[ExI] Exploring perceived acceleration of time as we age

[Bryan Bishop]

I found this insightful (quoted below):

http://everything2.com/user/Professor+Pi/writeups/Why+time+appears+to+speed+up+with+age

Please note that it's from the Journal of Irreproducible Results
before you invest much thought into it. I was wondering if the kind
fellows in the Gerontology Research Group could mention whether or not
supercentenarians report a perception of acceleration of time at age,
say, 115, greater than at age 100 or age 80, etc.

"""
In a groundbreaking article, T. L. Freeman discusses the relationship
between actual age and effective age1. His conclusion is that the
passing of the years goes faster as we grow older. This makes sense;
for instance when you are 10 years of age, a year represents 10% of
your life, and seems like a very long time. However, when you are 50
years old, one year has reduced to only 2% of your life, and hence
seems only one-fifth as long.

Summarizing this work, Freeman comes to the conclusion that the actual
age (AA) needs to be corrected for the apparent length of a year (AY).
The apparent length of a year is inversely proportional to one
person's actual age:

          AY= α/AA

The constant of proportionality α is rather loosely defined by Freeman
as the age at which a year really seems to last a year, and it was
arbitrarily set at 20 years (α=20).

Now Freeman determines the concept of Effective age, which is simply
the integral over time of the Apparent Year from age 1 to the actual
age (AA) of interest:

               AA            AA
          EA =  ∫ AY d(AA) =  ∫ 20/AA d(AA) = 20 ln(AA)
                1             1

Although this formula results in some interesting conclusions, there
are several flaws with this concept. As mentioned above, the choice of
the proportionality constant is rather arbitrary. There is no rational
justification for the choice of this age, but it was solely chosen
based on Freeman's own perception of (the passing of) time. Next, the
evaluation of the integral seems incorrect, since its lower limit was
set at 1, and not at 0. Obviously, the choice of zero as lower
integration boundary yields can not be evaluated due to the
logarithmic term in the expression. Because of the obvious problems
with Freeman's concept of time perception, it is necessary to redefine
the Effective Age on a sounder basis.

In the traditional concept of time perception, one person's Actual Age
is proportional to the passing of time (t).

          AA = βt + γ

Note the occurrence of two parameters β and γ that are traditionally
set to one and zero, respectively. However, each has a clear (though
usually underappreciated) function in time perception. The β-parameter
describes the rate at which one person ages; some persons remain
annoying little crybabies during their life, while others become
boring old farts at 20. The γ-parameter describes the origin of one
person's time perception. Did you ever meet those proud parents
boasting about their little one who is only x months old, and already
walks, writes obfuscated C, or recently sold his first dot.com? No,
these youngsters aren't bright for their age; they simply have a high
γ-factor.

It is clear that with this definition, one person's Actual Age may
already be non-synchronous with time. However, analogous to Freeman's
work, the apparent length of a year (AY) is not constant:

          AY= α/AA = α/(βt + γ)

We can remove one of the parameters by defining two parameters δ and ε.

          AY= α/(βt + γ) = (α/β)/(t + γ/β) = δ/(t + ε)

The actual values of δ and ε will become clear from the boundary conditions.

In order to obtain the Effective Age, the integral of AY is evaluated.
Note that the integral is evaluated over time, and not over Actual
Age, since AA is a function of time:

               t           t
          EA = ∫ AY d(t) = ∫ δ/(t + ε) d(t)
               0           0


          EA = δ ln(t + ε) - δ ln(ε)


The lower boundary condition (t=0) should yield an Effective Age of
zero years (EA=0). Therefore ε = 1.

The upper boundary is less apparent. It should be chosen so that at
t=tmax, EA = t. At death, the Effective Age and real time are again
equal. However, no person knows for sure his or her personal life
expectancy. This is clearly an issue for molecular biologists to
address. However, if we assume for a person a life expectancy of 80
years (t=80, EA=80), we obtain:

          δ = 80/ln(81)

               80 ln(t + 1)
          EA = ----------
                 ln(81)

This formula can now be used to calculate the Effective Age (and the
Effective percentage Completion of Life) as a function of time. This
is shown in the following table:

time (yrs.)  EA (yrs.)  Life%
0            0.0      0
1           12.6      16
2           20.0      25
3           25.2      32
4           29.3      37
5           32.6      41
10          43.7      55
15          50.5      63
20          55.4      69
30          62.5      78
40          67.6      85
50          71.6      89
60          74.8      94
70          77.6      97
80          80.0      100

And thus, the bold statement in the title is justified. Life is half
over at age ten, and three quarters over at age thirty. Note the rapid
increase at very young ages: in the initial stages of life, life
itself makes big strides forward. For instance, consider the concepts
of speech, eating and walking; skills that are learned at a young age
and are carried on throughout a person's life.

Another interesting observation that we can make is the age at which
one year really seems to last one year. This can be calculated quite
easily from the derivation above. For a life expectancy of 80 years,
it is equal to 80/ ln(81) - 1 = 17.2 years. Quite close to Freeman's
original assumption of 20 years.
Consequences:
The concept of Effective Age has far stretching implications. Some of
these I have summarized below:

    * "Summer vacations lasted almost forever when I was in grammar school":
      True, they did. In fact, when you were six years old, an
Apparent Year would be close to three years. That would make a three
week summer vacation feel like almost nine weeks!
    * "Now that I am older, I can communicate better with my parents"
      Right. As you can see, you're catching up with them! Closing the
"generation gap", so to speak.
    * "Life starts after 65"
      The credo of many people close to their pension age. Wrong: at
65, you only have about 5% of your Effective Age left. Choose your
time wisely; start working late, and retire early.
    * "Old people are slow"
      That is such an insensitive comment. Old people aren't slow at
all, they simply have a different time perception.
    * "Those annoying birthdays seem to roll around faster every year
      True, they do. Better start celebrating your Effective Age.

T. L. Freeman, Why it's later than you think, J. Irr. Res., 1983.
"""

Now, what happens when you throw in longevity escape velocity? The
perception of the duration between SENS-like-treatments goes down.
According to Freeman's model, or Professor Pi's model, your aging of
"effective age" (EA) slows down considerably once you hit 80. At 1k
years, your EA is 120, and at 10k years, your EA is 160, and at 100k
years your EA is just at 210. Perceiving time like you're (at most)
200 years old for 100k years is a pretty neat deal, if you don't
account for (1) any benefits that SENS-like treatments give to
perception of time (i.e. maybe it's accumulative damage on a molecular
scale in the brain that causes the peculiar perception of time), or
(2) any sort of neurological intervention. I suspect though that the
actual results will not follow this Irreproducible Result/model, but
it's still interesting as an extrapolation and something to bounce
questions off of.

Of course, different people likely have different considerations of
what they consider to be a normal set point for time perception. So
far, in my life, I am just now hitting 20 years in January, and I
already feel ancient- half of my life has been spent on the internet,
for instance, and according to that Irreproducible Result, this has
been 70% of my perception of my lifespan if I was to live to 80.

I'll have to play with this some more. Just some fun, don't read too
much into this.

- Bryan
http://heybryan.org/
1 512 203 0507