Both experimentally and theoretically, the curved spacetimes of general relativity are explored by studying how test particles and light rays move through them. This chapter derives and analyzes the equations governing the motion of test particles and light rays in a general curved spacetime. Only test particles free from any influences other than the curvature of spacetime (electric forces, for instance) are considered. Such particles are called free, or freely falling, in general relativity. In general relativity, free means free from any influences besides the curvature of spacetime. We begin with the equations of motion for test particles with nonvanishing rest mass moving on timelike world lines, and revisit the equations of motion for light rays.

This chapter expands a little on the idea that gravity is geometry, and then describes how the geometry of space and time is a subject for experiment and theory in physics. In a gravitational field, all bodies with the same initial conditions will follow the same curve in space and time. Einstein’s idea was that this uniqueness of path could be explained in terms of the geometry of the four-dimensional union of space and time called spacetime. Specifically, he proposed that the presence of a mass such as Earth curves the geometry of spacetime nearby, and that, in the absence of any other forces, all bodies move on the straight paths in this curved spacetime. We explore how simple three-dimensional geometries can be thought of as curved surfaces in a hypothetical four-dimensional Euclidean space. The key to a general description of geometry is to use differential and integral calculus to reduce all geometry to a specification of the distance between each pair of nearby points.

In this second introductory chapter, the concept of gravitational potential is presented and then developed up to the level usually encountered in applications of stellar dynamics, such as the computation of the gravitational fields of disks and heterogeneous triaxial ellipsoids, the construction of the far-field multipole expansion of the gravitational field of generic mass distributions, and finally the expansion to orthogonal functions of the Green function for the Laplace operator.

This chapter presents the basic properties of quasi-circular orbits in axisymmetric stellar systems. In fact, axisymmetric models are often sufficiently realistic descriptions – beyond the zeroth-order spherical case – of elliptical and disk galaxies. The associated potentials (with a reflection plane, the equatorial plane) admit circular orbits, and in this chapter we focus on the properties of orbits slightly departing from perfect circularity, describing in some detail the second-order epicyclic approximation. We also derive, in a geometrically rigorous way, the expression of the Oort constants, important kinematical quantities related to the rotation curve of disk galaxies, and in particular to the orbit of the Sun around the center of the Milky Way.

In this chapter, we show how the (infinite) set of equations known as the Jeans equations is derived by considering velocity moments of the collisionless Boltzmann equation (CBE) discussed in Chapter 9. The Jeans equations are very important for physically intuitive modeling of stellar systems, and they are some of the most useful tools in stellar dynamics. In fact, while the natural domain of existence of the solution of the CBE is the six-dimensional phase space, the Jeans equations are defined over three-dimensional configuration space, allowing us to achieve more intuitive modeling of directly observable quantities. The physical meaning of the quantities entering the Jeans equations is also illustrated by comparison with the formally analogous equations of fluid dynamics. Finally, by taking the spatial moments of the Jeans equations over the configuration space, the virial theorem in tensorial form is derived, complementing the more elementary discussion in Chapter 6.

With this chapter, the final part of the book, dedicated to collisionless stellar systems, begins. As should be clear, in order to extract information from the N-body problem, we need to move to a different approach than direct integration of the differential equations of motion, and a first (unfruitful) attempt will be based on the Liouville equation. In fact, the basic reason for the “failure” of the Liouville approach is that, despite its apparent statistical nature, the dimensionality of the phase space Γ where the function f(6N) is defined is the same as that of the original N-body problem. Suppose instead we find a way to replace the 6N-dimensional R6N phase space Γ with the six-dimensional one-particle phase space γ: we can reasonably expect that the problem would be simplified enormously, and in fact along these lines we will finally obtain the collisionless Boltzmann equation, one of the conceptual pillars of stellar dynamics.

Armed with the power of the Jeans theorem, we now proceed to formulate and discuss the so-called direct problem of collisionless stationary stellar dynamics. This approach is best suited for systems where empirical/dynamical arguments can lead to a plausible ansatz for the form of the underlying distribution function, expressed in terms of the relevant integrals of motion. In the absence of such an ansatz and in the presence of specific requirements (in general motivated by observations) for the density and velocity dispersion profiles, a different and complementary approach based on the use of the Jeans equations is often followed, which is the subject of Chapter 13.

In astronomy in general, and in the study of stellar systems in particular, one is often led to consider the effects of an “external” gravitational field on a body of some spatial extension: examples are satellites around planets, binary stars, open and globular clusters in galaxies, and galaxies in clusters of galaxies. The general problem can be mathematically very difficult; however, when the extension of the body of interest is small compared to the characteristic length scale of the external gravitational field (i.e., when the system is in the tidal regime), the problem becomes more tractable. In this chapter, we provide the basic ideas and tools that can be used in stellar dynamics when dealing with tidal fields. Among other things, we will find that tidal fields are not always expansive (as in the familiar case of the Earth–Moon system), as they can be also compressive (e.g., for stellar systems inside galaxies or for galaxies in galaxy clusters).

In this last chapter, we discuss a final theoretical step of the moments approach illustrated in Chapter 13: under the assumption that the macroscopic profiles (e.g., density and velocity dispersion) of each component are known, there is a possibility of recovering the phase-space distribution function (DF) of a model and checking its positivity (i.e., verifying the model consistency). The problem of recovering the DF is in general a technically difficult inverse problem, and even when it is doable, unicity of the recovered DF is not guaranteed, so that a simple consistency analysis is quite problematic. Fortunately, there are special cases when (in principle) the DF can be obtained analytically (generally in integral form), and in these cases a few general and useful consistency conditions can be proved, such as the so-called global density slope–anisotropy inequality. The student is warned that this chapter is somewhat more technical than the others; however, the additional effort needed for its study will be well repaid by the understanding of some nontrivial results allowing for the construction of phase-space consistent collisionless stellar systems.

The N-body problem, the study of the motion of N point masses (e.g., stars) under the mutual influence of their gravitational field, is one of the central problems of classical physics, and the literature on the subject is immense, starting withNewton’s Principia (e.g., see Chandrasekhar 1995). Conceptually, it is the natural subject of celestial mechanics more than of stellar dynamics; however, experience suggests that some space should be devoted to an overview of the exact results of the N-body problemin a book like this. In fact, due to the very large number of stars in stellar systems, stellar dynamics must rely on specific techniques and assumptions, and one may legitimately ask which of the obtained results hold true in the generic N-body problem; for example, these exact results represent invaluable tests for validating numerical simulations of stellar systems. The virial theorem is onesuch result, and in this chapter we present a first derivation of it, while an alternative derivation in the framework of stellar dynamics will be discussed in Chapter 10.