Is Matter Shrinking?
It is often asked "If the space of the universe is expanding, and,
if matter is mostly space, then isn’t matter expanding as well?". The
conventional answer as well as my own is no. But, the explanation I have is much different
than the answer you might read about in textbooks. See Do
Atoms Get Bigger as the Universe Expands? for the textbook explanation.
Here is my side:
You are in deep space with a friend. You are facing each other. Neither
one of you is aware of the very, very small rocket that I’ve placed on one of your
space suits. After some random time interval, it fires. The acceleration from the rocket
is very, very tiny, too small to be noticeable. Eventually you notice that your friend is
moving away from you. But, to your friend, you are moving away. You might ask, who is
actually receding from who?
This is the essential point of Galileo’s
Relativity (Excellent link!), that the motion of an object can only be determined with
respect to another object. (BTW, Einstein’s relativity
does not replace Galileo’s. It extends it with a few new principles with some very
dramatic consequences. )
Apply Galileo’s principle to galactic recession. From any point
on our planet we can look in any direction into the universe and observe that all
observable galaxies at distances greater than about 300 million light years look like they
are receding from us. If you can see a galaxy moving away from our planet, you might
conclude that, from the perspective of a viewer in one of these distant galaxies she must
see you as moving away. If we can see all distant galaxies receding from us. then,
according to Galileo’s principle, observers in these surrounding galaxies should see
us as receding from them. That is, with respect to the rest of the universe we must be
shrinking! Kind of weird huh? Well, lets call this the "shrinking matter
hypothesis" and see if it makes sense.
First, lets look at what must be your first question: "if we are
shrinking, then how small can we get?" This is an easy question to answer, but not
easy to understand.
Take a look around your surrounding environment, it is obvious to you
that you and everything around you are not shrinking! But, how should your
environment appear if it were shrinking?
Imagine that you are in the middle of a football stadium that is filled
to capacity. You take a growing pill and start to slowly grow. To you, the stadium, the
people and the whole earth will look as if they were shrinking. To the people in the
stands, you will seem to be growing. Which is the true statement? Intuition says that the
stadium and the people in it are not shrinking, only you are growing. But this leads to a
contradiction of Galileo’s principle of relative motion. Furthermore, it violates one
of Einstein’s principles that says "there are no favored frames." That is,
according to Einstein’s principle, we can’t accept the view from one frame and
reject the view from another.
Can we resolve these paradoxes? Sure! By allowing relative expansion. We
don’t need to take a position with respect to the question. Both are correct! We can
allow both interpretations! …How?
Think of distance as a measurement then look at your meter stick and
consider the shrinking matter hypothesis. The distances that we are capable of measuring
depend on the "length" of your material meter stick! That is, with respect to
this piece of reference length we are not shrinking.
Say you are two meters tall on day 1 and on day 2 you are one meter
tall. You shrink by a factor of two. So has the meter stick – since it too is made of
matter. It is only one half of a meter on day 2. But is it? Clearly, since you and
everything around you is made of matter (if matter is actually shrinking with respect to
space) you would not be able to detect it – at least not with another piece of
matter! So, maybe a meter doesn't become a one half a meter?
This raises an interesting question about a meter. How long
is it? Really!? If it has shrunk, how long is it now?
This may be hard to believe, but, there is no "real" meter. A
meter is just a definition. That is, when we define a unit length we always refer to
(stand by) this definition even if it is shrinking (or growing) with respect to something
else. This choice defines not only a physical frame of reference, but an intellectual one
as well. That is, In order to be able to relate a "physical length" to an
abstract length we require a "semantic alignment" that allows us to join
physical reality to the power of mathematical abstraction. In some sense, with this union,
we can translate the language of the universe into the language of mathematics.
With respect to our meter sticks, then the universe is growing. But, if
we choose a couple of receding galaxies as the endpoints of our reference length, this
length is by definition always a unit length and our material meter sticks are
contracting. Physically, this is not a problem because all material proportions remain
constant.
"Ok", you say, "but what about mass? If we are shrinking,
shouldn’t we be getting less massive? Clearly we don’t see this!" Well if
there were a geometric connection to mass then we might be able to use such a relationship
to deduce how mass should change with respect to an expanding universe. As it turns out
there is such a relationship that comes from Einstein’s special relativity:

(1) 
and 

(2) 
These equations relate one measurement to another when
one frame is moving with respect to another. Physicists, cleverly enough, usually call
these two frames the primed and unprimed frames. Where, l represents a length in
the unprimed frame and l’ represents an equivalent length in the primed frame.
m’ represents the mass in the primed frame and m represents the
equivalent mass in the unprimed frame. v is the speed of one frame with respect to
another and c is the speed of light. (don’t worry about understanding these
equations they are pretty confusing. But if you'd like to take a shot at it
visit: http://www.math.washington.edu/~hillman/relativity.html
)
Lets rewrite these as:

and 

Then, by equating and rearranging these two expressions
we completely eliminate the velocity dependence, we have l’m’=lm Where l’
represents a length in space and l represents an equivalent material length. And m’
represents a mass in the spatial frame and m represents a standard terrestrial
mass.
This equation tells us that the product of length and mass (lm)
for our material frame remains constant for all l’m’. This means
that if the length between two objects differs by some proportion a, then the
mass of that object should differ by the proportion 1/a. In otherwords, if
a space is dense with respect to some reference length, then it will be more massive.
"Ah!" you say, "If we are talking about space where does
mass enter the picture? Space is space and has nothing to do with mass!".
Actually, there are two very strong physical theories that suggest that
space is filled with mass. The first is the theory about Zero Point Energy (you really should read
this!). According to this theory, space is filled with energy. Consequently, if
E=mc^2 then space is filled with mass.
The second theory is called the Cold Dark Matter
hypothesis. It says that there are indications of unseen matter in the rotational behavior
of galaxies. The observed rotational velocities of the arms of the galaxies cannot
possibly rotate as fast as they do unless there is a gravitational source that we cannot
yet explain. Otherwise, rotating galaxies as large as we have observed them would fly
apart. A possible explanation is that there is matter in these galaxies that is not
radiating any detectable radiation. Thus, it is assumed that if this matter exists, it is
cold.
There is a third theory, (my own) that equates mass to the scale of a
frame of reference. I call this theory a Deeply Unified Field
Theory since it unifies the units of measurement rather than the four fundamental
forces. The motivation for this theory is to explain the relationship of mass to
field. Unifying the four fundamental forces (the holy grail of physics) seems an
impossible task if we don’t understand the fundamental significance (via first
principles) of f=ma (force), p=mv (momentum) or E=mc^2. However, when we consider how
length, and especially a reference length is contracting or expanding, it seems very
rational to examine the scale of one reference length with respect to any other. In my
work, in the cases that I have considered so far, the scale of a frame appears in the same
place and in the same way as mass does in some of the most fundamental equations known to
physics.
Now, what about time? The same equations apply to time. That is, for
time we have.

which we can rewrite as 

Then using the above, we have l’/l=t’/t
which says that l’/t’=l/t. Or that the proportion of length to time in
the expanding space can change in anyway at all and the equivalent proportions in our
matter frame will remain constant for all l and t.
Apparently, even though the universe is expanding (or matter is
contracting) the effects are not easily detected  locally. But then, so many
things in physics depend on some kind of a change or a difference – energy, field,
work, motion etc.. And, in our universe, we can see a difference in the expansion
of space and/or the contraction of matter. Might there be physical effects that we be
could detect? (Ans: yes!)
Einstein found that if he represented space as a curved space in four
dimensions that he could calculate a field potential. The form of his field equation
matched Newton’s gravitational field exactly. That is, he found that space itself can
account for gravity – a causal effect! So then, if a curved space can have a causal
effect, what effects, other than a Doppler shift, might we be able to detect as a result
of an expanding space?
A question for the Experts: Consider
equation 1. If you were to look at meter stick in your hand, how would you know if
it was contracted with respect to any other length in the universe or not? If you
measured it with another meter stick, you'd learn nothing. (That would be like
looking at your hand to find out how fast you were going.)
A question for the layman: Recently,
astronomers have found that, not only is the universe expanding (But wait! There is no
"outside" of the universe!), the rate of expansion is accelerating. Now,
if we choose two distant galaxies as our reference length, then not only is matter
shrinking with respect to this reference, its rate of shrinkage should be
accelerating. Whenever an object accelerates there is an associated force....
Can we detect the expanding universe locally? Have you ever seen an apple fall to
the earth?
Chase, Scott I., "What is Dark Matter?" (updated 11MAY1993),
http://math.ucr.edu/home/baez/physics/dark_matter.html
Puthoff, H. E. "Quantum Vacuum Fluctuations: A New
Rosetta Stone of Physics?", http://www.best.com/~dolphin/zpe.html
THE CLASSICAL VACUUM [ZEROPOINT ENERGY] From Scientific American Magazine, Aug.
1985, pp 7078. http://www.padrak.com/ine/ZPESCIAM2.html
Weiss, Michael "What Causes the Hubble Redshift? Are the
LightWaves "Stretched" as the Universe Expands, or is the Light DopplerShifted
Because Distant Galaxies are Moving Away From Us?", 1994,
http://math.ucr.edu/home/baez/physics/hubble.html

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