It is a well known phenomenon, that as we grow older, life seems to get by faster. Many remember those infinite days in their teenage years, and wonder why a month passes so quickly now.
Some people tend to explain this in terms of the density of bright and extraordinary events happening, and it seems to be little doubt, that the density of such events has quite a bit of influence on both the current and retrospective perception of the speed of physical time.
However, let us consider the situation when the outside life flows uniformly. It is natural to assume, that "value" of a unit of physical time, t, for an individual, is inversely proportional to his/her perception of internal age, T. We are going to derive from this, that T = C*SQRT(t), where the choice of C is just a matter of the choice of units for physical and internal time, so assuming that at an age of 1 physical year, the person has an internal age of 1 internal year, we obtain T = SQRT(t), where t is measured in years, and T is measured in our artificial units of internal years.
We would like to express the hope that this model reasonably describes the relationships between physical and internal time in more realistic cases of non-uniform life flow as well, although a more accurate analysis of probability distributions is needed to confirm this.
Our assumption, that "value" of a unit of physical time, that is, its contribution to our internal time, is inversely proportional to one's internal age, can be expressed via a differential equation dT/dt = D/T, where D is some constant. Integrating this equation, we obtain that T = SQRT(2*D*t) + E, where E is another constant. Choosing the physical time coordinate so that t = 0 in the moment of birth, assuming that at the moment of birth internal age, T, is also zero, and denoting C = SQRT(2*D), we obtain T = C*SQRT(t), and T = SQRT(t), when T is measured in our internal years.
Hence, our expected lifespan can be thought of, as 8-10 internal years (64-100 years), of which initial physical years are especially valuable, and thus should not be sacrificed for "successes in future life". Indeed, assuming even the lifespan of 100 physical years (and the shorter this lifespan, the more is the value of initial physical years), we obtain, that the first year corresponds to 10% of life, the years 2-4 for another 10%, they years 5-9 for another 10%, yielding 30% of life by the age of 10. Years 10-16 yield the next 10%, and years 17-25 another 10%, making a half-life at the age of 25 for a person who expects to live a hundred years.
Say again, it is quite stupid to sacrifice the quality of really meaningful initial years of our lifes, in order to supposably insure better quality of "future" or "adult" life. An hour of a 9-year old is worth 2 hours or 36-years old, an our of a 16-year old is worth 2 hours of 64-years old, etc.
REMARK: It is important not to make an error of deriving the equation dT/dt = D/t, and obtaining an incorrect logarithmic answer. In some sense, the results we obtained are less pessimistic than one might have expected, after one had agreed that time passes faster as we age.
I've made this derivation sometime in late 70-s or early 80-s (that is at the internal age of 4), and wrote it down in December 1997 (I'll be 6 internally in the year 2000).
(November 2015 update: It turns out this model had been discovered at least as early as 1975: Subjective Acceleration of Time with Aging, Perceptual and Motor Skills, 1975, 41, 235-238. The topic is further discussed in "Time Warped: Unlocking the Mysteries of Time Perception", by Claudia Hammond, 2012-2013.)
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