Time Flies (faster when you are old)

We all know the saying: "time flies when you are having fun". The reverse is also true, when you wait out an hour it goes tantalizingly slow. Paradoxically, when after a year you recall these events, the hour wait seems like a minute, while the busy hour feels much longer. So our perceived sense of time is not the same as the time that actually passed. In analogy with a feel-temperature, let's introduce a "feel-time".

Many have proposed that one psychological effect has to do with the fact that you measure duration relative to the time you have already lived. Your yardstick grows over time and hence time intervals feel shorter. Assuming your yardstick (L) grows linearly in time, L = a*t, then a feel-time-interval as measures in units L is ds = dt/L = b*dt/t (with b=1/a.) If you didn't follow this, it says that time intervals feel shorter as the reciprocal of your age: a year at 20 feels twice as long as a year at 40.

A somewhat orthogonal explanation that one can find in the literature is that it has to do with the number of new events experienced in a time interval. If each new experience means an increment of 1 unit in your feel-time, then you spend a lot of feel-time in your youth and very little in your old age. In fact there are two effects at play: when you are young everything is new, while when you get older many everyday experiences have become routine and do not increment your feel-time as much. Add to that the fact that your memory is much better when you are young and thus many more experiences are stored (In fact, research has shown that you mostly seem to remember the time when you were around your twenties.)

The second explanation seems more plausible to me. I have worked out the math for the highly simplified case where there is a bag if N experiences from which the world samples experiences uniformly at random. The assumption is that after you have seen an experience once, it will no longer count as new and not increment your feel-time. This model predicts that feel-time-intervals become exponentially smaller with a rate of 1/N (with N the total nr. of experiences.) This exp(-t/N) form is very different than the 1/t predicted by the yardstick theory. Most importantly, if you have reached N, time stops! Of course, this model was too simple to be of any practical use because there is a very large amount of possible experiences and they are not at all uniformly sampled (e.g. going to the restroom is sampled thrice a day while marrying your spouse hopefully only once.)

In Orange County people seem to be obsessed with staying young judged by the number of plastically enhanced individuals. But perhaps a better strategy is to invent a pill that stretches out the perception of time. I have heard this possibility is quite seriously being researched. Take this pill before you go on an adventurous trip and you can buy yourself many years of life. Or maybe memory implants such as played out Total Recall will be the easier solution for the future.