Geometric invariants for initial data sets: analysis, computer algebra and numerics

Juan A. Valiente Kroon

The construction of a geometric invariant measuring how a given initial data set for the Einstein field equations deviates from initial data for the Kerr solution is used to exemplify the interaction between analytical, computer algebra and numerical methods in General Relativity. The construction of the invariant was motivated by the need of providing a rigorous meaning to the idea that a given spacetime is "close" to the Kerr solution. The construction of the invariant combines ideas from the study of exact solutions to the Einstein equations with methods from mathematical General Relativity and computer algebra. The construction is, in principle, completely amenable to a numerical evaluation. Obtaining further insight into the nature of the invariant, specially in what concerns its behaviour upon time evolution may require input from numerical Relativity. In turn, this numerical information is expected to suggest what kind of analytical results can be proved.