Inertial Frames of Reference

As the name implies, Relativity describes
how two people can relate their observations of the world, each to the other's.
More specifically, we use the theory of Special Relativity to relate
measurements that observers occupying different inertial frames of reference
make between pairs of events. And we actually use the centerpiece of the theory,
the Lorentz Transformation, to translate one observer's measurements of a pair
of events into copies of another observer's measurements of the same events.
Named for the Dutch physicist, Hendrik Antoon Lorentz, it comprises four
equations by which we perform the transformation of measurements. Here I will
derive it, as Einstein did, and present it first as a series of rules that will
make Relativity much more transparent than we have it in its refined form, then
I will convert those rules into the transformation equations and show you how
physicists use those equations. But first we must determine what we mean by the
phrase "inertial frame of reference".

Hermann Minkowski, one of Einstein's math
teachers, showed in 1908 that Relativity constitutes a kind of Euclidean
geometry that we work out in four dimensions, the fourth dimension (time) being
related to the three dimensions of space through a strange analogue of the
Pythagorean theorem. 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. Thus, with Relativity we want to relate the
measurements of distance and duration between some pair of events, which
measurements come from two observers occupying two different inertial frames of
reference. But if we make an analogy between Relativity and geometry and thus
study the analogues of points, then we should expect that we might in some way
make an analogy between inertial frames of reference and the coordinate grids
that geometers use as frames for measurements in analytic geometry. To give
ourselves a clearer picture of how we might make that analogy work let's imagine
measuring the distance and duration that Reality puts between two events.

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. We take as one good candidate 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, the
pop of a second firecracker.

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: we measure simply through the act of assigning a number to
something that we do not find obviously countable. Obviously countable things
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,
come before us as continuous, unbroken, and intangible entities. How can we
assign numbers to such things? Simply enough, we span them with things that we
can count.

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 laying 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 call the distance between the rock and the
tree twenty-one feet. We see clearly no subtlety in that process and Relativity
doesn't introduce any.

Duration, the temporal analogue of
distance, gives us another example of an entity whose measurement we must
consider. In one way measurement of time gives us more difficulty than does
measurement of space: we can't merely cut rods to span temporal intervals. Yet
in another way Reality eases the measurement of time for us by giving us an
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 we will make our
standard unit of time from the duration of some action that we can repeat
exactly and indefinitely. Among the kinds of actions that people have 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 want to observe.
That concept, of the union of a means to produce a repeated action with a means
of counting the repetitions, gives us 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. Imagine the laser and the
mirror mounted at opposite ends of a long glass tube in which we have dispersed
a small amount of smoke and imagine the photocell mounted next to the laser. To
make such a clock count time we need only 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, one that will help us particularly well in 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 comprises a vast array of straight
lines that fall into three groups: you have oriented one group east-west (which
I will call the x-direction), one group north-south (which I will call the
y-direction), and the remaining group 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 you have synchronized all of the ghostly clocks that you place in
this array with each other. Now you have constructed in your imagination 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 all float
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 we would find such grids 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 make 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 have the ability 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 that we can consider it an accurate
reflection of that Reality. Imagine that you have before you a length of
perfectly straight railroad track that someone has 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 come 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 moves
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 motion of my speeder will carry
that smudge 25 feet before the second firecracker pops, so the smudges will lie
75 feet apart. I must say, then, that in my frame the pops come 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 motion of my speeder
carries the eastern firecracker 25 feet further east after the western
firecracker pops before it pops, so in your frame the pops still come 100 feet
and one second apart. This alternate version emphasizes that we must say that
the events and not the bodies that participate in them comprise the proper
objects of our measurements: the bodies merely serve as convenient markers
against which we make 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 we call the Galilean Transformation. If, in the preceding imaginary experiment, you had used upper-case letters to represent your measurements, I had used lower-case letters to represent my measurements, and we had both used lower-case vee to represent the relative velocity between our frames, then those equations would be

x=X-vT

(Eq'n 1)

y=Y

(Eq'n 2)

z=Z

(Eq'n 3)

t=T

(Eq'n 4)

To ensure that those equations yield the
proper results, I must further specify that we must calculate the time interval
by subtracting the reading on the western clock from the reading on the eastern
clock (for their respective events), just as I would calculate the distance
along the x-axis by subtracting the position of the western event from the
position of the eastern event on that axis.

Physicists named the Galilean
Transformation after Galileo Galilei because it reflects the rule of Relativity
that he introduced into classical physics in 1633: that rule corresponds simply
to 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. We make very little of the Galilean Transformation
in classes in classical physics because it's actually quite trivial, but in
courses in Relativity we introduce it as the classical equivalent of the Lorentz
Transformation; that is, we conceive the Lorentz Transformation as 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
modification must fade away and the transformation equations that the observers
in those frames use upon each others' measurements must shade smoothly from
Lorentzian to Galilean. That last requirement provides one of the tests that
Relativity had to pass before physicists would accept it into modern physics.
Now we want to go in the opposite direction and convert our Galilean rules into
their Lorentzian counterparts.

So far we have contemplated situations in which the relative speeds involved are small relative to the speed of light, the realm of Isaac Newton's rational mechanics (it was Wilhelm Gottfried Leibnitz who named it dynamics, much against Newton's wishes). Now we want to extend that contemplation to velocities close to the speed of light. Because we have described the speed of light as a finite speed with the character of an infinite speed, we may expect that in inertial frames moving at such rapid speeds we will find the relation between space and time deformed relative to the same relation in frames moving much more slowly. We gain that expectation from the observation of how the conservation laws act as constraints that determine the shape of space and time, giving us the postulates of Relativity as a by-product. Relativity gives us 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 a clearer picture of the shape of space, clues to the destiny of the Universe, and the foundation of a fully Rationalist physics.

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