1.3 Inertia and Relativity

In his treatise "On the Revolution of Heavenly Spheres" Copernicus argued for the conceivability of a moving Earth, noting that

Similar to the ideas of Aristarchus, this is a purely kinematical conception of relativity, based on the simple proposition that we judge the positions (and changes in position) of objects only in relation to the positions of other objects. Even in the time of Copernicus, most scholars still believed they had irrefutable reasons for rejecting this idea. Nevertheless, the relativistic model attracted supporters, including Kepler and Galileo, during the century following Copernicus' death. It was Galileo who re-interpreted the principle of relativity in a more profound sense based on the principle of inertia, i.e., the proposition that inertial phenomena are isotropic with respect to any uniformly moving frame of reference. To illustrate this concept, he considered the behavior of objects inside a ship moving at some arbitrary speed, and pointed out that

Galileo's approach based on dynamical rather than merely kinematic analysis soon led to the modern principle of inertial relativity, although Galileo himself never achieved a full and clear criterion for distinguishing between accelerated and unaccelerated motion. He believed, for example, that circular motion was a natural state that would persist unless acted upon by some external agent. This shows that the resolution of dynamical behavior into inertial and non-inertial components (which we generally take for granted today) is more subtle than it may appear. There is an inherent circularity in the concept of inertia, because it essentially tells us that objects move inertially except when subject to external (non-inertial) influences. In other words, objects always behave inertially - except when they don't. The concept of inertial motion is a heuristic tool to help in organizing and analyzing our observations, with the aim of isolating all identifiable interactions between objects that may affect their motions. When all those interactions have been removed, what remains is called inertial motion. The difficulty for Copernicus and Galileo was that the gravitational interaction between bodies (such as between the Earth and the Moon) had not been clearly recognized, so the (roughly) circular motions of celestial objects could only be regarded as "inertial".

The first explicit statement of the modern principle of relativity and inertia was apparently made by Pierre Gassendi, who is most often remembered today for reviving the ancient Greek doctrine of atomism. In the 1630's Gassendi repeated many of Galileo's experiments with motion, and interpreted them from a more abstract point of view, consciously separating out gravity as an external influence, and recognizing that the remaining "natural states of motions" were characterized not only by uniform speeds (as Galileo had said) but also by rectilinear paths. Of course, the circularity in the meaning of inertia was still present, because the very ideas of uniform speed and linear paths are ultimately derived from observations of inertial motion. The task was to review the whole range of observable motions, focus on those that represent the movements of material objects (as opposed to changes in the direction of vision, for example), conceptually remove the effects of all known external influences, and then from this resulting set of ideal states of motion to identify the largest possible "equivalence class" of relatively uniform and rectilinear motions. These ideal motions and configurations then constitute the basis for inertial measurements of space and time, i.e., inertial coordinate systems. Naturally inertial motions will then necessarily be uniform and rectilinear with respect to these coordinate systems.

Shortly thereafter (1644), Descartes presented essentially the same idea in his "Principles of Philosophy":

Similarly, in Huygens' "The Motion of Colliding Bodies" (composed in the mid 1650's but not published until 1703), the first hypothesis was that

Ultimately Isaac Newton incorporated this principle into his masterpiece, "Philosophiae Naturalis Principia Mathematica" (The Mathematical Principles of Natural Philosophy), as "the first law of motion"

This expresses the modern principle of relativity, asserting equivalence between the conditions of "rest" and "uniform motion in a right line". In addition, since no distinction is made between the various possible directions of uniform motion, the principle also implies the equivalence of uniform motion in all directions in space. Thus, if everything in the universe is a "body" in the sense of this law, and if we stipulate rules of force (such as Newton's second and third laws) that likewise do not distinguish between bodies at rest and bodies in uniform motion, then we arrive at a complete system of dynamics in which, as Newton said, "absolute rest cannot be determined from the positions of bodies in our regions".

Beginning with Galileo's inertial interpretation of motion, the relativistic idea came to be recognized as a tremendously powerful and simplifying principle for understanding the workings of Nature. In their efforts to discover the detailed rules of dynamics, Galileo and his successors realized the need to express those rules in the form of mathematical equations involving the spatial positions and velocities of objects at different times, but as believers in the Copernican theory that the Earth itself is in constant motion, they also knew that the spatial positions and states of motion of objects relative to the Earth's surface cannot be regarded as the "true" positions and velocities of those objects. In fact, they knew that we have no way of determining the "true" state of motion of any object, so if the laws of physics could be coherently formulated only in terms of the "true" positions and velocities, we would face an apparently impossible task. Fortunately the principle of inertial relativity implies that the same physical laws of motion must apply when the positions and velocities of objects are evaluated with respect to any uniformly moving inertial reference system. Einstein expressed this principle in 1905 as follows

(It's unfortunate that Einstein omitted the crucial word "inertial" in this statement, but the meaning is clear from the context, since he refers to systems of coordinates in terms of which Newton's laws hold good - to the first approximation.) However, the principle of relativity does not tell us how the descriptions with respect to one such system of reference relate to the descriptions with respect to another. It merely tells us that the same "laws" will be satisfied in both systems. Of course, for this statement to be meaningful, it's necessary to place constraints on the form of the "laws", because a single set of non-relativistic laws can always be made applicable to an infinite class of coordinate systems simply by allowing the laws to contain an "absolute velocity" parameter v. A more rigorous statement of the relativity principle would stipulate that no such "absolute velocity" parameter appears in the statement of the laws, and yet the same laws are applicable to physical phenomena when expressed in terms of any member of a suitable class of relatively moving reference systems.

It's interesting that all the originators of our classical mechanics (including Einstein in 1905) approached the subject from this same direction, i.e., they tacitly pre-supposed a common system of reference with respect to which the common words such as "uniform" and "straight" were assumed to have unambiguous meanings. In these terms they expressed the law of inertia by saying that material objects naturally move at uniform speed in a straight line unless acted upon by some other objects. It would probably not have been helpful, in those early stages, to dwell on the fact that (as noted above) our intuitive concepts of uniform speed and straight paths are ultimately derived from our observations of natural unforced motions and configurations of objects. In retrospect we can see that Newton's laws of motion are not only defined in terms of inertial coordinate systems, they also effectively serve to define inertial coordinate systems. From this point of view, we might re-phrase the law of inertia as the proposition that for any material object in any state of motion there exists a system of space and time coordinates in terms of which the object is momentarily at rest and Newton's laws are (at least quasi-statically) valid. In place of Newton's laws, it would be sufficient to specify isotropic inertia, i.e., to require that two identical objects, initially at rest with respect to the coordinates and exerting a mutual force on each other, recoil by equal distances in equal times.

Of course, this is precisely what Galileo stated in his illustration of inertial motion on-board a moving ship, and it is crucial to understanding how inertial coordinates are constructed. It might seem that we could fully specify a spacetime coordinate system simply by the requirement for a particular object to be at rest with respect to that system, but in fact that only suffices to determine the direction of the "time axis", i.e., the loci of constant spatial position. Galileo and his successors realized (though not always explicitly) that it is also necessary to specify the loci of constant temporal position, and this is achieved by imposing the assumption of inertial isotropy. When Galileo wrote that an object thrown with equal force will reach equal distances [in the same time], he was proposing the principle of inertial isotropy as the basis for defining simultaneity at separate locations.

Einstein's theory of special relativity effectively uses this same definition of simultaneity based on inertial isotropy. In special relativity, just as in Newtonian physics, an inertial coordinate system is defined as a system of space and time coordinates in terms of which free material objects move uniformly in straight lines, and such that the inertia of material objects at rest is the same in all spatial directions. Equivalently, inertial coordinate systems are those in terms of which Newton's laws of motion are quasi-statically valid. Based on this definition, Galileo's fundamental principle of relativity, adopted essentially unchanged by Einstein, can be expressed as follows:

(G1) For any massive body, in any state of motion, there exists a system of space and time coordinates (called "inertial coordinates") with respect to which that body is instantaneously at rest, free objects move uniformly in straight lines, and inertia is the same in all spatial directions.

To be clear, the statement that "inertia is the same in all spatial directions" means that the inertia of objects does not depend on any absolute reference direction (although it may depend on the velocity of the objects). In particular, this independence implies that for two identical bodies initially at rest with respect to a system of inertial coordinates, the accelerations imparted to those two bodies (in opposite directions) by another stationary body maintaining its state of rest will be identical and independent of the direction of their axis of motion. This particular case is sufficient to establish the operational content of the relativity principle G1.

We note that G1 asserts the existence of an inertial coordinate system corresponding to each material object, which is all that is strictly required from an operational point of view. However, for completeness, we can further assume that the principle applies not just to the paths of actual massive bodies, but to any timelike worldline that could be the path of a massive body. For convenience we will sometimes refer to an inertial coordinate system as a "frame of reference", or simply a "frame".

For the purpose of characterizing the mutual dynamical states of two material bodies, the associated inertial rest frames are actually more meaningful than the mere distance between the bodies, because an inertial coordinate system possesses a fixed spatial orientation with respect to any other inertial coordinate system, enabling us to take account of tangential motion between bodies whose mutual distance is not changing. For this reason, the physically meaningful "relative velocity of two material bodies" is best defined as their reciprocal states of motion with respect to each others' associated inertial rest coordinate systems.

To conceive of an operational procedure for actually establishing a complete system of space and time coordinates based on inertial isotropy, imagine that at each point in space there is an identically constructed cannon, and all these cannons are at rest with respect to each other. At one particular point, which we designate as the origin of our coordinates, there is a clock and numerous identical cannons, each pointed at one of the cannons out in space. The cannons are fired from the origin, and when a cannonball passes one of the external cannons it triggers that external cannon to fire a reply back to the origin. Each cannonball has identifying marks so the observer at the origin can correlate each reply with the shot that triggered it, and with the identity of the replying cannon. Then the ith reply event is assigned the time coordinate t_i = [t_return(i) - t_send(i)]/2 seconds, and it is assigned space coordinates x_i, y_i, z_i based on the angular direction of the sending cannon and the radial distance r_i = t_i cannon-seconds. This procedure would have been perfectly intelligible to Newton, and he would have agreed that it yields an inertial coordinate system, suitable for the application of his three laws of motion.

Needless to say, inertial isotropy is not the only possible basis for constructing a spacetime coordinate system. We could impose a different constraint to determine the loci of constant temporal position, such as a total temporal ordering of events. However, if we do this, we will find that mechanical inertia is generally not isotropic according to our coordinate systems, resulting in a physics in which Newton's laws of motion are not generally valid - at least not if restricted to ponderable matter. Indeed this was the case for the ether theories developed in the late 19th century, as discussed in subsequent sections. Such coordinate systems would be highly unintuitive in most circumstances, but not logically inconsistent. The choices we make to specify a coordinate system and to resolve spacetime intervals into separate spatial and temporal components are largely conventional. The real physical content of the structure of space and time is expressed not by any single coordinate system, but by the quantities that are invariant with respect to different coordinate systems, and by how the measures of intervals with respect to two different coordinate systems are related to each other. In order for the results of our measurements to have any meaning for systems of reference other than the particular one in which they were made, we need, in addition to the principle of relativity, knowledge of how to translate the coordinates of moving events from one system of reference to another. Based on experience over a very small range of relative velocities, Galileo (tacitly) adopted as his second principle

(G2) If a material object B is moving at the speed v with respect to the inertial rest frame coordinates of A, and if an object C is moving in the same direction at the speed u with respect to the inertial rest frame coordinates of B, then C is moving at the speed v + u with respect to the inertial rest frame coordinates of A.

This assumption may seem plausible, but it's important to realize that we are not free to arbitrarily adopt this - or any other - condition on the set of inertial coordinate systems, because they are already fully defined (up to insignificant scale factors) by the requirements for free motion to be linear, for inertia to be isotropic, and for every material object to be at rest with respect to one of these coordinate systems. These three properties suffice to determine the set of inertial coordinate systems and the relationsips between them. Given these conditions, the relationship between relatively moving inertial coordinate systems, whatever it may be, is an empirical fact. It turns out that the assumption of additive composition of speed is approximately valid if the velocities involved are not too great, but as the relative velocities increase, the assumption of simple additivity breaks down. Alternatively, if we take G2 as fundamental, then G1 breaks down. In other words, if we define our coordinate systems so as to satisfy G2, then empirically we find that they do not satisfy G1.

Historically the realization that G1 and G2 are empirically incompatible was prompted mainly by considerations of light, or, more generally, electromagnetic phenomena. The physical principles described above refer to material objects, so it was not clear, a priori, whether they apply to the phenomenon of light. Is the propagation of light necessarily isotropic with respect to a system of coordinates in which mechanical inertia is isotropic? Is light to be considered a "material object" in the sense of the principle of inertia, i.e., does there exist a system of inertially isotropic coordinates in terms of which a pulse of light is at rest? We consider these questions in the next section.