CONFORMAL ANALYSIS OF ANTI-DE SITTER-LIKE SPACETIMES
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In this thesis, several aspects of anti-de Sitter-like spacetimes are treated under conformal methods. More speci cally, the analysis is based on Friedrich's metric conformal formulation of the Einstein eld equations. First, it is proved that the conformal Einstein equations coupled to a tracefree matter model imply a system of wave equations for the conformal elds. Under an appropriate gauge choice, these relations are cast as a system of quasilinear wave equations. The analysis is supplemented with a set of homogeneous wave equations for the subsidiary variables. The problem of the existence of continuous symmetries in vacuum anti-de Sitter-like spacetimes is also considered. Following an approach based on the construction of wave equations for the relevant elds, the problem is reduced to the existence of a Killing vector on the conformal boundary. A necessary and su cient condition is found to be given by the so-called obstruction tensor. More speci cally, the spacetime possesses a Killing vector if and only if the conformal boundary has vanishing obstruction tensor. Next, a systematic construction of vacuum anti-de Sitter-like spacetimes is carried out by means of the quasilinear system previously obtained. Suitable initial and boundary data for this system are constructed via the conformal constraints. An analysis of the geometric subsidiary variables yields a local result for the existence of this class of spacetimes. The previous analysis serves as a prelude to the tracefree matter case. Following an analogous construction, three explicit matter models are considered: the conformally invariant scalar eld the Maxwell eld and the Yang-Mills eld. For each one of these, suitable boundary data sets are constructed and their relation to the corresponding subsidiary variables is established. This leads to a local result for the existence of anti-de Sitterlike spacetimes coupled to any of the aforementioned matter models.
AuthorsCarranza Ortiz, DA
- Theses