*(English)*Zbl 1057.83002

The book is devoted to some class of gravitational theories which are alternatives to Einstein’s general relativity. Scalar-tensor theories describe gravitation by a Lorentzian metric $g$ together with a scalar field $\varphi $; the latter is interpreted as the strength, variable in space and time, of coupling between metrical gravitation and matter. General relativity is just the case where $\varphi $ is constant; it is then reciprocal to the classical gravitational constant. Because of the non-minimal coupling, Mach’s principle is incorporated in the ansatz. The first theory of this type appeared as a by-product of Kaluza-Klein theory. The latter originally had no use for the component ${g}_{55}$ of the 5-dimensional metric and set ${g}_{55}=1$. Jordan et al. realized that ${g}_{55}={\varphi}^{2}$ leads to a meaningful theory, where the 4-dimensional scalar $\varphi $ is a part of gravitation. Brans and Dicke inserted a new constant $\omega $ into the Lagrangian and started the study of the resulting family of theories. It turns out that the Einstein effects in the solar system require unpleasently great values of $\omega $. Another generalization made the theories more flexible: a generalized scalar-tensor theory, in the terminology of the present book, follows from a Lagrangian of the form $L=f(\varphi ,R)-\omega \left(\varphi \right)\nabla {\varphi}^{2}$ with two functions $f$ and $\omega $, and $R$ denoting the scalar curvature of $g$. Recently, the interest for such theories renewed, because scalar fields appeared in various parts of modern physics: cosmological Higgs field, dilaton field of string theory, ...

Here the author introduces and motivates both Brans-Dicke and generalized theories. He then focusses on cosmology and gravitational waves in this framework. The headlines of the chapters read as follows. 1. Scalar-tensor gravity. 2. Effective energy-momentum tensors and conformal frames. 3. Gravitational waves. 4. Exact solutions of scalar-tensor cosmology. 5. The early universe. 6. Perturbations. 7. Non-minimal coupling. 8. The present universe.

Scalar-tensor theories in the generalized scheme can be gauged by conformal transformations. Faraoni calls the original form of the theory “Jordan frame” and the conformally gauged form which looks very much like general relativity “Einstein frame”. He discusses at length the pros and cons and interpretations of the two frames; ultimate answers are not yet available. Usually, cosmological inflation is driven by a minimally coupled scalar field. One expects that the non-minimally coupled $\varphi $ of the theories under consideration might do the job as well or even better. Indeed, the variety of theories leads to a richness of possible effects: slow-roll and fast-roll inflation, superinflation, accelerated and superaccelarated expansion, cosmological birefringence, coasting universe, vacillating universe. There may be big bang (begin) or big smash (end) singularities or not. Gravitational waves, particularly primordial ones of great wave-length, may disturb gravitational lensing and so become detectable. The author uses general physical arguments as well as exact (cosmological) solutions and weak-field approximations. Special attention is devoted to the question whether the $\varphi $ field is a candidate for dark energy or quintessence. Again, quintessential effects can be modelled by a suitable choice of a particular theory.

The book is fairly complete on its themes. The list of references has 925 entries; many citations are of the years 2000–2003. The book is both, a useful survey and a contribution to current discussions.