Most of classical and quantum physics can be expressed in terms of variational principles, and it is often when written in this form that the physical meaning is most clearly understood. Moreover, once a physical theory has been written as a variational principle it is usually straightforward to identify conserved quantities, or symmetries of the system of interest, that otherwise might have been found only with considerable effort. Conversely, by demanding that the variational principle be invariant under some symmetry, one ensures that the equations of motion derived from it also respect that symmetry. In this final chapter, we therefore present an introductory account of variational principles and the Lagrangian formalism. Our ultimate aim will be to derive afresh the field equations of general relativity from this new perspective. This will require us to consider some general aspects of classical field theory in flat and curved spacetimes. As a result, this chapter lies somewhat outside the mainstream discussion presented in preceding chapters and may be omitted on a first reading. Nevertheless the variational approach that we shall outline is extremely powerful and provides the basis for most current research into the formulation of classical (and quantum) field theories, including general relativity and other candidate theories of gravitation.

Hamilton's principle in Newtonian mechanics

To begin, let us remind ourselves of a familiar example of a physical variational principle, namely Hamilton's principle in Newtonian mechanics.

The coordinates with which one labels points in a manifold are entirely arbitrary. For example, we could choose to parameterise the surface of a sphere in terms of the coordinates (θ, φ), taking any point as the north pole, or we could use any number of alternative coordinate systems. It is also clear, however, that our description of any physical processes occurring on the surface of the sphere should not depend on our chosen coordinate system. For example, at any point P on the surface one can say that, for example, the air temperature has a particular value or that the wind has a certain speed in a particular direction. These respectively scalar and vector physical quantities do not depend on which coordinates are used to label points in the surface. Thus in, order to describe these physical fields on the surface, we must formulate our equations in a way that is valid in all coordinate systems. We have already dealt with such a description for scalar and vector quantities on manifolds, but now we turn to the generalisation of these ideas to quantities that cannot be described as a scalar or a vector. This requires the introduction of the concept of tensors.

Tensor fields on manifolds

Let us begin by considering vector fields in a slightly different manner. Suppose we have some arbitrary vector field, defining a vector t at each point of a manifold.

We now consider how to solve the Einstein field equations and so discover the metric functions gμν in any given physical situation. Clearly, the high degree of non-linearity in the field equations means that a general solution for an arbitrary matter distribution is analytically intractable. The problem becomes easier if we look for special solutions, for example those representing spacetimes possessing symmetries. The first exact solution to Einstein's equations was found by Karl Schwarzschild in 1916. As we shall see, the Schwarzschild solution represents the spacetime geometry outside a spherically symmetric matter distribution.

The general static isotropic metric

Schwarzschild sought the metric gµν representing the static spherically symmetric gravitational field in the empty space surrounding some massive spherical object such as a star. Thus, a good starting point for us is to construct the most general form of the metric for a static spatially isotropic spacetime.

A static spacetime is one for which some timelike coordinate x0 (say) with the following properties: (i) all the metric components gµν are independent of x0; and (ii) the line element ds2 is invariant under the transformation x0 → −x0. Note that (i) does not necessarily imply (ii), as is made clear by the example of a rotating star: time reversal changes the sense of rotation, but the metric components are constant in time. A spacetime that satisfies (i) but not (ii) is called stationary.

Our discussion of special relativity has led us to model the physical world as a four-dimensional continuum, called spacetime, with a Minkowski geometry. This is an example of a manifold. As we shall see, the more complicated spacetime geometries of general relativity are also examples of manifolds. It is therefore worthwhile discussing manifolds in general. In the following we consider general properties of manifolds commonly encountered in physics, and we concentrate in particular on Riemannian manifolds, which will be central to our discussion of general relativity.

The concept of a manifold

In general, a manifold is any set that can be continuously parameterised. The number of independent parameters required to specify any point in the set uniquely is the dimension of the manifold, and the parameters themselves are the coordinates of the manifold. An abstract example is the set of all rigid rotations of Cartesian coordinate systems in three-dimensional Euclidean space, which can be parameterised by the Euler angles. So the set of rotations is a three-dimensional manifold: each point is a particular rotation, and the coordinates of the point are the three Euler angles. Similarly, the phase space of a particle in classical mechanics can be parameterised by three position coordinates (q1, q2, q3) and three momentum coordinates (p1, p2, p3), and thus the set of points in this phase space forms a six-dimensional manifold. In fact, one can regard ‘manifold’ as just a fancy word for ‘space’ in the general mathematical sense.

We now discuss the application of general relativity to modelling the behaviour of the universe as a whole. In order to do this, we make some far-reaching assumptions, but only those consistent with our observations of the universe. As in our derivations of the Schwarzschild and Kerr geometries, we begin by using symmetry arguments to restrict the possible forms for the metric describing the overall spacetime geometry of the universe.

The cosmological principle

When we look up at the sky we see that the stars around us are grouped into a large-density concentration – the Milky Way Galaxy. On a slightly larger scale, we see that our Galaxy belongs to a small group of galaxies (called the Local Group). Our Galaxy and our nearest large neighbour, the Andromeda galaxy, dominate the mass of the Local Group. On still larger scales we see that our Local Group sits on the outskirts of a giant supercluster of galaxies centred in the constellation of Virgo. Evidently, on small scales matter is distributed in a highly irregular way but, as we look on larger and larger scales, the matter distribution looks more and more uniform. In fact, we have very good evidence (particularly from the constancy of the temperature of the cosmic microwave background in different directions on the sky) that the universe is isotropic on the very largest scales, to high accuracy.

Let us now follow Einstein's suggestion that gravity is a manifestation of spacetime curvature induced by the presence of matter. We must therefore obtain a set of equations that describe quantitatively how the curvature of spacetime at any event is related to the matter distribution at that event. These will be the gravitational field equations, or Einstein equations, in the same way that the Maxwell equations are the field equations of electromagnetism.

Maxwell's equations relate the electromagnetic field F at any event to its source, the 4-current density j at that event. Similarly, Einstein's equations relate spacetime curvature to its source, the energy–momentum of matter. As we shall see, the analogy goes further. In any given coordinate system, Maxwell's equations are second-order partial differential equations for the components Fµν of the electromagnetic field tensor (or equivalently for the components Aµ of the electromagnetic potential). We shall find that Einstein's equations are also a set of second-order partial differential equations, but instead for the metric coefficients gµν of spacetime.

The energy–momentum tensor

To construct the gravitational field equations, we must first find a properly relativistic (or covariant) way of expressing the source term. In other words, we must identify a tensor that describes the matter distribution at each event in spacetime.

General relativity is one of the cornerstones of classical physics, providing a synthesis of special relativity and gravitation, and is central to our understanding of many areas of astrophysics and cosmology. This book is intended to give an introduction to this important subject, suitable for a one-term course for advanced undergraduate or beginning graduate students in physics or in related disciplines such as astrophysics and applied mathematics. Some of the later chapters should also provide a useful reference for professionals in the fields of astrophysics and cosmology.

It is assumed that the reader has already been exposed to special relativity and Newtonian gravitation at a level typical of early-stage university physics courses. Nevertheless, a summary of special relativity from first principles is given in Chapter 1, and a brief discussion of Newtonian gravity is presented in Chapter 7. No previous experience of 4-vector methods is assumed. Some background in electromagnetism will prove useful, as will some experience of standard vector calculus methods in three-dimensional Euclidean space. The overall level of mathematical expertise assumed is that of a typical university mathematical methods course.

The book begins with a review of the basic concepts underlying special relativity in Chapter 1. The subject is introduced in a way that encourages from the outset a geometrical and transparently four-dimensional viewpoint, which lays the conceptual foundations for discussion of the more complicated spacetime geometries encountered later in general relativity.

In our discussion of the Schwarzschild geometry, we have thus far used the coordinates (t, r, θ, φ) to label events in the spacetime. In this context, (t, r, θ, φ) are called the Schwarzschild coordinates. Moreover, until now we have been concerned only with the exterior region r > 2µ. We now turn to the discussion of the Schwarzschild geometry in the interior region r < 2µ, and the significance of the hypersurface r = 2µ. We shall see that, in order to understand the entire Schwarzschild geometry, we must relabel the events in spacetime using different sets of coordinates.

The characterisation of coordinates

Before discussing the Schwarzschild geometry in detail, let us briefly consider the characterisation of coordinates. In general, if we wish to write down a solution of Einstein's field equations then we need to do so in some particular coordinate system. But what, if any, is the significance of any such system? For example, suppose we take the Schwarzschild solution and apply some complicated coordinate transformation xµ → x′µ. The resulting metric will still be a solution of the empty-space field equations, of course, but there is likely to be little or no physical or geometrical significance attached to the new coordinates x′µ.

One thing we can do, however, is to establish whether at some event P a coordinate xµ is timelike, null or spacelike.

The Schwarzschild solution describes the spacetime geometry outside a spherically symmetric massive object, characterised only by its mass M. In the previous chapter we derived further spherically symmetric solutions. Most real astrophysical objects, however, are rotating. In this case, a spherically symmetric solution cannot apply because the rotation axis of the object defines a special direction, so destroying the isotropy of the solution. For this reason, in general relativity it is not possible to find a coordinate system that reduces the spacetime geometry outside a rotating (uncharged) body to the Schwarzschild geometry. The non-linear field equations couple the source to the exterior geometry. Moreover, a rotating body is characterised not only by its mass M but also by its angular momentum J, and so we would expect the corresponding spacetime metric to depend upon these two parameters.

We now consider how to derive the metric describing the spacetime geometry outside a rotating body. Since the mathematical complexity in this case is far greater than that encountered in deriving the Schwarzschild metric (or the other spherically symmetric geometries discussed in the previous chapter), we shall content ourselves with just an outline of how the solution may be obtained.

The general stationary axisymmetric metric

In our derivation of the Schwarzschild solution, we began by constructing the general form of the static isotropic metric. We are now interested in deriving the spacetime geometry outside a steadily rotating massive body.

In the preceding three chapters, we have considered in some detail the Schwarzschild geometry, which represents the gravitational field outside a static spherically symmetric object. We also considered the structure of the Schwarzschild black hole, in which the empty-space field equations are satisfied everywhere except at the central intrinsic singularity. In this chapter, we consider solving the Einstein equations for a static spherically symmetric spacetime in regions where the presence of other fields means that the energy–momentum tensor is non-zero. In particular, we will concentrate on two physically interesting situations. First, we discuss the relativistic gravitational equations for the interior of a spherically symmetric matter distribution (or star); in this case the energy–momentum tensor of the matter making up the star must be included in the Einstein field equations. Second, we consider the spacetime geometry outside a static spherically symmetric charged object; once again this is not a vacuum, since it is filled with a static electric field whose energy–momentum must be included in the field equations.

The form of the metric for a stellar interior

Most stars in the sky are nowhere near dense enough for general-relativistic effects to be important in determining their structure. This is true for main sequence stars (of which our Sun is an example), red giants and even such high-density objects as white dwarfs.