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:20260916T104011Z UID:6696b7c3-d244-4b63-acbe-3e8314c3c28a SEQUENCE:2 CREATED:20260721T103615Z DESCRIPTION:Future null infinity is the location where idealized observers infinitely far from gravitationalfield sources reside. It is also the pla ce where gravitational radiation can be defined unambiguously. Since numer ical relativity is widely used to model gravitational-wave sources\, havin g direct access to future null infinity within numerical simulations is hi ghly desirable. This thesis addresses the challenge of incorporating futur e null infinity into numerical relativity simulations. The approach adopte d combines the Generalized Harmonic Gauge formulation of General Relativit y with the Dual-Foliation technique to construct hyperboloidal slices that extend to future null infinity. To investigate whether the asymptotic pro perties of the Dual-Foliation Generalized Harmonic Gauge formulation can b e captured using standard numerical methods\, the first part of this work develops a three-dimensional numerical implementation of a toy model desig ned to reproduce its asymptotic behavior. The resulting simulations are st able and convergent\, successfully recovering the analytically predicted d ecay rates of the evolved fields. The thesis then presents a numerical imp lementation of the full nonlinear Einstein equations under the assumption of spherical symmetry. This constitutes the first successful implementatio n of the present approach and represents one of the main achievements of t his work. Numerical evolutions were performed for gauge perturbations\, co nstraint-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 numer ical solutions\, including the quasi-normal mode ringing and late-time pow er-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 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. Nul l vectors play a fundamental role in this context because they determine t he characteristic propagation speeds of the system\, whose behavior is cru cial for the successful inclusion of future null infinity. The resulting f ormulation yields compact expressions that are well suited for numerical i mplementation. LAST-MODIFIED:20260721T103630Z LOCATION:Online URL:http://df.vps.tecnico.ulisboa.pt/pt/eventos/hyperboloidal-evolution-of -the-dual-foliation-generalized-harmonic-gauge-formulation-of-general-rela tivity/ X-ALT-DESC;FMTTYPE=text/html:

Future null infinit y is the location where idealized observers infinitely far from gravitatio nalfield sources reside. It is also the place where gravitational radiatio n 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 add resses the challenge of incorporating future null infinity into numerical relativity simulations. The approach adopted combines the Generalized Harm onic Gauge formulation of General Relativity with the Dual-Foliation techn ique to construct hyperboloidal slices that extend to future null infinity .

To investigate whether the asymptotic properties of the Dual-F oliation Generalized Harmonic Gauge formulation can be captured using stan dard numerical methods\, the first part of this work develops a three-dime nsional 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 evo lved fields. The thesis then presents a numerical implementation of the fu ll nonlinear Einstein equations under the assumption of spherical symmetry .

This constitutes the first successful implementation of the pr esent approach and represents one of the main achievements of this work. N umerical evolutions were performed for gauge perturbations\, constraint-vi olating initial data\, and constraintsatisfying initial data\, all of whic h exhibited stable and convergent behavior. Moreover\, several well-establ ished physical phenomena were recovered directly from the numerical soluti ons\, including the quasi-normal mode ringing and late-time power-law deca y 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 respe ct to two null vectors\, without imposing any additional assumptions.

Null vectors play a fundamental role in this context because they de termine the characteristic propagation speeds of the system\, whose behavi or is crucial for the successful inclusion of future null infinity. The re sulting formulation yields compact expressions that are well suited for nu merical implementation.

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