The Case of
The Inertial Frame
describes a long-enduring model of space, time, and motion in the physical world which systematically loses certain relations and highlights others. This story illustrates how models in science, like models elsewhere, are interfaces which select and discard aspects of the systems they connect. Science uses use this selectivity and isolation to gain tremendous descriptive, predictive, and manipulative power. The inertial frame, first formalized by Isaac Newton, links the fall of an apple to the dance of the planets to the path of a rocket flying through the air.
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Much of the science of physics was and still is based on a brilliant model known as the "inertial frame." The scope and power of this model is astonishing. The utility of this model was such that scientists in the early 1900s abandoned our conventional notions of space and time in order to preserve it, as we will see in The Case of The Special Theory.

Inertial frames were first formalized by Isaac Newton based on the observations and principles described by Galileo Galilei and Christiaan Huygens. Galileo started with a way of converting changes in time into visible patterns, drawing on a single sheet the different positions which an object occupies at different moments, as in:

where the motion of the ball from left to right is represented in a series of images combined into one.

Looking at experiments and measurements in this way was already a model combining a series of observations into a single figure. With this model as a tool, he demonstrated that we can see a cannonball's path as a combination of a horizontal motion (at a uniform velocity) and a vertical fall (at a constant acceleration). The striking thing about this "way of seeing" was that the moving cannonball's fall, when the horizontal motion was ignored, looked exactly like the path of a ball dropped straight down from a height, as shown in this figure from Galileo's original paper and my sketchy clarification.

Through this method, two very different motions could be identified with one another: the arc of a cannonball across a plain and the toss of a ball vertically into the air. This reduction of a series of positions into a single figure ignores time in a particular way and highlights time in another. The kind of time highlighted by Galileo's invention is the same external objective time which the calendar describes; the kind of time ignored by Galileo's invention is the more relational subjective time which we glimpsed in the utility of the child's systematic forgetfullness.

Newton expanded this way of seeing into an eighteenth century "Theory of Relativity" telling us that we could choose to look at physical systems as though we were "moving along with them" rather than describing them only as they moved relative to us. The notion of an inertial frame asserts that we can group together objects which are all moving with the same speed and direction, considering it as a single system whose internal interactions are exactly the same if it were moving twice as fast or sitting still. Whether it is a complex piece of clockwork, a human nervous system juggling clubs, or a complex physics experiment, it makes no difference whether it is sitting still on the ground or set in the passageway of an airliner going 1000 kilometers per hour. The clockwork works the same, the clubs are no harder (or easier) to juggle, and the experiment yields the same results.

The airliner story is an idealization and turbulent weather or the subsonic murmur of engines might disturb it in practice. But the idealization holds up remarkably well and one aspect of its extraordinary usefulness is that we can design and troubleshoot a machine in one inertial frame and feel confident that it will also function in another inertial frame. This extends our ability to think about the world because we can figure out a theory or construct a mechanism in one context and apply it or use it in a different context.

Inertial frames allow us to see the world in simpler ways and make possible solutions which would have otherwise been inconceivable. And today, when physicists or physics students solve problems, they still start by identifying the relevant frames of reference and then proceeding to a solution.

On the surface, being able to describe part of a single experiment in different ways seems more complicated than having a single description. Our desire for simplicity might object to the complication of "multiple ways of seeing." However, one of the paradoxes of using models is that the combination of multiple models may be much simpler --- altogether --- than a single account in a single model. What is important, however, is the ability to divide situations into these multiple descriptions and to put them back together afterwards.

Pick a frame, any frame...

If one part of Newton's innovation was the characterization of inertial frames, another component was the assertion that you could choose the frame from which you described a problem, solve the problem, and then convert back. As a concrete example, consider the following combination of cinematic and physics-book cliches:

Chris and Robin are running towards each other across a sun-dappled meadow, separated by 350 feet. Chris is running at 4 feet per second while Robin is running at 3 feet per second. How long will it take them to meet?

which might be represented schematically like this:

and which Newton teaches us we can transform to a more convenient description:

where Chris is standing still and Robin is running at 7 feet per second towards him. When described in this way, it is immediately clear that it will take 50 seconds to close the distance, just long enough for the background music to reach a satisfying finale. Note that Newton's transformation and solution drastically transforms the scene's dramatic impact: Chris is standing still and Robin is doing all of the running. But the transformation makes the physical problem much easier to solve.

This was one of Newton's most important contributions, a mathematical way of combining different inertial frames which were moving in straight lines at uniform velocity. Galileo, knowing the Earth was a sphere, understood that the cannonball's horizontal motion could be seen as movement along a small arc of a huge circle. Galileo actually thought that constant motion in circles was quite natural. Newton's "inertial frames," on the other hand, only went in straight lines and Newton's mathematics and physics was able to describe how considerations from different inertial frames combined to create the circular motion which Galileo had taken for granted.

Rewriting the Laws

Newton's combinational trick was based on the idea that distances (differences between positions) and intervals (differences between moments) remained the same in separate inertial frames. The distance between Chris and Robin (to use our example) is the same for Chris, Robin, and the camera operator filming the scene. This preservation of intervals is the "distance version" of the common-sense "calendar logic" we encountered earlier in the Case of the Missing Calendar. The "calendar logic" asserted that five weeks is five weeks whether you are sitting at home or flying around the planet, so the homebody and the world traveller can successfully arrange and execute a meeting. We might use the term "ruler logic" to describe the "distance version" of this reasoning, since it says that no matter how we move or turn our point of view, a ruler will measure distances as the same. Together, calendar and ruler logic were the basis of Newton's innovation.

With these assumptions about distances and intervals, Newton could take descriptions based on positions and moments in one frame and rewrite them in terms of distances and intervals. He could then take these distances and intervals to another frame --- whose position, orientation, and velocity might be different --- and show that the laws looked the same because the distances and intervals were the same. Because of these assumptions and the rewriting which they made possible, Newton was able to describe laws which could be applied to different inertial frames and then combined the laws' consequences in each.

This combinational trick both made description easier and captured the similarity of the falling ball and the arcing projectile at the heart of Galileo's observations. This is often the way that a model works, taking some observed fact across different contexts --- like the invariance of a ball's rise or fall --- and creating a way of describing it where the descriptions of the distinct contexts are actually the same. The new kind of description actually loses the information which doesn't matter, allowing theory and prediction to focus on the things which do matter.

Galileo and Newton

Galileo described many of the phenomena which Newton explained with his theory. In particular, Galileo described the "uniform acceleration" (which is different from the uniform motion of an inertial frame) which characterized falling objects. Newton went further by proposing a new kind of abstract entity, the force, which was the direct cause of all uniform acceleration. If we see something accelerating, Newton says, we must look for the force impelling it.

Galileo would have probably been initially skeptical of Newton's force. He rejected Johannes Kepler's suggestion that some "force" from the sun might be moving the earth and planets and had an even stronger response to Kepler's suggestion that a force from the moon might cause tides:

But among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he has nevertheless lent his ear and his assent to the moon's dominion over the waters, and to occult properties, and to such puerilities.

Looked at without the help of Newton's theory, we can understand the "occult" dismissal of the idea that some mysterious force nudges objects from natural straight lines into circular orbits. Galileo's own proposal, that motion in circles was natural, seems more satisfying. But the genius of Newton's new way of describing the situation was to change the inertial frame to one which was continually tangent to the satellite's motion around planet:

In this transformed set of frames, we can see that the same thing is happening all the time: the planet is accelerating towards the earth. Indeed, we can also see that this is the exact same phenomenon which Galileo recorded for the falling body, but extended to include the motion of planets. It was this consistency of description that made it reasonable to propose an "unseen" force causing the acceleration, just as Galileo had readily accepted another "unseen force" sustaining the cannonball's horizontal motion.

It is striking to juxtapose Galileo's earthbound and and cosmic observations. One of Galileo's other outstanding accomplishments (and a source of his personal fame and inspriration) was his examination of the heavens through the telescope he constructed based on accounts from Holland. Among these observations, he spied four of the moons of Jupiter orbiting around the giant planet; this discovery led to his own assertion regarding the motion of the Earth which so complicated his later life (the Church forced him to recant). Galileo did not understand why the moons moved as they did. But Newton, given the full-blooded model of inertial frames which he developed from Galileo's "way of seeing," was able to explain the motion using the same uniform acceleration which Galileo had observed in his earthbound experiments.

The power of this simple model is astonishing, linking the fall of a ball to the dance of the planets. And the model underlying it is the source of that power, allowing us to understand the invariant identity of laws despite changes in time, place, and motion. Models in science act as interfaces which transfer some observations, discard some, and rearrange still others.

We now move forward to the second half of the 1800s, where the quest to retain Newton's powerful model for other phenomenon led scientists to refute Newton's first assumptions and eventually abandon part of our common sense understanding of space and time.

Copyright (C) 1997, 1998 by Kenneth Haase
Draft, not for citation or circulation
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