This is part of my Objectivity essay “Space, Rotation, Relativity” written over the years 1995–98. The endnotes are here omitted. I will first list the sections of the essay that preceded the sections on electromagnetism and special relativity. The entire essay is available at the Objectivity Archive .
V. Kant – Precritical Period
VI. Kant – Critical Period
VII. Galilean Invariance
Consider, as had Galileo, “an observer in the main cabin below decks of a large ship. Everything—flying insects, incense smoke, dripping water—will act the same whether the ship moves uniformly or is at rest. With no view to the outside and no acceleration [of the ship], such an observer could not tell whether he, his cabin, and his ship move or remain at rest” (Jetton 1991, 89-90). Similarly, in Newton’s mechanics, “the motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. . . . A clear proof of this we have from the experiment of a ship; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line” (P 20-21, Cor. V).
We have seen also Huygen’s requirement that a valid law of impact in some frame of reference S must hold also in any other frame S' moving with uniform velocity v with respect to S (sup §II, 27). A law of physics—to fully merit the name law—should obtain in any inertial frame of reference. Our choice of which inertial frame is taken as at rest, like our choice of coordinate system (e.g., rectangular or polar, oriented this way or that) seems subjective, purely a matter of our convenience. What is really out there in the physics should be so regardless of our choice of inertial frame. A fundamental physical law should be the same in every inertial frame. Before Einstein the invariance of law in different inertial frames was not much talked about, but it was probably presumed to be so by every physicist, especially after Newton.
To determine whether a putative law obtains in all inertial frames, we write the law in coordinates (say, rectangular coordinates) of the rest frame and see what becomes of the law when expressed in coordinates (again rectangular, say) of the moving frame. To do this, all we need to know is (i) how coordinates specifying locations and times in an (approximately) inertial frame S taken as at rest translate into coordinates specifying locations and times in a frame S' moving uniformly with respect to S, (ii) how the components of velocities translate between these frames (not the relative velocity v between the frames; v is of the same magnitude, and is constant, in either frame) and (iii) how components of accelerations translate between the frames. Such transformations are given purely by kinematics. Until the relativity work of Lorentz, Poincaré, and Einstein in the late-nineteenth and early-twentieth centuries, the translation, across inertial frames, of locations and times, velocities, and accelerations was assumed by everyone to be that translation which today we call the Galilean transformation.
Are Newton’s laws of mechanics invariant in their form under a Galilean transformation across inertial frames? Yes. Is Newton’s law of gravitation invariant in its form under such a transformation? Yes. What about Hooke’s law (1676) of elasticity? Yes.
Are the laws of free fall, discovered empirically by Galileo (c. 1590), invariant under the Galilean transformation across inertial frames? Consider a marble rolling off the edge of a table and falling to the floor in an elevator. Compare this fall in an elevator at rest with the same fall in an elevator moving uniformly downward at a uniform speed v. Yes, the equations for free fall are invariant under the Galilean transformation across these two inertial frames. We have no laws of dynamics under mathematical test here. The equations of free fall are kinematical, with the gravitational acceleration g providing an instance of constant acceleration. In showing that the equations for free fall are invariant under Galilean transformation across inertial frames, we are exhibiting the self-consistency of Galilean kinematics and, to some extent, the empirical validity of that kinematics.
Since the nineteenth-century refinement of the concept of energy and verification of its conservation, we also assert of the free fall of the marble in the elevator that the marble’s change in kinetic energy is balanced by its change in potential energy. Is the equation expressing this balance invariant under the Galilean transformation across the two elevator frames (one at rest, the other moving uniformly downward)? Yes.
Euler’s (1755) equation for flow (level flow, possibly unsteady, but having no appreciable viscosity) of a nearly incompressible fluid, such as water in a pipe (the pipe being approximately inertial in its state of motion), says that the pressure gradient at a fixed point in the stream (fixed relative to the pipe) will be balanced by the time rate of change of the flow-field velocity at that point plus the convective rate of change of the flow-field velocity at that point. Is this, Euler’s fundamental equation of hydrodynamics, invariant under Galilean transformations across inertial frames? Yes. That is what we might have expected since Euler’s equation is based on Newton’s second law, which we already know to be invariant under the transformation.
Euler’s flow equation for hydrodynamics was a first-order, though nonlinear, partial differential equation. The partial differential equation known as the wave equation is second order and is linear. The wave equation was formulated and initially explored in the eighteenth century by Daniel Bernoulli, d’Alembert, Euler, and LaGrange, primarily in analyses of the vibrations (especially musical vibrations) of sound sources (such as strings, membranes, and plates) and in analyses of the subsequent propagation of sounds through the air. Traveling sound waves are waves of compression (and complementary rarefaction) of a medium, compressive waves traveling through that medium. A sound wave is a longitudinal wave; the orientation of the compressions is along the direction the compressive wave is propagating. Gases, liquids, and solids are all media that can support compressive waves. That statement coincides with statement of the familiar fact that sound can be propagated through all those media. Only solids can support waves that are not compressive: shearing waves and torsional waves. Shearing and torsional waves in an elastic solid became important for nineteenth-century modeling of the luminiferous ether. (A medium is elastic insofar as it deforms under the action of a force and recovers its original shape when the force is removed.) Shearing and torsional waves are transverse waves; the orientations of the stresses are perpendicular the direction the stress wave is propagating.
The wave equation says, for an ideally continuous medium in which mechanical waves are propagating, that the gradient of the elastic strain in the disturbed medium is (i) directly proportional to the time rate of change of the time rate of change of the material displacements from equilibrium positions and (ii) inversely proportional to the square of the velocity of the waveform through the medium. This equation is derived from Newton’s second law, the force being specified by Hooke’s law.
Is the wave equation invariant under Galilean transformations across inertial frames? No.
That is moderately disturbing. If we strike a tuning fork (c. 1750) in the cabin below decks of Galileo’s ship, surely what goes on inside the metal constituting the fork is the same whether the ship rests or moves. Indeed, surely we should hear the fork give the same note to the air whether the ship rests or moves. (In either case, ship resting or moving, we are here taking the listener to be approximately at rest relative [to the cabin and thence] to the air in the cabin.)
Although the preceding gedanken might incline one to think the wave equation should be invariant across inertial frames, there is a fairly ordinary experience (available prior to the technological advances that made Doppler’s  frequency effects a part of ordinary experience) that might lead one to expect variance: high winds can throw our voices. A given volume of one’s voice will reach farther downwind than upwind. That is, intensities of the sound waves will be attenuated to an inaudible level farther from the source, if the listener is downwind of the source; nearer, if upwind. That suggests the sound waves are drifting somewhat with the wind. Then, since a wind blowing on a stationary observer can be recast as a scene in which an observer (a material, or at least geometrical, reference frame) moves through still air in the upwind direction, we have a reason for thinking that the wave equation should not transform invariantly across moving frames, even frames moving uniformly. One might then expect what we do find in transforming the wave equation under the Galilean transformation across inertial frames: new terms arise in the wave equation, terms that are functions of the relative speed between the frames (speed of the wind), and these terms . . . [see original text in V2N6 with endnote 7].
The failure of Galilean invariance for the wave equation gives us our first doubt about the correctness of Galilean kinematics. Perhaps a new family of transformations of locations, times, and accelerations can succeed in transforming the wave equation invariantly across inertial frames (while holding onto all the successes of the Galilean transformations). Such a new family was formulated as early as 1887, though its full profundity would not be perceived until Einstein’s special theory of relativity (1905).
We should observe, in anticipation of general relativity, that we might be mistaken about the class of frames across which fundamental physical laws should take the same form. The proper equivalence class might be wider than the equivalence class: inertial frames. The proper class might be, for example: inertial frames plus any frames in uniform but nonrotating accelerations relative to one another. Throughout the era of classical mechanics, then classical electrodynamics, then even through Einstein’s special relativity, the proper equivalence class was assumed to be: inertial frames as defined in Newton’s mechanics, that is, frames moving at most with constant speed in a straight line in Euclidean space.
(to be continued)
Jetton, M. 1991. Imagination and Cognition. Objectivity 1(3):57–92.
Newton, I. 1934 [1687/1713/1729]. Mathematical Principles of Natural Philosophy. U CA.