Special Relativity: Where and When

In Aristotle's space-time there as an absolute notion of positions and motions. In Galileo's space-time we lost absolute positions and motions, but we retained absolute times. Saying when two events occurred with respect to one another was objective and absolute. In my last post I had two example diagrams featuring two astronauts flying past each other in empty space. The diagrams were equivalent in the sense that they depicted the same state of affairs, only differing by a coordinate transformation. There was no objective answer to the question whether you were moving past your stationary friend, or vice versa. But there was an objective answer to the question of when (for example) you two crossed paths, or indeed, the order of any sequence of events either of you witnessed.

This latter part---questions about when things happen, and their order, will lose their objective status in Einstein's relativity. And we'll see why this causes trouble for faster-than-light travel in science-fiction stories.

To begin we need to come up with a concrete understanding of what "happens at the same time" means, or even better, how to test it. For this we need a protocol for determining when events too distant for us to see "happen" relative to our own lives. To give a feel for this I will shamelessly self-plagiarize from my dissertation:

Imagine sending an RSVP to a friend, Ali, and you'd like to determine when, exactly, they get the invitation. We first make three simplifying stipulations: (1) It takes no time for your friend to compose their a response, (2) they will do so immediately upon receiving the invitation, and (3) the invitation spends no time sitting in the mailbox (i.e. the mail carrier immediately picks it up at both ends). Now, you send the invitation at noon on Monday, and receive Ali's response at noon on Wednesday. From this information we can easily calculate the total elapsed time: 48 hours. From symmetry assumptions (specifically, that there is no difference in the average speed of the mail carriers nor is there a difference in distance traveled on the outbound and return routes) we can then calculate that the letter arrived at Ali's house at noon Tuesday (that is, exactly 1/2 the round-trip time). You can now say with confidence that Ali received notification of your party exactly when you were enjoying your lunch on Tuesday. Now, suppose that you sent out invitations (at noon) in the same manner on the previous Saturday and Sunday to Boris and Claire respectively. And further suppose that you received a response from Claire on Thursday at noon, and from Boris on Friday at noon. By the same methods described above, you would calculate that they too both received their letters at noon Tuesday (a remarkably coordinated mail service!). Thus, the events of Ali, Boris, and Claire receiving invitations, and your eating of lunch on Tuesday all occupy the same "simultaneity space"---we say they are simultaneous with one another.

Let's summarize the protocol---to figure out when a distant event happened simultaneous with some event in your life, first, send out a signal that can be returned by the event in question. Second, start tracking to time, and record the elapsed time it takes to receive the returned signal. Finally, divide that total time by two---whatever you were doing half-way between sending and receiving the signal was simultaneous with it.

Luckily, pictures help, and again I will  copy from my dissertation:


What we see here is a space-time diagram of an object alpha. At event s alpha sends a flash of light off---shown above by the dotted line. After that event alpha waits, eventually receiving a flash of light from event e at r. Looking back on the time passed between the sending and receiving of the signal, alpha reasonably divides the total time in half, and concludes that their signal was received by e at the same time that alpha was experiencing event o. Of course, there was no way for alpha to know that events o and e were simultaneous at the time, but that's no impediment to concluding post-facto when the two events occured.

That's all well and fine, and it comports perfectly well with our common sense notion of 'simultaneous'. Things change, however, then we compare two observers in motion relative to one another:


In the diagram above, we see an observer beta in motion relative to our inertial frame---specifically moving from left-to-right. Notice that beta is performing the same measurement protocol for determining simultaneity that we described for alpha. However, the principles of relativity begin to have a noticeable effect. Recall that one of the fundamental assumptions which undergirds special relativity is that the speed of light is constant under all inertial reference frames. What this means for the diagram above is that even though beta is moving from left-to-right from our vantage point, when she sends out a flash of light at event s, that light signal doesn't receive a 'boost' from her velocity, like we might naively expect. This is represented in the diagram by noting that the light beam remains at a 45-degree angle relative to the diagram's overall orientation.

As beta carries out the measurement protocol we see that her 'simultaneity-space' (the collection of events mutually simultaneous) is 'tilted' relative to our perspective. She will determine that e happened simultaneous with an event noticeably prior to event o. It's due to this sort of behavior of measurement by light signals which results in our claim that the time-ordering of events loses objective meaning in special relativity.

We can see this explicitly by considering two observers in relative motion in one space-time diagram:


Here we have both alpha and beta, with alpha stationary in our reference frame, and beta move to the right. Suppose they are collocated at event s and both agree to send off light signals. For alpha, the protocol determines that events e and e' are simultaneous, as will be every event falling in the simultaneity space Aα. However, for beta the time ordering of events will be quite different. Event e will happen a significant time earlier than e'. Additionally, it's not hard to see that a third observer moving to the left, and also collocated at s will see e' happening before e!

Next time we'll see how this feature (namely, relative motion changes your simultaneity space) will cause trouble for FTL fleets looking to invade distant solar systems.