Hyperboloidal Evolution of the Dual-Foliation Generalized Harmonic Gauge Formulation of General Relativity

Future null infinity is the location where idealized observers infinitely far from gravitationalfield sources reside. It is also the place where gravitational radiation 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 addresses the challenge of incorporating future null infinity into numerical relativity simulations. The approach adopted combines the Generalized Harmonic Gauge formulation of General Relativity with the Dual-Foliation technique to construct hyperboloidal slices that extend to future null infinity.

To investigate whether the asymptotic properties 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 designed to reproduce its asymptotic behavior. The resulting simulations are stable and convergent, successfully recovering the analytically predicted decay rates of the evolved fields. The thesis then presents a numerical implementation of the full nonlinear Einstein equations under the assumption of spherical symmetry.

This constitutes the first successful implementation of the present approach and represents one of the main achievements of this work. Numerical evolutions were performed for gauge perturbations, constraint-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 numerical solutions, including the quasi-normal mode ringing and late-time power-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 respect to two null vectors, without imposing any additional assumptions.

Null vectors play a fundamental role in this context because they determine the characteristic propagation speeds of the system, whose behavior is crucial for the successful inclusion of future null infinity. The resulting formulation yields compact expressions that are well suited for numerical implementation.