As we move away from an isolated body, its gravitational field decreases and tends to zero as we approach infinity. In a spacetime picture, where gravity is described by curvature, it would therefore seem entirely reasonable to model an isolated body on a spacetime that is, in some sense, asymptotically flat. If the body possesses some sort of symmetry – it might, for example, be an axisymmetric or spherically symmetric star – then it would also seem reasonable to model it on spacetime with the same type of symmetry. But what exactly do we mean by asymptotic flatness, and what do we mean by a spacetime symmetry? In this chapter we shall attempt to answer these two questions.

Asymptotically Flat Spacetimes

In order to use general relativity to study the gravitational field of an isolated body, such as a star, it is necessary to have some well-defined notion of asymptotic flatness. An asymptotically flat spacetime represents the idealized situation of a gravitating body that, to all intents and purposes, is totally isolated from the rest of the universe by virtue of its great distance from all other bodies. As we move away from such an object, its gravitational field should decrease, and thus we expect that spacetime should become flat at asymptotic distances. Of course, no physical system truly can be isolated from the rest of the universe, but for a system such as a star the gravitational influence of all other matter is so slight that it is entirely reasonable to consider it as being totally isolated, essentially a single system in an otherwise empty universe.

As we have seen, a (charged) matter distribution on flat spacetime is described by the fields Ja, Fab, and Tab, where Ja determines the charge density, Fab describes the electromagnetic field, and Tab determines the four-momentum density. The background geometry of M determines ∇a and the constant tensor fields gab and εabcd. Nature does not, of course, allow Ja, Fab, and Tab to vary in an arbitrary manner, but imposes constraints in the form of conservation laws and equations of motion, which may be expressed in the form of field equations. In this chapter we shall discover the field equations satisfied by Ja, Fab, and Tab.

Conservation Laws

Consider a congruence whose curves are the particle world lines of a uniform dust distribution with particle-number current ja. Let us select one particular curve, l0, and three neighboring curves, l1, l2, and l3, which are joined to l0 by three space-like connecting vectors aa, ba, and ca (Fig. 8.1).

According to an observer with four-velocity va orthogonal to aa, ba, and ba, the volume V = εabcd vaabbccd will always contain the same number of particles. Note that va is determined uniquely by the condition that it is orthogonal to aa, ba, and ba, but this does not mean that it is Liepropagated along the congruence.

One of the greatest intellectual achievements of the twentieth century is surely the realization that space and time should be considered as a single whole – a four-dimensional manifold called spacetime – rather than two separate, independent entities. This resolved at one stroke the apparent incompatibility between the physical equivalence of inertial observers and the constancy of the speed of light, and brought within its wake a whole new way of looking at the physical world where time and space are no longer absolute – a fixed, god-given background for all physical processes – but are themselves physical constructs whose properties and geometry are dependent on the state of the universe. I am, of course, referring to the special and general theory of relativity.

This book is about this revolutionary idea and, in particular, the impact that it has had on our view of the universe as a whole. From the very beginning the emphasis will be on spacetime as a single, undifferentiated four-dimensional manifold, and its physical geometry. But what do we actually mean by spacetime and what do we mean by its physical geometry?

A point of spacetime represents an event: an instantaneous, pointlike occurrence, for example lightning striking a tree. This should be contrasted with the notion of a point in space, which essentially represents the position of a pointlike particle with respect to some frame of reference. In the spacetime picture, a particle is represented by a curve, its world line, which represents the sequence of events that it “occupies” during its lifetime. The life span of a person is, for example, a sequence of events, starting with birth, ending with death, and punctuated by many happy and sad events.

In this chapter we shall consider the intrinsic geometric structure of flat spacetime M. In particular, we shall use the results of the previous chapter to show that M has a natural physically defined affine structure (i.e., the notion of a displacement vector makes sense) and, most importantly, that the space of displacement vectors possesses a natural, physically defined metric.

Spacetime Vectors

A spacetime displacement is simply an ordered pair of spacetime points. We write OP = - PO and OP + PQ = OQ. If P lies inside N(O), the null cone of O, we say that OP is timelike, in which case O and P lie on the world line of an observer or a massive particle. There are two types of timelike displacements: if P lies in the future of O (according to an observer whose world line passes through O and P), we say that OP is future-pointing otherwise, of course, we say it is past-pointing. If P lies on N(O), we say that OP is null, in which case O and P lie on a null ray or the world line of a massless particle. Again, there are two types of null vectors, future-pointing and past-pointing, depending on whether they lie on N+(O) or N−(O). If OP is neither timelike nor null, that is, if P lies outside N(O), we say that OP is spacelike (Fig. 4.1).

For a timelike displacement OP, the points O and P lie on a world line l with proper time t. If Q also lies on l and t (Q) – t (O) = a[t (P) – t (O)], we write OQ = α OP.

An isolated physical body such as a star may be described in terms of a spacetime picture by an asymptotically flat spacetime, (M, g), containing a world tube representing the region of M occupied by the body. Inside the world tube, the energy–momentum tensor, Tab, will be nonzero and should describe some physically reasonable matter distribution. Outside, in the vacuum, matter-free region, Tab=0, and hence, by Einstein's equation, Rab = 0. The curvature, and hence the gravitational field, in the vacuum region is thus completely given by the Weyl tensor, that is, Rabcd = Cabcd.

While the situation described above serves as a framework for the discussion of isolated bodies in general relativity, it is too general to provide much real information; what we need is some simplification to make it more tractable to calculation. Fortunately, most stars are more or less spherically symmetric and stationary. We can thus obtain a useful, but considerably simpler, model by imposing spherical symmetry and stationarity. The problem of finding the most general spacetime satisfying these conditions was, in fact, essentially solved by Schwarzschild in 1916, not long after Einstein first proposed his general theory of relativity. In particular, he showed that the vacuum region outside a spherically symmetric and stationary body has a simple geometric structure, which we shall refer to as a Schwarzschild geometry, that depends on only one number, m, which may be identified with the total mass of the body. This result is similar to the Newtonian case, where the external gravitational field of any symmetric body is given simply by the inverse square law.

We come now to the physical interpretation of the curvature tensor Rabcd. As we have seen, Rabcd arises from the spacetime metric, which in turn arises from the properties of inertial world lines and null rays. If there are no gravitational tidal effects, then spacetime is flat and hence Rabcd = 0. Conversely, if Rabcd = 0, then, apart from the possibility of topologies other than ℝ4, spacetime is flat and there are no gravitational tidal effects. A nonvanishing curvature tensor is thus an indication and a measure of gravity. In this chapter we shall show that a gravitational field is completely described by Rabcd. But first we need the concept of a geodesic.

Geodesics

The world line of an inertial particle has the remarkable property of being uniquely determined by its initial four-velocity vector. If we take proper time to be the parameter along the world line, then its tangent vector at any point will be a four-velocity vector. Similarly, a null ray is uniquely determined by specifying a null vector at some point, O say, and the set of all null rays through O forms a null cone N(O). We recall that it was precisely these properties of inertial world lines and null rays that allowed us to define a unique spacetime metric. Furthermore, the metric led to a unique connection and, finally, to the curvature tensor.