Aspects of Penrose limits and spacetime singularities

The present thesis is divided into two parts. The first one is devoted to open questions directly related to the geometrisation and interpretation of the Penrose limit. Accordingly, in chapter 2 we provide the technical prerequisites concerning Penrose limits, plane waves and string theory in these backgrounds. The necessary equipment at hand we go to work in chapter 3 where we identify the Penrose limit [20] as the lowest order of a covariant metric expansion around a null geodesic, i.e. extend the result obtained in [20]. To this end we introduce the null analogue to Fermi coordinates which are usually based on an expansion around a timelike geodesic. Eventually, this will lead us to what we dubbed a Penrose-Fermi expansion of the metric. In addition to its aesthetically appealing properties this novel prescription comes with different technical advantages which we will discuss. Building on these results in chapter 4 we compare the Penrose-Fermi expansion of the metric on the one hand with a Riemann coordinate expansion of the string embedding variables on the other in order to clarify the usual statement that exact string theory in a Penrose limit plane wave background might be interpreted as a lowest order approximation to string theory in the original background. Indeed, as we will see both expansions agree order by order after imposing the light cone gauge which acts as a pivotal point between space-time and world-sheet expansions in this context. In the second part of this thesis we embed the observed universality for Penrose limits of various space-time singularities in the broader context of different space-time probes. To this end in chapter 5 we shortly review the results obtained in [20,25] underlining the interpretation in terms of null geodesic congruences. Then in chapter 6 we replace the null congruence by what we consider as their nearest relative, namely a massless scalar field. After observing a completely analogous universal behaviour in this context, we concentrate on related topics as the uniqueness of time evolution near space-time singularities. We conclude in chapter 7 with a short outlook and discussion of open questions. Three publications are embedded into this thesis and form the larger part of its body, namely chapters 3, 4 and 6 are identical to [26], [27] and [28] respectively up to minor changes w.r.t. layout, (page, section, equation and citation) numbering and in some rare cases symbols, which were adjusted in favour of overall readability. For the same reason we also combined the different tables of content and the bibliographies. All alterations as well as errata are indicated individually at the end of each chapter/publication.

Thèse de doctorat : Université de Neuchâtel, 2008 ; Th.2004