Chapter 10

Conclusion of Volume 2

10.1 The Legacy of Volume 1

In the first volume we have presented the theory of General Relativity comparing it at all times with the other gauge theories that describe non-gravitational interactions. We have also followed the complicated historical development of the ideas and of the concepts underlying both of them. In particular we have traced back the origin of our present understanding of all fundamental interactions as mediated by connections on principal fibre-bundles and emphasized the special status of gravity within this general scheme. While recalling the historical development we have provided a, hopefully rigorous, exposition of all the mathematical foundations of gravity and gauge theories in a contemporary geometrical approach. In the last two chapters of Volume 1 we also considered relevant astrophysical applications of General Relativity that provide some of the most accurate tests of its predictions.

10.2 The Story Told in Volume 2

Volume 2’s mission was the historical and mathematical analysis of the further conceptual developments in the Theory of Gravity up to contemporary times. The first part of the Volume concentrated on two main issues:

1.Black Holes,

2.Cosmology.

Black Holes are probably the most profound implication of General Relativity. They occupy a distinguished and outstanding position both in Mathematics and in Physics, being an endless source of inspiring views and of challenging problems, like that of the information loss and the thermodynamical formulation of their dynamical laws which leads to the question of the statistical interpretation of their entropy. The historical/conceptual path going from the interpretation of the Schwarzschild radius as an event horizon to the proper definitions of causal structures and Penrose diagrams was carefully described and the necessary mathematics

was step by step developed. With rotating black holes and the area-entropy law we opened the first window on a more profound level of gravitational theory that leads to superstring and supergravity.

Cosmology is equally important in the development of gravitational theory since gravity is the only relevant interaction at very large scales. Einstein theory leads necessarily to the view of an expanding universe, notwithstanding the original philosophical enmity of Einstein himself to this very idea. The mathematical formulation of the Cosmological Principle requires the notion of homogeneous coset manifolds and that was the occasion to develop this important chapter of Differential Geometry whose uses are ubiquitous in Theoretical Physics and in particular in Supergravity. The resolution of the conceptual problems raised by the Standard Cosmological Model leads to the paradigm of the inflationary universe whose predictions seem to be generically in agreement with the observations of the anisotropies in the Cosmic Microwave Background radiation. The simplest mechanism of inflation requires scalar fields that are abundant in all versions of the supergravity/superstring modeling of the fundamental laws of physics. Scalar fields are anyhow required by symmetry breaking and their experimental revelation is by now overdue.1

The second part of the Volume was conceived as an introduction to Supergravity, Branes and Strings. Once again our approach was conceptual and historical. The goal was explicitly that of continuing the logical development initiated in Volume 1 introducing supergravity as the dynamical theory of the super-Poincaré connection, in the same way as General Relativity is the dynamical theory of the Poincaré connection. From the mathematical point of view, new concepts enter the stage: the extension of (super-)Lie algebras to Free Differential Algebras and the generalized principle of analyticity named rheonomy. On one hand Free Differential Algebras are canonically induced by their normal Lie subalgebra, by means of its cohomology, on the other hand the new generators of Free Differential Algebras are p-forms that naturally couple to the world volume of (p − 1)-branes. This is the core of the Bulk-Brane dualism which we tried to illustrate in some detail in a dedicated chapter. The other important legacy of the supersymmetric extension of General Relativity is the incredible wealth of new geometrical structures that are contributed by the scalar sector of the various supergravities. Without any claim to completeness we tried to provide the reader with an overview of this richness and with and an introduction to the Supergravity Bestiary.

From the point of view of Gravity Theory, which is that adopted in this book, the main interest of supergravity, namely of the Beyond GR World, is its contribution of an impressive variety of new classical solutions of quite different type and with quite different interesting properties. A taste of that variety is provided to the reader in Chap. 9 through the presentation of a series of examples.

Repeating what was already stressed at the end of that chapter, the author considers his mission fulfilled if the reader could follow his arguments from the first intuitions about Lorentz symmetry at the end of XIXth century to some of the frontiers

1By the publication date of this book the Higgs boson, that is a scalar particle, has already been discovered at CERN with very high likehood.