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Electromagnetism to Special RelativitySubmitted by Stephen Boydstun on Tue, 20080902 11:50
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.
I. Descartes II. Huygens III. Newton IV. Leibniz 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, 8990). 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 2021, 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 latenineteenth and earlytwentieth 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 selfconsistency of Galilean kinematics and, to some extent, the empirical validity of that kinematics.
Since the nineteenthcentury 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 flowfield velocity at that point plus the convective rate of change of the flowfield 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 firstorder, 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 nineteenthcentury 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 [1842] 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)
References Jetton, M. 1991. Imagination and Cognition. Objectivity 1(3):57–92. Newton, I. 1934 [1687/1713/1729]. Mathematical Principles of Natural Philosophy. U CA.
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New Test
Test over cosmic distance of the invariance of light speed with increasing energy upholds special relativity. This test also provides a new constraint for theories of quantum gravity, which is to be desired.
GammaRay Observations Shrink Known Grain Size of Spacetime
Ron Cowen – Science News (11/21/09)
Testing Einstein’s Special Relativity with Fermi’s Short Hard GammaRay Burst
Sylvain Guiriec (8/09)
Einstein  SR  Dynamics
XII. Einstein – Special Relativity – Dynamics
Continuing in his 1905a paper, Einstein applied the Lorentz transformations for coordinates, which transformations he had deduced purely from his new postulates for kinematics, to Ampère’s corrected law and to Faraday’s law (in their field formulations by Maxwell) for the case of a region of space free of electric charges and currents (but, of course, not free of electric and magnetic fields). Thence he deduced how the components of electric and magnetic fields are transformed across inertial frames. Notably, components of the electric (or magnetic) field transverse the direction of frame motion are in each frame composed of what are transverse components of the electric and magnetic fields in the other frame (1905a, §6).
Having the transformation equations for the components of electromagnetic fields and having the transformation equations for velocities, Einstein returned to the laws of Ampère and Faraday, this time for the case in which electric charges and currents are also present in the region of space under consideration. Remember that current is charge in motion; hence the relevance of velocity transformations. Transforming the fields and velocities in those laws, Einstein demonstrated that with his “kinematic principles taken as a basis, the electrodynamic foundation of Lorentz’s theory [Maxwell’s theory applied to models of electrified bodies, particularly, models of the electron] of the electrodynamics of moving bodies agrees with the principle of relativity,” specifically, the principle of invariance of the form of laws of electrodynamics under transformation across inertial frames (1905a, 16667).
Moreover, we now know that the laws of Ampère (corrected) and Faraday and the law of Magmononicht can be deduced roughly from charge conservation and Coulomb’s law (or Gauss’ law), joined with Einstein’s kinematics, the kinematics of special relativity (Rosser 1968; Lucas and Hodgson 1990, 195205). With some subtle reservations, we may say that “the magnetic interaction of electric currents can be seen as the inevitable corollary of Coulomb’s law. If the postulates of relativity are valid, if electric charge is invariant, and if Coulomb’s law holds, then the effects we commonly call magnetic are bound to occur. They will emerge as soon as we examine the electric interaction between a moving charge and other moving charges” (Purcell 1963, 173).
I have indicated that as of 1905a Einstein was not yet able to show that the new, Lorentz transformations fully satisfy our requirement (1), our requirement that the new transformations succeed wherever the Galilean transformations had succeeded in transforming laws of nature invariantly across inertial frames. The problem was Newton’s second law, which says that the force exerted upon a body is equal to the time rate of change of that body’s momentum.
Newton’s second law is a definition of force. However, it should not be thought to be a definition purely stipulative, uninformed by the physical world. The quantity timerateofchangeinmomentum is a salient and important quantity in nature. Newton and his followers (including Einstein and followers in special relativity) have a concept externalcauseofchangesinspeedordirectionofmotion. We call that force, in partial coincidence with the ordinary parlance of taking force to designate cause of change. Newton’s second law says that the strength of external causes of changes in speed or direction of motion should be gauged by (equal to) the time rate of changes in quantity of motion (momentum) in the subject physical system. (Should be in order to arrive at a proper analysis of motions.) Momentum is a salient and important quantity in nature. Momentum is inertial mass times velocity. Where the amount of mass in a system subjected to a force is constant, time rate of change in momentum becomes mass multiplied by acceleration (time rate of change in velocity). Mass is the capacity of a system to resist changes in its velocity (speed and direction); mass is the ratio of applied force to consequent acceleration.
To obtain equations of motion, one applies Newton’s second law. One writes that mass times acceleration equals some expression for the specific force at hand: expression for the restoring force of a spring (Hooke’s law), for the pull of gravity (Newton’s law of gravitation), for the repulsion or attraction of static electric charges (Coulomb’s law), or for the sweeping of an electron in an electromagnetic field (Lorentz force law). This is a differential equation; its solution yields an equation of motion (locations and momenta of a body as functions of time).
We define energy as the ability to do work. Like force, the term work in physics is in partial coincidence with ordinary parlance, but physics specifies, or narrows, those concepts further than in ordinary vocabulary. Work is defined in physics as force applied through distance. Energy of motion acquired through work is called kinetic energy. From the definitions (full mathematical definitions) of force and work, it follows that kinetic energy is one half the quantity: mass times squared velocity acquired. That is the outline of elementary mechanics at midnineteenth century.
In the final section of 1905a, Einstein attempted to examine the dynamics of an electron in an external electromagnetic field. He examined how application of Newton’s second law to this specific case (an instance of the Lorentz force law) transforms under a Lorentz transformation across (momentarily, approximately) inertial frames; how the equation that makes this application transforms. He does not display for us how accelerations are transformed under Lorentz transformations across inertial frames (Zahar 1989, 3067; French 1968, 15254), but he employs those transformations as they are simplified when the acceleration of the electron is slow (external electromagnetic field is weak). And he employs his Lorentz transformations for components of the electromagnetic field. From the form of this transformed application of Newton’s second law, he exhibits the result that the electron has an inertial mass along the line of constant frame velocity equal to the ordinary inertial mass of the electron times the cube of gamma, where “the ordinary inertial mass” is the old, simple, frameindependent, constant mass (which we now call rest mass) and where gamma is {1 / square root of [1 – (v x v)/(c x c)]}. At the same time, he exhibits the result that the electron has a mass in directions transverse the line of uniform frame motion equal to the rest mass times the square of gamma. (Einstein made a subtle error on this transverse expression, which was corrected by Planck in 1906; the correct expression is rest mass times gamma; Miller 1981, 329.) The peculiarities that electron mass (electron resistance to change in its velocity under the action of an applied force) should be dependent on electron velocity (v in the gammafactor) and should differ by direction (longitudinal vs. transverse) were not very peculiar to physicists in 1905. These were features that had already appeared in electrodynamics (Miller 1981, 32728). But they were features of electrodynamics, and it seems to me that already in 1905a, Einstein is insinuating that these features attach to inertial mass in general, not just to mass of electrons or electrified bodies (cf. 1907, §8). At any rate, in Einstein’s exhibition, the root of the difference between transverse and longitudinal mass is in kinematics, specifically the difference in Lorentztransformed transverse and longitudinal components of acceleration (deeper root: Lorentz transformations of velocities, locations, and times).
I suspect that the generalizing leap from electron mass to mass per se was not thought particularly radical at the time; not so radical as Einstein’s generalizing leap from the velocity c of electromagnetic waves to all occasions of the velocity c, his incorporation of c into basic kinematics. I say this because in those days prominent theoretical physicists were engaged in a research programme aiming to establish electrodynamics as more basic than mechanics (Miller 1981, 4586). That was opposite the hierarchy in physics before or since; mechanics (specifically, dynamics: mass, force, etc.) is more basic than electrodynamics. Some physicists (not Einstein) around the turn into the twentieth century were trying to explain inertial mass from electrical properties of elementary matter. That programme was not successful.
Einstein’s generalizing leap from electron mass to inertial mass per se was a profound shift. On Lorentz’s view of the electron at rest, the electron was a minute spherical volume of charge (negative charge), nothing more. (That is problematic, on the face of it, since regions of the sphere should then all repel each other; the sphere should fly apart.) The electron possessed only charge, no inertial mass at all. On Lorentz’s account, the moving electron “engenders a field which, by acting back on its source, decelerates its motion; this capacity for resisting change of motion is the electromagnetic mass, which varies with the speed and accounts for the total inertia of the particle [the sphere]” (Zahar 1989, 83; further, 244, 31217). Einstein and his early followers, over the years 19058, made a profound change to this picture. They took the electron to be a point particle possessing its own inertial mass in its rest frame. They took the “electromagnetic mass” of the moving electron to be simply intrinsic inertial mass as presented to a frame in relative motion. Moving inertial mass, all moving inertial mass, is then a function of speed. “Electromagnetic mass” became totally identified with inertial mass per se (ibid., 23751).
Naturally one expects that this successful identification must have given Einstein a major suggestion as he turned to create the theory of general relativity: Identify gravitational mass with inertial mass. They are the same thing, truly, totally, the same.
In 1905a Einstein computed the energy of motion, kinetic energy, acquired by the electron in the electromagnetic field and found that it is rest mass times the square of c times the quantity (gamma minus 1), which differs from the old expression for kinetic energy (one half the mass times the square of v), but which reduces to the old expression in the limit of (v x v) much smaller than (c x c). Einstein’s new formula for kinetic energy implies that as a body is accelerated ever closer to c, its kinetic energy approaches infinity. Newtonian mechanics is in for some corrections.
After publication of 1905a, Einstein had a jolting insight. “A consequence of the work on electrodynamics has suddenly occurred to me, namely that the principle of relativity in conjunction with Maxwell’s fundamental equations requires that the mass of a body is a direct measure of its energy content—that light transfers mass. An appreciable decrease in mass must occur in radium” (letter of 1905 in Miller 1981, 353). In a brief article in autumn of 1905, Einstein concluded: “If a body releases the energy E in the form of radiation [electromagnetic waves], its mass decreases by E/(c x c). Since obviously here it is inessential that the energy withdrawn from the body happens to turn into energy of radiation rather than into some other kind of energy, we are led to the more general conclusion: The mass of a body is a measure of its energy content” (1905b, 174; argued further in 1906a, 1906b, and 1907; see also French 1968, 1618, 2728). E = m(c x c). How big is the velocity of light squared? Forty years later, the earth shook. Then for decades, the nuclear shadow lay over humankind. Now passing over.
Mass can be converted to kinetic energy. Can kinetic energy be converted to mass? Yes, whenever electromagnetic radiation is absorbed by matter and whenever there is an inelastic collision of matter with matter.
Taking up development of Einstein’s theory of special relativity, Max Planck (19067) modified Newton second law to make it invariant under Lorentz transformations across inertial frames. This required replacement of the old, constant inertial mass with Einstein’s transverse mass (corrected by Planck), which is a function of [(v x v)/(c x c)]. New definition of inertial mass implies new definition of momentum. New definition of momentum and revision of concept of time yields new definition of time rate of change in momentum, which is the new definition of force in the new second law of mechanics (Zahar 1989, 22734; Miller 1981, 36062; French 1968, 21419). On the surface, words in the statement of the relativistic second law are the same as in the old—time rate of change of momentum—but the precise mathematical meanings of the words have changed.
We can now see how special relativity came round to fully meeting our requirement (1), our requirement that the new, Lorentz transformations succeed wherever the Galilean transformations had succeeded in transforming laws of nature invariantly across inertial frames. The Lorentz transformation does not transform the old second law of mechanics invariantly across inertial frames. Einstein and his followers nevertheless chose not to abandon the new kinematics. They instead supposed that the second law was not fully correct and proceeded to correct it, by asking: What must be the correct form of the second law such that it is invariant under the Lorentz transformations and such that it reduces to the old form of the law when v is small in comparison to c?
I have spoken so much of kinematic transformations across inertial frames. Given the transformation, Galilean or Lorentzian, we can use it to project quantities and laws not only onto other inertial frames, but onto other, noninertial frames. In general we expect laws of nature to change their form under transformations to noninertial frames; we do not expect invariance here. An important class of noninertial frames is the class rotating frames. Galilean transformation of Newton’s old second law to a rotating frame gives us the two familiar “fictitious” forces: the Corriolis force and the centrifugal force. Those are the extra force terms that show up under the Galilean transformation of inertial force (ma) to rotating frames (Marion 1965, 34548). The Lorentzian (specialrelativity) transformation of the new, corrected second law yields the two old, extra force terms and a correction to the old centrifugal force (Misner, Thorne, and Wheeler 1973, 175, 332).
In 19078, G. N. Lewis and R. C. Tolman, using the new, exact definitions of mass, momentum, and energy, rendered the conservation laws of mechanics relativistic. Thereby, as we know by later experience with elementary particles, they rendered those conservation laws truer to fuller reality (Zahar 1989, 23754; French 1968, 168213; Wehr and Richards 1967, 16570). As it turned out, under the relativistic definitions of energy and momentum, conservation of one implies conservation of the other. That was a new result for mechanics, not known prior to special relativity. In special relativity, just as times (one dimension) and spatial locations (three dimensions) stand under a fourdimensional invariance relation with each other, so, in special relativity, energy (one dimension) and momentum (three dimensions) stand under a fourdimensional invariance relation with each other. Moreover, just as simple proper time gauges (equals) an invariant spacetime magnitude across inertial frames, so rest energy (rest mass times velocity of light squared) gauges (equals) an invariant momentumenergy magnitude across inertial frames (Zahar 1989, 25154).
Einstein’s kinematics and his equivalence of mass and energy, especially kinetic energy, signal a reformation of ontology. They do not support the vulgarities “everything is relative” nor “everything is in flux.” The kinematics of special relativity leads us to spacetime, to its surprising interlocking structure. Our concepts of space and time stand reformed. In their relativistic form, space and time stand as real, as real as spacetime.
Equivalence of mass and energy (subsuming momentum under the term energy) leads us to see mass and energy as equally basic with each other. Matter and motion are equally real. To say that mass and energy are equivalent, however, is not to say that mass and energy are wholly the same thing; they are no more identical than time and space are identical in special relativity.
In concluding that mass and energy are equally basic in ontology, I am swimming upstream the natural current of thinking that solid objects and other material substances (such as oatmeal and milk) are more basic than motions and activities. Yes, I realize that matter is primary in the genesis of physical schemas and concepts in childhood. That is genesis. Genesis and history of our ideas is one thing; perfection of fit between our ideas and widest reality is another. Perfection of fit is the goal. Approaching the goal requires imagination (Jetton 1991) and close analysis. I think, however, that intellectual history can awaken us to conceptual subtleties and alert us to old mistakes. Of course, even when informed by consciousness of past thought, imagination and close analysis cannot securely bring us nearer perfect truth unless they be joined to close looking at the world.
References
Einstein, A. 1905a. On the Electrodynamics of Moving Bodies. In Stachel 1989.
―――. 1905b. Does the Inertia of a Body Depend Upon Its Energy Content?
―――. 1906a. On the Inertia of Energy Required by the Relativity Principle.
―――. 1906b. The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy.
―――. 1907. On the Relativity Principle and the Conclusions Drawn from It.
French, A.P. 1968. Special Relativity. Norton.
Jetton, M. 1991. Imagination and Cognition. Objectivity 1(3):57–92.
Lucas, J. R., and P.E. Hodgson 1990. Spacetime and Electromagnetism. Clarendon.
Marion, J.B. 1965. Classical Dynamics of Particles and Systems. Academic.
Miller, A.I. 1981. Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911). AddisonWesley.
Misner, C., Thorne, K., and J.A. Wheeler 1973. Gravitation. W.H. Freeman.
Purcell, E.M. 1963. Electricity and Magnetism. McGrawHill.
Rosser, W.G.V. 1968. Classical Electromagnetism via Relativity. Plenum.
Stachel, J. 1989. The Collected Papers of Albert Einstein. Princeton.
Wehr, M.R., and J.A. Richards 1967. Physics of the Atom. AddisonWesley.
Zahar, E. 1989. Einstein’s Revolution. Open Court.
Epilogue
(1946)
Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation, and I soon noticed that many a man whom I had learned to esteem and to admire had found inner freedom and security in devoted occupation with it. The mental grasp of this extrapersonal world, within the frame of the given possibilities, swam as highest aim half consciously and half unconsciously before my mind’s eye. Similarly motivated men of the present and of the past, as well as the insights that they had achieved, were the friends that could not be lost.
—Albert Einstein
Einstein  SR  Kinematics
XI. Einstein – Special Relativity – Kinematics
In the year Maxwell died, Einstein was born. He was a quiet child who preferred to play alone. He attended public schools in Munich. He was an excellent student (contrary to a popular myth), although selfstudy was more important to him than his formal schooling. A regular visitor to the Einstein home, Max Talmud, introduced the youth Albert Einstein to popular books on science and to the writings of Kant (Kritik der Reinen Vernunft). At age twelve, Einstein was captivated by Euclidean geometry, and over the next four years, on his own, he studied differential and integral calculus. (Ampère was even faster in this respect; by the age of twelve, he had mastered all extant mathematics.) Einstein attended college, in Zurich, from 1896 to 1900, with his focus on physics. One of his mathematics teachers was Hermann Minkowski: “Oh, that Einstein, always cutting lectures—I really would not have believed him capable of it [special relativity]” (c. 1908). During those college years, Einstein read Ernst Mach’s book on mechanics.
After graduation, Einstein sought a teaching position at his university, but to no avail. He eventually obtained steady employment at the Swiss national patent office in Bern. He assessed patents, often patents for electric generators. During his years there, he did important research in a number of areas of theoretical physics. I count twentyfive published papers in the year 1905. One of them was “On the Electrodynamics of Moving Bodies,” which announced his theory of special relativity (1905a).
Why special? In 1905 it was simply relativity (see further, Stachel 1989a, 254). Special relativity is relativity amongst inertial frames of reference. The special theory comprehends laws of physics (the laws of dynamics [for both particles and media], the laws of electromagnetism [including optics], and the laws of thermodynamics) under the assumption that all inertial frames are equivalent; in objective, physical reality, no inertial frame is privileged above any other inertial frame. Beyond the special theory, Einstein gives us the general theory of relativity, which proposes a broader, or more general, equivalence class of reference frames and in that broadening, or generalization, radically reconceives the nature of gravitation.
The special theory clarified our understanding of the relativity of motion, recast our concepts of space and time, and corrected Newtonian mechanics. Thereby, it enabled our fundamental physics to encompass the highest possible velocities and to comprehend the convertibility of energy and inertial mass. The general theory unified inertial frames with freely falling frames and formulated a profound relation between matter and physical geometry, revising and subsuming Newton’s theory of gravitation. Thereby, it prepared our fundamental physics to fathom the densest celestial objects and the largescale structure of the universe.
We have come to suspect (sup. §§VIIX) that the Galilean transformations for locations, times, velocities, and accelerations are flawed. We seek new transformations. What possible transformations will (1) succeed in all the contexts in which the Galilean transformations have been successful in transforming laws of physics invariantly across inertial frames, (2) transform all occasions of the velocity c invariantly across inertial frames, and (3) transform all laws for which the Galilean transformations failed (such as Faraday’s law of induction and the MaxwellHertz electromagnetic wave equation) invariantly across all inertial frames?
Requirement (1) implies that our new transformation equations will be functions of the relative velocities v between frames. That is a dependence we (and Galileo, . . .) expect from common sense; more specifically, from common physicalmathematical intuition. Requirement (2) is not dicta from common sense, but dicta from the uncommon sense of Albert Einstein. Requirement (2) is radical and is original with Einstein (1905a). The constant we denote by c was a creature of electromagnetic phenomena only. Einstein imported it into our fundamental kinematics. He took it to be a fundamental factor in all our physics (including our basic dynamics, from Newton), not only in electromagnetism. Einstein took the velocity c to be an element in the kinematical structure of the world, an element at the level of space and time themselves. Requirement (2) was a stroke of genius.
Now the magnitude of v, the relative velocity between frames, is something found to be the same, by measurement, in either frame. (That was so already in the Galilean kinematics and remains true in Einstein’s new, surpassing kinematics.) And any occasion of the velocity c in either frame must be also found c in the other frame. (That is new with Einstein’s kinematics.) From those two propositions, it would seem to follow that relative velocities v between material frames must not be greater than the velocity c. If the relative velocity v were greater than c, how could the propagation of a light wave (for example) be found to be only c in both frames? It boggles the mind. (See further, Lucas and Hodgson 1990, 11213; Friedman 1983, 15963; Zahar 1989, 14042.)
Furthermore, if a frame were moving with a relative velocity v not greater than but equal to c, then in such a frame, a light wave would not propagate at c; it would not propagate relative to such a frame at all; it would, so to speak, just stand still and bop up and down. Yet in material frames with v not yet fully c, but merely approaching c, we have that light is always found propagating and at the velocity c. Then, suddenly, in passing from v < c to v = c, a material frame no longer finds light traveling at c, but at zero? In special (and general) relativity, no material frame, no piece of matter (the electron and positron being the smallest such pieces), can reach v = c and remain material. This proposition has been strongly confirmed by experiment (e.g., Bertozzi 1964; French 1968, 611).
What transformations for locations and times will show a sphere expanding at the velocity c to be such in each of two material frames having between them the relative velocity v? (See Einstein 1905a, 14549; see further, Friedman 1983, 14042, and Lucas and Hodgson 1990, 15167.) That transformation that Woldemar Voight in 1887 had shown (indirectly) to transform the electromagnetic wave equation in free space (free ether) invariantly across inertial frames (Miller 1981, 114n50, 215, 218n2; Lucas and Hodgson 1990, 16566). Voight was working on theory of the Doppler effect, “which can be regarded as the problem of observing a wave motion from different inertial frames” (French 1968, 270). These transformations we know today by the name Lorentz.
In 1892 Hendrik Antoon Lorentz published the paper “The Electromagnetic Theory of Maxwell and Its Application to Moving Bodies.” He took Maxwell’s field equations to be valid in a frame at absolute rest (viz., at rest with respect to the electromagnetic ether permeating all space and all ponderable bodies). Those are partial differential equations. Applying a Galilean transformation to them, Lorentz obtained the Maxwell field equations (again, partial differential equations) in the form they must have (if the Galilean transformations were the correct transformations) in a frame of reference moving uniformly with respect to the ether. (Over sufficiently short time intervals—such as the time required for a beam of light to cross a laboratory—an accelerating body, such as the earth, will have an approximately uniform velocity.) Solutions to Maxwell’s partial differential equations, including the electromagnetic wave equation, in the rest frame were known. To obtain solutions to those partial differential equations in the form they assume (upon Galilean transformation) in the earth frame (for short time intervals), Lorentz found a change of variables (a common mathematical technique for rendering some partial differential equations solvable) that would bring them into the form they had in the rest frame, where solutions were known. Having the solutions in terms of the changed variables, he could then (final step of the mathematical technique) change back to the original variables in those solutions. That gives him the solutions, in terms of the original variables, for Maxwell’s equations in the moving, earth frame. In forming his change of variables, Lorentz was forming what (except for an extra multiplicative factor gamma in the time variable) we today call the Lorentz transformations for locations and times (and for partial derivatives with respect to those variables) (Zahar 1989, 5257).
Lorentz, in this 1892 paper, was not seeking invariance of the form of Maxwell’s equations across inertial frames. He was simply trying to solve the variant, Galileaninduced form of those equations (including the wave equation) in the moving frame. Had Lorentz been seeking invariance, he might have achieved it by removing the extra gammafactor in his transformation of the time variable. As it occurred, Lorentz did not initially try to give a physical interpretation to his set of transformations, his change of variables. It was simply an uninterpreted mathematical device (ibid., 5758). Very soon, however, he did begin to try to give the transformations a physical interpretation, one that could be squared with electromagnetic and optical experiments. He continued to revise and refine his interpretation for several years (ibid., 5883). None of his interpretations—all of them physical interpretations against a background of absolute space and time—were the interpretation we give his (corrected) transformations today, the interpretation given them by Einstein (1905a).
In Einstein’s mind, the Lorentz transformations were seen as new kinematical relations that should replace the old Galilean transformations. Under Einstein’s interpretation, the Lorentz transformations are the kinematical relations needed for transforming all occasions of the velocity c invariantly across inertial frames (our requirement [2]). They transform also the wave equation and all of Maxwell’s equations invariantly across inertial frames (our requirement [3]). Moreover, in the limit of velocities low in comparison to c, the Lorentz transformations reduce to the Galilean transformations. That is a necessary but not sufficient condition that must be met in order to satisfy our requirement (1). Einstein (1905a) was not yet able to fully satisfy requirement (1). Without further development of the special theory of relativity, beyond 1905a, the Lorentz transformations could not succeed in all the places the Galilean transformations had succeeded in transforming laws of nature invariantly across inertial frames.
I want to linger, however, in 1905. Einstein and Planck will button up our condition (1) soon enough. The Lorentz transformations, under Einstein’s construal, will stand. They stand today. Their implications for the nature of space and time were and are revolutionary.
A light source moving uniformly due east sends a pulse of light due north. . . .
[Derivation of time dilation formula: V2N6 154–55]
We derive time dilation, the proposition that “moving clocks run slow.” Newton’s view that time flows at the same pace in all frames was incorrect (though, as we readily see, time does flow at very nearly the same pace in the two frames when v x v is very much smaller than c x c). Time dilation has received strong experimental confirmation (Frisch and Smith 1963; French 1968, 1015). A good thing to remember about time dilation: in the moving frame S', the elapsed time is measured by a single clock (a clock at rest in S, but moving in S'), whereas, in the stationary frame S, the elapsed time is measured by two clocks spatially separated.
The frame in which the moving light source is at rest is called the “proper” frame. Every piece of matter has its own frame, its proper frame, the frame attached to that matter. Times in the proper frame are called the proper times. The timedilation equation tells us that the time intervals between two events will be minimal (time creeps most slowly) in the proper frame, that is, in the frame in which the time interval is gauged by a clock attached to that frame. Time intervals between those same two events, if gauged by clocks in other frames (not comoving with the proper frame), will be greater. Expressing time dilation by the statement that “moving clocks run slow” is ambiguous. We say better, “proper clocks run slow” (or “nonproper clocks run fast”). And in relativity theory, when we speak of clocks, we mean any periodic physical system, not only contrived clocks, such as a pendulum clock, a springdriven wrist watch, or an atomicbeam clock.
It is customary in relativity to speak of what an observer attached to suchandsuch reference frame might measure in that frame and to then compare those measurements to like measurements, of the same physical items, made by an observer attached to another frame. This customary talk of observers (with minds) is now clearly understood to be not essential to the theory of relativity, special nor general. Observertalk in relativity is, in the final analysis, simply a convenient, storybook way of talking about the geometry, kinematics, and dynamics of the physical world.
There is, however, a serious point to the customary observertalk in relativity. By attending closely to the essence of what we do (and to what we assume doable in principle) when we measure time or distance, Einstein was able to capture better than had anyone before him the character of those physical items. Attention to what we do in measuring those items precised our concepts of those items. Notice that doing and measuring are physical activities. I am not retracting my prior assertion (sup §III) that observers (their minds) are not an essential constituent of the subject matter of the theory of relativity.
. . . .
Another striking result from special relativity is length contraction. We cannot arrive at length contraction by considering only the oneway trip of a light pulse. We must consider a round trip, and that must lie along the line of motion of the light source (the source being, for example, a flashbulb).
A light source moving due east sends a pulse of light due east. . . .
[Derivation of length contraction formula: Obj V2N6 157–58]
We derive length contraction, the proposition that nonproper lengths are contracted (or proper lengths are dilated) along the line of frame motion (French 1968, 1079).
Lengths transverse the direction of frame motion are invariant across frames. Length contraction arises only along the line of motion between the frames.
Length contraction coincides with the earlier LorentzFitzgerald contraction, which, before Einstein, served to explain the null results of the famous MichelsonMorely experiments to detect drift of the earth through the ether. However, the LorentzFitzgerald contraction was a shortening of physical rods, thought to be due to an electromagnetic effect among the molecules composing rods. Einstein’s length contraction is simply in the character of space and time.
In the kinematics of special relativity, it turns out that not only is c an invariant across frames; the spacetime interval between two events is also an invariant across inertial frames (Geroch 1978, 80108). A light source moving due east sends a pulse of light a distance l' (= l) due north to a mirror posted there in S', the proper frame of the light source. The light pulse then returns home to its source. Consider a clock attached to that moving light source, a clock which registers the duration 2 x deltat' between departure of the light pulse and return of the light pulse. Consider just the ticking of that clock, a clock remaining at the location of the light source in the proper frame. The duration registered on that clock between the tick of lightpulse departure and the tick of lightpulse return (which is just the proper time between those two events) is an instance of the invariant spacetime interval. In the proper frame, the spacetime interval between those two ticks is purely temporal; the clock is just resting at a single point in the proper frame. In the earthstationary frame, that instance of the invariant spacetime interval is partially spatial (it has a deltax component) and partially temporal (it has a deltat component). Specifically, we have (from Minkowski 1908, 85) the following invariance:
(proper time interval in S') = (nonproper spacetime interval in S)
deltat' = square root of [ –(deltax/c)(deltax/c) + (deltat)(deltat)].
A pure time interval (or a pure space interval) between two events does not remain a pure time interval (or a pure space interval) between those same two events across frames. (Of course, they do remain so approximately if v << c.) However, the spacetime interval between two events is invariant across frames for all possible v.
In composing the invariant spacetime interval, we parallel in spacetime the theorem of Pythagoras for space. To parallel Pythagoras strictly, we should have a positive sign in front of the space component as well as in front of the time component. The fact that we have to place a negative sign in front of the space component (i.e., the fact that the space component and the time component are antisymmetric) to get a true physical invariance tells us that in spacetime physics, time is not simply a fourth spatial dimension (Lucas and Hodgson 1990, 24, 83, 216, 238).
[Derivation of framerelativity of simultaneities for spacelike separations: Obj V2N6 159–60]
If the spacetime interval between two events has a larger space component than time component, we say the interval is spacelike. If the spacetime interval between two events has a larger time component than space component, we say the interval is timelike. If the spacetime interval between two events has equal space and time components, we say the interval is lightlike; for the very good reason that a light pulse would connect those events physically. Events with timelike separations can be connected physically by bits of matter in motion. Spacelike separations cannot be connected physically (causally). Nevertheless, they are perfectly real, as real as a purely spatial separation. Separation of two events spacelike in one frame will be spacelike in any other frame. Lightlikeness is lightlikeness across all frames (invariance of all occasions of c; sphericity of cpulses in all frames; invariance of ccone across Minkowski diagrams). Timelikeness is timelikeness across all frames. This last implies that propertime intervals will be timelike in all frames.
. . . .
In the “Transcendental Aesthetic” of The Critique of Pure Reason, Kant had contended that our ideas about space and time are synthetic a priori sensory intuitions and that they are the fundamental forms of sensory experience in general. I should say that I think special relativity weighs against the a priori aspect and the sensoryintuition aspect that Kant conferred on these notions, space and time. Kant took a prioricity of a notion to entail unrevisability (apodeictic certainty). Yet in special relativity, Euclidean space is superseded by Minkowskian spacetime. The relativity of simultaneities for spacelike intervals is not something that Kant, any more than Newton, would have found acceptable. It is very counterintuitive. Kant’s “fundamental forms of sensory intuitions” have been corrected by new experience, and most of all, by new thinking upon that experience.
I have jumped ahead of Einstein 1905a by speaking of spacetime and of the invariant spacetime interval. These conceptions were formulated by Poincaré and Minkowski in 19068 (Miller 1981, 7882, 23843).
Einstein 1905a does derive, as kinematical relations, the Lorentz transformations, including new rules for adding velocities (§5). With the Galilean transformations for locations and times, only space coordinates for the frame axis along the direction of frame velocity v were functions of that velocity. Then, considering the velocity u of some object in arbitrary direction, only that component of u parallel the direction of v will be different in the two frames. With the Lorentz transformations for locations and times, it is not only spatial coordinates for the frame axis along the line of frame velocity v that are functions of v (and c); times are also functions of v (and c). Because of the latter (and the mundane fact that velocities are time rates of change in position), under the Lorentz transformations, not only that component of u parallel the direction of v will be different in the two frames; nonzero components of u transverse the direction of v will also differ between the frames.
More famously, for the case of frame S' moving in some direction D at speed v nearly c and with an object moving in S' in direction D at speed u' nearly c, the Lorentz transformations imply that in the stationary frame S, the object speed u (which in the stationary frame S is v added to u') will be only nearly c, never c nor greater than c. The addition of velocities in special relativity (i.e., in reality wider than that found in ordinary experience) is not simple addition, contrary to the addition of velocities according to the commonsense, Galilean transformations.
(to be continued)
References
Bertozzi, W. 1964. Speed and Kinetic Energy of Relativistic Electrons. Am. J. Phys. 32:551–55.
Einstein, A. 1905a. On the Electrodynamics of Moving Bodies. In Stachel 1989.
French, A.P. 1968. Special Relativity. Norton.
Friedman, M. 1983. Foundations of SpaceTime Theories. Princeton.
Frisch, D.H., and J.H. Smith 1963. Measurement of the Relativistic Time Dilation Using μMesons. Am. J. Phys. 31:342–55.
Geroch, R. 1978. General Relativity from A to B. Chicago.
Lucas, J. R., and P.E. Hodgson 1990. Spacetime and Electromagnetism. Clarendon.
Miller, A.I. 1981. Albert Einstein’s Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911). AddisonWesley.
Stachel, J. 1989. The Collected Papers of Albert Einstein. Princeton.
Zahar, E. 1989. Einstein’s Revolution. Open Court.
Maxwell
X. Maxwell
James Clerk Maxwell was a mathematical physicist. In 1856, two years after obtaining a degree in mathematics from Cambridge, he published a paper, “On Faraday’s Lines of Force.” Maxwell gave mathematical representation to Faraday’s picture. Lines of electric and magnetic force could be represented as lines of fluid flow had been represented in hydrodynamics. The intensity of electric and magnetic force at each point along a line could be represented as fluid velocity had been represented in hydrodynamics. Maxwell’s electric and magnetic fluid of 1856 was not “a hypothetical fluid which is introduced to explain actual phenomena” (quoted in Gillespie 1990, 462). Rather, it was a collection of imaginary properties that made possible a coherent mathematical representation of actual phenomena, such as the production of electric currents by changes in the magnetic or electric state of the region surrounding a conductor or production of magnetic states in regions surrounding electric currents.
In 1861 and 1862, Maxwell published a series of papers, “On Physical Lines of Force.” Therein the word field was introduced into our physics vocabulary. “Magnetoelectric phenomena are due to the existence of matter under certain conditions of motion or of pressure in every part of the magnetic field, and not to direct action at a distance between the magnets or currents” (ibid., 470–71). Maxwell supposed an insensible elasticsolid medium permeating all space and all ponderable matter. This medium had to have certain complicated elastic properties to account for all electric and magnetic phenomena (ibid., 466–72; Schaffner 1972, 79–80). Longitudinal waves cannot be propagated through Maxwell’s elastic medium, and that is as it should be if waves propagating in his elastic medium are to be identified with light waves.
Wave modeling of light began with Huygens (1690), who held that “light is propagated in some manner, an understanding of which we may obtain from our knowledge of the manner in which sound travels through air” (quoted in Schaffner 1972, 9). He proposed an elastic medium, which he called the ether, in which waves of light propagate. Huygens was able to explain refraction and even double refraction with this model (using his principle of secondary wave propagation, which we still use today). Naturally, he supposed that light waves were longitudinal, like sound waves, which have their variation (in pressure) in the same direction as their propagation. Huygens died in 1695, and in 1717 it was discovered that in certain double refractions, light comes to possess what Newton called sides, or what is (since E. L. Malus [1808]) now called polarization. Polarization stood unexplained under Huygen’s wave model. In 1817 Thomas Young proposed the general solution: in order to explain polarization, light waves must be transverse (their variations perpendicular to their direction of propagation, as in the vibrations of a taut string), rather than longitudinal. Augustin Jean Fresnel (d. 1827) proceeded to make systematic all known optical phenomena and to predict new optical phenomena on the transversewave model. The wave model of light then prevailed over the particle model (see Whewell’s midnineteenthcentury “Inductive Table of Optics” in Butts 1989) until the twentieth century.
Maxwell showed, in his papers of 1861–62, that transverse waves could propagate through his proposed elasticsolid medium, his medium for conveyance of electric and magnetic forces through otherwise empty space. He demonstrated that the velocity with which a transverse wave would propagate in his hypothetical medium was a certain electromagnetic, constant property of the vacuum. This constant had been measured in electromagnetic experiments of Wilhelm Weber and Rudolph Kohlrausch in 1856. Maxwell pointed out that the value of their electromagnetic constant (which had the units of a velocity) was the same, within experimental accuracy, as the velocity of light, measured by Fizeau in 1849. (Actually, Gustav Kirchoff had already in 1857 noted the coincidence between the electromagnetic and optical measurements.) Maxwell then identified his electromagnetic ether (elasticsolid medium for conveyance of electric and magnetic forces) with the luminiferous ether, and he identified his electromagnetic transverse waves with light.
In 1865 Maxwell delivered the paper “A Dynamical Theory of the Electromagnetic Field.” His characterization of the electromagnetic field no longer included detailed hypothetical mechanism to explicate the ether. The ether remained in his theory, but the equations we now call Maxwell’s field equations had themselves come to be seen as a sufficient characterization of the ether.
“I have on a former occasion attempted to describe a particular kind of motion and a particular kind of strain, so arranged as to account for the phenomena. In the present paper I avoid any hypothesis of this kind; and in using such words as electric momentum and electric elasticity in reference to the known phenomena of the induction of currents and the polarization of dielectrics, I wish merely to direct the mind of the reader to mechanical phenomena which will assist him in understanding the electrical ones. All such phrases in the present paper are to be considered as illustrative, not as explanatory.
“In speaking of the Energy of the field, however, I with to be understood literally. All energy is the same as mechanical energy, whether it exists in the form of motion or in that of elasticity, or in any other form. The energy in electromagnetic phenomena is mechanical energy. The only question is, Where does it reside? On the old theories it resides in the electrified bodies, conducting circuits, and magnets, in the form of an unknown quality called potential energy, or the power of producing certain effects at a distance. On our theory it resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, and in two different forms, which may be described without hypothesis as magnetic polarization and electric polarization, or, according to a very probable hypothesis, as the motion and the strain of one and the same medium.” (1865; quoted in Schaffner 1972, 82–83)
As every schoolchild knows, the ether would, about forty years later, be dispelled from our ontology, as had been the music of the spheres, the celestial spheres themselves, phlogiston, caloric, and vital forces. Continual further specification of the nature of the ether by the followers of Maxwell (such as H. A. Lorentz), in the last quarter of the nineteenth century, gradually deprived the ether of all mechanical properties. “As to the mechanical nature of the Lorentzian ether, it may be said of it, in a somewhat playful spirit, that immobility is the only mechanical property of which it has not been deprived by H. A. Lorentz” (Einstein 1920, 10–11).
The ether faded into pure absolute space, and as a result, the electromagnetic field came to be regarded as an independent entity, rather than as the state of a medium (Earman 1989, 51). “The electromagnetic fields are not bound down to any bearer, but they are independent realities which are not reducible to anything else, exactly like the atoms of ponderable matter” (Einstein 1920, 12). The electromagnetic field remains in our ontology. (See, however, Mundy 1989.) In our understanding, as in Maxwell’s, the electromagnetic field is real: it contains and transports momentum and energy; electric and magnetic sources have their effects across space only after a lapse of time, not instantaneously; and the propagation of effects are affected by material changes in the intervening space (cf.. Hesse 1965, 197–98, 210–12, 220–22).
Maxwell’s equations taken together imply that disturbances in electromagnetic fields will propagate in vacuum according to the wave equation and with the particular speed c in all directions. As we know already, the wave equation is not invariant under Galilean transformations across inertial frames. We learned that result in our look at laws of fluid mechanics, but it is simply true of the wave equation in general, in all its occasions. The peculiarity of Maxwell’s (and Hertz’s) wave equation is its specification of a particular speed for propagation of light waves in vacuum. From the Galilean transformations, we know that that speed would have to be different in inertial frames having different states of uniform motion. That is, we know that that speed would have to be different in inertial frames for which the constant relative velocity between them is not zero. It seemed natural to take the frame in which light waves propagate at the particular value c (and c in all directions) as a very special, fundamental frame of reference. After all, Maxwell’s field equations seem very fundamental, and the lightwave propagation equation is a direct consequence of Maxwell’s field equations. And according to the Galilean transformations, in only one inertial frame can the light wave propagate with perfect spherical symmetry about its source location (location of some electric charge undergoing an acceleration). It was natural to take such a special inertial frame as at absolute rest (whether there is an ether at absolute rest or only Newton’s empty absolute space). Which inertial frame is at rest then seemed to be not a matter determined by our subjective choice (according to convenience), but determined by objective reality.
This is an inconsistent physics. The Galilean transformations are again putting us into conflict with Galilean relativity. The latter says that all inertial frames are created equal. The former tells us, in the context of electrodynamics and electromagnetic wave propagation, that one inertial frame is special: it is truly at rest, objectively so.13 Sensitive experiments to detect any inertialframe dependence of the speed of light failed. The speed was found to be simply the unique value c, in every direction in the vacuum, whatever the (approximately inertial) frame of reference. In the frame of reference in which the location of the light source (say, a flashbulb) is at rest and in all frames in which the source is moving uniformly, the light wave will propagate at the unique value c in all directions; perfect spherical symmetry in all frames, even though the frames are moving with respect to each other. This is the straightest implication of Maxwell’s equations, and this is what is found experimentally. (A most stunning experimental demonstration is Alväger, Farley, Kjellman, and Wallin 1964; see also Brecher 1977.) All inertial frames are created equal. The Galilean transformations for locations, times, velocities, and accelerations have got to give way to new transformations.
(to be continued)
References
Alväger, T., Farley, F.J.M., Kjellman, J., and I. Wallin 1964. Test of the Second Postulate of Special Relativity in the Gev Region. Physics Letters 12:260.
Brecher, K. 1977. Is the Speed of Light Independent of the Velocity of the Source? Physics Reviews Letters 39(17):1051–54.
Butts, R.E. 1989 [1968]. William Whewell: Theory of Scientific Method. Hackett.
Earman, J.1989. World Enough and SpaceTime. MIT Press.
Einstein, A. 1920. Ether and Relativity. In Sidelines on Relativity. 1922. G.B. Jeffery and W. Perret, trans. Methuen.
Gillespie, C.C. 1990 [1960]. The Edge of Objectivity. Princeton.
Hesse, M.B. 1965. Forces and Fields: The Concept of Action at a Distance in the History of Physics. Greenwood.
Mundy, B. 1989. Distant Action in Classical Electromagnetic Theory. British Journal for the Philosophy of Science. 40:39–68.
Schaffner, K.F. 1972. NineteenthCentury Aether Theories. Oxford.
Faraday
IX. Faraday
Turn back now to 1804, the year of Kant’s death. A thirteenyearold boy in London was then beginning his first employment, apprentice to a bookseller and bookbinder. The boy’s name was Michael Faraday. He had no formal education. In the hours after work, young Faraday read books, especially books about science. He attended public lectures on science, some by the eminent chemist Humphrey Davy. At age twentyone, Faraday sent an application for employment to Davy, enclosing some notes he had made on Davy’s lectures. Davy took Faraday as his laboratory assistant at the Royal Institution.
Reflection on Ørsted’s discovery led Faraday to discover, in 1821, that a wire hoop bearing an electric current will rotate about an axis lying in its plane when the hoop is placed in a magnetic field (see further, Gillespie 1990, 443). This was the first demonstration of the principle of the electric motor. This discovery of Faraday’s may have stimulated Ampère to the idea that macroscopic magnets are congeries of minute current loops.
Ørsted had shown that electric currents produce magnetic forces. Faraday had shown that magnets have an effect on electric currents. Faraday conjectured that magnetism could produce electricity, and in 1831 he demonstrated that this is the case. He discovered that by starting or stopping a magnet’s motion through a stationary closed coil of conducting wire one can induce an electric current in the wire. That the amount of magnetic flux through the conductor be changing with time was necessary for the effect (ibid., 443–46). The principle in operation here is one of the fundamental laws of electromagnetism, the one we know as Faraday’s law of induction.
There are three other fundamental laws: Gauss’ law (descendant from but deeper than Coulomb’s law), the law contending that there are no magnetic monopoles (conveniently denominated Magmononicht), and Ampère’s law in the generalized form given it by Maxwell. These four laws are expressed by what we call Maxwell’s field equations. Maxwell set them down rightly (1865) as the laws that together cover all the classical (vs. quantum) statics and dynamics of electromagnetism. (The law of continuity of charge and current and the wellknown Lorentz force law are implicit in Maxwell’s field equations [and in the force laws antecedent to Maxwell’s equations].)
Faraday’s law says (in modern, Maxwellian expression) that a changing magnetic field causes an electric field having a vorticity proportionate the time rate of change of the magnetic field. Is Faraday’s law invariant under Galilean transformations across inertial frames? No! The form of the law as applied to its component along the direction of frame motion is not invariant. An extra term appears, a term proportional to the frame velocity.
If Faraday’s law is a fundamental law of electromagnetism, truly a law of nature—and we certainly think it is so—then the Galilean transformations must be wrong. Galilean transformation of Faraday’s law implies a contradiction. It implies that Faraday’s law enables us to find frames of reference that are at rest absolutely. This contradicts Galilean relativity. Moreover, although the implication that Faraday’s law should enable us to find frames of reference at absolute rest is concordant with Newton’s (inconstant) conviction that there is such a thing as objectively given absolute rest, the implication is discordant with Newton’s mechanics.
There is even more trouble. We have seen that application of a Galilean transformation adds a frame velocity term to Faraday’s law. Electrical experiments (such as Trouton and Noble 1903; Rosser 1967, 32, 296–97) indicate that in Earth’s orbital travels the additional velocity term is zero (and zero at all points of the orbit!). It seems the earth happens to be in the absolute rest frame (all the time!). That is enormously implausible.
By the 1840’s, Faraday and others had shown that electricity and magnetism are intimately related. Faraday then succeeded in showing that the polarization of light can be affected by a magnetic field. He thought, furthermore, that electricity and gravity must be intimately related. (Euler had this thought as well, in the preceding century.) Electrostatic (Coulomb) and magnetostatic (BiotSavart) attractions and repulsions had been found to obey an inversesquare law, like gravity. In 1850 Faraday tested for connections experimentally between electromagnetism and gravity. He found none. Faraday retired in 1858. In the succeeding forty years—principally through the work of Maxwell, Lorentz, and Hertz—the unity of electricity, magnetism, and light would be thoroughly established and comprehended. To this day, no one has succeeded in establishing a unity of electromagnetic and gravitational forces (although, some beautiful and illuminating parallels have been discerned; Ciufolini and Wheeler 1994, 315–24, 71–77).
I have spoken of flux, and vorticity and these suggest some sort of fluidity. In the nineteenth century, concepts borrowed from fluid mechanics could be taken more literally in application to magnetism or electricity than in application to gravity. For gravity, every point around and within a massive body, one thought of the force that would be exerted on a testparticle mass were it present at that point. One spoke sensibly of a field of gravitational force. Alternatively (following work of Lagrange, Legendre, Laplace, Poisson, Green, and Gauss), one spoke of the field of the gravitational potential. (Potential in this context means a scalar function whose partial derivatives with respect to spatial coordinates are components of force.) Both the gravitational force and its corresponding potential are field functions—functions of spatial locations—but there is a great difference between these fields and the velocity or pressure fields of a fluid. The fluid, the medium for the properties of velocity and pressure, has other detectable properties which lead us to identify it as material. We can then say that the velocity and pressure fields are properties of (or relations over) the medium. In contrast the gravitational field seemed to have no properties other than potentiality for gravitational forces on testparticle masses. The gravitational field was most reasonably taken to be a mere mathematical device. Action of gravity could be reasonably conceived as action at a distance, not action through a medium, even if one spoke of a gravitational field (Hesse 1965, 196–97, 222–25).
On experimental grounds, Faraday had become convinced, beginning in the 1830’s that electric and magnetic forces must act through a medium intervening between electric and magnetic bodies. Concerning electric force, Faraday pointed out that it had been established that induction of electrostatic charge from one conductor to another, an insulating material between them, varies in strength according to the nature of the intervening insulator. Moreover, lines of electrostatic induction can be curved, even across a gap, as seen in sparks of electrical discharges. These sorts of facts strongly suggested that an electrostatic field, even across apparently empty space, acts through an invisible medium. Magnetic force, too, seems to lie along curved lines, as illustrated by iron filings on a sheet of paper held over a magnet. Faraday pictured and wrote of lines (or tubes) of force emanating from charged conductors and from magnets; lines physically real (ibid., 198–206; Gillespie 1990, 452–58). Though Faraday speculated that there is a subtle medium in all space, conveying electric and magnetic lines of force and light waves (which he suspected were electromagnetic), he shied away from identifying the medium with an elasticsolid luminiferous ether (mechanical posit to undergird wave optics) such as had been explored mathematically by Fresnel, Cauchy, Green, and MacCullagh (Schaffner 1972, 77–78, 14–19, 45–66).
(to be continued)
References
Ciufolini, I., and J.A. Wheeler 1995. Gravitation and Inertia. Princeton.
Gillespie, C.C. 1990 [1960]. The Edge of Objectivity. Princeton.
Hesse, M.B. 1965. Forces and Fields: The Concept of Action at a Distance in the History of Physics. Greenwood.
Rosser, W.G.V. 1967. Introductory Relativity. Plenum.
Schaffner, K.F. 1972. NineteenthCentury Aether Theories. Oxford.
Trouton, F.T., and H.R. Noble 1903. The Forces Acting on a Charged Condenser Moving through Space. London Royal Society Proceedings 72:132–34.
Ampere
VIII. Ampère
Development of the theory of electromagnetism in the nineteenth century generated our modern Age of Electricity. It also impelled deep new thinking on the nature of space and motion.
In 1785 Charles Augustin de Coulomb published results of his electrical experiments. Between two charged bodies, there is a force mutually attractive or repulsive along the line joining the bodies, a force whose strength varies directly as the magnitude of the charge on each body and inversely as the square of the distance between the bodies. This we know as Coulomb’s law. Is Coulomb’s law invariant under Galilean transformations across inertial frames? Yes.
An electric current is electric charge in motion. Conservation of electric charge requires that the charge density at any point of space be related to the current density in that neighborhood by what is known as the continuity equation (analogous to the massdensity continuity equation in fluid mechanics). The continuity equation expresses the proposition that a change of charge with time inside a minute volume is balanced by a flow of charge through the surface of that volume. Is the continuity equation invariant under the Galilean transformations across inertial frames? Yes, provided we take the charge densities as identical in both frames. It is thought in our modern physics, since around the turn into the twentieth century, that that is indeed a true taking, provided the charge speeds and the frame speeds are small compared to the velocity of light.
In 1820, Hans Christian Ørsted established a connection between electricity and magnetism. He demonstrated that a straight wire conveying along its length an electric current deflects a magnetic compass needle (a needle lying in a plane perpendicular the wire length) brought near the wire. More specifically, magnetic needles brought to the currentbearing wire became aligned into axial circles about the wire. If the direction of the current flow in the wire is reversed, all of the compass needles about the wire reverse their orientations endforend.
Ørsted’s discovery in Copenhagen animated physicists in Paris. AndréMarie Ampère, at the École Polytechnique, set upon Ørsted’s phenomenon with mathematicalphysical modeling and further laboratory investigations. Ampère became convinced that magnetism itself was somehow really, basically an electrical phenomenon. He conceived of all magnets, permanent and impermanent, as congeries of minute magnets caused fundamentally by minute electric current circuits within the macroscopic magnet. He had hit upon the view we hold today: magnetic effects are caused by electric currents (else by, as we know since Maxwell, timevarying electric fields). This readily explains why we find no magnetic monopoles anywhere.
Ampère’s primary theoretical result was Ampère’s law. Translated into our modern terminology, that law says that a static and circular field of magnetic force is set up about a steady electric current, the strength of the force at a point being directly proportional to the amount of the source current and inversely proportional to the distance of the point from that current. Field? The axial circles of magnetic force remind one of a vortex in a liquid. One arrives naturally at the idea that there is a field of magnetic force about the wire; analogous to the fields of velocity and fields of pressure in a fluid, as articulated by Daniel Bernoulli and Euler in the eighteenth century. Note that magnetic fields and electric fields can be produced not only in the atmosphere, but in vacuum chambers.
Is Ampère’s law invariant under Galilean transformations across inertial frames? We have reason to suspect that Ampère’s law as we received it from Ampère is not yet worthy of that question. That is, we have reason to suspect that Ampère’s law as we have it so far is not sufficiently general to be a law of nature deep enough, prima facie, to span all inertial frames of reference. The problem is that the law is restricted to steady source currents. A steady current in a frame at rest with respect to the conducting wire can be an unsteady current in a frame moving uniformly with respect to the rest frame. That is why, above, when we assessed Galilean invariance for hydrodynamics, we considered Euler’s flow equation (relating fluid pressure and velocity), rather than Bernoulli’s flow equation. Bernoulli’s is Euler’s restricted to steady flow. Only Euler’s, the more general equation, stood a chance of being invariant under (true or approximately true) transformations across inertial frames.
Bernoulli’s equation stands to Euler’s as Ampère’s law stands to Maxwell’s generalization of Ampère’s law, the generalization being necessary to accommodate unsteady current sources. Let us reach out of proper history here and accept a gift from Maxwell (1865). We want Maxwell’s generalization of Ampère’s law, but we want to give its reference slightly less extension than Maxwell would give it (cf. Rosser 1968, 23743). Maxwell had in hand the results of Coulomb, Ampère, and Faraday. Through consideration of Faraday’s work, Maxwell was able to surmise that source terms in Ampère’s law should allow for not only unsteady currents of electric charge, but unsteady electric fields in the absence of any charge current. (Steady electric fields have no role here.) Maxwell called the latter “displacement currents.” We want Ampère’s law generalized to allow for unsteady charge currents only. It seems to me that the generalized form of Ampère’s law, if restricted in the reference of its source terms to charge currents, would be reasonably comprehensible for theoreticians of electricity and magnetism in Ampère’s day (d.1836).
Is Ampère’s law, generalized to allow for unsteady charge currents, invariant under Galilean transformations across inertial frames? Yes, but only if (pace Einstein [1905a] or, less clearly, Lorentz [1904]) we say that certain components of the magnetic field in the moving frame are composed of magnetic and electric components in the rest frame. I think Ampère would be pleased. With some ahistorical help from Maxwell, Lorentz, and Einstein, we have the following result, if I am not mistaken: Ampère’s law, suitably generalized, is invariant under Galilean transformations across inertial frames, provided we restrict charge speeds and frame speeds to values small compared to the velocity of light [my demonstration is in endnote 11]. Experience with electricity and magnetism in the days of Ampère fell within that restriction (see further, Earman 1989, 5154).
(to be continued)
References
Earman, J.1989. World Enough and SpaceTime. MIT.
Einstein, A. 1905a. On the Electrodynamics of Moving Bodies. In Stachel 1989.
Rosser, W.G.V. 1968. Classical Electromagnetism via Relativity. Plenum.
Stachel, J. 1989. The Collected Papers of Albert Einstein. Princeton.