Maxwells Equations

As we have seen, light or, in other words, electromagnetic radiation plays a central role in spacetime physics. It should come as no surprise, therefore, that the theory of electromagnetism finds its most natural form in such a setting and that, when expressed in tensorial form, its equations assume their simplest and most elegant form. Indeed, as we shall show, Maxwell's equations can be expressed as a single tensorial equation. A very basic observation is that electric charge is an additive,...

The Spacetime Metric

If OP is timelike and t(P) - t(O) velocity vector. In other words, a four O and P that are one second (proper Figure 4.2. Displacements OP and O'P', lying on parallel world lines, are equivalent if the time interval between O and P is equal to that between O' and P'. 1, we say that v OP is a four-velocity vector connects two points time) apart and are such that P lies to the future of O. If an observer's world line passes through O and P, we say that he has four-velocity v. If he sets his clock...

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...

The Abstract Index Notation

In the previous section we represented a tensor by a single letter. This has the serious drawback of giving no indication of the tensor's type and involves the introduction of new symbols to indicate its various contractions. For example, a tensor t of type (2,1) admits two contractions, which we denoted by cra t and cr21, the first contraction being with respect to the first upstairs slot and the second with respect to the second upstairs slot. This sort of notation works well for tensors of...

Solutions and Hints to Selected Exercises

2.3 S is the intersection of N+(p) and N+ (q). Since dim N+(p) dim N+ (q) 3 and dim M 4, we have dim S 2 if S is nonempty. If p occurs before q to some observer, then N+(p) lies inside N+(q) and S is empty. 3.1 Referring to Fig. 3.10, we have t'2 K-1t2 and t Kt , where K-1t2 - Kti _ K-1(t + x) - K(t - x) 2 2 x vt 3.2 Both Fig. S.1(a) and (b) lead to In the case of (b), where the particles are moving towards each other, the Doppler shift is given by 3.3 From Fig. S.2 we see that Peter will be K...