BEGIN:VCALENDAR VERSION:2.0 PRODID:-//linuxsoftware.nz//NONSGML Joyous v1.4//EN BEGIN:VEVENT SUMMARY:Hyperboloidal Evolution of the Dual-Foliation Generalized Harmonic Gauge Formulation of General Relativity DTSTART:20260722T140000Z DTEND:20260722T160000Z DTSTAMP:20260928T071311Z UID:6696b7c3-d244-4b63-acbe-3e8314c3c28a SEQUENCE:1 CREATED:20260721T103606Z DESCRIPTION: Future null infinity is the location where idealized observer s infinitely far from gravitationalfield sources reside. It is also the pl ace where gravitational radiation can be defined unambiguously. Since nume rical relativity is widely used to model gravitational-wave sources\, havi ng direct access to future null infinity within numerical simulations is h ighly desirable. This thesis addresses the challenge of incorporating futu re null infinity into numerical relativity simulations. The approach adopt ed combines the Generalized Harmonic Gauge formulation of General Relativi ty with the Dual-Foliation technique to construct hyperboloidal slices tha t extend to future null infinity. To investigate whether the asymptotic pr operties of the Dual-Foliation Generalized Harmonic Gauge formulation can be captured using standard numerical methods\, the first part of this work develops a three-dimensional numerical implementation of a toy model desi gned to reproduce its asymptotic behavior. The resulting simulations are s table and convergent\, successfully recovering the analytically predicted decay rates of the evolved fields. The thesis then presents a numerical im plementation of the full nonlinear Einstein equations under the assumption of spherical symmetry. This constitutes the first successful implementati on of the present approach and represents one of the main achievements of this work. Numerical evolutions were performed for gauge perturbations\, c onstraint-violating initial data\, and constraintsatisfying initial data\, all of which exhibited stable and convergent behavior. Moreover\, several well-established physical phenomena were recovered directly from the nume rical solutions\, including the quasi-normal mode ringing and late-time po wer-law decay of a massless scalar field\, as well as the evolution of the Bondi mass. Finally\, a new geometric formalism is developed with the goa l of extending the present approach to fully three-dimensional simulations . The formalism is based on a decomposition of spacetime geometry with res pect to two null vectors\, without imposing any additional assumptions. Nu ll vectors play a fundamental role in this context because they determine the characteristic propagation speeds of the system\, whose behavior is cr ucial for the successful inclusion of future null infinity. The resulting formulation yields compact expressions that are well suited for numerical implementation. LAST-MODIFIED:20260721T103606Z LOCATION:Online URL:http://df.vps.tecnico.ulisboa.pt/en/events/hyperboloidal-evolution-of- the-dual-foliation-generalized-harmonic-gauge-formulation-of-general-relat ivity/ X-ALT-DESC;FMTTYPE=text/html:

Future null infini ty is the location where idealized observers infinitely far from gravitati onalfield sources reside. It is also the place where gravitational radiati on can be defined unambiguously. Since numerical relativity is widely used to model gravitational-wave sources\, having direct access to future null infinity within numerical simulations is highly desirable. This thesis ad dresses the challenge of incorporating future null infinity into numerical relativity simulations. The approach adopted combines the Generalized Har monic Gauge formulation of General Relativity with the Dual-Foliation tech nique to construct hyperboloidal slices that extend to future null infinit y.

To investigate whether the asymptotic properties of the Dual- Foliation Generalized Harmonic Gauge formulation can be captured using sta ndard numerical methods\, the first part of this work develops a three-dim ensional numerical implementation of a toy model designed to reproduce its asymptotic behavior. The resulting simulations are stable and convergent\ , successfully recovering the analytically predicted decay rates of the ev olved fields. The thesis then presents a numerical implementation of the f ull nonlinear Einstein equations under the assumption of spherical symmetr y.

This constitutes the first successful implementation of the p resent approach and represents one of the main achievements of this work. Numerical evolutions were performed for gauge perturbations\, constraint-v iolating initial data\, and constraintsatisfying initial data\, all of whi ch exhibited stable and convergent behavior. Moreover\, several well-estab lished physical phenomena were recovered directly from the numerical solut ions\, including the quasi-normal mode ringing and late-time power-law dec ay of a massless scalar field\, as well as the evolution of the Bondi mass .

Finally\, a new geometric formalism is developed with the goal of extending the present approach to fully three-dimensional simulations. The formalism is based on a decomposition of spacetime geometry with resp ect to two null vectors\, without imposing any additional assumptions.

Null vectors play a fundamental role in this context because they d etermine the characteristic propagation speeds of the system\, whose behav ior is crucial for the successful inclusion of future null infinity. The r esulting formulation yields compact expressions that are well suited for n umerical implementation.

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