This paper is devoted to the study of stationary trajectories of free particles. From a classical point of view, this appears to be an almost trivial problem: Free particles should follow straight lines as predicted by Newton’s first law, and straight lines are indeed the stationary trajectories of the standard action integrals in the classical theory. In the following, however, a general relativistic approach is studied, and in this situation it is much less evident what action integral should be used. As it turns out, using the traditional Einstein–Hilbert principle gives us stationary states very much in line with the classical theory. But it is suggested that a different action principle, and in fact one which is closer to quantum mechanics, gives stationary states with a much richer structure: Even if these states in a sense can represent particles which obey the first law, they are also inherently rotating. Although we may still be far from understanding how general relativity and quantum mechanics should be united, this may give an interesting clue to why rotation (or rather spin, which is a different but related concept) seems to be the natural state of motion for elementary particles.

In this paper, different aspects of the concept of spin are studied. The most well-established one is, of course, the quantum mechanical aspect: spin is a broken symmetry in the sense that the solutions of the Dirac equation tend to have directional properties that cannot be seen in the equation itself. It has been clear since the early days of quantum mechanics that this has something to do with the indefinite metric in Lorentz geometry, but the mechanism behind this connection is elusive. Although spin is not the same as rotation in the usual sense, there must certainly be a close relationship between these concepts. And, a possible way to investigate this connection is to instead start from the underlying geometry in general relativity. Is there a reason why rotating motion in Lorentz geometry should be more natural than non-rotating motion? In a certain sense, the answer turns out to be yes. But, it is by no means easy to see what this should correspond to in the usual quantum mechanical picture. On the other hand, it seems very unlikely that the similarities should be just coincidental. The interpretation of the author is that this can be a golden opportunity to investigate the interplay between these two theories.

In this paper, the concept of causality in physics is discussed. Causality is a necessary tool for the understanding of almost all physical phenomena. However, taking it as a fundamental principle may lead us to wrong conclusions, particularly in cosmology. Here, three very well-known problems-the Einstein-Podolsky-Rosen paradox, the accelerating expansion and the asymmetry of time-are discussed from this perspective. In particular, the implications of causality are compared to those of an alternative approach, where we instead take the probability space of all possible developments as the starting point.

In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in R-3, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However, in the case of Lorentz geometry, the curvature will instead initially decrease. In other words, rotating metrics can, in this case, be said to be less curved than non-rotating ones. This phenomenon seems to be very general, but because of the enormous amount of computations required, it will only be proved for a class of metrics which are close to the flat one, and the main (symbolic) computations have been carried out on a computer. Although the results here are purely mathematical, there is also a connection to physics. In general, a deeper understanding of Lorentz geometry is of fundamental importance for many applied problems.

In this paper, a probabilistic approach is used to derive a kind of abstract candidate for a natural Lagrangian in general relativity. The methods are very general, and the result is in a certain sense unique. However, to turn this abstract Lagrangian into an ordinary one, expressible in terms of the Riemann tensor, is so far an open problem. Some possible cosmological consequences are discussed.

In this paper, the relationship between the thermodynamic and historical arrows of time is studied. In the context of a simple combinatorial model, their definitions are made more precise and in particular strong versions (which are not compatible with time symmetric microscopic laws) and weak versions (which can be compatible with time symmetric microscopic laws) are given. This is part of a larger project that aims to explain the arrows as consequences of a common time symmetric principle in the set of all possible universes. However, even if we accept that both arrows may have the same origin, this does not imply that they are equivalent, and it is argued that there can be situations where one arrow may be well-defined but the other is not.

In this paper, simple models for the multiverse are analyzed. Each universe is viewed as a path in a graph, and by considering very general statistical assumptions, essentially originating from Boltzmann, we can make the set of all such paths into a finite probability space. We can then also attempt to compute the probabilities for different kinds of behavior and in particular under certain conditions argue that an asymmetric behavior of the entropy should be much more probable than a symmetric one. This offers an explanation for the asymmetry of time as a broken symmetry in the multiverse. The focus here is on simple models which can be analyzed using methods from combinatorics. Although the computational difficulties rapidly become enormous when the size of the model grows, this still gives hints about how a full-scale model should behave.

In this paper, I suggest a possible explanation for the asymmetry of time. In the case that I study, the dynamical laws and the boundary conditions are symmetric, but the behavior of time is not. The underlying mechanism is statistical and closely related to the idea of multiple histories in quantum mechanics, but otherwise rather independent of the particular framework.

The purpose of this thesis is to develop a theory for certain extreme value problems, depending analytically on some set of parameters. In particular it aims at a description of the singularities of the extreme value functions. In this context, a central role is played by the concept of subanalytic functions (i.e. functions whose graphs are subanalytic in the sense of Hironaka). 

The content of this paper naturally divides into three parts. After some preliminary material in the three first chapters, the first main topic comes up in Chapter IV, where it is proved that singular supports of subanalytic functions are subanalytic. As a consequence, it follows that the singular set of a subanalytic set is again subanalytic. Chapter V is then devoted to finding sufficient conditions on an extreme value problem for its extreme value function to be subanalytic. In Chapter VI finally, two applications of the general theory are presented. The first is to prove that cut loci on analytic Riemannian manifolds are always stratifiable and triangulable. The second concerns the theory of phase transitions in a certain model in statistical mechanics (the so called Van der Waals limit). In particular, the piecewise analytic behaviour of the free energy as a function of temperature (and density) is explained in this context.