Foundations of the Age-Area Hypothesis

A useful tool in understanding the roots of the world geography of culture is the Age-Area-Hypothesis. The Age-Area Hypothesis (AAH) asserts that the point of geographical origin of a group of related cultures is most likely where the culture speaking the most divergent language is located. In spite of its widespread, multidisciplinary application, the hypothesis remains imprecisely stated, and has no theoretical underpinnings. This paper describes a model of the AAH based on an economic theory of mass migrations. The theory leads to a family of measures of cultural divergence, which can be referred to as Dyen divergence measures. One measure is used to develop an Age-Area Theorem, which links linguistic divergence and likelihood of geographical origin. The theory allows for computation of the likelihood different locations are origin points for a group of related cultures, and can be applied recursively to yield probabilities of different historical migratory paths. The theory yields an Occam’s-razor-like result: migratory paths that are the simplest are also the most likely; a key principle of the AAH. The paper concludes with an application to the geographical origins of the peoples speaking Semitic languages.

Black hole thermodynamics

The chapter presents the Penrose process, Hawking radiation, entropy and the laws of black hole thermodynamics. The Penrose process is derived and the area theorem is stated. A heuristic argument for the Hawking effect is given, emphasising a correct grasp of the concepts and the nature of the result. The Hawking effect and the Unruh effect are further discussed and linked together in a precise calculation. Evaporation of black holes is described. The information paradox is presented.

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Some applications

The aim of this chapter is to present key applications of causality theory, as relevant to black-hole spacetimes. For this we need to introduce the concept of conformal completions, which is done in Section 3.1. We continue, in Section 3.2, with a review of the null splitting theorem of Galloway. Section 3.3 contains complete proofs of a few versions of the topological censorship theorems, which are otherwise scattered across the literature, and which play a basic role in understanding the topology of black holes. In Section 3.4 we review some key incompleteness theorems, also known under the name of singularity theorems. Section 3.5 is devoted to the presentation of a few versions of the area theorem, which is a cornerstones of ‘black-hole thermodynamics’. We close this chapter with a short discussion of the role played by causality theory when studying the wave equation.

Elements of causality

A standard part of studies of black holes, and in fact of mathematical general relativity, is causality theory, which is the study of causal relations on Lorentzian manifolds. An essential issue here is understanding the influence of energy conditions on the causality relations. The highlights of such studies include the incompleteness theorems, known also as singularity theorems, of Penrose, Hawking and Geroch, the area theorem of Hawking, and the topology theorems of Hawking and others. The aim of this chapter is to provide an introduction to the subject, with a complete exposition of those topics which are needed for the global treatment of the uniqueness theory of black holes. In particular we provide a coherent introduction to causality theory for metrics which are twice differentiable.