Inertial Frames of Reference

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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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