Many people think that as we age, time appears to fly by faster. We have countless memories from our early years, and later on, it gets tough to tell one year apart from another. But is this just how we perceive things? Could there be an explanation for this mind-boggling phenomenon?

Event Density and Time Perception

When I discussed this with several people, they claimed that as children, they had many more interesting and new experiences, while today, life seems dull. This idea is fascinating, but it doesn't entirely reveal why childhood events would impact our perception of time in later years!

Many people also suggest that the time we perceive is relative to the time we've experienced. For example, a year perceived by a 4-year-old child is a whopping 25% of their entire life, while for an adult, it would be less than 5%.

Mathematical Model of Relative Time

Let's rephrase the above statement to facilitate writing it as an equation: The perceived speed of time passing decreases with age.

In scientific publications, I discovered a proposal that this relationship was proportional to the absolute passage of time (objectively measured, e.g., in years) [Doob (1971), Janet (1877)]. However, according to our assumptions, we experience relative time, not objective time! Let's attempt to express this mathematically.

Assume that objective time is denoted by O, and subjective time (perceived by us, relative) is represented as S.

dO and dS are infinitesimal time periods – dS signifies the passage of time dO as experienced by us.

As previously mentioned, the perceived passage of time decreases with the amount of time lived (also subjective!), so dS decreases as S increases. Consequently:

$$dS = dO\frac{K}{S}$$

where K is some constant that we'll get rid of shortly. Additionally, we assume that at the moment of birth, both O and S are equal to 0:

$$O_0=S_0=0$$

The remaining task is to transform the equation and perform integration:

$$\begin{aligned} dS &= dO \cdot \frac{K}{S} \\\\ SdS &= K dO \\\\ \int_0^S SdS &= K \int_0^O dO \\\\ \frac{1}{2} S^2 &= KO \\\\ S^2 &= 2KO \\\\ S = \sqrt{2KO} &\ \ \ \ \ S \geq 0;\ \ O \geq 0 \\\\ \\\\ dS &= dO \frac{K}{S} \\\\ dS &= dO \frac{K}{\sqrt{2KO}} \\\\ dS &= dO \frac{\sqrt{K^2}}{\sqrt{2KO}} \\\\ dS &= dO \sqrt{\frac{K^2}{2KO}} \\\\ dS &= \sqrt{\frac{K}{2O}}\ dO \\\\ \end{aligned}$$

Now, let's assume the same time period dO is perceived at two different moments in life, O1 and O2, and determine the proportion between the relative perceptions:

$$\begin{aligned} dS &= \sqrt{\frac{K}{2O}} dO \\\\ \\\\ dS_1 &= \sqrt{\frac{K}{2O_1}} dO \\\\ dS_2 &= \sqrt{\frac{K}{2O_2}} dO \\\\ \\\\ \frac{dS_1}{dS_2} &= \frac{\sqrt{\frac{K}{2O_1}} dO}{\sqrt{\frac{K}{2O_2}} dO} \\\\ \frac{dS_1}{dS_2} &= \frac{\sqrt{\frac{K}{2O_1}}}{\sqrt{\frac{K}{2O_2}}} \\\\ \frac{dS_1}{dS_2} &= \sqrt{\frac{O_2}{O_1}} \\\\ \end{aligned}$$

Isn't it fascinating that, for an average person, their subjective perception of the same time period shifts in a way that's inversely proportional to the square root of their life's length?

Application

Now let's consider how the perception of a year in one's life changes for two people who are 10 and 40 years old. By substituting into the derived formula, we obtain:

$$\sqrt{\frac{40}{10}} = \sqrt{4} = 2$$

This means that for a 40-year-old, a year will pass twice as fast as a 10-year-old!

Relative perception of time

We can also take it a step further – let's consider what part of our life (relatively) we have already experienced. If we go back to one of the equations for a moment…

$$S = \sqrt{2KO} \ \ \ \ \ S \geq 0;\ \ O \geq 0$$

Let's denote the moment of death as Sd and Od (perceived relatively and objectively, respectively), substitute into the equation above, and divide by the same equation:

$$\begin{aligned} S_d &= \sqrt{2KO_d} \\\\ S &= \sqrt{2KO} \\\\ \\ \frac{S}{S_d} &= \frac{\sqrt{2KO}}{\sqrt{2KO_d}} \\\\ \frac{S}{S_d} &= \sqrt{\frac{O}{O_d}} \end{aligned}$$

To simplify the calculations, let's first assume that a person lives for 100 years (Od=100), and the person being analyzed is 25 years old (O=25):

$$\sqrt{\frac{25}{100}} = \frac{5}{10} = 0,5$$

This means that 25-year-olds have already experienced 50% of their life, at least in their perception, even though they still have 75% of their life measured in objective years ahead of them.

Now let's analyze me. Assuming that the average life expectancy in Europe is 80 years, and I am 31, then:

$$\sqrt{\frac{31}{80}} = 62\%$$

Well, most of my life is already behind me 🙃

Relative time vs. objective time

Fortunately, as rightly pointed out, the subjective time we have left will never be less than half of the objective time until death. Let's try to prove it!

If the relative and objective times until death are given by the equations:

$$\begin{aligned} 1 - \sqrt{\frac{O}{O_d}} \\\\ 1 - \frac{O}{O_d} \end{aligned}$$

Let's start with the two previously discussed examples:

$$\begin{aligned} 1 - \sqrt{\frac{25}{100}} &= 0.5 \\\\ 1 - \frac{25}{100} &= 0.75 \end{aligned}$$

If we divide one by the other, it turns out that the result is greater than half:

$$\begin{aligned} \frac{0.5}{0.75} &= 0.\overline{6} \end{aligned}$$

Similarly, for the second example:

$$\begin{aligned} 1 - \sqrt{\frac{31}{80}} &\approx 0.3775050201 \\\\ 1 - \frac{31}{80} &= 0.6125 \\\\ \frac{0.4193027447}{0.6627906977} &\approx 0.6163347267 \end{aligned}$$

The theorem seems to hold true. Let's try to generalize:

$$\begin{aligned} \frac{1 - \sqrt{\frac{AGE}{DEATH}}}{1 - \frac{AGE}{DEATH}} \geq \frac{1}{2} \end{aligned}$$

AGE is the variable here, and DEATH is the constant. In that case, we can substitute:

$$\begin{aligned} x = \frac{AGE}{DEATH}\ \ \ \ \ 0 \leq x \leq 1 \end{aligned}$$

$$\frac{1 - \sqrt{x}}{1 - x} > 1/2 \ \ \ \ \ \ \ \ \ \cdot \frac{1 + \sqrt{x}}{1 + \sqrt{x}}$$

$$\frac{(1 - \sqrt{x})(1 + \sqrt{x})}{(1 - x)(1 + \sqrt{x})}$$

$$\frac{1 - x}{(1-x)(1+\sqrt{x})}$$

$$\frac{1}{1 + \sqrt{x}}$$

$$\frac{1}{1 + \sqrt{x}} \geq 1/2$$

$$1 + \sqrt{x} \leq 2$$

$$\sqrt{x} \leq 1$$

$$0 \leq x \leq 1$$

$$\therefore\ \ \ 0 \leq \sqrt{x} \leq 1$$

$$\blacksquare\ QED$$

Summary

Interestingly, the above theories and calculations have also been preliminarily confirmed experimentally [Robert Lemlich – Subjective acceleration of time with aging, 1975]. How much of your life have you already experienced in relative units?

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