Lectures on String Theory in Curved Spacetimes

Recent progress on string theory in curved spacetimes is reviewed. The string dynamics in cosmological and black hole spacetimes is investigated. The different methods available to solve the string equations of motion and constraints in curved spacetimes are described. That is, the string perturbation approach, the null string approach, the τ -expansion, and the construction of global solutions (for instance by inverse scattering methods). The classical behaviour of strings in FRW and inflationary spacetimes is now understood in a large extent from the various types of explicit string solutions. Three different types of behaviour appear: unstable, dual to unstable and stable. For the unstable strings, the energy and size grow for large scale factors R → ∞, proportional to R. For the dual to unstable strings, the energy and size blow up for R → 0 as 1/R. For stable strings, the energy and proper size are bounded. (In Minkowski spacetime, all string solutions are of the stable type). Recent progress on self-consistent solutions to the Einstein equations for string dominated universes is reviewed. The energy-momentum tensor for a gas of strings is then considered as source of the spacetime geometry and from the above string behaviours the string equation of state is derived. The selfconsistent string solution exhibits the realistic matter dominated behaviour R ≃ T 2/3 for large times and the radiation dominated behaviour R ≃ T 1/2 for early times (T being the cosmic time). We report on the exact integrability of the string equations plus the constraints in de Sitter spacetime that allows to systematically find exact string solutions by soliton methods and the multistring solutions. Multistring solutions are a new feature in curved spacetimes. That is, a single world-sheet simultaneously describes many different and independent strings. This phenomenon has no analogue in flat spacetime and follows to the coupling of the strings with the geometry. Finally, the string dynamics next and inside a Schwarzschild black hole is analyzed and their physical properties discussed.