniqueness of the solution to the Cauchy problem for Einstein vacuum field equations is proved. Another part of this work consists in the study of the causal structure of space-time. The definitions of strong causality, stable causality and global hyperbolicity are given and the relation between the property of global hyperbolicity and the existence of Cauchy surfaces is stressed. The last chapter concerns the study of gravitational radiation in axisymmetric black hole collisions at the speed of light as an useful application of Riemann kernel method to solve a second-order hyperbolic equation in two variables.

gr-qc↗