What is an Inertial Coordinate System?

Arguably the most single important insight leading to the development of modern science was the recognition of a class of spatio-temporal coordinate systems in terms of which the motions of bodies satisfy (to high precision) 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 understanding that quantities involving position and time are referred to one of these special systems of coordinates. This is, of course, a necessary restriction, because the laws are obviously not valid with respect to arbitrary coordinate systems. However, there is an almost universal confusion in the literature (at all levels) as to the necessary and sufficient conditions for the laws of mechanics to be satisfied with respect to a given system of coordinates. This confusion is exemplified by the following discussion from E. T. Whittaker’s historical introduction to relativity:

According to Newton's First Law of Motion, any particle which is free from the action of impressed forces moves, if it moves at all, with uniform velocity in a straight line…We can assert as a fact of experience that certain systems of axes exist such that free particles move in straight lines with reference to them. Moreover, we can assert that there exist certain ways of measuring time such that the velocity of free particles along their rectilinear paths is uniform. A set of axes in space and a system of time-measurement, which possess these properties, may be called an inertial system of reference…. In Newtonian mechanics, if S is an inertial system of reference, and if S' is another system such that the axes of S' have any uniform motion of pure translation with respect to the axes of S, and if the system of time-measurement is the same in the two cases, then S' is also an inertial System of reference:  the Newtonian laws of motion are valid with respect to S' just as with respect to S.

The problem here, as in almost the entire literature on this subject, is the invalid extrapolation from Newton’s first law to all of Newton’s laws. Whittaker correctly notes that the first law is satisfied only in terms of space and time coordinates that are not accelerating, but then he jumps to the assertion that the Newtonian laws (plural) of motion are valid with respect to these same systems – an assertion that is demonstratably false. Compatibility with Newton’s first law is not sufficient to ensure compatibility with the second and third laws. To cement the confusion, the phrase “inertial coordinate system” is commonly used interchangeably to refer to (1) coordinate systems compatible with Newton’s first law, and (2) coordinates systems compatible with all three of Newton’s laws, despite the fact that the latter are only a subset of the former. There really ought to be two different terms for these distinct sets of coordinate systems, but unfortunately a single term is used for both.

A further source of confusion when attempting to unravel the overlapping definitions is due to the fact that Newton’s second and third laws, in their usual formulations, entail not just the essential symmetries of inertia but also, implicitly, the assumption that relatively moving systems of fully symmetrical coordinate systems are related by Galilean transformations, an assumption now known to be false. Therefore, we need to extract the essence of Newton’s three laws and separate them from assumptions about coordinate transformations. The factual essence of the Newtonian and Galilean concept of inertia is that there exists a system of space and time coordinates in terms of which mechanical inertial is homogeneous and isotropic. Homogeneity corresponds to the fact that free objects move uniformly in straight lines (i.e., Newton’s first law), and isotropy corresponds to the fact that two identical resting objects acting against each other acquire equal and opposite speeds in equal times (representing the factual content of Newton’s second and third laws). By rights such coordinate systems deserve the name “inertial”, because they are the unique coordinate systems in terms of which inertia is maximally symmetrical, but unfortunately the word “inertial” carries connotations from its use as an adjective for material objects. A material object possesses only an individual world-line within spacetime, and the object may be called “inertial” if its space coordinates are linear functions of the time coordinate. In other words, an object is inertial if and only if it is unaccelerated. In contrast, a system of coordinates is much more extensive than a single worldline, and it is not fully specified merely by requiring the absence of acceleration. Coordinate systems can be defined in infinitely many ways, but if we are to define our coordinate system in terms of inertia, we need to stipulate not only the homogeneity of inertia, but also the isotropy of inertia. Without both of these requirements, the coordinate system is under-specified. Hence the term “inertial” when applied to coordinate systems means more than just that free objects are unaccelerated, it means that the resistance to acceleration of a resting body is the same in all spatial directions.

Despite this, the most common definition of the term “inertial coordinate system” appearing in the literature is simply a coordinate system compatible with Newton’s first law. It is, of course, permissible to define inertial coordinate systems in this way, but then we recognize that (1) the systems so defined are not fully specified, and (2) inertia is not generally isotropic in terms of coordinate systems that satisfy only this partial definition. Unfortunately, almost every published reference on the subject (at all levels) not only defines inertial coordinate systems purely in terms of Newton’s first law, they then immediately assume these coordinate systems are thereby fully defined (which they aren’t) and that Newton’s second and third laws hold good in terms of these coordinates (which they don’t).

In order to fully specify the class of coordinate systems in terms of which the laws of mechanics (corresponding to Newton’s three laws of motion) are valid, we must not only stipulate that the coordinates are defined such that free particles move at uniform speed in straight lines (i.e., Newton’s first law) , we must also impose a suitable definition of simultaneity at spatially separate points. Without this, Newton’s third law is not generally satisfied (even approximately). This is the crucial aspect of the classical definition of what we may call “Newtonian” coordinate systems that is often overlooked. To understand the significance of this condition, consider the figure below, which shows 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 a Newtonian coordinate system, then Newton's first law (and arguably his second law) is satisfied with respect to each of these three coordinate systems. However, Newton's third law is not satisfied with respect to all of these coordinate systems. 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 resting 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 reference frames and special relativity. The coordinate systems in terms of which the resistance to acceleration of a resting object is the same in all directions are not fully specified by the requirement to be unaccelerated, i.e., by the requirement to satisfy Newton's first law. The space and time components of a “Newtonian” inertial coordinate system must satisfy all three of Newton's laws, and 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 Newtonian 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 single 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, and then going on to claim that all of Newton’s laws (or their relativistic counterparts) are valid in terms of these coordinate systems. A coordinate system is compatible with Newton’s laws, 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 – if, by “inertial coordinate system” we mean a system of space and time coordinates in terms of which the laws of mechanics are valid. 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. Needless to say, we can define terms any way we like, and it would be permissible to define the phrase “inertial coordinate system” as simply a characterization of a large class of coordinate systems, a small subset of which are compatible with the laws of mechanics, but this cannot be what the text books are doing, because they immediately assert that their definition based solely on Newton’s first law is sufficient to ensure compatibility with all the laws of mechanics, which is clearly false. So either they are defining the term “inertial coordinate system” incorrectly, or they are incorrectly describing the properties of those systems. It seems most reasonable to reserve the expression “inertial coordinate system” to those systems of space and time coordinates in terms of which inertia is homogeneous and isotropic, because this is sufficient to unambiguously define a unique reference frame for each state of motion. On this basis, the definition of inertial coordinate systems given in all existing modern text books (at least all I have seen) is wrong. (I restrict this to “modern” texts, because clearly Galileo, Newton, and the other 17th century originators of modern physics understood the need for inertial isotropy, but this understanding seems to have been lost in the intervening centuries. If it had not been forgotten, scientists would never have regarded the concept of an operational definition of simultaneity to be novel when it was re-introduced by Poincare and Einstein.)

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.  ...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.

As mentioned previously, it is permissible to define “inertial frame” in this limited sense, but unfortunately Halliday and Resnik go on to assert that all three of Newton’s laws are (approximately) valid in terms of inertial frames, an assertion that would be true only if we define “inertial frame” in the full sense, stipulating isotropy as well as homogeneity. Of course, this reduces all of Newton’s laws to tautologies, albeit extraordinarily useful ones. They essentially amount to the definition of Newtonian inertial coordinate systems, and the usefulness of this definition arises from the fact that such coordinate systems actually exist.

By the way, in addition to the deficiency of Halliday and Resnik’s 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]

This list could be extended indefinitely. Essentially every text book perpetuates this fundamental error, which consists of defining inertial coordinate systems (and frames) only using Newton’s first law, but then subsequently assuming inertia is isotropic in terms of those coordinate systems – which is generally false. It was the failure to appreciate the operational definition of simultaneity (based on mechanical inertia) implicit in Newtonian physics that caused scientists to regard as novel Poincare’s concept of an operational definition of simultaneity. They had forgotten that simultaneity in Newtonian mechanics had always been operationally based on the isotropy of inertia, which is contained in the full definition of inertial coordinate systems.

Perhaps not surprisingly, some of the modern textbook 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.

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