Elements of classical and quantum gravity

Abstract

This chapter has two purposes; to describe a few elements of differential geometry that are required in different places in this work, and to provide, for completeness, a short introduction to general relativity (GR) and the problem of its quantization. A few concepts related to reparametrization (more accurately, diffeomorphism) of Riemannian manifolds, like parallel transport, affine connection, or curvature, are recalled. To define fermions on Riemannian manifolds, additional mathematical objects are required, the vielbein and the spin connection. Einstein–Hilbert s action for classical gravity GR is defined and the field equations derived. Some formal aspects of the quantization of GR, following the lines of the quantization of non-Abelian gauge theories, are described. Because GR is not renormalizable in four dimensions (even in its extended forms like supersymmetric gravity), at present time, a reasonable assumption is that GR is the low-energy, large-distance remnant of a more complete theory that probably no longer has the form of a quantum field theory (QFT) (strings, non-commutative geometry?). In the terminology of critical phenomena, GR belongs to the class of irrelevant interactions: due to the presence of the massless graviton, GR can be compared with an interacting theory of Goldstone modes at low temperature, in the ordered phase. The scale of this new physics seems to be of the order of 1019 GeV (Planck s mass). Still, because the equations of GR follow from varying Einstein–Hilbert action, some regularized form is expected to be relevant to quantum gravity. In the framework of GR, the presence of a cosmological constant, generated by the quantum vacuum energy, is expected, but it is extremely difficult to account for its extremely small, measured value.