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NATS 1800 6.0 SCIENCE AND EVERYDAY PHENOMENA
Lecture 12: Space, Time, and SpaceTime
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… because mathematicians frequently make use of time, they ought to have a distinct idea of the meaning of that word, otherwise they are quacks …
Isaac Barrow (1630  1677) Quoted in P Coveney & R Highfield, The Arrow of Time. Ballantine, NY 1990
Topics

… not just 'time,' but 'space' too; not just mathematicians, but
everyday people too!
Despite the fact that we can not but inhabit
time and space, be embedded in them, our notions about these concepts
are rather primitive. Even Isaac Newton (1642  1727), whose laws of
motion represent probably the best distillation of our everyday experience,
defined time as follows:
"Absolute, true, and mathematical time, of itself and from its own
nature, flows equably without relation to anything external, and by
another name is called 'duration;' relative, apparent, and common time
is some sensible and external (whether accurate or unequable) measure of
duration by the means of motion, which is commonly used instead of true
time, such as an hour, a day, a month, a year." [ from Absolute and Relative Space, Time, and Motion ]
Netwon's definition of space is in the same vein:
"Absolute space, in its own nature, without relation to anything
external, remains always similar and immovable. Relative space is some
movable dimension or measure of the absolute spaces, which our senses
determine by its position to bodies and which is commonly taken for
immovable space; such is the dimension of a subterraneous, an aerial, or
celestial space, determined by its position in respect of the earth.
Absolute and relative space are the same in figure and magnitude, but
they do not remain always numerically the same. For if the earth, for
instance, moves, a space of our air, which relatively and in respect of
the earth remains always the same, will at one time be one part of the
absolute space into which the air passes; at another time it will be
another part of the same, and so, absolutely understood, it will be
continually changed."
[ ibidem ]
After defining "place [as] a part of space which a body
takes up and is, according to the space, either absolute or relative,"
Newton then defines "absolute motion [as] the translation
of a body from one absolute place into another, and relative motion
the translation from one relative place into another." [ ibidem ]
Notice that Newton's definitions imply a complete separation between
space and time : you can have one without the other.

What's wrong with this framework? Nothing, apparently! It seems to fit our
experience quite adequately, and it permits the development of of the laws
of motion, which are good enough to allow us to send spacecrafts to Mars
and beyond. The definitions of space and time, however, smell a bit of metaphysics.
In fact they assign to time and space attributes that are usually reserved in many
religions for the definition of God, which, by definition, transcends our daily
experience. Are such attributes needed? Can they lead to unexpected, and/or
unwarranted, consequences? Can we experience 'absolute'
space and 'absolute' time?
The first crack in Newton's framework appeared quite early. It is best illustrated
by a letter Newton wrote in 1692 to Richard Bentley (1662  1742), in which the
following famous passage appears:
"It seems to me, that if the matter of our sun and planets, and all
the matter of the universe, were evenly scattered through all the
heavens, and every particle had an innate gravity towards all the rest,
and the whole space throughout which this matter was scattered, was
finite, the matter on the outside of this would by its gravity tend
towards all the matter on the inside, and by consequence fall down into
the middle of the whole space, and there compose one great spherical
mass. But, if the matter were evenly disposed throughout an infinite
space, it could never convene into one mass, but some of it would
convene into one mass and some into another, so as to make an infinite
number of great masses, scattered great distances from one to another
throughout all that infinite space. And thus might the sun and fixed
stars be formed, supposing the matter were of a lucid nature."
[ from Cosmlogy: Newton Cosmology ]
As Stephen Hawking puts it in A Brief History of Time: From the Big Bang to Black Holes
(Bantam, NY 1988, p. 5): "This argument is an instance of the pitfalls
that you can encounter in talking about infinity." If Newton were right,
all the stars in his 'infinite' universe would collapse into one fireball.
Not only Bentley, but also Descartes (1596  1650) and Leibniz (1646  1716),
among others, did not share Netwon's view. Leibniz, for example, declared:
"I hold space to be something purely relative, as time is." [ ibidem ]
He also persuasively argued that the universe had to be finite, though,
perhaps, unbounded (how can that be?).
Be as it may, this issue was not resolved until the beginning of the
20th century. A very interesting chapter in this story is represented
by Heinrich Olbers (1758  1840), who in 1823 formulated what is now known
as Olber’s paradox, which in plain English runs as follows:
"Suppose that the universe is (1) static, (2) infinite, (3) eternal and
(4) uniformly filled with stars (or galaxies, which are made of stars).
If we look in any direction, our line of sight must eventually run into
a star (galaxy), just as a frictionless arrow shot in the middle of an
infinite forest will eventually stick itself into a tree. Therefore,
the night sky should be as bright as the average star (galaxy) and certainly
should not be dark."
[ from Olber's Paradox: Why is the Night Sky Dark? ]
Olber's paradox is an example of a simple but astute piece of reasoning that demands
a satisfactory answer, since the might sky is definitely not as bright as the day sky.
The resolution of this paradox is far from simple. In fact we are only now beginning
to understand what a satisfactory answer requires. Here is a comment that shows how
far an answer would take us:
"It is interesting that in asking and answering the seemingly trivial question,
'Why is the night sky dark?' one could have inferred that the Universe was expanding
and that the Universe had a finite age (or at the least the stars and galaxies had
finite ages)." [ from Olber's Paradox ]

Contrast now Newton's approach to time and space with Einstein's.
In 1905, Albert Einstein (1879  1955) published his special theory
of relativity. Here is how, in a later nontechnical exposition,
he proposed to tackle the issue of time and space (note the simplicity
and clarity of his language):
"The purpose of mechanics is to describe how bodies change their position
in space with 'time.' I should load my conscience with grave sins against the
sacred spirit of lucidity were I to formulate the aims of mechanics in this
way, without serious reflection and detailed explanations. Let us proceed
to disclose these sins.
It is not clear what is to be understood here by 'position' and 'space.'
[ … ] In the first place we entirely shun the vague word
'space,' of which, we must honestly acknowledge, we can not form the slightest
conception, and we replace it by 'motion relative to a practically rigid body
of reference.' [ … ] In order to have a complete description
of the motion, we must specify how the body alters its position with
time; i.e. for every point on the trajectory it must be stated at
what time the body is situated there. These data must be supplemented by such
a definition of time that, in virtue of this definition, these timevalues can
be regarded essentially as magnitudes (results of measurements) capable of
observation."
[ from Albert Einstein, Relativity. H Holt & Co, NY 1920, p. 9 ]
In other words, time is simply, but not trivially, that which a clock measures.

In a very real sense, the difference in the two approaches lies in this:
both appeal to our everyday experience, and both reflect the historical
and cultural period in which they were formulated. In Newton's times,
religion and metaphysics were considered as real and relevant as the
natural world, and the two often mixed, quite freely. In Einstein's times,
the danger of such hybrids was felt much more strongly, just as a growing
trend to rely only on that which can be empirically observed and measured
began to establish itself.
These considerations are best illustrated by the two principles (or postulates)
that Einstein introduced in 1905 as the foundation of his theory of special relativity :
 Relativity Principle : The laws of physics have the same form in all inertial frames of reference.
 Constancy of the Speed of Light : Light propagates through empty space at a constant rate,
c = 299,792,458 m/s.
An inertial frame of reference is defined as a frame in
which nonaccelerated objects move in straight lines at constant velocity
(i.e. at constant speed and in a constant direction). Notice also that
c is short for celeritas, the Latin word
for speed.
The assumption of the constancy of the speed of light was based on a large number
of astronomical measurements, particularly in the 19th century, which left
little doubt about its reality. The history of such measurements is fascinating.
See for example A Case History in Astronomy and Physics: The Speed of Light .
The amazing consequence of these two assumptions was that it was possible to
derive, without any further assumptions, using only some rather elementary mathematics,
a complete theory of motion. This theory differed in some fundamental and surprising
ways from Newton's. Before summarizing the major features of the theory of special
relativity, it is important to emphasize the basic difference between Einstein's assumptions
and Newton's.
Postulate 1 is identical in Newton and in Einstein (in fact Galileo (1564  1642)
had already considered it). You can verify it, for example, by comparing the results of
simple experiments carried out on a train moving at constant speed and in a constant
direction, with similar ones carried out on land, near the train's tracks.
Disagreement arises with regard to Postulate 2. In practice if not in theory, Newton
assumed the speed of light to be infinite, while Einstein accepted
the empirical evidence that it was finite. In fact it is possible
to show that if one replaces the value c = 299,792,458 km/s in Einstein's theory with
infinity, the special theory of relativity reduces to Newton's theory. Notice also
that the speed of light was first measured in 1676 by the Danish astronomer Ole
Christensen Rømer (1644  1710), using careful observations of the motions of
Jupiter's moons [ see, for example, Ole Rømer ].
It would take more than two centuries before Einstein realized the fundamental
importance of such a result.

Here, quite concisely, are the major consequences of the special theory of relativity:
 Two events, simultaneous by the clock in one inertial frame of reference, in general
are not simultaneous by the clocks of other inertial frames in motion relative to
the first one.
 Given two identical, synchronized clocks in motion relative to each other, each clock
will be observed (measured) to be slow by an observer moving with the other clock.
 The length of an object (say, a rod) at rest in an inertial frame is measured as shorter
by an observer in another inertial frame moving parallel to the object (the rod).
 For an observer at rest in an inertial frame, the mass of a moving object increases
with its speed.
 Mass and energy are two forms of the same substance, and can be transformed, under
suitable conditions, into each other: E = mc^{2}.
Notice that each of these statements has been verified experimentally over and over again.
Why is it then that our daily life seems not to have been affected by relativity, and that
the consequences described above are nowhere to be seen around us? The answer is simple
and revealing: the speed of light is so great that, in the context of our daily experiences,
it is practically infinite. The speeds which are experientially meaningful
to us are puny in comparison. That's why we can not really say that Newton was wrong. Think:
when we are told that a spaceship is traveling towards, say, Mars, at some 60,000 km/hr,
we must remember that that means approximately 1.64km/s. That's only 1/18,000th the speed
of light! Relativistic effects become noticeable when we have to deal with speeds which
are substantially comparable to that of light.
At the same time it is important to realize that to extrapolate from our earthly experience
to the universe at large is a risky business. We may be wrong—as Einstein has shown us.
Only in this sense was Newton 'wrong.'

There is one important feature that both Newtonian mechanics and Special Relativity
have in common: the mathematical equations describing motion are timereversible.
Roughly, this means that they are not sensitive to the direction of time. That's
rather strange, since our everyday experience amply demonstrates that time runs
only one way—towards the future. Specifically, by 'future' I mean here the
tendency of all closed or isolated systems to
become more and more disordered (the socalled second law
of thermodynamics ). Equivalently, one can say that usable energy
can only decrease in isolated systems. This situation is often claimed to apply
to the universe as a whole, but that's a questionable hypothesis (why?).
To explore this issue, however, lies well outside the the scope of this course.
I will simply note that the inability of dynamical equations to account for
time irreversibility has preoccupied scientists for a long time, and that no satisfactory
solution has been found yet. I will mention here the work of Nobel Prize Ilya Prigogine
(1917  2003) and his collaborators, who made important contributions that may
lead to a solution. Visit the Educational
Features at the Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems.
"Until the second half of the 19th century the ideal of the natural
sciences were the laws of dynamics, which are symmetrical in respect of
time (for a dynamic system, the past and the future are absolutely
equivalent). The Second Law of Thermodynamics, discovered more than a
hundred years ago, created some confusion in the minds of scholars by
proclaiming that, left to itself, a system will tend from order towards
chaos, this process being irreversible, i.e. 'timeoriented.' But the
'arrow of time' and the concept of chaos did not become fully
incorporated in the natural and social sciences until the second half of
the century that has just ended. And their incorporation was wrought
first and foremost by the works of Ilya Prigogine, who showed that
timeorientation is a fundamental property of all natural systems
(physical, chemical, biological and social) and that the 'natural
tendency' towards chaos by no means entails loss of harmony. Prigogine
succeeded in explaining in the language of mathematics that chaos can be
constructive—that it is precisely chaos that gives birth to the new
order."
[ from The Rediscovery of Time ]
We will deal with chaos in another lecture. What is important
here is to note that Prigogine showed that time may be, not a general property of
the universe, but a property of each individual system. In other words, each
system is the origin of its own time, and that the basic behavior of each system is
governed by dynamical laws which are not timeirreversible.

What about time in quantum mechanics? We can barely scratch the surface in this course.
Here is a few important points:
Time measurements are defined by the type of clocks used:
 There is always some uncertainty due to the very act of measurement, but in the
case of quantum clocks there is additional uncertainty.
 The time measured by clocks may depend on the physical situation, and there is
no guarantee that the clock is not influenced by some nearby field.
 Measurements may depend on the path of the clock, not only on the start and end points.
Notice that, despite the above qualifications, the equations of the
standard theory of quantum mechanics are also timereversible, and in that sense they
too are 'unrealistic,' just like Newton's and Einstein's equations. We are still awaiting
a theory that embodies the irreversibility we so easily observe in our everyday world.
The success of Einstein's theory of special relativity and of the general theory of
relativity (which lies outside the scope of this course) points to a new conception
of space and time. What Einstein showed is that the really meaningful concept is that
of spacetime (also spelled spacetime or space
time). Space and time are inextricably woven together. We can not speak of one
without using the other. For example, within the framework of special relativity,
the measurement of speed (the rate of displacement in space as a function of time)
requires both clocks and rulers. But the behavior of these depends on the state of
motion of the observer relative to what he is measuring. For a very clear, fairly
elementary, and nicely illustrated explanation, visit Spacetime 101
Readings, Resources and Questions

A good resource for horology, the
science of time measurement, can be found in the Internet Public Library.

A very helpful and entertaining illustration (also in a literal sense)
of the special theory of relativity was published sometime before 1966 in
an issue of Time Magazine, under the title The Great Relativity Bomb Plot.
It was reprinted in 1966 as part of a TimeLife book by S A Goudsmit entitled Time.
(Toronto Public Libraries have copies). You can find a scanned copy at The Great Relativity Bomb Plot
(click on each section to see the corresponding scan of the original, illustrated pages).
Visit Han Erim's Alice Law, where you can download
a free program that will teach you, in a very original and interesting way, the basics of Special and General Relativity.

The number of books on the nature of space and time, on the evolution of our ideas
about space and time, and current research trends, is enormous. I will mention here
only a few, accessible ones. Notice thay they all seem to be about 'time' only. In fact,
because time is really part of 'spacetime,' all these books also deal with 'space.'
 A Einstein and L Infeld, The Evolution of Physics: From Early Concepts
to Relativity and Quanta. Simon & Schuster, NY (1938), 1966.
 A Einstein, Relativity. H Holt & Co, NY 1920.
 S Hawking, A Brief History of Time: From the Big Bang to Black Holes.
Bantam, NY 1988.
 C A Pickover, Time: A Traveler's Guide. Oxford UP, NY 1998.
 P Coveney and R Highfield, The Arrow of Time: A Voyage Through Science to Solve
Time's Greatest Mystery. Fawcett, NY 1990.
 B Russell, The ABC of Relativity. Unwin, London 1958.
 G Smoot and K Davidson, Wrinkles in Time. W Morrow & Co., NY 1993.
© Copyright Luigi M Bianchi 20032005
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Last Modification Date: 06 March 2006
