The Nature of Space, Time, and Matter

FORWARD THINKING IN ASTRONOMY

[A series of eight lectures specially prepared for Compu-
Serve Information Systems (CIS), for presentation in ASTROFORUM.
Copyright 1990 by Tom Van Flandern of Washington, DC [CIS ID code
71107,2320]. Please seek the author's permission before
reprinting more than two paragraphs. If redistributed in
electronic form, must include this statement of source and
copyright.]

CHAPTER II. THE NATURE OF SPACE, TIME, AND MATTER

******** We will be using Scientific Notation for large or small
numbers. For example, 1E5 = 10 to the power 5 = 10,000; 2E-7 = 2
times 10 to the power -7 = 0.000 000 2. Ask about this, or
anything else you aren't familiar with. Use private messages, if
you wish. But your questions are probably shared by others.

A. Introduction

Last week we saw the value of deductive reasoning for
determining the nature of reality, provided that a suitable
starting point can be found. This week we will reason
deductively about the nature of the physical universe starting
with a minimum of assumptions. In fact, I propose that we start
out with nothing whatever: a universe completely empty of
everything which exists.

Is space an absolute thing, existing even without matter in
it? Or does it depend upon the existence of matter to give it
meaning? Let us define "substance" broadly as anything which
exists, whether it takes the form of matter, energy, or "other".
In order to answer the question of whether space itself exists in
the sense of having substance of any kind, we need to introduce
some additional useful properties of substance.

B. The One Particle Universe

Let our starting universe remain empty of everything except
a single infinitesimal "stationary" particle. Now imagine the
same particle in motion. How fast is it going, and in what
direction? There is nothing for it to move relative to, and
nothing to provide orientation. All directions are equivalent,
and all distances are equivalent. The only way it can be
otherwise is if space itself has a sort of "structure" to it, a
framework to provide meaning to orientation and scale and motion.

However we have postulated an empty universe. In it, there
is no matter, no energy, no substance of any kind except the
single particle. How can there be "structure" without substance?

In the real universe there is a frame of reference to
provide meaning to distance and direction. The reference frame
is provided both by the presence of distant matter in the
universe, as well as by seas of rapidly moving "agents", such as
photons and neutrinos. The essential point is that the reference
frame is provided by the presence of substance in the universe.
I would not insist that MATTER is needed; but I take it as self-
evident that some sort of substance is required, or there can be
no reference frame in space. In the absence of other substance
in the universe, our lone particle would be incapable of motion,
for motion could have no meaning.

Moreover (and this is something to note), the size of the
universe would be indeterminate, even if our lone particle has
"finite" dimensions. Indeed, it is impossible to say whether the
particle has infinite dimensions, finite dimensions, or is
infinitesimal (without size), since there is no scale to measure
by. The number of such particles which can fit into the universe
around it is infinite in any case.

Our lone particle would even be incapable of spin. If it
had parts, they might move relative to one another. But a
uniform spherical lone particle cannot spin about any axis,
because there is nothing outside the particle to spin relative
to. By extension, the particle could not be made to exhibit the
properties of spin, such as centrifugal force -- a tendency to
hurl objects off itself due to spin; nor would it tend to flatten
from very rapid spin. The origin of these "inertial forces" is
surely rooted in the substance which defines the framework of
space. Without a framework, without substance (except for the
particle), without "agents" to produce forces, surely there could
be no meaning to, nor consequences of, "spinning". (The idea
that the presence of distant matter in the universe is the origin
of inertial forces is known as "Mach's Principle".)

Our example may start to seem a little less hypothetical if
we postulate a finite limit to all of the substance in the real
universe, with nothing beyond. (The "Big Bang" Theory in its
simplest form is such a case, in which all substance remains
inside a sphere whose surface consists of photons moving outward
at the speed of light since the instant of the original
explosion.) Under this assumption, the entire substance of the
universe would be like our single particle; and all remarks about
its size or motion in a larger infinity of space and time would
be fully applicable; i.e. they would be indeterminate.

C. The Two Particle Universe

Consider again our simple lone particle in an empty
universe. Now let us imagine a second particle just like the
first at another location, not touching. Now, for the first
time, we have "scale" in our universe, and can measure the
dimensions of the particles themselves as a fraction of the
distance between them. There is no such thing as "absolute
length" in this universe -- we cannot tell if the two particles
are "close together" or "far apart". Their separation is
indeterminate relative to the universe beyond. It can only be
measured in terms of the number of particle diameters.

We have also introduced meaning to motion, since the
separation measured in particle diameters can vary. But with
only two particles, if the separation "varied", we could not tell
whether the particles had moved, or perhaps only changed diameter
(shrunk or expanded) -- either would give the same result. We
can also now detect spinning. Note that the two particles cannot
"see" or influence each other in any way except by collision,
since our otherwise empty universe definitely contains no photons
or agents to produce forces or actions at a distance, such as
electromagnetism or gravitation.

Consider a hypothetical pendulum suspended at the "north
pole" of one of the two particles, taken as spinning. In the
real universe, a suspended pendulum would continue swinging back-
and-forth in the same direction in the universe, ignoring the
spin of the body (e.g. the Earth) underneath it (as many museum
exhibits of the Foucault Pendulum demonstrate). But our two-
particle universe can have no such properties, because there can
be no framework to provide a "remembered" preferred orientation
for the pendulum. Indeed, the pendulum could not swing at all,
because there is no gravity in this imagined universe.

Now if the particle on which the pendulum is suspended is
imagined to have local gravity only, so that the pendulum can
swing; but gravity which does not reach out to influence the
second particle, so that no framework is provided to the
universe; then clearly the pendulum must keep its orientation
with respect to the particle it resides on, since that is the
only framework it has. But as soon as we imagine a sort of
universal gravitation, this immediately provides a framework for
the pendulum. The proximity of the pendulum to a spinning
particle is then no longer relevant, since the pendulum "senses"
only the universal gravitational framework, and must maintain its
orientation in that frame. Nothing about the forces acting on
the pendulum would tell it the particle, above which it is
suspended, is spinning.

By these constructions, we begin to see the origins of the
what are called inertial forces, and the importance of a frame of
reference to the properties of the universe we live in. We also
begin to see why it must be that scale and motion are relative,
not absolute, in nature.

We have just seen that absolute motion has no meaning
without a frame of reference; and that such a reference frame
must logically be provided by some sort of substance. This gives
us a basis for looking at a very famous dilemma called "Zeno's
Paradox".

D. Zeno's Paradox

Zeno's Paradox deals with the ultra-small structure of space
and time. In its essence, the paradox notes that, if a moving
body is in a specific place at every instant, then there is no
instant when it is in transition from one place to another; and
therefore motion is impossible. Since this contradicts everyday
experience, it is called a paradox.

The same paradox can be expressed in a different form: to
move from point A to point B one must first complete the trip to
the mid-point. Having reached that far, one must next reach the
new mid-point of the remaining distance. But however far one has
travelled, one must first travel half the remaining distance
before one can travel all of it. Hence one can never reach point
B, because an infinite number of "half-the-distance" steps are
required.

It might be, of course, that space is not infinitely
divisible -- that there is a smallest possible increment of
distance. But this leads to all sorts of conceptual problems.
Consider points X and Y, separated by the smallest possible
increment of distance. Now consider another point Z, also
separated from X by the minimum possible distance, but in a
slightly different direction. Then the distance between points Y
and Z is less than the minimum possible distance, contradicting
the starting assumption. But if space were "grid-like", so that
adjacent cells had no overlap, then motion in any desired
direction would not be possible, unless one took a zigzag path
from grid-point to grid-point! Clearly, the postulate of a
"minimum possible distance" is problematical.

If time is treated like just another dimension (a "fourth
dimension" of space), the same remarks might be extended to
include the concept of a "minimum possible time unit". Or we may
make a separate argument about time. If there were a minimum
possible time unit, then all existing substance would have one
condition at one time moment, and some slightly different
condition at the next time moment. By hypothesis, there is no
possible interval in time, nor any moment in between when
anything could have happened to provide a transition from the
first condition to the second. It is therefore just exactly as
if everything existing at the first time moment ceased to exist,
and then was created from nothingness in its new condition at the
next time instant.

We conclude then that space and time must be infinitely
divisible in order to avoid these dilemmas. But is this not also
ruled out, by Zeno's argument? The problem is with our
intuitions: while it is easy for us to imagine a whole as
composed of an infinite number of parts, it is difficult for us
to imagine an infinite number of components being assembled into
a finite whole. As is well known in mathematics, an infinite
series CAN have a finite sum. For example, there are an infinite
number of possible fractions or decimal numbers between zero and
one, yet obviously only a finite interval.

In Gamow's book, "One, Two, Three ... Infinity", we learn
how to count and compare things made up of an infinite number of
parts, using one-to-one correspondences. Such a one-to-one
correspondence can be set up between points in a space interval,
and decimal numbers between zero and one. Since the interval
from zero to one is finite by definition, the one-to-one
correspondence shows us that the space interval is finite also.
With another one-to-one correspondence we also conclude that it
is possible to traverse the space interval in a finite time as
well. This is an important point, even though our intuitions do
not deal with it easily. The infinite mathematical series (1/2 +
1/4 + 1/8 + 1/16 + ...), where each new term is half the
preceding one, has a finite sum of 1.0. It is clear that, in
mathematics, there exist finite intervals with an infinite number
of points between, and infinite series with finite sums. By
placing these in one-to-one correspondence with the physical
concepts of space, time, and mass, we can reason by extension
that FINITE intervals and masses may actually be composed of an
infinite number of divisions; and conversely, that an infinite
number of divisions may have a finite sum.

What about Zeno's objection, that if a moving body is
someplace specific at EVERY instant, then at no instant is it
moving, making motion impossible? One way to see the resolution
of this paradox is by considering time to be another dimension,
just like the three dimensions of space (although admittedly not
exactly like a space dimension; e.g. we cannot travel both ways
in time). Then a body travelling at a uniform velocity from
point A at time 1 to point B at time 2 is travelling on a
straight line in this space-time universe. To clarify this
picture, suppose the body is at rest in space. It nonetheless
takes a straight "line" in space-time to connect its position at
one time with the same position at a later time -- the "line"
representing an interval in time, instead of space.

Viewed in this way, it may be seen that the body is at every
instant at some specific point on a space-time line. And once
again the points in the interval can be put into a one-to-one
correspondence with numbers between zero and one. So even though
the distance travelled by the body in zero time is zero, it is
nonetheless possible to traverse a finite distance in a finite
time, each interval consisting of an infinite number of time
instants and space points.

To put this conclusion more strongly, it is possible for
substances to be unchanging at every instant, yet changed after a
finite interval, ONLY if there are an infinite number of steps in
the interval!

******** This "one-to-one correspondence" may be the toughest
concept to understand in this entire course. Does anyone have
any ideas on how to better explain it?

E. Zeno-like Paradox for Matter
There is another form of Zeno's Paradox which applies to
masses: if bodies are infinitely divisible, then contact is
impossible. For example, when macroscopic bodies seem to touch,
they actually consist of mostly empty space at the atomic level;
so it must be their atoms which actually touch. But atoms are
themselves composed of smaller particles and mostly empty space,
so it must be these smaller constituents which actually touch.
But if matter is infinitely divisible, this argument can be
prolonged indefinitely, and nothing can ever actually touch.

One might use this argument to conclude that there is a
smallest possible unit of matter or substance. Imagine such a
"unit particle". It must be utterly uncomposed. It therefore
cannot be broken or divided, nor even deformed by spin or
collision -- since these are properties of bodies composed of yet
smaller particles. What then are we to assume will happen when
two such unit particles collide? What density will the unit
particle have? Indeed, will there be anything inside it at all?
What would the unit particle's "surface" be like? Could it be
hollow? With what thickness of shell? Would two colliding unit
particles have to stick, since they can't rebound elastically?
If they rebounded, with what resultant velocity? Would the unit
particles be spherical in shape? Why would they have finite
space dimensions, yet infinite dimension in time? Or do they
come into and go out of existence constantly? Where and when
would they appear and disappear?

It should be apparent from these considerations that
postulating a "minimum possible unit of substance" is no more
logically palatable than a "minimum possible unit of space or
time". Substance must be infinitely divisible, as must space and
time; or else the paradoxes quickly lead to unresolvable logical
dilemmas. But how then can matter ever experience "contact", if
everything which might experience contact is itself composed of
smaller substances? The resolution of this paradox would seem to
be analogous to that for space-time. If the substance of bodies
always gets denser (more substance per unit volume) at smaller
and smaller scales, then in the limit as dimensions approach
zero, density approaches infinity and substances approaching each
other must make "contact" (i.e., at infinite density, they cannot
be "transparent" to other substance). In the real universe, the
density of matter greatly increases as scale decreases. Hence
the ratio of mass to volume in electrons is enormously greater
(about 1E10 g/cc) than the same ratio for matter in ordinary
human experience (of order 1 g/cc), which in turn is enormously
greater than the ratio for the entire visible universe (1E-31
g/cc). "Contact" is therefore possible for infinitely divisible
matter, as long as the smaller and smaller particles continue to
increase in density with sufficient rapidity, without limit.

I appreciate that it is very difficult for the intuition to
grasp this concept. Consider the approach of one minute particle
of substance to another. As the outer surfaces approach, the
lesser particles (call them "second level" particles) of which
each is composed begin to approach each other. After the
original particles traverse only a very small distance, the third
level particles of which the second level particles are composed
begin to approach each other. After an even more minute traverse
of distance, and after an ever smaller lapse of time, the fourth
level particles begin to interact. Although this continues
without limit, as we have already seen, the process takes place
in a FINITE time and a FINITE distance. The penetration of each
level of particle into its counterparts in the approaching
particle continues until the density of matter in the approaching
particle is too great for it to penetrate deeper. Then the
smaller particles at the next level penetrate until the density
becomes too great for them to make further progress, and so on.
By one-to-one correspondence with terms in our infinite series
with a finite sum, we see that the depth of penetration has a
finite limit, and requires a finite time, after which the
original particles react with resistance to the intrusion of new
substance into their ranks JUST EXACTLY AS IF THERE HAD BEEN A
COLLISION!

By analogy with the proposed resolution of Zeno's paradoxes
for space and time, the paradox for mass is resolved, apparently
necessarily, by the conclusion that substance must be infinitely
divisible, and that it must approach infinite density as size
decreases toward zero dimensions. This conclusion is reached by
reasoning alone; it is reinforced by the observation that matter
does in fact increase rapidly in density as scale becomes smaller
over a range of 40 orders of magnitude in the observable
universe.

From the preceding considerations it seems altogether
reasonable, and in a way compelling, to deduce that space, time,
and substance are all infinitely divisible; because the
consequences of the alternative are logically absurd. But if
they are infinitely divisible on the smaller scale, what about
the larger scale? Recall our earlier argument that the entire
visible universe would have undefined scale in space, time, and
mass, unless such scale is provided by the presence of other
substance in the greater universe beyond. That argument must
remain true without limit. The upper limits to the structure of
substance, the dimensions of the universe, and the extent of
time, must all be as unbounded on the high side as they need to
be on the small side. This will become even clearer as we
further examine the nature of substances.

F. Meaning of Space and Time

Let us return again to our empty universe which contains no
substance, and therefore no frame of reference, except for a
single uniform particle of substance. But as we have just seen,
the particle must itself be composed of an infinitely divisible
variety of sub-particles. We could have chosen a single particle
at any of an infinite number of sub-levels to be our single
particle. To avoid the issue of the arbitrary size of the
particle we select, let us conceive of it as having zero radius.
Although it does not, this conception will allow us to introduce
one scale of distance at a time.

As remarked earlier, motion and orientation have no meaning
for a single particle in an empty universe. Now introduce a
second infinitesimal particle. This gives meaning to
orientation, since angles can be measured from the line joining
the two particles. It also provides a single measurement of
length, the distance between the particles. It does not, as
before, provide a scale for the empty universe, since the
distance cannot be measured in units of particle diameters, which
are still being assumed to have no dimensions. Therefore there
is no way yet to determine whether our particles are separated by
a microscopic or a macroscopic distance. There is as yet still
no meaning to motion in this two particle universe. The two
particles cannot change direction, since all directions have
meaning only relative to the particle-to-particle direction. And
the two particles cannot change distance, since all distances
have meaning only relative to the particle-to-particle distance.

In a very real sense, this universe without the possibility
of motion or change has no time. Time can have no meaning if
there cannot be events or change to mark its progress. Put
differently, if there were such a thing as an absolute time which
existed somehow in addition to our two particles, the lapse of a
microsecond or a million years would be just the same and utterly
indistinguishable. But the existence of something with
substance, such as an absolute time scale, violates the
assumptions of our construction, that nothing exists except our
two infinitesimal particles in an empty universe. Remember, we
refer to "substance" rather than "matter" to cover ANYTHING which
exists. An absolute scale of time, just as for a structure or
framework in space, would have substance in this broad
definition.

Perhaps you have thought about one possible event or change
which might occur in our two-particle universe up to this point.
We might imagine that the two particles coincide, which is a
distinguishable condition from non-coincidence. It might be fair
to say that the first coincidence of the two particles marks the
beginning of time; and that the interval between any two
coincidences marks an interval of time. This interval still has
arbitrary and indeterminate length. We cannot tell if the
interval to the next coincidence is longer or shorter than the
last (that implies an absolute scale of time to measure against).
We can merely mark the progression of time by counting
coincidences.

This brings us to an important point of our mental
construction. In an empty universe consisting of two elementary
units of substance, the ordinary properties of the universe
(time, space, matter) do not exist outside of the particles and
between events of coincidence. It can therefore be said in a
logically meaningful way that space and time which are empty of
particles and events DO NOT EXIST! This eliminates a logical
fallacy we have been skirting around up to now about whether the
empty space and time surrounding our particles exist. In our
construction they do not. Therefore our use of "substance" to
mean "anything which exists" is logically correct, since a true
void would not exist (in either space or time), in the
operationally-defined meaning of the word "exist" as used here.

Of course, for actual particles with finite dimensions,
events of coincidence do not occur. Instead we have what may be
operationally described as "collisions", in the sense already
discussed. Two particles interact "collisionally" when their
sub-particles at all levels approach the infinite density
limitation and are forced to retreat. Notice, however, that if
we were to imagine an infinitesimal volume of space IN OUR REAL
UNIVERSE within which there were only two uncomposed
infinitesimal particles and nothing else (including forces), then
all that we have concluded about distance and time not existing
between events of coincidence would still be true. No time or
time interval would exist until an event occurred, with the only
possible events being collisions with other elementary particles
of substance.

Therefore, on the most microscopic levels, time must proceed
"instantly" from one collision event to the next. Reflection on
this construction, which implies the non-existence of space and
time between events in a region, begins to provide some insight
into why the universe seems to behave as if space and time were
relative, not absolute. We have reasoned to the conclusion that
they must be.

To emphasize the point that true vacuum implies non-
existence, we are asserting that every point in the perceptible
universe is at every moment of time filled with contiguous
substance at some infinitesimal level. If substance could be
imagined to become absent anywhere at any time, time there would
cease and the perceptible universe would collapse until the
"vacuum" was filled. Put another way, a particle reaching one
edge of a "vacuum" would skip instantaneously to the opposite
edge, just as if the "vacuum" had zero dimensions, because there
is no substance to mark the passage of time inside of the
"vacuum", and no absolute time without substance.

G. Implications and Discussion

Pausing for a moment to digest some implications of our
reasoning, it would be fair to conclude that the only logically
imaginable way in which substance can come into, or pass out of,
existence (in this model) is for it to "enter" or "leave" the
region of collisional interactions with other substance. But if
there were such regions where matter density is so low that no
collisional interactions between units of substance occurred,
then all substance on the edge of such regions would instantly
dissipate itself into the non-interacting regions, followed by
substance slightly further in, and so on. All substance in this
universe would dissipate instantly into the void. We suppose
that even solid bodies are held together by the action of agents
which would disperse if not continually held together by the
presence of other substances, so that even solids would dissolve.
Since this does not happen, we conclude that this universe has no
such regions where collisional interactions between units of
substance do not occur.

The same reasoning applies to time. A cessation of
collisional events would bring a cessation of time; but with
matter existing everywhere with sufficient density for
collisions, it follows that time continues forever, in both the
future and the past. But couldn't substance redistribute itself
so that densities no longer approach infinity anywhere, thereby
ending collisional events? By analogy with the dissipation of
substance in space, if it could so dissipate (for example, if the
amount of substance in the universe were finite), it would have
already happened, virtually instantaneously. Conversely, if
substance does not start out with density which approaches
infinity as dimension approaches zero, it could not assemble
itself into such an infinite-density configuration in a finite
time. We may therefore be reasonably certain that the "universe"
(in our model) is infinite in space, time, and mass or scale.

It must be the case that every bit of space is occupied at
all times by a continuum of substance; and that wherever
substance is not, existence of time, space, and matter is not.
The substances whose presence "define" space-time must be
infinitesimal compared to the substances in our experience, such
as baryons or photons, or even neutrinos.

Distance scales must be purely relative, with no absolute
meaning to "large" or "small". Likewise we should not be
surprised by very large velocities. If distance and time scales
are unlimited on the large side, then velocities must be also.
We will discuss in future weeks how this can be reconciled with
Special Relativity, which postulates that the speed of light is a
maximum speed in the universe.

Our conclusions are deductive, not inductive. So they can
be invalidated only by faulty reasoning or an incorrect starting
point or assumptions. They do not at first glance appear to lead
to descriptions of the REAL universe; for example, they do not
easily reconcile with the sort of universe inferred using the Big
Bang as a starting point. But we will see in coming weeks that,
although they do imply changes in some of our theories, the
descriptions from this new starting point seem entirely
reconcilable with REALITY. If they add understanding and make
successful predictions, I argue that is sufficient for them to be
worthy of consideration AS HYPOTHESES in the field of astronomy.

******** If this model is unclear, ask questions to help clarify
it. If the model is clear, it must lead to a description of the
real universe and make successful predictions to be of value. It
is still too early to compare this model with observed reality.
But let's see some discussion of the implications of the model to
this point, and perhaps anticipating the next step by guessing
the nature of FORCE using this model.

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