Section II, Chapter Three: Galilean Invariants and Inertia
If the principle was to survive, something fundamental had to replace the apparently shapeshifting concepts of inertia and impetus.
Note—this is a draft of Chapter Six of a book I’m writing. The book is going to cover humanity’s deepest ideas from philosophy, physics, epistemology, and economics. Each chapter is meant to be short and digestible. Most of the chapters will explain just one or two ideas.
‘Chapter []’ indicates a future chapter that I’ve not yet written.
Section II, Chapter Three: Galilean Invariants and Inertia
As I alluded to in Chapter Four, while Copernicus’ heliocentric model and Ptolemy’s geocentric model offered radically different pictures of what the universe was really like, they made the same predictions for the movements of astronomical bodies across the night sky. Both were consistent with the intuitive sense that the Earth was fixed beneath our feet, even though they disagreed about whether or not our planet actually stood still. More generally, it seemed that one could place any astronomical object at the center of the solar system (or even the cosmos) and, with the right model, yield the same predictions as those who regarded any other heavenly body as the motionless center.
Fortunately for Copernicus, he discovered arguments for the heliocentric model on other grounds, thereby evading this issue. Still, it was surely not a coincidence that one could construct a model of the universe with any arbitrary point as its center around which the rest of the cosmos moved.
In an effort to show that Copernicus’ heliocentric model is fully consistent with our experience of an unmoving Earth, Galileo conjectured that experiments conducted under uniform motion yield the same results as those conducted under no motion at all—in other words, one cannot conduct an experiment that tells you whether you are at rest or moving at constant speed (an idea we now call Galilean invariance or Galilean relativity). For example, you would observe the same vertical motion of a tennis ball you toss in the air whether you are standing still in a train station or whether you are standing still in a moving train (provided it’s not accelerating). This is because in the frame of reference of the moving train, you are, in fact, standing still.
But it’s not just the outcome of an experiment that is the same in all inertial (ie, non-accelerating) reference frames. Although Galileo wouldn’t have expressed it this way at the time, we now know that the laws of physics are the same across all inertial reference frames. While observers in different reference frames might use different coordinates and move at different velocities relative to each other, the laws of physics are reference frame-agnostic. And because of that, certain measurable quantities will be the same for all of them. That is, there exist in Nature invariants upon whose magnitude all observers agree (see Chapter []), no matter their frame of reference.
A couple decades after Galileo proposed his idea, Dutch mathematician and physicist Christiaan Huygens (1629 - 1695) worked out one of its implications that had particular significance for the debate over the shape of the solar system. As he wrote in his 1656 book, “The motion of bodies [is] to be understood respectively, in relation to other bodies which are considered as at rest, even though perhaps both the former and the latter are subject to another motion that is common to them.” [Emphasis mine]
In other words, because the boy and the falling tennis ball both travel along with the moving train, their motions can be measured relative to one another just as if the train was not moving at all.
Intimately bound to the concept of reference frames is that of inertia. Johannes Kepler (1571 - 1630), a German astronomer who improved upon Copernicus’ original heliocentric model (see Chapter Seven), introduced the concept to explain why planets move in the first place (recall that, in the older scheme, astronomical bodies were simply carried along the rotating spheres in which they were embedded). Kepler posited that planets ‘naturally’ sought a state of rest in space and that something must have been overpowering this inertia.
Huygens and Galileo then took inertia not to be an object’s resistance to motion but to resistance to a change in motion. Kepler was wrong that objects ‘wanted’ to come to rest. Rather, their inertia kept them from changing their speed and direction of travel in the absence of external influence.
The concept of impetus was a kind of mirror opposite to inertia: while inertia explains why objects resist change, impetus explains why objects change course. As Newton wrote in his manuscript, De gravitatione et aequipondio fluidorum, “Impetus is force in so far as it is impressed on another…Inertia is the internal force of a body, so that its state may not be easily changed by an external force.”
With this distinction in mind, it was only natural that subsequent scientists would guess that the magnitude of an object’s impetus correlated with its, say, velocity, while its inertia was a constant. Leibniz attributed an object’s inertia to its passive power and its velocity-dependent impetus to its active power.
Newton came to realize that his distinction between impetus and inertia could not be fundamental if the laws of physics were uniform across all reference frames. As he wrote in his 1687 magnum opus, Principia Mathematica (see Chapter []), “The exercise of [a force on a body] is, under different aspects, both resistance and impetus: resistance insofar as the body, to maintain its state, opposes the impressed force; impetus insofar as the same body…strives to change the state of that obstacle. Resistance is commonly attributed to resting bodies and impetus to moving bodies; but motion and rest…are only relatively distinguished from each other; and bodies commonly seen as resting are not always truly at rest.” [Emphasis mine]
In short, Newton explained that impetus and inertia were not intrinsic attributes of an object but depended on one’s frame of reference: one frame’s impetus could be another frame’s inertia! But the principle of relativity tells us that observers of all reference frames must agree on the underlying physics of the situation—that there must be some invariant quantity across all frames. If the principle was to survive, something fundamental had to replace the apparently shapeshifting concepts of inertia and impetus.
Newton conjectured that every object had an inertial mass whose magnitude was frame-invariant. As philosopher Robert DiSalle writes, “Newton…understood that inertia has three inseparable aspects: the tendency to persist in motion, the resistance to a change in motion, and the power to react against an impressed force. All are essential to the explication of inertial mass as a measurable theoretical quantity.”
Shedding the appeals of Aristotelian physics was hard work, and with each rejection of his teachings and the acceptance of new concepts, new problems emerged. How could heliocentric and geocentric models make similarly accurate predictions? What did and did not change from reference frame to reference frame? How could a principle like Galilean invariance come to bear on theories of motion? Unlike in the ivory towers of the Middle Ages, in which students pored over the same ancient texts generation after generation, the thinkers of the Scientific Revolution surely delighted in the fact that, with each problem they solved, new questions presented themselves that would have so far been inconceivable. In short, they were making progress—and they were noticing.
Thanks to Moritz Wallawitsch and Dennis Hackethal for early feedback.