The Geometry of Spacetime

Already thoroughly familiar to the seasoned science fiction fan, hyperspace is that realm which enables a spaceship captain to take his ship on a physics-defying shortcut (or "wormhole") to the outer shores of the Galaxy in less time than it takes a 747 to fly from New York to Tokyo. But might such notions be more than science fiction? Some physicists suggest a 10-dimensional hyperspace may actually exist, albeit at a scale almost too small to comprehend, smaller even than a quark; and that in spite of its tiny size, it may be the basis on which all the forces of nature will be united.

This concise book reviews methods used for gluing space-time manifolds together. It is therefore relevant to theorists working on branes, walls, domain walls, concepts frequently used in theoretical cosmology, astrophysics, and gravity theory. Nowadays, applications are also in theoretical condensed matter physics where Riemannian geometry appears. The book also reviews the history of matching conditions between two space-time manifolds from the early times of general relativity up to now.

The Zamolodchikov c-theorem has led to important new insights in the understanding of the Renormalisation Group (RG) and the geometry of the space of QFTs. The present primer introduces and reviews the parallel developments of the search for a higher-dimensional generalisation of the c-theorem and of the Local RG (LRG).