We review modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions. In four-dimensional N = 2 compactifications of string theory or M-theory these modifications are crucial for finding agreement between the macroscopic entropy obtained from supergravity and the microscopic entropy obtained by counting states in string or M-theory. Our discussion is based on the effective Wilsonian action, which in the context of N = 2 supersymmetric theories is defined in terms of holomorphic quantities. At the end we briefly indicate how to incorporate non-holomorphic corrections. March 1999 a cardoso@phys.uu.nl b bdewit@phys.uu.nl c mohaupt@hera1.physik.uni-halle.de 1 Introduction It is one of the most intriguing properties of black holes in general relativity that one can derive a set of laws, called the laws of black hole mechanics, which are formally equivalent to the laws of thermodynamics [1]. For instance, the first law of ...

We investigate the Church-Kalmar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be dierent.) We also look at various "obstacles" to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the "design" of our future computer). We also ask ourselves, how all this reects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.

We present a conformal field theory – obtained from a gauged WZW model – that describes a closed, inhomogeneous expanding and recollapsing universe in 3 + 1 dimensions. A possible violation of cosmic censorship is avoided because the universe recollapses just when a naked singularity was about to form. The model has been chosen to have c = 4 (or ĉ = 4 in the supersymmetric case), just like four dimensional Minkowski space.

hep-th/0211290 It has recently been shown that the uniqueness theorem for stationary black holes cannot be extended to five dimensions. However, uniqueness is an important assumption of the string theory black hole entropy calculations. This paper justifies this assumption by proving a uniqueness theorem for supersymmetric black holes in five dimensions. Some remarks concerning general properties of non-supersymmetric higher dimensional black holes are made. It is conjectured that there exist new families of stationary higher dimensional black hole solutions with fewer symmetries than any known solution. 1

We review physical motivations, phenomenological consequences, and open problems of the so-called pre-big bang scenario in superstring cosmology. Contents 1

In this paper the whole geometrical set-up giving a conformally invariant holographic projection of a diffeomorphism invariant bulk theory is clarified. By studying the renormalization group flow along null geodesic congruences a holographic version of Zamolodchikov’s c-theorem is proven. 1

Solutions to low energy string theory describing black holes and black strings are reviewed. Many of these solutions can be obtained by applying simple solution generating transformations to the Schwarzschild metric. In a few cases, the corresponding exact conformal field theory is known. Various properties of these solutions are discussed including their global structure, singularities, and Hawking temperature.

ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.

1.1 Classes of domain walls....................... 5 1.2 Walls in N = 1 supergravity..................... 6 2 Supergravity theory 9

A set of simple rules for constructing the maximal (e.g. analytical) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, maybe, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. -- The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity-Yang-Mills systems introduced in Part I [1], comprising, e.g., all generalized (linear and nonlinear) dilaton theorie...