Galileo's Forgotten Leap
Arguably the most important insight leading to the development of modern science was the recognition of a class of spatio-temporal systems of reference with respect to which the motions of bodies satisfy a very simple set of mathematical relations, namely, Newton's three laws of motion. Newton's original statement of these laws was based on the tacit understanding that quantities involving position and time are referred to an inertial system of coordinates. This is, of course, a necessary restriction, because the laws are obviously not valid with respect to more general coordinate systems. However, if we're asked to define an "inertial coordinate system", we can only say that it is a system of space and time coordinates with respect to which the three laws of motion are valid. Hence the laws are, in a sense, tautological. The real significance of these laws is due to the existence of inertial coordinate systems (and their equivalence classes, called inertial reference frames), which enables the laws to serve as useful organizing principles.
A crucial aspect of inertial reference frames, and one that is often overlooked, is the fact that they represent the imposition of an operational definition of simultaneity. The figure below illustrates three different systems of space and time coordinates, denoted by (x,t), (x',t'), and (x",t").
Any path that is a straight line with respect to one of these systems is a straight line with respect to all of them, so if one of them (say, x,t) is an inertial coordinate system, then Newton's first law (and arguably his second law) is satisfied with respect to each of these coordinate systems. However, Newton's third law is not satisfied with respect to all of them. The transformation from (x,t) to (x',t') is of the form
for constants a, b, and e. Hence any straight line x = at + b maps to the straight line
Now suppose identical particles initially at rest at the common origin of (x,t) and (x',t) exert a mutual impulse on each other, causing them to accelerate away from the spatial origin. According to Newton's third law, the net impulse exerted on these particles is equal in magnitude and opposite in direction, so they acquire the velocities v and -v with respect to the (presumed inertial) x,t coordinate system. (Note that this symmetrical impulse form of the third law is unambiguously applicable, even in a relativistic context). The paths of the two particles after the impulse are therefore described by the equations x = vt and x = -vt. Thus we have b = 0 and a = v, so the paths of the two particles with respect to the x',t' system are
In other words, when described in terms of the x',t' coordinate system, two identical particles initially at rest and exerting a mutual impulse on each other depart from the origin at speeds whose magnitudes are in the ratio (1 + ev) / (1 - ev), so Newton's third law (in combination with the second) is violated unless e equals zero.
This isn't just a minor detail. It is crucial for a meaningful understanding of inertial reference frames and special relativity. Most anti-relativity kooks fail to recognize that inertial reference frames are not fully specified by the requirement to be unaccelerated, i.e., by the requirement to satisfy Newton's first law. They fail to realize that the space and time components of an inertial coordinate system must satisfy all three of Newton's laws, and that this amounts to the imposition of an operational simultaneity. The widespread failure to recognize this crucial fact may be due partly to the impression that Poincare (around 1900) was the first person to suggest an operational definition of simultaneity. In truth, this aspect of inertial reference frames can be found at the very beginnings of the modern science of dynamics, in the writings of Galileo. When he wrote (in "Dialogue on the Two Chief World Systems") about leaping in different directions on the deck of a moving ship, noting that with equal force we will reach equal distances (implicitly in equal times), regardless of the direction, he was describing an early form of Newton's third law and the conservation of momentum, which necessarily entails a specific operational definition of simultaneity, namely, inertial simultaneity. The contribution of physicists in the early 1900's was to recognize that electromagnetic simultaneity (i.e., synchronization based on light signals) and inertial simultaneity (i.e., synchronization based on mechanical inertial isotropy) are identical.
Surprisingly, given the central importance of inertial coordinate systems (and the equivalence classes known as inertial reference frames), a review of several modern textbooks reveals that physicists have fallen into the habit of giving seriously deficient definitions of inertial coordinate systems. Every text book that I checked presents essentially the same definition, claiming that a necessary and sufficient condition for a reference frame to be inertial is simply that it is unaccelerated. A coordinate system is inertial, they say, if and only if, with respect to that system, every object not subject to an external force moves at uniform speed in a straight line. As explained above, this is false. Satisfaction of Newton's first law of motion is not sufficient to define an inertial coordinate system. Sometimes these texts invoke the second law instead of the first, but they are used for the same purpose, i.e., simply to establish that the reference frame is unaccelerated. The formal definition of inertial reference frames given in every one of these sources fails to require that the third law be satisfied, despite the fundamental importance of this requirement. The sources do, of course, acknowledge that all three of Newton's laws are satisfied with respect to inertial coordinate systems, but they inexplicably assume that satisfaction of the first two laws ensures satisfaction of the third, which is certainly false.
As a typical example of the (deficient) definition of inertial reference frames, here is how the standard college text, Physics, by Halliday and Resnik defines them:
...it is possible to find a family of reference frames in which a particle [free of applied forces] has no acceleration. The fact that bodies stay at rest or retain their uniform linear motion in the absence of applied forces is often described by assigning a property to matter called inertia. Newton's first law is often called the law of inertia and the reference frames to which it applies are called inertial frames.
Later they say
...an inertial frame... is a reference frame that is either at rest or is moving at constant velocity with respect to the average positions of the fixed stars; it is the set of reference frames defined by Newton's first law, namely, that set of frames in which a body will not be accelerated if there are no identifiable force-producing bodies in its environment.
By the way, in addition to the deficiency of this definition due to its failure to invoke the third law, we should mention that the phrase "observations made from an inertial frame" is pedagogically awful. Such statements have no real meaning, and serve only to mislead and confuse students, because everything is "in" an inertial frame. In fact, everything is in infinitely many inertial frames, so to talk about making observations "from" an inertial frame is sloppy at best, and at worst it's indicative of a real lack of clarity in understanding. The same type of awful terminology appears throughout the literature, even in books on the theory of relativity, where it is crucially important to be clear and precise about the meaning of statements made in terms of specific systems of reference.
In any case, Halliday and Resnik are not alone in presenting an erroneous definition inertial reference frames. To substantiate the claim that the failure to invoke Newton's third law in the definition of inertial reference frames is widespread, the following is summary of the definitions given in several well-known texts:
A reference frame is said to be inertial when... every test particle that is initially at rest, and every test particle that is initially in motion, continues that motion without change in speed or in direction. [Spacetime Physics, Taylor and Wheeler]
...in order not to introduce effects due to the acceleration of the observer, we must take care to apply [the second law] in a frame that is itself unaccelerated. We refer to these as inertial frames. In many situations, one can often effectively assume that an inertial frame of reference is one at rest with respect to the earth. [Philosophical Concepts in Physics, Cushing]
...let us assume that we have found an inertial reference frame, and therefore that Newton's laws apply for motions relative to this frame. It can be shown that any other reference frame that is not rotating but is translating with uniform velocity relative to an inertial frame is itself an inertial frame... For example, if system B is translating with constant velocity with respect to an inertial system A, then... observers on systems A and B see identical forces, masses, and accelerations, and therefore [the second law] is equally valid for each observer. [Principles of Dynamics, Greenwood]
In order to fix an event in space, an observer may choose a convenient origin in space together with a set of three Cartesian coordinate axes. We shall refer to an observer's clock, ruler, and coordinate axes as a frame of reference... there exists a privileged set of bodies, namely those not acted on by forces. The frame of reference of a co-moving observer is called an inertial frame. [Introducing Einstein's Relativity, D'Inverno]
The reference frame attached to a [free-falling] spacecraft simulates an inertial reference frame: a test particle at rest relative to the spacecraft remains at rest, a test particle in motion remains in motion with uniform velocity. [Gravitation and Spacetime, Ohanian and Ruffini]
Inertial reference frame, defined by uniform velocity of free test particles... [Gravitation, Misner, Thorne, and Wheeler]
Newton's first law serves as a test to single out inertial frames among rigid frames: a rigid frame is called inertial is free particles move without acceleration relative to it. [Essential Relativity, Rindler]
Perhaps not surprisingly, some of these definitions seem to have been carried over from one text to another. For example, Taylor and Wheeler introduce their formal definition (quoted above) by discussing at length a spaceship in free-fall. They say "we call such a space ship that rises and falls freely an inertial reference frame...", and then they go on to talk about the motions of "test particles", terms and images that reappear almost verbatim in Ohanian and Ruffini. (The first endorsement on the book cover of the latter is from Wheeler, who wryly comments that it is the best gravitation book on the market "of 500 pages or less". Wheeler's own Gravitation is over 1200 pages.) To their credit, Ohanian and Ruffini do refrain from repeating the ridiculous statement that a space ship is an inertial reference frame (is it any wonder that students presented with such statements becomes confused, and begin to talk in terms of measurements performed in an inertial reference frame?), but they carry over the fundamentally deficient definition, failing to ever mention the necessity of imposing Newton's third law in order to give a complete definition of inertial coordinates, and never acknowledging the crucial fact that this represents the imposition of a definite operational simultaneity. This illustrates how difficult it is - even professional scientists who, on some level, know better - to free our minds from the Galilean assumption of absolute simultaneity.