To Another Dimension!

May 21, 2014

One of the new visitors to the forums brought up a neat topic the other day: Dimensions. Namely, what would gravity look like, if instead of 3 spatial dimensions, we had 2? Or 4? Whoa.

I’ve talked about something similar before with the Electric Shocktopus, but this topic warrants more.  And it needs some swanky gifs from the new Gravity Simulator.

3D

Let’s start off simple.  The world we live in, for all practical purposes, is in three spatial dimensions.  You can go up-down, left-right, or forward-backward.  And in this world, Newton’s Law of Universal Gravitation tells us that two objects with mass will attract each other, according to:

If you only care about the distance, the force goes as 1/R^2, which is why we call this an inverse-squared law.  Double the distance between the earth and the sun, and gravity will pull them together with 1/4 the strength.  Inverse square laws are beautiful, because they lead to cool things like closed and stable orbits.

1/R^2 Force law. Ah, beautiful, closed orbits. Home.

All GREAT things for life!

Unfortunately, Newton’s Law of Gravity can’t help us when we’re in a world with a different number of dimensions.  The law only works in 3D.

Gauss’s Law

Instead, our starting point instead is Gauss’s Law.  Take a point mass, make an imaginary spherical surface around it.  Imagine that instead of gravity, the point mass is just shooting out 100 bullets in random directions.

How many bullets pass through the sphere around the point mass?  No matter how big or small you make your sphere, the answer will always be 100.  That’s the core of Gauss’s Law.  The number of bullets passing through a sphere doesn’t depend on its radius.  When we’re talking about gravity, the bullets represent the Flux, which is simply the strength of gravity times the area of your sphere.

So then:

If the total flux is always the same, that means the Force of Gravity is proportional to 1/(Total Area of a Sphere).  In 3D, a sphere has a surface area of 4PI*R^2.  So the Force goes as the inverse of that, or 1/R^2.

2D

Suppose we were like the Flatlanders, and lived on a 2D plane.  We could move forward-backward and left-right, but not up-down.  How would that change gravity?

Well, Gauss’s Law still holds.  But this time our ‘sphere’ that we draw around the point mass is actually just a circle.  (Remember, we can’t leave our 2D surface!)  The ‘surface area’ of the circle is just its circumference: 2PI*R.  Which means that the Force of Gravity goes as 1/R.  And we get orbits that look like this:

1/R Force Law. Notice that the orbits don’t match up nicely anymore.

4D

What if there were an extra dimension?  Suppose we lived in a world where we could move up-down, left-right, forward-backward, and… and… uh… 4Up-4Down.  We don’t have a word for those last two of course, since we never move through a fourth spatial dimension.  But nonetheless, we are armed with all the information we need to figure out how gravity would look.

This time the ‘sphere’ we make around a point charge is in 4D, which is hard to draw, but we can figure out what it must be like with a simple pattern.  In 2D, the ‘surface’ of a sphere is a line.  In 3D, the ‘surface’ of a sphere is an area.  So in 4D the ‘surface’ of a sphere should be a volume.  Its units should be length cubed.  No surprise, then, that the surface ‘area’ of the 4D sphere is 2*PI^2*R^3.  Which means that the Force of Gravity must go as 1/R^3.  So we get orbits like this:

A 1/r^3 Force Law. Yikes, I’d hate to be on *any* of those planets.

Beyond

You’ve now got everything you need to ponder the force of gravity in any number of dimensions.  And with the Gravity Simulator, you can check out what any of these worlds might look like.  Try to make a 5D solar system, I challenge you!

This was just a quick introduction to some interesting topics — so if you wanna know more, there are plenty of resources out there you can use.

-Andy