EVENTS IN INERTIAL
FRAMES

As the name implies, Relativity is the
theory devoted to describing how two people can relate their observations of the
world, each to the other's. More specifically, the theory of Special Relativity
is concerned with relating measurements made between events by observers
occupying different inertial frames of reference. The Lorentz Transformation is
the part of the theory that is actually used to translate one observer's
measurements into copies of another observer's measurements of the same events.
It consists of four equations, but I'm going to derive it and present it as a
series of rules that will make Relativity much more transparent than it is in
its raw algebraic form.

Hermann Minkowski, one of Einstein's
math teachers, showed in 1908 that Relativity is a kind of Euclidean geometry
that's worked out in four dimensions, the fourth dimension (time) being related
to the three dimensions of space in a way that makes the Pythagorean theorem
look a little strange. As Euclid's plane geometry begins with a consideration of
points, so Relativity begins with a consideration of the spatio-temporal
analogue of points - events. What we want to do with Relativity, then, is to
relate the measurements of distance and duration made between some pair of
events by two observers occupying two different inertial frames of reference.

What kind of events should we
consider? We want something analogous to a geometric point. We recall that the
ideal point has no extent (and is thus invisible to us), so we must infer that
the ideal relativistic event has no extent and no duration (and is thus equally
invisible to us). When we work out the theorems of plane geometry we draw dots
on paper to stand in for ideal points and we do so with the understanding that
the theorems that we deduce through the use of such drawings remain valid when
we apply them to the actual ideals. In like manner, then, we can imagine using
some cartoonish approximation to the ideal event as a means of working out the
theorems of Relativity. One good candidate is the crisp pop of a small
firecracker: for our crude senses it will adequately mark a point in space and
an instant in time, from which we can measure the distance and duration to
another event.

Does measurement have any special
meaning in Relativity? Did Einstein discover some subtlety in it that surpasses
common understanding? No, he didn't. Though measurement occupies a position of
special importance in the theory, it has the same meaning that it has in more
mundane circumstances: it is simply the act of assigning a number to something
that is not obviously countable. Things that are obviously countable include
such collections of discrete objects as pebbles on a beach, birds in a flock,
and cattle on the range. But space and time, the objects of Relativity, are
continuous, unbroken, and intangible. How can we assign numbers to such things?
Simply enough, we span them with things that __are__ countable.

Distance in space gives us the
opportunity to work out a clear example of measurement. How can we describe the
distance between a rock and a tree, for example? We must start by defining some
standard unit of distance, describing it in such a way that anyone else can
reproduce it. Imagine that you have called together twelve men, as for a jury,
and asked them to stand with their right feet heel to toe in a straight line. If
you then mark the distance between the last man's heel and the first man's toe
on a straight rod and then cut the marked section of that rod into twelve
smaller rods of equal length, you will have obtained a standard of distance
known, naturally enough, as one foot. Alternatively, you could have measured one
ten-millionth of the distance from the North Pole to the Equator along the
meridian passing through Paris, France, and called that distance one meter, the
basis of the metric system. Whichever system you choose, the rods that you have
cut enable you to cut even more rods of the same length. Thus, you measure the
distance between the rock and the tree by laying one rod with one end against
the rock and lay other rods, end to end, from it to the tree in a straight line
and then counting the number of rods that you laid down. If you find that you
laid down twenty-one rods, then you would say that the distance between the rock
and the tree is twenty-one feet. There's clearly no subtlety in that process and
Relativity doesn't introduce any.

Duration, the temporal analogue of
distance, is another example of an entity whose measurement we must consider. In
one way measurement of time is more difficult than measurement of space: we
can't merely cut rods to span temporal intervals. Yet in another way measurement
of time is eased for us by our innate familiarity with the concept: from our
intimate association with our mother's heartbeat and breathing and then with our
own we intuit the concept of counting time and know what it means, all the more
so if we have made any significant acquaintance with music. Thus we already know
that our standard unit of time will be the duration of some action that can be
made to repeat exactly and indefinitely. Among the kinds of actions that have
been used to count time we find the swing of a pendulum, the shudder of a quartz
crystal, and the pulse of an electric circuit. Any simple, repetitive motion
will do the job for us.

Because our temporal standard involves
something that moves, we might well contemplate the possibility of harnessing
that motion to a device that will automatically count for us the number of times
that our standard action repeats between the two events that we are observing.
That concept, of the union of a means to produce a repeated action with a means
of counting the repetitions, is the fundamental idea of the clock. We now wish
to consider the realization of that aetherial Platonic form: in what array of
matter shall we clothe it? For the purpose of teaching Relativity, physicist
Richard Feynman conceived a clock that comprises a laser, a mirror, and a
photocell attached to an electrically-driven counter. The laser and the mirror
are mounted at opposite ends of a long glass tube in which a small amount of
smoke has been dispersed and the photocell is mounted next to the laser. To make
such a clock count time it is sufficient to stimulate the laser into emitting
one brief pulse of light. The smoke in the tube scatters a small amount of the
light in the pulse, thereby enabling us, in our imaginations at least, to follow
the pulse as it travels from the laser to the mirror and thence back to the
laser. Upon its return to the laser the pulse illuminates the photocell,
generating a pulse of electricity that makes the counter advance one unit, and
illuminates the laser, causing it to emit another pulse of light. Thus we have a
clock and, as you will see, it is one that is particularly well suited to
exploring the relationship between space and time.

For the convenience of making
imaginary measurements, we now assemble the ghosts of those two entities,
measuring rods and clocks, into what physicists call an inertial frame of
reference. Construct in your imagination a transparent jungle gym that extends
in all directions as far as you can see. What you have in mind is a vast array
of straight lines that fall into three groups: one group is oriented east-west
(which I will call the x-direction), one group is oriented north-south (which I
will call the y-direction), and the remaining group is oriented vertically
(which I will call the z-direction). Where three lines cross each other each
line crosses each of the other two at a right angle, so the lines effectively
subdivide space into little cubes. At each such intersection imagine placing a
ghost of a clock and imagine further that all of the ghostly clocks that you
place in this array are synchronized with each other. What you have constructed
in your imagination is a coordinate frame of reference. To use it, imagine that
two events occur at or near two of the intersections of lines; count along three
lines from one intersection to the other and use the three-dimensional
Pythagorean theorem to calculate the spacial distance between the events; and
subtract the reading of one intersection's clock from the reading of the other
intersection's clock to calculate the temporal interval between the events.

That coordinate frame of reference
becomes an __inertial__ coordinate frame of reference if we specify that it
does not accelerate. That means that any two inertial frames will always have
the same velocity between them. It also means that if I change my velocity, I
leave my original inertial frame and enter another one, passing through a whole
continuum of other inertial frames while I am accelerating. Looked at the other
way, any object that does not accelerate occupies and thus marks an inertial
frame and any object that has a uniform, unchanging velocity relative to that
first object occupies and marks another inertial frame. If we ignore the effect
of gravity, usually by considering only horizontal motions, and also ignore its
acceleration toward the sun as it follows its orbit, by considering events
separated by short intervals of time, then Earth can mark an inertial frame for
us, one that conforms more or less to our preconceptions of Reality.

With that definition in mind, we can
now refine our definition of space: instead of saying that space comprises an
infinity of points, we can say more accurately that space comprises an infinity
of inertial frames, each of which comprises an infinity of points that are all
motionless relative to each other. In all directions these ghostly grids move
through each other at speeds from zero up to the speed of light, providing the
means to measure all possible events by covering every point and every velocity
that a material body may occupy.

In Reality such grids would be highly
impractical to use, even if we could build material versions of them, so we use
more conventional surveying and timing methods to measure the spatial and
temporal intervals between events. But in the laboratories of our minds those
ghostly images of inertial frames allow us to ignore the mechanics of
measurement and to concentrate our attention upon the measurements themselves
and what they can tell us about the nature of Reality. In asserting this
proposition we are making the tacit assumption that our mental pictures of
rulers and clocks behave much as real rulers and clocks would do (not much of a
stretch there) and that real rulers and clocks actually represent to us, within
the limits of the errors inherent in material things, the intangible foundations
of Reality; that is, space and time.

If that last assumption is true to
Reality, then we should be able to test it and verify it through an imaginary
experiment. We will imagine measuring the intervals between two events with
rulers and clocks and we shall convince ourselves that what we imagine does,
indeed, mimic Reality closely enough to be considered an accurate reflection of
it. Imagine that you have before you a length of perfectly straight railroad
track that has been laid due east-west. To create the two events that we wish to
study, you have set two short poles 100 feet apart in the ground next to the
track and then you have taped to the top of each pole a small firecracker. When
you light the fuses, you do so in such a way that the western firecracker pops
precisely one second before the eastern one does. In your inertial frame, the
one occupied and marked by Earth in this short experiment, the two pops are 100
feet and one second apart.

Now imagine that you see me driving a
track speeder eastward at 25 feet per second (a tad over 17 miles per hour);
that is, that you see me occupying and marking an inertial frame that's moving
eastward at 25 feet per second relative to your frame. Imagine further that I
have mounted a long, perfectly white plank on the side of my speeder and that I
have done so in such a way that the pops of your firecrackers will each leave a
gray smudge upon it. How far apart will I measure the centers of those smudges
to be? After the first firecracker pops, the smudge that it makes will be
carried 25 feet before the second firecracker pops, so the smudges will be 75
feet apart. I must say, then, that in my frame the pops are 75 feet and one
second apart.

Clearly that experiment would yield
the same results if I were to mount the firecrackers 75 feet apart on my plank
and so light the fuses that the western firecracker pops one second before the
eastern one does. In that version of the experiment the eastern firecracker is
carried 25 feet further east after the western firecracker pops before it pops,
so in your frame the pops are still 100 feet and one second apart. This
alternate version emphasizes that it is the events and not the bodies that
participate in them that are the objects of our measurements.

Analysis of that experiment leads us readily to three simple rules that we know will tell us how our measurements will be related if we repeat the experiment with different spacings and timings between the events: we have

**GALILEAN RULE 1; **If
you measure a distance between two events in the direction of relative motion
between your frame and mine, I will measure the same distance plus or minus
(depending upon which event occurred first) the distance that my frame moves
relative to yours in the duration between the events.

**GALILEAN RULE 2;** The
distances that we measure between the same two events in the directions
perpendicular to the direction of the relative motion between our frames are the
same for both of us.

**GALILEAN RULE 3;** The
duration that we measure between the same two events is the same for both of us.

If we were to translate those rules
into their algebraic counterparts, we would obtain four equations (one for the
direction parallel to the relative motion , two for the two directions
perpendicular to the relative motion, and one for time) that comprise what is
called the Galilean Transformation. It's named after Galileo Galilei because it
reflects the rule of Relativity that he introduced into classical physics in
1633: that rule is simply Einstein's first postulate and Galileo introduced it
in his book, "Dialogue on the Two Chief World Systems", the book that was the
focus of his infamous trial before the Inquisition. Very little is made of the
Galilean Transformation in classes in classical physics because it's actually
quite trivial, but in courses in Relativity it is introduced as the classical
equivalent of the Lorentz Transformation; that is, the Lorentz Transformation is
taken to be a modified form of the Galilean Transformation and when the relative
velocity between two inertial frames becomes extremely small relative to the
speed of light the transformation equations that the observers in those frames
use upon each others' measurements must shade smoothly from Lorentzian to
Galilean. That last requirement is one of the tests that Relativity had to pass
before it could be accepted into modern physics. Now we want to go in the
opposite direction and convert our Galilean rules into their Lorentzian
counterparts.

But why do we bother ourselves with
this bizarre parody of the surveyor's art? Wasn't our experience of plane
geometry in high school bad enough to warn us off this heavier stuff? What do we
hope to gain from a study of Relativity? You've already seen how the
conservation laws act as constraints that determine what the shape of space and
time must be, giving us the postulates of Relativity as a by-product. Relativity
is just an extension of those constraints, one that determines the permitted
shape of the other laws of physics. Exploiting that constraint will enable us to
refine our knowledge of the relation between matter and motion and will gain us
an astonishing result, one that both deepens our understanding of Reality and
has practical applications as well. That's what we gain from our study of
Relativity.

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