# Causality and the Light Cone

Dr Lim Zhi Han

Koh Zhi Yong

Gerald Kang Joon Kiat

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Published 2022-09-25

# 6.9 Causality and the Light Cone

Without any constraint on how fast information can travel, the causal structure of space and time can be illustrated in Figure 1.

Figure 1. The causal structure of space and time before relativity.

Two events p and q will fall in either of the following mutually exclusive cases: (1) q is to the past of p, thus possible for q to influence p ; (2) q is to the future of p , thus possible for p to influence q; (3) p and q are simultaneous events, thus impossible for them to influence each other.

However, the above description of causality is incorrect! Information does take time to travel, and without information transfer, one event cannot influence another. As we have learnt from special relativity, the speed limit at which information can travel is the speed of light c=2.998\times10^{8}\text{ m/s}. By taking into account this speed limit, the new causal structure of space and time is as illustrated below. (Figure 2)

Figure 2. The more accurate causal structure of space and time. The events in the past and future of p that have possibilities of cause and effect are constrained within the past and future light cones. — Figure 2. The more accurate causal structure of space and time. The events in the past and future of p that have possibilities of cause and effect are constrained within the past and future *light cones*.

Note that the light cones in Figure 2 are the three-dimensional generalisation of the world lines of a light signal in the Minkowski space-time diagram (Section 6.8.1). Here the events p and q will fall in one of the following categories.

(i) q is in the definite past of p, thus possible for its information to reach p ;

(ia) q is on the past light cone of p. Nothing other than light can reach p from q.

(ii) q is in the definite future of p, thus possible for its information to reach p;

(iia) q is on the future light cone of p. Nothing other than light can reach q from p.

(iii) Events p and q are separated in such a way that no material body nor light can go from one event to another. Some observers may see p then q, some may see q then p and some may even see p and q simultaneously.