It may seem surprising that, after this insightful analysis of the concept of inertial frame and its role in electrodynamics, Einstein should have turned almost immediately to call that concept into question. But he had a compelling combination of physical and philosophical motives to do so. On the physical side, he realized (along with many others) that special relativity would require some fundamental revision of the Newtonian theory of gravity. On the philosophical side, he became convinced, largely by his reading of Mach (1883), that the central role of inertial frames was an “epistemological defect” that special relativity shared with Newtonian mechanics. (Einstein 1916, pp. 112–113.) Only relative motions are observable, yet both of these theories purport to identify a privileged state of motion and use it to explain observable effects (such as centrifugal forces). Coordinate systems are not observable, yet both of these theories assign a fundamental physical role to certain kinds of coordinate system, namely, the inertial systems. In either theory, inertial coordinates are distinguished from all others, and the laws of physics are said to hold only relative to inertial coordinate systems. In an epistemologically sophisticated theory, both of these problems would be solved at once: the new theory would only refer to what is observable, which is relative motion; it would admit arbitrary coordinate systems, instead of confining itself to a special class of system. Why, after all, should any genuine physical phenomenon depend on the choice of coordinate system?
Another way of putting the same point is to say that, in Newtonian mechanics and special relativity, rotation is “absolute” because the transformations between inertial frames (Galilean or Lorentzian) preserve rotational states. Thus the “absoluteness” of rotation arises precisely from singling out one type of frame, by one type of transformation, instead of allowing arbitrary transformations and arbitrary frames. Einstein held that this epistemological insight had a natural mathematical representation in the principle of general covariance, or the principle that the laws of nature are to be invariant under arbitrary coordinate transformations. More precisely, what this means is that coordinate transformations are no longer required (as in the affine spaces of Newtonian mechanics and special relativity) to take straight lines to straight lines, but only to preserve the smoothness of curves (i.e. their differentiability). The general theory of relativity was intended to be a generally covariant account of spacetime, and its general covariance was intended to express the general relativity of motion. And the theory came into being because Einstein perceived a deep connection between this project and that of finding a relativistic theory of gravitation.
2.6 The Equivalence of Inertia and Gravity
The philosophical motivations and implications of Einstein's view are dealt with elsewhere. (See, for example, the entries on Einstein's philosophy of science; the hole argument; and early philosophical interpretations of general relativity.) We will consider here only the bearing of general relativity on the notion of an inertial frame. It is questionable whether Einstein succeeded in establishing the general relativity of motion, but it is clear that general relativity undermines the concept of inertial frame in important respects. This arises from the equivalence principle: that inertial mass—the quantity that enters into Newton's second law, and that is a measure of a body's resistance to acceleration—is equivalent to gravitational mass, the quantity that enters into Newton's law of universal gravitation. A more empirical way of expressing it is that all bodies fall with the same acceleration in the same gravitational field, or, the trajectory of a body in a given gravitational field will be independent of its mass and composition. This is the principle that Newton tested by constructing pendulums with wooden boxes as their bobs, which he would fill with different materials in order to see whether those differences made a difference to the speed of falling; they didn't. Eötvös made more precise tests in the late 19th century, and established the principle to much greater accuracy; these are the results on which Einstein would have relied. Newton also tested the principle for bodies whose masses differ greatly, by observing that Jupiter and its four moons all received precisely the same acceleration from the sun's gravitational field.
The equivalence principle suggests, however, that a freely-falling frame of reference is physically indistinguishable from an inertial frame. Newton had already noticed this, and indeed he stated it, more or less, in Corollary VI to the laws of motion:
If bodies are moving in any way whatsoever with respect to one another and are urged by equal accelerative forces along parallel lines, they will all continue to move with respect to one another in the same way as they would if they were not acted on by those forces. (1726, p. 423.)
words have meaning.