This chapter completes the description of the Einstein equation by finding the correct measure of energy density and the more general measure of spacetime curvature. A density is a quantity per unit spatial volume, such as rest-mass density, charge density, number density, energy density, and so on. The chapter begins by discussing how densities are represented in special and general relativity – for example, the densities of energy and momentum, and their conservation.

This introductory chapter gives a brief survey of some of the phenomena for which classical general relativity is important, primarily at the largest scales, in astrophysics and cosmology. The origins of general relativity can be traced to the conceptual revolution that followed Einstein’s introduction of special relativity in 1905. Newton’s centuries-old gravitational force law is inconsistent with special relativity. Einstein’s quest for a relativistic theory of gravity resulted not in a new force law or a new theory of a relativistic gravitational field, but in a profound conceptual revolution in our views of space and time. Four facts explain a great deal about the role gravity plays in physical phenomena. Gravity is a universal interaction, in Newtonian theory, between all mass, and, in relativistic gravity, between all forms of energy. Gravity is always attractive. Gravity is a long-range interaction, with no scale length. Gravity is the weakest of the four fundamental interactions acting between individual elementary particles at accessible energy scales.

Mass produces spacetime curvature – that is a central lesson of general relativity. The static spherical mass of the Sun produces the Schwarzschild geometry outside it. Mass in (nonspherical, nonuniform) motion is the source of ripples of curved spacetime, which propagate away at the speed of light. These propagating ripples in spacetime curvature are called gravitational waves. Their free propagation will be discussed in this chapter. There are many important sources of gravitational waves in the universe – binary star systems, supernova explosions, collapse to black holes, and the Big Bang are all examples. Gravitational waves provide a window for exploring these astronomical phenomena that is qualitatively different from any band of the electromagnetic spectrum. However, the weakness of the gravitational interaction in everyday circumstances means that gravitational waves are not easily detected.

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).