By W. Grenier

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**Additional resources for Classical Mechanics - Point Particles and Relativity**

**Sample text**

For further details of the causal properties of space-times, see Carter (1971a), Hawking and Ellis (1973), Penrose and Rindler (1986) or Garc´ıaParrado and Senovilla (2005). 6 It is as a consequence of this property that space-times with vanishing Weyl tensor are said to be conformally ﬂat. The metrics for such space-times can always be expressed as some conformal multiple of the metric for Minkowski space. 3 Minkowski space-time The simplest space-time is that which is ﬂat everywhere. Such a space-time contains no matter and no gravitational ﬁeld.

2 Horizons The past light cone with vertex at a particular event covers all the points in the space-time that could be seen at that event. Thus, the inﬁnite family of past light cones with vertices on a complete timelike worldline covers all the events that could theoretically be “seen” by an observer with that worldline over its entire history. In a ﬂat space, as will be shown in Chapter 3, this normally spans the complete space-time. However, there are exceptions (such as for uniformly accelerated observers), and even more exceptions occur in a curved space-time.

This introduces a global repulsion which exactly balances the gravitational attraction between particles of matter in the universe. 17). This characterises the closed space-time known as the Einstein static universe. 10). However, although this initially seemed to represent a reasonable model of a static universe, it was later shown to be unstable. More signiﬁcantly, it subsequently became generally accepted that the universe is expanding, and a more general family of the Friedmann–Lemaˆıtre–Robertson–Walker space-times (see Chapter 6) was introduced and adopted.