On the Dynamics of the Perception of Time

Every morning I try to write three pages in my diary. I've noticed that the third page always takes less time to write than the first page. It reminds me of "time dilation" (the apparent slowing down of time) from Einstein's theory of relativity. Might there be something analogous here? When you wake up, your brain is coming up to speed, so your first diary page will be done somewhat slowly. Then, as your brain warms up, the second page will be done faster, and finally, when you are fully awake, the third page will be done fastest of all. But to you, your writing speed seemed to be constant throughout. Time seemed to slow down for you in the morning.

We might expect the opposite to happen in the evening: time speeds up. So as your brain starts shutting down in the evening, things take longer for you to do, though you don't notice it. Outside agents like clocks and TV shows seem to go by faster. In other words, time seems to speed up.


Formalizing the Nature of Perceived Time

Let's call the speed of perceived time s. In an empty room with no windows, s is constant. If you add a window or a clock, time will seem to slow down in the morning and speed up in the evening. That is, as your ability to think improves, s decreases; when your mental capacity drops, s increases. One might suppose that s = 1 / k, where k is mental capacity.

This explains the old saying that "time speeds up as you get older". As your mental capacity gradually decreases (k↓), the perceived time will speed up (s↑). Conversely, when you are a child, your mental processes are at their fastest (k↑) and time seems slow (s↓). Recall the long days of elementary school.

But this isn't the whole story; grown-ups can have long days as well! And what about the observation that "time flies when you're having fun"? Is this not experienced by young and old alike?


Resolution to the Anomaly

When your mind is engaged in something fun and stimulating, you devote most of your mental capacity k to that activity; your brain is too busy to check outside agents like clocks and TV shows very often. Because these external markers of time are checked with less frequency, long periods of time will elapse between each reading, and thus time will appear to be moving quickly. We can generalize our earlier result to

            s = a / k

The speed of perceived time s increases with mental activity a but decreases with mental capacity k.

Thus, "time flies when you're having fun" (s↑) because of the increased mental activity (a↑). Conversely, if your workday is extremely boring, your mental activity will be low (a↓) and time will seem to go by slowly (s↓).

s = a / k has some interesting implications for the manipulation of perceived time. If you are in the middle of a lecture that you find boring and you want to speed up the time (s↑), try sleeping (k↓), i.e., decrease your mental capacity. If you find you've been missing deadlines consistently and want to slow down the time (s↓), try learning, to increase your overall mental capacity (k↑).

Want your vacation to go by more slowly (s↓)? Do as little activity as possible: sit in a chair and stare at a corner of a room (a↓). On the other hand, if you visit Las Vegas, your mental activity will likely be highly stimulated (a↑) and time will speed up (s↑).