Notation and sign conventions adopted for the rest of the book are explained. The book employs index notation, but not abstract index notation. The metric signature for GR is taken as (-1,1,1,1). Terminology such as “local inertial frame” and “Rieman normal coordinates” is explained.
We now embark on the full theory, beginning with the concept of a manifold in differential geometry. The meaning of coordinates and coordinate transformations is carefully explained. The metric and its transformation between coordinate frames is discussed. Riemann normal coordinates are described. The concepts of a tangent space and local flatness are discussed and derived. It is shown how to use the metric to calculate distances, areas and volumes, and to describe submanifolds.
In this chapter, which forms the central chapter of Part II, the effective action of quantum matter fields in curved spacetime is formulated in terms of functional integrals. A qualitative, albeit incompletely conclusive, analysis of divergences and renormalization in curved space is given. Both non-covariant and covariant methods of calculations are discussed. Normal coordinates and local momentum representation are used to derive the effective potential. The basic elements of the Schwinger-DeWitt technique are elaborated in detail, resulting in the general formulas for one-loop divergences. The heat-kernel technique is also discussed.
A planar restricted elliptic three-body problem is considered. The motions close to the triangular libration points are studied. The problem parameters (the eccentricity of the orbit of the main attracting bodies and the ratio of their masses) are assumed to lie inside the linear stability region of the libration points. The magnitude of eccentricity is considered small. A linear canonical, periodic in true anomaly transformation is obtained analytically up to the second degree of eccentricity inclusive that reduces the Hamiltonian function of the linearized equations of perturbed motion to real normal form in the vicinity of the libration points. This form corresponds to two harmonic oscillators not connected to one another, with frequencies depending on the problem parameters. In constructing the normalizing canonical transformation, the Depri-Hori method of the perturbation theory of Hamiltonian systems is used. Its implementation in the problem under study relies heavily on computer systems of analytical calculations.
The singular structure of the lower vibrational states of the difluorosilylene molecule (up to four quanta of total excitation) was studied by expanding the energies of each state in the series of high-order Rayleigh-Schrödinger perturbation theory and analyzing their implicit multivalued properties using the fourth degree Padé-Hermite approximants. The quartic potential energy surface in dimensionless normal coordinates was calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of the values of multivalued approximants reproduces the variational solution with high accuracy, while other values, starting from the fourth polyad, in many cases coincide with the energies of other states of the polyad. The Fermi and Darling-Dennison resonances are analyzed on the basis of the coincidence of the singular complex branch points of the approximants for interacting states inside or near a circle of unit radius on the complex plane. It was found that a pair of states can have several coinciding branch points of solutions, including those inside the unit circle. It is concluded that this approach is an effective method for determining the polyad structure of vibrational states. The calculation parameters are selected, which are necessary for the reproducibility of key results. The calculations were carried out using a software package in the Fortran language using a package of arithmetic calculations with a long mantissa of real numbers (200 digits).
On normal coordinates in the vicinity of the Lagrangian libration points of the restricted elliptic three-body problem
