The Universe on its Terms

The idea of an expanding space (or contracting matter) raises an
interesting question about length and its measurement in general. In order to know how far
away something is or how long something is we have to take a measurement. To do this, We
need to choose a reference length. Which one do we choose? What is it about a ruler that
allows us to construct those mathematical objects that we call numbers? And how can we get
rid of the arbitrariness? 

For mathematicians, it’s easy. They just define an abstract
unit of length. For the physicist, it’s not quite so easy. An arbitrary choice
for a unit length, a ruler, can lead to dead ends, unusual complexity, special cases or
just plain weirdness. The physicist must be absolutely certain that that which
characterizes the mathematical notion of length has a physical counterpart in the real
world  a ruler. The question to address is how does a ruler  something physical
 represent something abstract  a unit of length? Or more precisely "what is
the minimal requirement for length?" (see "What is Occam's razor")
You might ask "can empty space alone define distance?". Empty space
would certainly qualify as far as the "minimal" part goes but...
Let's just say you and God made a deal to remove all matter from the
universe so that you could measure empty space. What questions might you have
about a distance say from point A to point B. You might ask, for example, how
many meters there are between point A and point B. But, what in the universe would
you choose to define a meter? In an empty universe there is nothing you can point to
and say "that is my unit length and it is one meter long.". What you might
do, however, is simply define the distance between A and B as one unit
length. If you can define a meter stick for a reference length, why not the AB
distance. A definition is an abstraction, why not choose a unit of space as an
abstract ruler. Actually, if you can figure out a way to use empty space as a ruler,
then you've got a way of measuring space with space. This would be far better than
measuring a spatial distance with a material object (a meter
stick). After all, a material unit length, like a meter, is based on matter.
And, if space is expanding (or matter is contracting (see "Is
Matter Shrinking")) we might want to consider that expanding spatial coordinates
will drift relative to our material coordinates. And, we might be very interested in
the physical consequences of these drifting coordinates in a more general context.
Anyway, back to my point...(more on this later...) In this experiment there is
nothing except empty space. There is nothing we can measure the velocity, or
acceleration of. No photons so there is no light. There's no mass or
energy. In an empty universe, there is nothing for us to sense at all! This
situation is kind of like the question "if a tree falls in the forest and there is no
one there to hear it, does it make a sound?" Actually, the idea of an empty
space or a universe with no matter or energy may not even be conceptually possible.
Of Einstein's last writings, one of the things he said is "there is no space without
field" (the Forward in Max Jammer's "Concepts of Space"). Of course
the only fields we know of are material fields. So if we are to have the situation
of an "empty universe" it would have to be just one particle with a field.
That is, this one existent would be (not represent) the entire universe. But,
then we have another "tree in the forest" kind of situation. If this is
the only particle/field in the universe, can we really say that there is a field (and thus
space) if it has nothing to act on? And should we even consider this case?... it
seems like a waste of time. Patience..., there are a couple of more questions to
ask. The first is, if this is the only existent there is, can it move? Can it
rotate? Not if Gallileo was right. If motion occurs only with respect to
another object, then this one existent universe cannot go anywhere. If you think
about it it makes sense. There is nothing outside of the universe for our single
existent to move through (like space) and no markers staked to the "outside (or
inside) of the universe" that would allow us to show "the universe had
rotated". It might seem that there is not a lot we can say about a one particle
universe. Except that you can say "there's not a lot we can say about a one
particle universe". Actually there is one more thing that we can say. A
one particle universe has no center. This probably seems pretty strange, but to show
that this particle surrounded by field was at the center of the universe we would have to
show that there was something we could call a boundary which was a constant distance away
from the particle for all angles. We can't do this. Why? Well, first
look at the universe as a venn diagram:
The blue represents everything that exists and the
black represents everything that does not exist. Of course there are no
nonexistents. There is nothing that we could imagine that could actually provide a
border for existence itself. So the black part, although I have represented it as a
nonexistence, is not there (well, maybe unicorns and dragons). In a logical sense,
there is nothing outside of existence. The definition of "center" depends
on a boundary equidistant from the "center". No boundary, no center.
OK. We'd like to measure the field strength of this
one particle. If we could show that the field strength was the same for each
constant r from a hypothetical center we would have something we could define as
a "boundary". Then we might infer that our particle was at the center of
the universe. So lets add a second particle  a test particle. Lets call our
first particle A and our new particle B. If the particle B moves toward particle A
then we have an attractive force. If they move away from each other then we have a
repulsive force. But we have another problem. How can we tell if there is
motion between these two particles at all if there is no reference object to compare the
distance between A and B against? This might seem kind of confusing.
Gallileo taught us that motion can and does only occur with respect to another
object. Here we have two objects. According to Gallileo, motion
should be possible. Why can't we say that they are moving with respect to each
other?
We can certainly imagine that these two particles might be
surrounded by some kind of field and move toward each other under, say, gravitational
attraction. To talk about a physical change in position we need to express this
physical change in physical units (to avoid defining nature) so we must have a ruler (like
Gallileo had) to define a static distance. Without
something physical that establishes physical "staticness" and physical
"lengthness" we have no criteria to judge what kind of distance or motion we are
"observing". So, to define our ruler, we add two more
points/particles. Call these C and D and let these define the endpoints of our
ruler. As we've seen before (in "Is Matter Shrinking") you can define any
two particles to represent the endpoints of a static reference length. Not only does
it define "length" it also defines "static" and "static"
implicitly defines "dynamic". These might seem like trivial semantic
details. And they are. But with these "observational semantics" we
can find the equivalent "theoretical semantics"  the math. And math is
what gives a theory its power.
Now, let's choose C and D as our reference length and call
the distance between C and D l_{0}. Note that however close or far
apart we might imagine C is from D it is still a unit length (Notice that coincidence is
not possible! Can you measure precisely without coincidence?... Food for thought...).
Keep in mind that a unit length is a just mathematical notion. It is a
conceptual piece of how we represent the physical. That is, it's part of a
conceptual technique that helps us represent the universe. (remember, a unit length
is not real, its just a definition. Face north say it three times then face east say
it three times again... )
Let's use our ruler to measure the distance from point A to
point B. Since our ruler defines physical "staticness" and
"lengthness" we have an enormous body of knowledge we can apply to the physical.
Mathtematical science... And in particular, geometry. But to actually
be able to apply geometry to our two "lengths" we need a way of using them to
construct numbers  or variables!. And that is just what a measurement does.
In, physics this is one of the most important steps. It is what gives a theory what
physicists call "physical significance". So lets take a minute and look at
how we turn a length into a number.
The Measuring process
Measuring the length of somthing is something we've all
done. But no one really looks too closely at the role of the ruler in the measuring
process. It works fairly well just as we've been using it for the last couple of
thousand years. But what we've found in studying quantum mechanics is that it seems
imposible, even in theory, to get the exact measurements we expect. Why?... We tend
to think of a unit of length like we think of single marble, or a chicken. We can
identify objects, like marbles, and count them exactly. In the universe,
however, if we look for a "true" reference unit length, we find that there are
an infinite number of choices. We do not find any single one that somehow identifies
itself as the one single "true" unit of length. In other words, length,
in a natural sense, is fundamentally ambigous. Whereas unit has both
a natural and sythetic exactness. If you have a philosopical bent, you might ask
yourself "Why do I need a reference length to quantify a length but I don't need a
reference chicken to count chickens".
We want to acheive, in a measurement, the same exactness
that you get from counting. The property that gives a count its exactness is that
objects (like chickens or marbles) have a unique and unambigous identity (a
physical uniqueness). If we assign a number to a length we give the length a mathematical
identity. The way we do this is to find the number of reference objects in
the target.
Lets start with two sticks call them r_{1} and
r_{0}. We'll define r_{0 }as the ruler length and
r_{1} as the target  the length we will translate into a number.
We use the following algorithm to measure (construct a number):
The Measuring Algorithm
digits is a string of digits we
define to be initially empty.
MLoop: Repeat the following for the number of decimal
places of desired accuracy.
Loop1: while I can see that r_{0} is
longer than r_{1} do the following:
Append a 0 to the right of the previous digits.
Break r_{0} into 10 equally sized fragments.
Redefine r_{0} to be one of these fragments an d
repeat Loop1.
Set fragment count to 0.
Loop2: While fragment count is less than 10 and r_{0} looks
shorter or "equal" to r_{1} do the
following.
Add 1 to fragment count.
Chop off from r_{1} a length "equal" to r_{0}.
Redefine r_{1} as this new length less r_{0}
and repeat Loop2.
Append the digit
represented by fragment count to the right of the previous digits.
Repeat MLoop. 
Notice there is no placement of the decimal point.
Why? Because we don't really know if r_{0} represents a legnth
that is 1.0 units long (it could be compressed  how would you know?). Although it
is a reference length we have not yet assigned it a numerical identity (...and may not
want to!). We can choose any number we want to. Note that where we place the
decimal point would depend on this choice. To amend our algorithm then, lets say
that at the top of our algorithm we define our reference length to represent 1.0
units. Then the first time we have to break r_{0} we place a
decimal point to the right of the accumulated digits and after we append the zero.
There are shortcuts we can take. You probably
already know them. You've probably never broken a ruler or the thing you were
measuring  probably because you've never had to write a measuring algorithm
yourself. I've broken lengths off because that is effectively what we do when we
measure things. As we look at greater and greater precission we ignore the results
of our previous observations as we attend to the smaller and smaller fractions and build
greater and greater precission. The salient point, however, is that this algorithm
sets up another view of a measurement that allows us to express a measurement in another
way. Notice that by breaking fragments off of r_{1} we are physically
subtracting r_{0} from r_{1}. There is a
mathematical shortcut we can use to represent an iterative subtraction: division.
In fact we can replace our measuring algorithm entirely with the mathematical
equivalent:

(1) 
Why is this good? Well, it emphasizes the
conceptual dependancy of Mathematical length on a reference object 
something that must exist. That is, without a Ruler we don't
have a Length. Length in not an intrinsic property of matter 
its one of our inventions. And secondly, this makes Length physically
significant  i.e., relating a length (an abstraction) to a physical object which
implyies physical properties. This is an important property to establish in the
begining of a physical theory. The third property is that equation (1) is
abstract. That is, it may be generalized for many kinds of measurements. With
respect to length there is one big problem though. If Target and Ruler
are shrinking with respect to some frame in exactly the same proportion Length will
remain constant (take a look around you) in some frames and change with respect to others.
The quality of the number we get as a length just does not compare to a count
of marbles. It doesn't matter if you are growing or shrinking, the number of marbles
in a bowl will always be the same number. Whereas if I see a length that is
shrinking with a rate v, observers of the shrinking frame will say it is always a constant
x. While we can accomodate this situation in a general way and for all observers,
the math turns out to be surprizingly difficult.
There is an alternative way, it turns out, of representing length that
is quite simple, frame independent and entails probably the most useful application of
equation 1. That is, measuring one ruler with another.
First, consider the "shrinking
matter hypothesis". If it's valid, then our material rulers have shrunk
with respect to the universe. Or put another way, the space represented by
our material rulers might be denser than that of intergalactic space. To
test this idea, we can use our measurement expression to measure this spatial density or
scale as:

(2) 
I prefer scale rather than spatial
density because it has a geometric connotation.
This kind of scale is similar to the conversion constants
you use to convert feet to meters or cubits. There is an important distinction here,
however, in that our ruler is exactly analogous to a geometric line  something bounded
by exactly two points. A meterstick, made out of metal or wood, has
zillions and zillions of "points" (particles) between its
"endpoints". While a conventional ruler is similar to a line it does not
fit the exact definition of a geometric line.
Take the measurement of a ruler of frame x with respect to
frame x':

or 

(3) 
and measuring of the scale of a second ruler:

or 

(4) 
combining (3), (4) and relabeling and , we have:

(5) 
This equation says that, if we multiply a ruler by
its scale, that the resultant value is the same for all rulers. In otherwords, the
product of scale and length is an observer independent representation of a static and
universal reference length. It expresses the exact quantity of space between two
points. In later pages we'll see that this is true even for every kind of moving
frame. And we'll see how to use this idea to show that the expansion of the universe
is relative to shrinking matter.
08/23/98
Copyright, © 1997Jack Martinelli

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