1% Smaller Every Time

2019-10-04

Since I found the McElroys and their several unconscionably good podcast
properties a few years ago, I've been trudging through the backlog in hopes of
one day getting caught up. I'm up to date on all of the McElroy properties I
consume, except one: *My Brother, My Brother, and Me*, that treasure trove of
good advice.

I recently listened to episode 177, *[Juicy
Crust](http://hwcdn.libsyn.com/p/2/5/9/2591527f751bc41b/MyBrotherMyBrotherandMe177.mp3?c_id=6460098&cs_id=6460098&destination_id=18443&expiration=1569955980&hwt=09c7d177ae4a378c61e0b9f9a7ce4bcf)*,
released 26 November 2013. Towards the end of the show, the boys consider a
question regarding the existence of double-ghosts---that is, people who have
died once as humans and also a *second* time as ghosts. Naturally, if such a
thing were to exist, there would be all sorts of supernatural implications, and
who better than three dudes from West Virginia to work them out.

### The Question Within the Question

At 41:55, Travis offers a side-question to the group:

> What if, each time the ghost died and came back, they were just 1% smaller
> [...] but also 1% denser?

That question gave me pause. Putting numbers to the issue, after all, would
make it easier to predict the consequences.

Unfortunately, there's a problem. Travis' question leaves open some possibility
for interpretation. Everyone can agree on how to make a ghost more dense---you
just add enough ghost mass to the ghost volume to make the density 1% more than
when you started---but how exactly do you make the ghost *smaller*? You see,
you *could* make the ghost smaller by chopping off a little bit of ghost meat,
taking the ghost mass with it (and probably angering the ghost), *or* you could
just pack *all* the ghost mass into a slightly smaller volume, thus increasing
the ghost density *even more*.

Travis goes on to say something about dark matter, which I tend to think means
he's thinking that the mass of the ghost will increase to celestial levels and
become a hilarious inconvenience. For this reason---and because the results of
the first interpretation are sort of boring, since the ghost mass would sort of
just decay into nothingness---we will consider the second interpretation above
as the canonical one.

Let's get started.

### The Math

Density is just the ratio of mass to volume.

$$ \rho = \frac{m}{V} $$

We'll assume that ghosts are of uniform density. For all I know, they are.

Let's use the variable $n$ to count the number of times the ghost has died and
been re-ghosted. That means we're looking for some function $\rho(n)$ that
tells us how much mass the ghost has after $n$ death-ghost cycles. We'll also
use $m_{i}$, $V_{i}$, and $\rho_{i}$ as symbols for the mass, volume, and
density of the ghost at the beginning of our horrible, horrible thought
experiment.

We want the ghost density to increase by 1% after each cycle, so let's add that
to our function.

$$ \rho(n) = 1.01^{n} \cdot \rho_{i} $$

We also want ghost volume to decrease by 1%, so we'll throw that in.[^1]

$$ \rho(n) = 1.01^{n} \cdot \rho_{i} + \frac{m_{i}}{0.99^{n} \cdot V_{i}} - \rho_{i} $$

[^1]: Note that we have to subtract a $\rho_{i}$ because the last two terms work together to provide the *additional* density contributed by the volume decrease, not the new density after that operation on its own. This bit took a while for me to figure out, and I hope it's right.

This can be rewritten more simply as the following.[^2]

$$ \rho_{n} = \rho_{i} [(1+0.01)^{n} + (1-0.01)^{-n} - 1] $$

[^2]: Recall that $\rho = \frac{m}{V}$, so $\frac{m_{i}}{V_{i}}$ can be reduced to simply $\rho_{i}$ and factored out with the others.

This function is scary. And not, like, *ooooo ghooooossts* scary, although
obviously that, too. No, this function is scary because it grows. You see,
[some](https://en.wikipedia.org/wiki/Factorial)
[functions](https://en.wikipedia.org/wiki/Graham%27s_number) like to explode,
but functions like this one start out slow. Over time, they gain speed, and
pretty soon things get cosmic.

After one cycle, the ghost has about 2% more mass than before. This makes
sense, since we added 1% more density and reduced the volume by 1%. It's not
quite exactly 2% because of how percentages work, but it's close enough for
now. If this ghost weighed as much as the average American, they'd be about 4
pounds (2 kg) heavier. Since their volume is decreasing, they'd also be about
two inches (5 cm) shorter.

By ten cycles, the ghost is now 21% heavier than when we started. They're 40
pounds (18 kg) heavier, but they're also about a foot and a half (45 cm)
shorter.

After 100 cycles, things start to get weird. Our ghost is now 4.4 times heavier
and 2.7 times smaller than when we started. Two of them put together would be
as heavy as a dairy cow, but they're only about a foot (30 cm) across. **They
are now more dense than human bone,** except their whole ghostly body is that
density.

Once we pass 1000 cycles, we leave the realm of reality (even where ghosts are
concerned). Ghost mass is now over 44,000 times greater than it was at the
beginning, but ghost volume is now only 0.004% what it was. **The ghost is now
denser than any material, known or theoretical, and may be more dense than
anything can possibly be.** The ghost became a black hole a long time ago, but
repeated deaths and re-ghostings (can black holes *die*?) have resulted in this
unholy concoction of our imagination.

### So there you have it.

From the parameters put forth by the middlest brother, we can conclude that
such ghosts would only have a few hundred cycles available to them before they
became unphysical insults to the universe. If the initial conditions were
reduced (less mass, less density, or more volume), then the ghost might have a
few more cycles to spare. In any case, its days (and all of our days, too)
would be numbered.