When little things matter

This post is a very simple lesson that can give us a profound deeper insight into how things work and why so much around us doesn't.

This arose from a very simple question I asked yesterday, to my daughter who is learning how to teach the concepts of physics to 4th graders. No number-crunching is involved. Relax.

Here's the situation. Some people, who work at the Daily News in New York, between sightings of Elvis, argue that the Earth is a hollow shell, like a tennis ball, and there is a whole world inside it, illustrated by my clever picture #1. Oh, yes, and they would add, there are holes at the poles where the flying saucers go in and out, which of course the military knows about but keeps secret from us all. The aliens from the UFO, shown here as "little green men" live on the inside as shown in the picture.

My question is, "Do you think that would work?" or paraphrased, "What's wrong with this picture?" Of course, there are many things wrong with the picture, but the one I wanted to focus on has to do with gravity. If you could make a hollow earth-sized space ship, unless it was spinning really, really fast, the problem is that the little green men wouldn't be able to stand up as shown. In fact, they would fall "up", towards the center of the earth.

See picture #2 for why that would happen.



So, here's the deal. In your mind, break up the earth into "billions and billions" of basketball sized pieces.

Some of these, "under" the alien's feet, would be close, and would pull on them with a lot of force.

Some of these, "above" the alien's head, would be far away, and would pull on them with just a little force.

So, you think, the close ones will win and the pull will be towards the local "ground".

The problem is that there are way more chunks of earth far away than there are close up. And even a little pull, multipled by billions and billions, adds up to a lot more than a huge pull, multipled by a small number.

As a result, the alien would fall towards the majority of the earth, which is over its head, instead of towards the little bit of earth beneath its feet. "Down" is still towards the center, even if you are inside the hollow shell, for the same reason "down" is towards the center if you are outside the hollow shell.

But, some one would ask, if the earth were spinning at a thousand miles an hour at the equator, which it is, wouldn't the centrigual force hold the aliens "down" onto the shell, just like in the movies of spinning circular spaceships? Yes, Timmy, it would - but, if the earth spun that fast, then the people, rocks, and buildings on the outside of the earth would also be going that fast, and we'd all be flung "upwards" and fly off the earth. We don't need any more math, just logic - because we aren't thrown "upwards", the earth isn't spinning fast enough to hold aliens "upwards" either.

What's the point? The point is that most people think that the first picture is right, and that local effects dominate the world. It's not true in this case - distant effects dominate the world, even though they are tiny, because there are just so many of them.

The conceptual mistake that "big" local things determine the outcome is carried over into our thinking of much of human society and our decisions about what things will "work" and what things won't work. Unfortunately, many of our social decisions, like picture #1, look fine at first glance, and seem fine, and "feel right", but they turn out to be wrong in the same way. That is, we throw the "tiny" effects out of the calculations before we have multiplied by how many places they occur, instead of afterwards. That gives us the wrong answer.

To decide whether an effect is "tiny" or "negligible" or not, we can't just compare it to some other effect locally. We have to multiply it out first, or compound it if there is feedback, and then decide which effect is "tiny" and which one is "big" - and whether our alien will fall "down" or "up".

Many of our social structures, like the alien, are built on the wrong model - and they keep trying to fall apart, and it's baffling to us why that is happening. It "should" work. If that is happening in your world, many the same thing is true there. Maybe, many tiny effects that you think "go away" actually turn out to dominate the answer.

This is true ten times over if there is one of those "feedback loops" I keep going on about. If an effect is "compounded", like interest on your credit card, it turns out to be way more powerful on the outcome than you would think at first glance.

These "feedback" systems that are used to control everything from elevators to airplanes, and the engineers who analyze them use a different lens than an inexperienced person would use, to account for this effect, before they do their calculations. They apply something called "The Laplace Transform", named after some guy who lived a long time ago whose name was "Laplace." That "operator" does two nifty things at once. First, it corrects everything for what it will add up to after you take "compounding" into effect. And second, it changes the loop into a straight line, so all our favorite statistics can be used on it again, but this time, giving us the right answer. (Warning -- If you look it up, don't mistake this for the "Laplace Operator", which is a whole different thing, named after the same guy!)

The math behind the Laplace Transform looks scary, but, just like the rest of statistics, that all gets hidden inside the calculator and all you need to do is push the correct button to use this, so it's not a big deal.

Actually, there's a third effect, but it's distant and subtle - the Laplace Transform means that this concept will not put biostatisticans out of a job, so they can stop the torch and pitchfork parade up to the castle to kill the idea. Without some way, like the Laplacian, to counter that distant and subtle effect, we'd expect the small resistance put up by a large number of established researchers to dominate the scene, and squash this idea from being taken seriously.

Anyway, now you have a better idea what I'm talking about when I say such things.

References & further reading
======================
A description of the Laplace Transform, and what it does, in something approaching English,
can be found in any "control system engineering" introductory textbook. I prefer the explanation in this one.

For anyone who enjoys the idea of big things and little things switching roles, stay tuned for a discussion of "Olber's Paradox", or the very serious question of why the sky is dark at night.
Any idiot knows that's because the sun has set. Olber, howerver, raised the point that the sun, where you can see it, will be the same brightness per area, regardless how far away from it you are. If you imagine holding up your thumb and first finger in a circle, and looking at the sun from close, you'd see a small portion of it, and that portion would be very bright. If you were twice as far away, you'd see more of it through that circle of your thumb and finger, but it would be farther away and less bright. Well, it turns out those two effects exactly cancel out.
The amount you see goes up as the square of the distance, and the brightness goes down by the square of the distance, so the total amout of light coming through that circle remains constant.

Sooner or later, you get far enough away that the sun doesn't fill the circle. But, no matter, because if there are an infinite number of stars scattered evenly around the universe, sooner or later any line you draw will run into one. And the little section of sky that one covers will also be as bright as the sun.

Which means, the entire sky should be as bright as the sun, and we should all be cooked.

The math, it turns out is correct. Which means, one of our other assumptions about what's out there is wrong. Fascinating...