# Special Relativity XII: Causality in Spacetime Diagrams

Hello, and welcome back to MPC! Last week, we celebrated the 25th blog post by reviewing what we have covered so far and discussing what we will be covering in the coming months. As promised, in this week’s post we will be relating three major concepts that we have been discussing: spacetime diagrams, light cones, and causality.

Recall from our discussion of spacetime diagrams that a light cone represents the region “where all events exist in spacetime.” Although this is what we said a few weeks ago, it is only partially true. To understand why, we have to reimagine the light cone.

Let’s imagine that the event of John eating dinner is currently located at the origin of a spacetime diagram:

Figure 1: John eating, plotted on a spacetime diagram

Remember, our reasoning for why we said the light cone represents “where all events exist in spacetime”: the boundaries of the light cone represent objects moving faster than the speed of light. For example, if an object was moving through spacetime as follows:

Figure 2: An object moving faster than the speed of light

it would be moving faster than the speed of light, which we know is impossible. For this reason, we said that all events in spacetime must be within the light cone.

However, we have to realize that the boundaries of the light cone represent the objects moving at the speed of light to John. If Frank eating dinner is located at some point in John’s past:

Figure 3: Frank eating, plotted on a spacetime diagram

he would have his own coordinate system and his own light cone boundaries (according to his coordinate system):

Figure 4:  Every event has its own light cone

**Note: If this is confusing, think back to our discussion of relativity. The 3.0 * 10^8 on John’s L-axis represents something that is 3.0 * 10^8 meters away from John (relative to John). This does not necessarily have the same meaning as something being 3.0 * 10^8 meters away from Frank (relative to Frank). A similar argument can be made for Frank’s ct-axis (because the event of Frank eating dinner is in the past, 1 second relative to his current time is not the same as 1 second relative to John’s time)

It is clear that Frank’s light cone extends into regions that are not in John’s light cone. This suggests that, according to Frank, events outside of John’s light cone can occur. Clearly, the light cone is not simply the region “where all events exist in spacetime.” It must mean something else!

This is where causality comes in. Each event’s light cone actually represents the region where all events that it can have a causal relationship with exist. To understand this, let’s go back to John eating dinner. Let’s imagine the event of John’s dinner being made (let’s say John is eating at a restaurant). It should be clear that these two events are causally related (if John’s food was never made, he could not be eating it). Let’s imagine that this event (John’s food being made) occurred at the point shown on the spacetime diagram below:

Figure 5: Food is made within the light cone

Let’s map out this food being brought to John:

Figure 6: Food being moved to John

**Note: This restaurant has very fast service!**

Is this possible? Looking at the angle of the line drawn in Figure 6 (in orange) makes with the ct-axis, we can see that the food is moving slower than the speed of light (look at this blog post for more information). Therefore we can conclude that, yes, this is possible.

However, what happens if the event of the food being made happened at the point shown on the spacetime diagram below:

Figure 7: Food is made outside of the light cone

For this food to make its way over to John, it would have to travel along the line shown below (in orange):

Figure 8: Food being moved to John

if the food were to travel along this line, it would be moving faster than the speed of light. This is not possible!

The important thing to note is that any event that is within one’s light cone can “reach” that event. For example, in Figure 6, the food was able to “reach” John. In general, if an event can “reach” another event, it can affect that event. In other words, if an event is inside of the light cone of another event, those two events can have a causal relationship. On the other hand, if an event is outside of another event’s light cone, those events cannot have a causal relationship (they cannot “reach” each other because nothing can travel faster than the speed of light; we saw this with the food in Figure 8 that was unable to reach John).

Overall, a person’s/event’s light cone is a region containing all events that it could have a causal relationship with. Recall that two events that have a causal relationship must have a certain order. That means in the following

Figure 9: Two events that have a causal relationship

A must happen before B (A is in B’s past light cone), while in the following:

Figure 10: Two events that do not have a causal relationship

A could happen before B to one observer, but B could happen before A to another observer.

But how do we know if one event is within another event’s light cone? We use the spacetime interval! For example, in Figure 6 or Figure 8 we can take the L-value  and ct-value for the food being made and plug them into the spacetime interval expression:

(**Note: When we were talking about the spacetime interval and light cones, we were taking all of our measurements relative to the origin. Because John is located at the origin, we don’t have to worry about his L-value and t-value at all. You can just pretend that nothing is at the origin and take the spacetime interval for the point representing the food being made like we have done in the past)

For Figure 6, we will get a spacetime interval greater than 0 (see this post), indicating that the “food being made event” is within John’s light cone, meaning that our two events could have a causal relationship. For Figure 8, we will get a spacetime interval less than (see this post), indicating that the “food being made event” is outside of John’s light cone, meaning that our two events could not have a causal relationship.

That’s it for this week! I hope that you have a better understanding of how the spacetime interval, light cone, and causality all fit together. Next week, we will be jumping into something new and exciting: general relativity. See you then!