Jason Shilling Kendall: Citizen Astronomer

William Paterson University
Amateur Astronomers Association of New York
Hunter College

What's with this cosmic homogeneity and isotropy? I don't get it.


2013-02-13:

Homogeneous means having a uniform density.

Isotropic means having a uniform appearance in all directions about a point.

For this little essay, I am assuming that you live near a college campus, such as mine at William Paterson U. (Go Pioneers!), or you live near a high-school football field. It does help. The first thing you'll need to do will require a bit of resources. Take a billion matchsticks and lay them out so they are aligned along the yard lines of the football field on campus, but make sure that they are all evenly spaced apart. That's homogeneous, but NOT isotropic. That is, if you're in the middle of them, some ways you see the sides of the matchsticks (looking toward the goalposts), and some ways you see narrow lines looking between them all (looking towards the sidelines). Now let's make it isotropic around one point. Stand at the 50 yard line in the center, and have all the other undegraduates turn the matchsticks so that the match head points at you in the center. Now every way you look from the center it looks exactly the same. You see matchstick heads. but it's not isotropic everywhere, just around the 50-yard line coin toss area. everywhere else on field if you look out from that new point, it looks different in different directions. (It's neither isotropic nor homogeneous near the edge of the football field for obvious reasons...)

Now take the matchsticks and put them so that they are both evenly spaced all over the field, and completely randomly oriented. Now we are homogeneous and getting very close to isotropic. It's close because matchsticks have a finite size so there will be some lines of sight that won't hit a matchstick. Those might be the 45 degree angles from the yard lines, if all is laid out perfectly. But it's a really good approximation. So now we have a football-field-universe that is filled with matchsticks at a uniform density, where it looks the same in every direction no matter where you are on the field (so long as you don't get too close to the sidelines or goal posts....)

Now what do we mean about "on a size scale"? Take our football-field-universe idea and now allow two things to happen, the first is easy. Make it big. REALLY BIG. Meaning make it larger than Wayne, New Jersey, where W.P.U. is located. Also, you need to wipe out all the hills and trees and buildings and roads and move everyone along. It's going to be one big flat open area. Perhaps we should move this experiment to Iowa or South Dakota or Wyoming?

The next step will take some thinking, and it helps to go back to our football-field-universe subset. We'll keep the Wyoming-sized football-field-universe handy with all the matchsticks, but now we need a set of boxes. So, let's now take string on our football field, and string it straight across the football field on each yard line, and the long way too, one yard apart. There are still matchsticks inside each box now demarcated by the strings laid on top of them. But let's be super-simple and call each yard-by-yard box one object. We'll say "in this yard-by-yard box there are X matchsticks." Now, we've laid it out so that EVERY box of yard-by-yard in our Wyoming-sized-football-field-universe has the same number of matchsticks. So the if we now judge the homogeneity and isotropy not based on individual matchsticks, but on the yard-by-yard boxes and their counts of matchsticks in each box, we'll still be homogeneous and isotropic. It doesn't matter if we make the boxes now 10-yard-by-10-yard or mile-by-mile or 20-mile-by-20-mile or even 6-feet-by-6-feet.

Now, it won't work if the box is 1-inch-by-1-inch, since the matchsticks are about an inch long or so, and we presume that we're spacing out the matchsticks by at least a couple of inches. This is our first hint about NON-isotropy and NON-homogeneity on small scales. That is to say not all boxes are the same or not all boxes have the same contents. But we want to know about the meaning of inhomogeneity and anisotropy on large scales.

We're missing something important: gravity. If each matchstick pulled on the others just like gravity, which they do anyway but the effect is REALLY small. The real world effect is so small that the gravity of one matchstick on another can't overcome the resistance due to the friction made by the grass holding them down. No problem. Just remove the entire Earth and float this whole thing in space. NOW the matchsticks' tiny masses can pull on each other with no resistance from air or grass or marching bands. We still have our adjustable grid lines laid out. Let's just for fun, keep it at yard-by-yard grid lines on our Wyoming-sized-football-field-universe-floating-in-space. Now a stray cat released from Schroedinger's Animal Shelter bumps one of the matchsticks in the middle of it all just a little bit, not even enough to move it very far. The Cat, being nice, tries to put it back, but doesn't get it quite right. (Or does he?) So, now that one matchstick pulls on one neighbor more than all the others. Thus we have a gravitational instability set up. Wait for a while, and the whole thing has moved a little bit in various directions. Places that are a little more empty become even more empty. Places where one matchstick is closer to another draw others near them, bunching them up further. So at some point the counts of matchsticks inside a yard-by-yard grid will no longer be the same in every box. Fine. So now we make the box bigger. We make it 10-yards-by-10-yards. Since the matchsticks are small and their gravity is weak and the bunching and empty spots are not huge, this 10-yard box might still have the same counts of matchsticks. This process continues for some time; matchsticks coming together, and us making the boxes bigger and bigger to make sure that the counts inside each box are the same. (Remember that we have to look at the counts of matchsticks in each box to judge our isotropy and homogeneity, without looking at the arrangements of matchsticks inside that box.)

Let's say we've let this process go for a while, and now our boxes are maybe 1-mile-by-1-mile on a side to keep the counts the same inside each box. That is the meaning of large-scale homogeneity and isotropy.

Notice I've dodged an expanding universe. This is basically a model of a static, flat, matter-filled universe. This matchstick model will eventually end up with all the matchsticks in one place smashed together. Much of the science of cosmology is trying to figure out the effect of expansion on this and the effect of other stuff inside the space we see and how far something can be before the influence of that distant something has no effect on us.

Here are a few helpful links I found...


William Paterson University Department of Physics American Astronomical Society Amateur Astronomers Association of New York Astronomical Society of the Pacific