# Special Relativity XI: Causality

Hello, and welcome back to MPC! Last week, we related the concept of spacetime intervals to light cones. Today, we will begin discussing the idea of causality.

Before we even begin, here is something to ponder: what word does causality sound like? I am sure that, for many of you, the word cause came to mind. Believe it or not, you are on the right track! Let’s start off by thinking about causality in terms of causes, specifically in terms of causes and effects.

The idea of cause and effect is everywhere in our lives. Some of you may have been introduced to cause and effect in your literature class (what caused this to happen in the story?), but we can think about cause and effect in a more general sense. A cause and effect relationship essentially describes a pair of situations where one situation is a result of the other one. In other words, because this event (the cause) happened, that event (the effect) happened. In physics, when two events have a cause and effect relationship, it is said that they have a causal relationship.

Let’s imagine a scenario in which we have two people, Albert and Richard. Let’s say that Albert eats lunch at 2:00 p.m. and Richard eats dinner at 6:00 p.m.

Figure 1: Ordering of Events (Meals)

**Note: We are representing events with circles just like we did when we first discussed spacetime diagrams**

Do these two events have a causal relationship? Does Albert eating lunch at 2:00 p.m. cause Richard to eat dinner at 6:00 p.m. (or vice versa)? Of course not (or at least most likely not, there could be some strange circumstances that we do not know about)! Here is an interesting way to think about it: if Albert chose to eat lunch at 1:00 p.m. or not eat lunch at all, would Richard still eat dinner at 6:00 p.m.? He most likely would — there is nothing to suggest that he would not. Therefore, Richard eating dinner at 6:00 p.m. cannot be the result of Albert eating lunch at 2:00 p.m., it must be the result of something else (perhaps the fact that Richard became hungry at 5:30 and his dinner took 30 minutes to make). This means that these two events are not causally related.

Let’s now imagine a second scenario involving Albert and Richard: Albert and Richard are playing football. Albert and Richard’s team is losing, but at the last second of the game, Albert throws a pass to Richard, Richard catches the ball, and they win the game.

Figure 2: Ordering of Events (Football)

Are the events of Richard catching the football and Albert and Richard’s team winning causally related? We can run the same test we ran before: if Richard did not catch the football, would Richard and Albert’s team still have won? No, it would not have (remember, their team was losing before Richard made the last second catch). Therefore, Albert and Richard’s team winning must be a result of Richard catching the football. To put it another way, Richard catching the football caused Albert and Richard’s team to win the game. Therefore, these two events are causally related.

Alright, so hopefully the concept of causality is starting to makes sense. To some of you, causality may seem simple. However, it has a very interesting subtlety. Let’s return to our first scenario (with the meals). What happens if we reverse the order of the two events: what happens if Richard eats his dinner first (say, at 3:00 p.m). and Albert eats his lunch after (say, at 4:00 p.m.)?

Figure 3: New Ordering of Events (Meals)

Does the situation as a whole still make sense? It seems as if it does — Richard just ate an early dinner and Albert just ate a late lunch. This is certainly possible/logic! So, regardless of who ate first, the situation makes sense.

Can the same be said about the second scenario (the football scenario)? If Albert and Richard’s team first won the football game and then Richard caught the pass at the last second, would the situation as a whole still make sense?

Figure 4: New Ordering of Events (Football)

No, it would not. We said that Albert and Richard’s team was losing prior to Richard’s last second catch, so how could the team possibly win before that catch was made (also, the fact that a catch was made after the game ended seems a little strange)? That makes no sense!

Now, we see something rather interesting. For two events that are not causally related, the order in which the events happen is not too important (the situation will make sense with either ordering). However, for two events that are causally related, the order of the events does matter (the situation only makes sense with one ordering).

That’s interesting, but who cares? Switching the order of events is just hypothetical — you cannot actually flip the order of two events. One event will happen first and one event will happen second, so why should we care about what would happen if the order of those two events were flipped? Well, yes, one event will happen first and one event will happen second. Remember, though, in relativity, time is relative. What this means is that, while one of the events may appear to happen first according to one observer, the other event may appear to happen first according to another observer (this idea is rooted in the concept of time dilation).

Yes, this may seem very unintuitive. Nonetheless, it is completely possible for one observer to see Albert eating lunch before Richard eats dinner while another observer sees Richard eating dinner before Albert eats lunch. And if this happens, so what? Either ordering of the events makes complete sense!

But what would happen if an observer saw Albert and Richard’s team win the football game before Richard catches the football? That would certainly be very strange! Luckily for us, though, the laws of physics dictate that this is impossible: two events that are causally related will always occur in the same order according to every observer? This means that two people may disagree on the order of meal eating in our first scenario, but all observers will agree that Richard caught the football before Albert and Richard’s team won the game.

We now have a way of determining if two events are causally related (if one event caused another event to occur): if it is logically possible for the two events to occur in different orderings the events are not causally related, if it is not logically possible for the two events to occur in different orderings the events are causally related.

That’s all for today. Next week we will relate what we discussed about causality today to the spacetime interval and light cones. See you then!