Size of the Universe

Date: 10/31/2001 at 19:53:00
From: Daniel Washburn
Subject: Size of universe

I have heard a lot of scientists say that the universe ends, but I 
don't see how its possible. If it does, is it possible to calculate 
the size of it?

- Daniel

Date: 11/01/2001 at 13:03:24
From: Doctor Ian
Subject: Re: Size of universe

Hi Daniel,

It's possible to calculate anything, once you've clearly defined what 
it is. But defining what you mean to calculate can be more difficult 
than it might at first seem. 

For example, what is the 'size' (length) of the coastline of the 
United States? It depends on how big your ruler is. You can take out a 
map of the U.S., and draw a curve along the coast, and measure the 
curve. That will give you one answer.  

But you could also move down the actual coastline with a 12-inch 
ruler, including every nook and cranny... and you would get a very 
different answer - a much larger one.  

And if you used a 'ruler' that was only a millimeter long, you would 
get an even larger answer. In fact, the smaller your ruler, the larger 
the measurement you'll end up with. So what do we really mean by 
'size' in this case?  (If you pursue this line of thought, it will 
eventually lead you to the area of math called 'fractals', or 
'fractional dimensions'.)  In order to discuss size, you have to 
discuss the shape of space... 

Find a globe, or a picture of a globe. Look at one of the circles of 
latitude, say the one that goes through Chicago. How large is that 
circle? Normally, by the 'size' of a circle, we mean its radius. Okay, 
what is the radius of that circle? Well, it depends on how many 
dimensions you use. If you measure along the the surface of the globe, 
you get one value. But if you measure through the earth, from its 
axis, you get a different value. So if moving from two dimensions to 
three can change the size of an object, what about moving from three 
dimensions to even more?  

Let's think a little more about our globe. Let's say we have a guy at 
some point, and he starts drawing circles around himself. Each circle 
is bigger than the previous one until he draws a circle that cuts the 
globe into halves. Now, as he continues on towards the point opposite 
where he started, each successive circle, although 'outside' the 
others, has a _larger_ radius, but a _smaller_ circumference. So 
whether the 'size' of the circle is increasing or decreasing depends 
on which measure of size he chooses to use. 

And if he decides to head off in search of the 'end' of the globe, 
will he find it? No. Does that mean that the globe is infinite in 
size? No.(And in fact, if he's clever, he can measure that size.) Is 
this just a bunch of crazy nonsense? No! But it _is_ an illustration 
of why scientists usually prefer to use mathematical symbols and 
equations (which they can precisely define) rather than words like 
'size' and 'ends'. 

But for now, let's ignore the subtleties we've been discussing, and 
simply note that we estimate the size of the universe by assuming that 
we can see all of it; and by measuring the distances to the farthest 
visible objects. 

How can we tell how far away something is? Well, we can't, really, but 
we can tell how fast it's moving away from us by using something 
called the Doppler shift. This is what makes the sound of a train seem 
higher when it's coming toward you, and lower when it's moving away 
from you. It works for light waves in the same way it works for sound 
waves.

So if we know that light is emitted from a star with a certain 
frequency, and we know that it has a different frequency when it 
reaches us, we can use the difference in frequencies to determine the 
speed of the star with respect to us. 

(In the same way, if someone is coming toward you in a car, playing a 
particular note on a trumpet, you could use the difference in 
frequency between that note and the note you actually hear to compute 
the speed of the car. This is more or less how the police use radar 
guns to tell whether you're speeding or not. The radar gun emits light 
of a certain frequency, which bounces off your car, changing the 
frequency of the light. The radar guns collects the reflected light, 
and uses the change in frequency to compute the speed of your car.)

How do we know the frequency of the emitted light?  It turns out that 
light from stars is emitted when electrons move between energy states 
in atoms in the stars. The differences between energy states are 
always the same; so when, for example, an electron in a lithium atom 
drops from energy state A to energy state B, it always emits a photon 
with a frequency C, and no other difference of energy states in 
lithium or other atoms corresponds that same frequency. So the pattern 
of frequencies in the light from a star is a kind of 'fingerprint' 
that can tell you which kinds of atoms are in a star. 

So when we see light from a star, and we recognize the pattern of 
frequencies, but find that they are shifted a little bit, all by the 
same amount, then we feel confident in assuming that this shift is a 
result of the movement of the star, and we can compute the speed of 
that movement. 

So now we can look out at stars, find the frequencies of light coming 
from them, and figure out how fast they are moving away from us. How 
does that tell us how far away they are? 
 
It turns out that, as far as we can tell, at a certain scale, 
everything seems to be moving away from everything else, and the 
faster something appears to be moving, the farther away it is. So that 
gives us a rough way to compute distance, given speed.  

So from a handful of formulas and assumptions, we make educated 
guesses about the size of the universe. Are these guesses correct? It 
depends on how reasonable you think the assumptions are. 

I'm reminded of a story I once heard about a famous Russian 
psychologist who set out to study uneducated peasants, to see how 
their reasoning skills compared to those of people educated in 
schools.  

After answering a series of questions about science, one of the 
peasants said that he was confused by something. The psychologist 
asked him what it was.  He said, "I can sort of understand how it 
would be possible to figure out the temperatures of the stars, and how 
far away they are, and how big they are, and other things like that.  
But what I can't figure out is, how do you find out what their names 
are?"

Anyway, I hope this helps.  

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/