On the electrodynamics of moving bodies · Capstone
Rebuild the electrodynamics of moving bodies
How much of this paper can you rebuild from two postulates and one definition, and which two steps does it choose rather than derive?
What to do with this page
Explain to a friend why “at the same time” needs a procedure, how that procedure changes the electrodynamics of a magnet and a coil, and which two things the paper chooses rather than derives.
Each claim below links to the passage it is read from. Follow the links and the argument is the paper's; read only this page and it is a summary of the paper, which is a different thing and says so.
The nine claims, in the order the paper makes them
The chain below fixes what must come before what, and 105 arrangements satisfy it. The paper prints one of them; the others are not mistakes.
The first five claims are the kinematical part and the last four are the electrodynamical one. The second half is where the paper does the work its title promises, and it is the half a summary usually drops.
An assumptionNeeds nothing before it
Two postulates. The same laws of electrodynamics and optics hold in every coordinate system in which the equations of mechanics hold, and a ray of light in empty space always travels at one definite speed, whatever the motion of the body that emitted it. The introduction announces both; the head of the kinematical part states them as numbered principles.
A definitionUses claim 1
Clocks at two places are brought into agreement by a stipulation rather than by a measurement. A ray leaves the first clock, is reflected at the second and returns, and the second clock is set so that the outward and the return times come out equal. The paper says in its own words that this is settled by definition.
A definition, not a result. It rests on the stipulation of equal light times each way and on setting the imprecision in simultaneity at one place aside, which is why it names those two and derives nothing.
A derivationUses claim 1 and claim 2
Simultaneity carries no absolute meaning. Clocks synchronized by that stipulation in the stationary system are not synchronous for an observer travelling with a rod, so the length of a moving rod and the time of a distant event are answers to a stated measuring procedure rather than properties a body carries with it.
A derivationUses claim 1 and claim 2
The two postulates, the definition of time, homogeneity, isotropy and reciprocity together fix the transformation between the systems. The paper prints a time equation and three position equations, carrying the factor it writes as beta, which is the modern gamma, and a further factor of the speed alone that the end of the section shows to equal unity. The auxiliary coordinate introduced earlier in the section, the distance from the moving origin reckoned in stationary-system terms, is not the moving system's own coordinate.
A derivationUses claim 4
Three consequences follow from the transformation. A sphere at rest in the moving system is measured from the stationary system as an ellipsoid shortened along the direction of motion by the square root of one less the squared speed ratio; a clock carried along runs slow by that same factor; and two collinear velocities compose so that their result stays below the speed of light.
A derivationUses claim 1 and claim 4
The Maxwell and Hertz equations for empty space keep their form in the moving system, with the electric and the magnetic components mixing under the transformation. What the magnet's rest system accounts for as a force on the conductor's moving charges, proportional to the charge's velocity across the magnetic field, is in the conductor's rest system an electric field. The asymmetry the paper opens with disappears, and the electromotive force is left as an auxiliary concept.
Read the direction with care. In the magnet's rest system there is a force on the conductor's moving charges; in the conductor's rest system there is an electric field. The two accounts agree, and the asymmetry was in the description rather than in the phenomenon.
A derivationUses claim 6
The transformed wave yields Doppler's principle and aberration, and the energy of a light complex changes with the observer's motion by the same law as its frequency. The pressure of radiation on a moving mirror follows from the same transformation.
A derivationUses claim 6
With convection currents present the equations again keep their form, provided the density of electricity and the velocity of electricity transform together. Lorentz's electromagnetic foundation therefore agrees with the principle of relativity, and the charge of a moving body comes out the same in both systems.
This extends the claim before it rather than repeating it. There the equations kept their form for empty space; here they keep it with electricity in motion, which is what lets the charge of a moving body come out the same in both systems.
A derivationUses claim 6 and claim 8
For a slowly accelerated electron the paper adopts a definition of force, comparing the force in the electron's momentary rest system with the acceleration reckoned in the stationary system, and obtains a longitudinal and a transverse coefficient which differ from each other and from the rest mass. The kinetic energy it derives grows without bound as the speed approaches that of light, and the section closes by saying that another definition of force would give other numbers for the masses.
The transverse coefficient depends on the definition of force this section adopts. The predictions an experiment could check, the potential difference and the radius of the deflection, do not.
What the argument is granted
Every claim above names the assumptions it uses. Two of them are the paper's own choices rather than things it establishes, and they are the two this capstone exists to separate out: the stipulation that light takes equal times out and back, which is how distant clocks are set, and the definition of force behind the electron's two masses, which the paper names as a choice in its closing paragraph. The rest are premises and idealizations.
Premise
A stationary system is one in which Newton's mechanical equations hold, and the second system moves uniformly with respect to it.
Premise
Space and time are homogeneous, and space is isotropic.
Premise
The equations of the transformation are linear, which the paper argues from that homogeneity rather than assuming outright.
Stipulation
Distant clocks are set by the stipulation that light takes equal times to travel out and to travel back between them. This is the first of the paper's two choices.
Premise
The speed of light in empty space is a universal constant in the stationary system.
Idealization
Rigid measuring rods and identical clocks, on which the introduction says every theory of this kind rests, because its assertions concern relations between rigid bodies, clocks and electromagnetic processes.
Idealization
Whether two events at approximately the same place are simultaneous is taken as unproblematic. The footnote to the first section sets that imprecision aside, and it is about simultaneity at one place rather than about the concept of the rigid body.
Premise
The principle of relativity is applied to the Maxwell and Hertz equations, so that they hold in the moving system if they hold in the stationary one.
Convention
Force is defined so that the force measured in the electron's momentary rest system is set against the acceleration reckoned in the stationary system. This is the second of the paper's two choices, and the paper names it as one.
Idealization
The electron is a point charge, slowly accelerated, and it radiates nothing.
The displays this argument turns on
The rule: out and back take equal times
The condition that sets a distant clock. Read it as the stipulation it is: the outward and the return light times are made equal, not found equal.
The outward assigned travel time equals the return assigned travel time.
Time in the moving system
The time equation of the transformation, in the paper's own notation and in modern symbols generated from the same expression tree.
The moving time is gamma times t minus v x over c squared.
Velocity along the motion
How two collinear velocities compose, with the denominator that keeps the result below the speed of light.
The velocity along x in the moving system is the velocity along x in the stationary system minus the speed of the moving system, divided by one minus that speed times the stationary velocity along x over the speed of light squared.
Electric field along y
One component of the field transformation, where a field that is purely electric in one system is a mixture of the two in the other.
The electric field along y in the moving system is gamma times the difference: the stationary electric field along y, minus v times the stationary magnetic field along z.
Where to watch the quantities move
Send a signal out and back between two stations and watch which time the agreement assigns to the reflection. The laboratory registers a second flash that leaves later, whose assigned time again falls midway through its own pair, which is what makes the rule a convention rather than a discovery.
Follow the teaching tape through one boost and watch which pairs of events count as a length and which do not.
Begin with a field that is purely electric in the stationary system and change system. The magnetic component that appears is not a second field; it is the same field described from a system in motion.
Change the definition of force and watch which numbers move. One transverse coefficient answers to the paper's convention and another to the later one, the longitudinal coefficient answers to neither, and the quantities an experiment could actually measure stay where they are.
What this does not claim
Spacetime diagrams and the invariance of the interval come from Minkowski's lecture of 1908. They are later aids and neither is a premise of anything above. The train and the embankment are Einstein's popular illustration of 1917 for the first two sections, not the argument of 1905. This page does not say what led Einstein to the paper: the unsuccessful attempts to detect the earth's motion through the light medium appear in the introduction among examples of a similar kind, and a reconstruction is not a cause. The two velocity-dependent masses are the paper's own language, and the modern account instead keeps one invariant mass and puts the speed dependence into energy and momentum; both are shown here, and neither is folded into the other.