The contraction the forces work out for themselves

The length that depends on when introduced length contraction as a disagreement about simultaneity. To measure a moving rod, both ends must be marked at the same time, observers in relative motion disagree about which events are simultaneous, and so they disagree about the length. That essay mentioned the older, dynamical account — Lorentz and FitzGerald’s proposal that motion through the ether physically compresses the forces holding a body together — and set it aside as the account that history discarded. The string that breaks between two rockets then found a case where the contraction has physical consequences that no amount of talk about simultaneity can dismiss.

This essay goes back to the dynamical account and finds that it was never wrong, only incomplete. It is the version John Bell urged physicists to teach alongside the geometric one, in a 1976 lecture titled How to teach special relativity. His point was simple and easy to test. Take an atom, a nucleus with an electron in orbit around it. Accelerate the nucleus gently. Compute the electron’s motion from nothing but Maxwell’s equations for the field of a moving charge and the relativistic law of motion. Do not assume anything about lengths or clocks. The orbit contracts, by exactly the Lorentz factor, and the time per turn stretches by exactly the same factor. Contraction is what the forces do.

An atom accelerated gently #

The calculation needs three ingredients. The nucleus’s field is the field of a charge moving at its current velocity, which Heaviside derived from Maxwell’s equations in 1888; if the nucleus accelerates slowly enough, the extra field caused by the acceleration is negligible. There is a magnetic field too, because a moving charge is a current, and it acts on the moving electron. And the electron obeys Newton’s second law with the relativistic momentum, $\mathbf{p} = \gamma m\mathbf{v}$. That is all.

The figure shows the result. Before the acceleration the electron moves on a circle of radius one around the nucleus. The nucleus is then accelerated smoothly along the horizontal over two hundred orbits, until it moves at 0.6 of the speed of light. Measured in the laboratory, at the laboratory’s instants, the electron’s orbit relative to the nucleus is now an ellipse: 0.810 wide along the direction of motion and 1.012 across it. Their ratio is 0.8003. The Lorentz factor at 0.6c is 1.25, and $1/\gamma$ is 0.8000.

The time the electron takes to go round has changed too. At rest it takes one period; after the acceleration it takes 1.2504 periods, against $\gamma$ = 1.2500. The atom has become a clock that runs slow by the time-dilation factor and a body that is shorter along its motion by the contraction factor, and neither was put in. They came out of Heaviside’s field and the relativistic momentum.

This is the calculation Lorentz’s programme was trying to do in the 1890s, for all of matter at once. What it needed and did not yet have was the relativistic law of motion, with its $\gamma$ in the momentum. Once that is included, a body whose parts are held together by electromagnetic forces rearranges itself in motion into the contracted shape, and processes inside it slow down, both by the factors special relativity predicts.

A shape that follows the speed #

The contraction is not a jump that appears at the end. It follows the speed continuously.

The figure measures the orbit’s width along and across the motion over successive stretches of the acceleration and plots them against the nucleus’s speed at the time. The width across the motion stays at one throughout. The width along the motion falls smoothly as the speed rises and stays close to $1/\gamma$ at every speed on the way, 0.938 at 0.36c against 0.933. The orbit is always, to good accuracy, in the contracted equilibrium for its current speed.

That is what “gently” buys. The acceleration takes two hundred orbits, so at every moment the electron has had many orbits to adjust to the slowly changing field. The orbit’s shape is then set by a slowly varying equilibrium rather than by its history, much as a pendulum whose string is shortened slowly keeps swinging with the same action. Pushed abruptly, an atom would ring, and the shape it ended with would depend on how it was pushed. Pushed gently, it ends in the shape its forces prefer. That is also the condition the string that breaks between two rockets could not meet: the rockets were forced to keep a fixed laboratory distance, not allowed to relax, and the string between them, which would have contracted if it could, broke instead.

One factor for shape and rate #

The agreement holds at every speed tried, and it is exact within the accuracy of the calculation.

The figure repeats the calculation for six final speeds. At 0.2c the width ratio and period ratio are 0.980 and 1.021, equal to $1/\gamma$ and $\gamma$ to three places. At 0.6c they are 0.800 and 1.250. At 0.7c they agree with $1/\gamma$ and $\gamma$ to within a thousandth. Only at 0.8c does a discrepancy appear, a one per cent drift in the width ratio. The orbit has picked up a slight eccentricity during the acceleration, because at the highest speed even two hundred orbits is not quite gentle enough.

Two consequences of special relativity have come out of one calculation with one factor. That is the strongest reason to take the dynamical account seriously. It is not a separate hypothesis for each effect. It is the observation that the laws governing the forces inside matter — Maxwell’s equations and the relativistic law of motion — share a symmetry, the Lorentz transformation, and that anything built from those laws inherits it. A moving atom is a Lorentz-transformed resting atom because its equations of motion are unchanged by the transformation, and the contraction and the dilation are what the transformation does.

The field that was flattened first #

The key ingredient was found before anyone thought matter might contract.

The figure draws the surfaces of equal electric potential around a point charge, at rest and moving at 0.6c and 0.9c. At rest they are spheres. In motion they are ellipsoids flattened along the direction of motion by exactly $1/\gamma$. Heaviside published this in 1888, and it was the direct inspiration for FitzGerald’s suggestion the following year that matter itself might contract. If the fields between the charges in a body are flattened in motion, the body’s equilibrium shape — the arrangement in which all the forces balance — should be flattened by the same factor.

The field that points where the charge is now met this field from the other side, finding that a uniformly moving charge’s field points at its present position rather than its retarded one. The flattening is the other half of that result. A field that is compressed along the motion and points at where the charge is now is exactly the Coulomb field of a charge at rest, seen through a Lorentz transformation. Heaviside had found the Lorentz transformation’s effect on a field seventeen years before Einstein found the transformation.

Forces that weaken along the motion #

The flattening shows up as a change in the forces between moving charges, and the change depends on how the charges are arranged.

The figure computes it for two equal charges moving together at the same velocity. One behind the other, along the line of motion, the electric field between them is weakened by $\gamma^2$ and there is no magnetic force, so the force falls as $1/\gamma^2$: 0.64 of its rest value at 0.6c. Side by side, the electric field is stronger by $\gamma$, but the charges are parallel currents and attract magnetically, and that attraction takes back a fraction $\beta^2$; the net force falls as $1/\gamma$, to 0.80 at 0.6c. The same cancellation, applied to a whole beam of charges, is what magnetism seen as electricity sideways explains and what makes a fast particle beam stop pushing itself apart.

A structure held together by such forces cannot keep its rest shape when it moves. The forces along the motion have weakened by $\gamma^2$ and those across by $\gamma$, and the two can only balance each other as they did at rest if the separations along the motion shrink by $\gamma$. Transverse forces also act on particles whose transverse inertia has grown with $\gamma$ and whose internal motions are slowed by time dilation, which is why the transverse spacings do not change. That accounting, done for every force and every inertia, is what the orbit figure does in one numerical calculation.

A warning the calculation contains #

The calculation has to be set up with care, and the care is informative. Done with a slow electron, moving at a fiftieth of the speed of light, it fails completely: the electron is stripped from its nucleus within a few orbits of the acceleration starting, however gently the acceleration is applied.

The reason is a special property of the inverse-square force. A Kepler orbit is degenerate: its shape and orientation are fixed by a conserved vector that the arrow that says which way the orbit points describes, and a steady push — the push the nucleus’s acceleration effectively applies in its own frame — makes that vector drift steadily, stretching the orbit’s eccentricity without limit. This is the classical version of the Stark effect. No amount of slowness cures it, because the drift adds up over the whole acceleration, and the total change in velocity was thirty times the electron’s orbital speed.

The fix is to break the degeneracy. An electron moving at a third of the speed of light has a relativistic mass that changes around the orbit, which makes the orbit precess: the orbit special relativity cannot close described exactly this rosette. Once the orbit precesses, the steady push averages away over each precession cycle, the orbit’s shape becomes an adiabatic invariant, and the gentle acceleration works. Real atoms are not degenerate either — their electrons’ mutual repulsion and quantum effects break the symmetry — so the dynamical argument applies to them. But it shows that “a body contracts because its forces rebalance” carries a condition. The body must actually be able to settle into its new equilibrium, and a system with an exact degeneracy can be driven somewhere else entirely.

Why the null experiments needed both effects #

The dynamical account also explains why the experiments designed to detect the Earth’s motion through the ether found nothing, and why it took two effects rather than one.

Michelson and Morley’s interferometer of 1887 compared the round-trip times of light along two perpendicular arms of equal length. Motion through a medium that carried light should have made the arm along the motion take longer than the arm across it, by a fraction $\beta^2/2$, and rotating the apparatus should have shifted the fringes. It did not. FitzGerald and Lorentz’s contraction of the arm along the motion by $1/\gamma$ shortens its round-trip time by exactly the amount needed. The clock that has to slow followed the same light-clock logic to its conclusion for time.

Contraction alone, though, fails a slightly different test. In 1932 Kennedy and Thorndike built an interferometer with arms of very unequal length and watched it for months as the Earth’s velocity changed with the seasons and the time of day. With unequal arms, contraction alone would leave a residual time difference that changed as the speed changed, and the fringes would have drifted. They did not drift, which requires the clocks — here, the frequency of the light source, set by atoms in the lamp — to slow by exactly $\gamma$ at the same time. Contraction and dilation together, by the same factor, make every such experiment null. That is precisely what the atom in the first three figures does: its size and its rhythm change together, by $1/\gamma$ and $\gamma$, because both are set by the same forces.

The Trouton–Noble experiment of 1903 is the sharpest case. A charged parallel-plate capacitor moving through the ether at an angle to its plates should, by a straightforward calculation of the forces between its moving charges, feel a torque trying to turn it perpendicular to the motion. None was found. The resolution took until the 1990s to state cleanly: the electromagnetic forces do produce a torque, but the stresses in the capacitor’s material, which hold the plates apart, carry a compensating momentum flow, and the net angular momentum does not change. Any consistent account has to include the forces that keep the plates in place, not only the electric ones — the same point the contracting atom makes. Forces between charges alone are not a body. A body is charges plus whatever holds them in equilibrium, and only the whole transforms simply.

These experiments are why the dynamical and geometric accounts stopped competing. Every null result is explained by both, and no experiment can distinguish them. Einstein’s approach replaced a growing list of dynamical effects, each derived separately, with one principle from which all of them follow. The atom shows that the list was correct. The pole that fits and does not fit and the spinning disc are cases where the geometric principle is much the easier way to see the answer, and the dynamical account would have to reconstruct it from the stresses in a pole or a wheel.

What the dynamical account cannot do by itself #

Bell’s argument has limits, and he stated them.

It covers only forces whose laws are Lorentz-invariant. The calculation works because Maxwell’s equations and the relativistic law of motion share the Lorentz symmetry. A body held together partly by a force that did not share it would contract by some other factor. The dynamical account therefore does not explain why every force in nature transforms the same way. It relies on that fact, which is the content of the principle of relativity.

It does not choose a frame. The calculation was done in the laboratory, where the atom ends up moving. It could equally have been done in the atom’s final rest frame, where the laboratory moves and is contracted. Both descriptions are consistent, and the dynamical account adds no preferred frame, whatever the ether theories that first suggested it assumed.

It is not quantum. A classical electron in orbit radiates and spirals in; the calculation ignores that, because the question was about the shape of the orbit, not its stability. A real atom’s contraction follows from the same symmetry applied to the quantum equations, and the classical calculation stands in for them.

Still open: which account explains which #

Special relativity’s predictions are not in dispute; the question is which account explains them. One view, associated with Harvey Brown’s Physical Relativity, holds that the dynamical account is the explanation: rods contract and clocks slow because of the forces inside them, and spacetime geometry is a summary of the fact that all those forces share one symmetry. The more common view holds that the geometry is fundamental and the forces are Lorentz-invariant because they live in Minkowski spacetime. The two agree on every measurement. They differ on which fact is prior, and the difference matters for how the theory is extended — for instance, in approaches to quantum gravity that start from matter fields rather than from a spacetime geometry. The argument is about explanation, not prediction, and it has not been settled.

The habit worth carrying away is to check a kinematic effect by computing it dynamically, when the forces are known. If every force holding a body together obeys laws with the same symmetry, the body’s equilibrium shape and internal rhythms must transform with that symmetry — so length contraction and time dilation can be derived from the forces as well as from the geometry, and the two derivations must agree. The atom did not know it was supposed to contract. It did anyway, because its forces were Lorentz-invariant, and its contraction was 0.8003 against 0.8000.