The lever that is twisted and does not turn

Take a lever bent at a right angle, pivoted at its corner, and hold it still with two equal forces, one at the end of each arm and each perpendicular to its arm. One force turns the lever anticlockwise, the other clockwise, with equal torques, and nothing happens. This is the most static situation mechanics has: a rigid object, at rest, in equilibrium, with every torque balanced.

In 1909 Gilbert Lewis and Richard Tolman asked what the same lever looks like to an observer it is moving past, and found that the torques no longer balance. Their lever has been a standing puzzle in every relativity course since — the right-angle lever paradox — because the arithmetic that unbalances it is the ordinary arithmetic of length contraction and the transformation of forces, with nothing exotic in it, and the conclusion is that a moving body can be twisted without turning. The answer, given by Max von Laue two years later, says something about what momentum is that the box of light and the hidden momentum of a magnet each say from another direction: a body under stress carries momentum that its motion does not account for.

Two torques that stop cancelling #

Let the lever move to the right, along one of its arms, at speed $v$. The observer it passes measures two things differently.

The arm along the motion is contracted: its length is $L/\gamma$ instead of $L$, for the reason the length that depends on when gave — the observer’s simultaneous positions of its two ends are a different pair of events. The arm across the motion keeps its length.

The force on the arm along the motion points across the motion, and a force across the motion is reduced by $\gamma$ in the frame in which its point of application moves: a transverse force on a body at rest in one frame is $F/\gamma$ in a frame where the body moves. That is the rule the push that does not point where the body goes used to show force and acceleration parting company; here it applies to forces on a body that does not accelerate at all. The force on the arm across the motion points along the motion, and a force along the motion is unchanged.

Both rules come from the same place. A force is a rate of change of momentum. Momentum across the motion is the same in both frames, since a boost along $x$ does not touch the $y$ component; but the observer’s clock runs $\gamma$ times as many seconds as the lever’s while the same impulse is delivered, because the lever’s clocks are the moving ones. So the same sideways impulse is spread over more of the observer’s time, and the sideways force is smaller by $\gamma$. Along the motion the momentum change and the time both pick up a factor of $\gamma$, and they cancel. The force transformation is not an extra postulate; it is time dilation applied to an impulse.

So in the observer’s frame the two torques are

$$\frac{L}{\gamma}\cdot\frac{F}{\gamma} = \frac{FL}{\gamma^2} \qquad\text{and}\qquad L\cdot F = FL,$$

and they do not cancel. The net torque is $FL(1/\gamma^2 - 1) = -FL\beta^2$, where $\beta = v/c$: clockwise, proportional to the square of the speed.

At 0.8 of light’s speed the twist is 0.64 $FL$, nearly two-thirds of either torque. The lever does not turn: it is at rest in its own frame, and an observer moving past it cannot set it rotating by moving. Every physical prediction in the lever’s frame — that the lever stays put — must hold in the observer’s frame as well. So the observer has an unbalanced torque on a body whose angular momentum, as the observer would naively compute it, is not changing. Something has been left out.

The orientations at which it vanishes #

The twist depends on how the lever is held relative to its motion. Turn it so that its first arm makes an angle $\alpha$ with the motion, contract each arm’s component along the motion, reduce each force’s component across it, and the net torque comes out as

$$\tau = -FL\beta^2\cos 2\alpha.$$

It is largest when one arm or the other lies along the motion, with opposite signs in the two cases — turning the lever through a right angle exchanges the roles of its arms — and it vanishes when both arms are at 45° to the motion, which is the orientation in which contraction and the transformation of force treat the two arms identically.

That pattern is itself a clue. A torque that depends on orientation through $\cos 2\alpha$ is the signature of something quadratic in the lever’s geometry — of the product of a direction with itself — and the stress inside a body is exactly such a thing. The forces at the ends of the lever compress or bend its arms, and the arms carry the forces to the pivot. In the lever’s own frame that is a static stress with no energy moving anywhere. In the observer’s frame it is not static, and the reason is the work the forces do.

Where the forces’ work goes #

In the observer’s frame both points of application are moving at $v$. A force whose point of application moves does work at the rate $\mathbf F\cdot\mathbf v$, the rule the floor that does no work insisted on for a jumper and a floor. The force on the arm across the motion points along the motion, so it does work at the rate $Fv$: energy is being fed into the lever at the end of that arm. The pivot pushes back with a force that does work at the rate $-Fv$: energy is being taken out at the corner. Nothing accumulates. So in the observer’s frame there is a steady current of energy, $Fv$ joules per second, flowing down the arm from the end to the pivot.

Energy that flows carries momentum. That is the content of $E = mc^2$ applied to a current rather than a lump: energy $E$ moving at speed $u$ carries momentum $Eu/c^2$, so a current of $P$ joules per second along a path of length $L$ carries momentum $PL/c^2$ along the path. The box of light that weighs something used the same accounting for the light inside a box; the lever’s arm is carrying energy through matter rather than through empty space, and the accounting is the same.

The size of that current is worth a moment’s arithmetic, because it is startling. A kilonewton at the end of a one-metre arm, on a lever going by at 0.8 of light’s speed, is doing work at $Fv = 2.4\times10^{11}$ watts — the output of a few hundred power stations — fed in at one end of the arm and taken out at the pivot. Nothing heats, nothing moves inside the lever, and in the lever’s own frame there is no current at all. The current exists only in the observer’s description, as the observer’s version of a static stress. Its momentum, $PL/c^2$, is $2.7\times10^{-6}$ kilogram-metres per second: small, because $c^2$ is large, but no smaller than the torque that needs it.

The arm’s energy current runs across the lever’s motion, from its end to its corner. So the moving lever carries momentum across its direction of motion, of size $FLv/c^2$ in this orientation. For a general orientation, adding the currents in the two arms,

$$G_\perp = -\frac{FLv}{c^2}\cos 2\alpha.$$

The torque that keeps the angular momentum still #

Now the bookkeeping closes. The angular momentum of a body about a point that moves with it changes at a rate equal to the torque about that point, minus the velocity of the point crossed with the body’s momentum:

$$\frac{d\mathbf J}{dt} = \boldsymbol\tau - \mathbf v \times \mathbf G.$$

For an ordinary body the last term is zero, because the momentum is parallel to the velocity. For the lever it is not: the momentum has a component across the motion, and $\mathbf v \times \mathbf G$ is exactly $-FL\beta^2\cos 2\alpha$, equal to the torque at every orientation. The torque is not unbalanced. It is precisely the torque needed to keep the angular momentum about the moving pivot constant while a sideways momentum is carried along with the pivot. A body whose momentum does not point where it goes has angular momentum about any moving point that grows steadily, and holding it still takes a torque.

Seen this way, the lever’s paradox is the mechanical version of the momentum of something that is not moving, where a current loop in an electric field carried momentum with nothing in it moving, and the centre-of-energy theorem forced it to. Here every part of the lever moves in a straight line at the same speed, and its momentum still has a sideways part. The parts are not where the momentum is; the energy currents in the stressed material are.

There is a third way to see the same thing, through the centre of energy. The centre that is not a place found that a body’s centre of energy shifts sideways when it is set moving, if energy circulates inside it — a spinning wheel’s centre moves towards its faster-moving rim. The lever has no rotation, but in the observer’s frame it has an energy current running down one arm, and a current displaced from the line of the motion carries the same sort of sideways shift. A body whose centre of energy sits off its geometric centre by an amount that depends on its stress has a momentum that is not simply its velocity times its total energy over $c^2$, and the torque is what keeps that offset from rotating. Every one of these descriptions is the statement of mass is a form of energy carried to its end: if energy has inertia wherever it is, then energy moving through a body has momentum wherever it moves, and the body’s momentum is the sum.

At everyday speeds the effect is minute but definite. A kilonewton on each end of a one-metre lever, carried at the Earth’s orbital speed of 30 kilometres a second, puts the sideways momentum at $3\times10^{-10}$ kilogram-metres per second and the twist at ten micronewton-metres, a part in a hundred million of either torque. Nothing would ever detect the lever’s twist directly. It is the logic, not the size, that matters.

The capacitor that should have turned #

The experiment that made the problem urgent came before the lever. In 1903 Frederick Trouton and H. R. Noble suspended a charged parallel-plate capacitor on a fine wire and looked for it to twist. The reasoning was the ether’s: a capacitor moving through the ether at an angle to its plates carries charges whose moving fields should exert a torque, turning the plates to face the motion. The Earth’s orbital motion supplied the speed, and the twist expected was large enough to see. None appeared.

A capacitor is a strut. Its plates attract with a force equal to its stored energy divided by their separation, and whatever holds them apart is compressed. Run the same arithmetic for a single straight strut of length $L$ squeezed by forces $F$ at its ends, moving at an angle $\alpha$ to its length, and the torque in the observer’s frame is

$$\tau = \tfrac12 FL\beta^2\sin 2\alpha,$$

largest at 45° and zero along and across the motion.

The answer to their null result is the lever’s answer. The torque on the electrical forces between the moving plates is real in the frame where they move, but the stressed material holding them apart carries an energy current and a sideways momentum, and the torque goes into keeping the angular momentum about the moving capacitor constant. In relativity the experiment’s result is predicted, not merely survived: a body in equilibrium in one frame is in equilibrium in all of them. The experiment was repeated with better sensitivity in the 1920s and again in 1994 with a modern torsion balance, each time finding nothing, and each null is a test of the same accounting. The contraction the forces work out for themselves told the half of this story in which the forces inside matter, being Lorentz-invariant, contract it by the right amount; the lever is the half in which their stresses carry the right momentum.

What Lewis and Tolman wanted it for #

Lewis and Tolman did not set the problem up to puzzle anyone. Their paper of 1909 was a defence of what they called non-Newtonian mechanics, and they used the lever to argue that a moving body’s resistance to a sideways push differs from its resistance to a push along its motion — the transverse and longitudinal masses then in use. The lever’s unbalanced torques followed from those masses, and they took the lever’s refusal to turn as a sign that the mechanics of moving bodies needed rebuilding from its foundations.

The rebuilding came, but not where they put it. Transverse and longitudinal mass were abandoned, and the resolution sits in the momentum rather than in the mass: the lever’s momentum gains a term that no assignment of mass to its parts can produce, because it belongs to energy flowing through the parts rather than to the parts. The question the lever was invented to raise — whether a moving observer can detect motion by watching a body that is balanced at rest — has the answer relativity requires. It cannot. A torque appears, and the body’s angular momentum about its moving pivot stays exactly constant, which is all that turning or not turning ever measured.

Stress has inertia #

Laue’s general result, from 1911, is that the momentum of a moving body under stress is not $\gamma m\mathbf v$. It contains an extra term, $\gamma$ times the velocity contracted with the stresses integrated over the body, divided by $c^2$. For a complete, isolated system in equilibrium the integrated stresses sum to zero — Laue’s theorem — and the term disappears; for a body held by forces from outside, like the lever or a capacitor’s spacer, it does not. The stresses then contribute to the momentum, and to the inertia: a rod compressed along its length and pushed along its length is harder to accelerate than the same rod unstressed, by its compression force times its length divided by $c^2$.

That is the same statement that pressure has weight, which the light that pulls twice as hard as its mass used for light in a box and the sound speed a neutron star needs for the matter in a star, where the inertia of pressure is what limits how stiff the matter can be. Energy, momentum and stress are three faces of one object — the stress–energy tensor — and a change of frame mixes them. What is pure stress in the lever’s own frame is partly an energy current and partly a momentum in the observer’s. The lever’s twist is the observable trace of that mixing in a body that, in its own frame, is doing nothing at all.

What the figures leave out #

The figures treat the lever as a mechanical object whose arms carry force without any account of how. A real lever’s stresses are carried by its interatomic forces, and the energy current flows through the electromagnetic fields between its atoms; the momentum density is distributed through the material rather than concentrated along a line. The resolution does not depend on those details, which is its strength: whatever carries the stress, the energy current is fixed by the work done at the ends, and the momentum it carries is fixed by $E = mc^2$.

The figures also take the lever to be in equilibrium and moving uniformly. If it were accelerated, or if the forces changed, the energy stored in its stresses would change, and the energy currents would not be steady. The domain of the drawings is a body in static equilibrium in its own frame, observed from a frame moving uniformly past it, with forces applied at points rather than over areas.

Still open: where exactly the momentum is #

The total is not in doubt. What has been argued for a century is the local question: where, in a stressed body or in a capacitor’s field, the momentum density sits, and whether the field’s momentum or the matter’s is the one to count. The answers depend on how the stress–energy of matter and field are separated, which is partly a convention — the same division between “field momentum” and “mechanical momentum” that makes the hidden momentum of a magnet look paradoxical in one bookkeeping and natural in another. Treatments that compute the full stress–energy of a model capacitor, with its dielectric and its charges, agree on the total and differ on how they describe the parts, and whether there is a preferred split is a question about the foundations of continuum mechanics in relativity rather than about any measurement.

The lever’s own answer is short. A right-angled lever balanced in its own frame and seen moving at $v$ has torques $FL/\gamma^2$ and $FL$, a net twist of $-FL\beta^2\cos 2\alpha$ — 0.64 $FL$ at 0.8c — and does not turn, because the forces’ work flows through its arms as an energy current whose momentum, $-(FLv/c^2)\cos 2\alpha$, lies across its motion; carrying that momentum along takes exactly the torque. The momentum of a body is not always its mass times its velocity. When the body is stressed, some of it is energy on the move.