Analysis of asymptotically flat and asymptotically de-Sitter spacetimes with conformal methods
PT-CERN Call 2022/2
Edgar Gasperin Garcia
Instituto Superior Técnico (Universidade de Lisboa)
CENTRA - Center for astrophysics and gravitation
The asymptotic properties of spacetimes play a central role in many physical aspects of General Relativity (GR), in particular for the concept of gravitational radiation. One of the crucial concepts introduced for the realisation of the latter, was the use of conformal compactifications introduced in General Relativity by R.Penrose to describe in a geometric way the notion of infinity. The formulation that takes Penrose’s idea of conformal compactification --in conjunction with the theory of partial differential equations-- to its ultimate consequences are Conformal Einstein Field (CEFE) equations introduced by H. Friedrich in . These equations have been used for establishing (semi)-global non-linear stability results de-Sitter and Minkowski spacetimes and some partial global non-linear stability results for black hole spacetimes such as Schwarzschild-de Sitter . Although these equations have been available since 1980 the numerical exploration of spacetimes using the CEFEs is still in its infancy compared with other formulations since they have not been systematically exploited as other formulations of GR. One of the alternatives to the CEFEs is the use of hyperboloidal foliations and scri-fixing techniques and more standard formulations of the Einstein Field Equations such as the Generalised Harmonic Gauge (EFE-GHG). These have been exploited for numerical implementations  and for mathematical analysis . By construction, hyperboloidal foliations reach the null-infinity but stays away from the region close to spatial infinity. The CEFEs have access to this region via the construction of the so-called cylinder at spatial infinity. These methods, hyperboloidal and conformal, being complementary and grating access to the different regions of spacetimes can be exploited for the analysis of global properties of spacetimes by means of mathematical and numerical methods or a combination of thereof.