Level
2

Given a clock for some observer (in this case Finn) we cannot intrinsically say anything about the behavior of a clock for any other observer. Finn's clock leaves Aaron's clock completely arbitrary. Hence just having a single observer with a clock is not enough to create a definition of time that holds everywhere. We also have to determine how we should relate clock ticks at one point with clock ticks at another. Let's first consider how to relate Finn and Aaron's clock ticks.

Let us denote the values of Finn's clock by a variable \(F\) and the values of Aaron's clock by a variable \(A\). Hence \(F_1\) might represent the moment when Finn's clock reads 10 AM, \(F_2\) 10:00:01 AM etc. etc. Note that Finn's clock does not have to tick in what we would call seconds, I'm just using that as a familiar example so you can get an intuition about the meaning of \(F_n\). Similarly, there is a set of \(A_n\) representing the ticks of Aaron's clock

To relate Finn's time to Aaron's time we need a method to synchronize the two observer's clocks. Synchronization here means relating the ticks of the two clocks in some consistent manner. This can be accomplished by sending a signal between Finn and Aaron and equipping Aaron with a mirror that reflects the signal back to Finn. We will choose light to be our signaling field.

Now for the synchronization procedure. There are a bunch of different ways to synchronize, we will choose the simplest. Let a flash of light be sent from Finn at tick \(F_1\), reflect off Aaron at tick \(A_r\), and be received by Finn at tick \(F_f\). We synchronize the clocks by adjusting Aaron's clock so that

\(A_r=\frac{F_1+F_f}{2}\)

i.e. the average of the sent and received ticks. This synchronization process has nice properties: a clock is automatically synchronized with itself, synchronization is transitive, etc.

Question 3: Assuming all clocks are synchronized with each other, how many clocks does it take to define a time *everywhere*?

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