Why is time flying by so quickly these days?

Preface

I think (nearly) everybody can relate to the question posed in the title. The older we get, the faster the years seem to fly by. Days become weeks become months -- and once again, another year has passed. Our perception of time seems to change as we age and every subsequent year seems to be shorter than the previous one. In this post, I'll introduce a hypothesis published by L. W. Doob in 1971 that tries to explain this observation. It should be noted that other alternative models exist, e.g. the one by R. Lemlich published in 1975 which are based on different assumptions. However, due to its simplicity we here will only examine the model by Doob. Other models generally produce similar, although less extreme results.

The Hypothesis

Doob suggested that the perceived duration of a time interval is influenced by our age. In particular, he proposed that the perceived duration of an interval corresponds to the fraction of its (real) length divided by the age of the perceiving person. Thus, considering only full years of life, we would perceive our second year of life to be half as long as our first one; the third year of life to be a third of the first one, and so forth.

Thus, the perceived length \(l_k\) of the \(k\)-th year of life would be given by

\(l_k = 1/k\),

and the perceived cumulative length up to year \(k\) can be calculated as the \(k\)-th partial sum of the harmonic series:

It is interesting to point out that the harmonic series diverges, meaning that the series will continue to grow without bound. This makes sense biologically, because we may very well expect that, although our perception of time might slow down, it will never run "infinitely slow."

Some numerical calculations

Focussing again only on complete years, we can use the series above to calculate the subjectively perceived lengths of the individual years of one's life. Fixing the first year to a subjective length of one unit, we get for the first 80 years of life:

Adding up all of these subjective lengths, we can calculate the relative fractions of the individual years (assuming the person lives for 80 years). From these relative fractions we can compute the relative fractions of the first \(x\) years of life. We get:

Thus, after 20 years of living, we have already experienced more than 70% of our subjectively experienced time. After 30 years, we're at approximately 80%, and after 50 years, less than 10% remains.

Another way of visualising this relationship is using the following timeline, which plots the perceived length of time on the x-axis, and marks the years of real time using vertical lines:

Indeed, the older we get, the shorter the years become; and indeed, at higher ages the years start to just fly by faster and faster.

Conclusion

Life is short, and time flies -- even more so as we age. And time is the only resource that is truly limited. Use it wisely :-).