The push that does not point where the body goes

The second law is usually remembered as $F = ma$ and was not written that way by Newton, who wrote it as a statement about the rate of change of the quantity of motion. That distinction is invisible in Newtonian mechanics, where the two forms are algebraically identical, and it becomes the whole of the subject at speeds near that of light.

A vector being rotated by a law of motion is not something a mass can do. Multiplying by a scalar changes a vector’s length and nothing about its direction, so no mass — constant, speed-dependent or otherwise — can produce this figure. Something else is going on, and it is worth noticing that the same effect appears in the composition of two boosts, where a product of two things that should be pure boosts turns out to carry a rotation.

What survives, and what it gives #

The law that survives is

$$\mathbf{F} = \frac{d\mathbf{p}}{dt}, \qquad \mathbf{p} = \gamma m \mathbf{v},$$

with $m$ the ordinary invariant mass of the body and $\gamma$ the usual factor. Differentiate the product and two terms appear: one in which $\mathbf{v}$ changes, and one in which $\gamma$ changes because the speed changes. The second term points along the velocity whatever direction the force points, and that is where the rotation comes from.

Carrying the differentiation through gives

$$\mathbf{a} = \frac{1}{\gamma m}\left[\mathbf{F} - \frac{(\mathbf{v}\cdot\mathbf{F})\,\mathbf{v}}{c^2}\right].$$

The component of the force across the motion is divided by $\gamma$. The component along it is divided by $\gamma$ and then reduced again by the projection term, which together make $\gamma^3$. The tangent of the acceleration’s angle is therefore $\gamma^2$ times the tangent of the force’s angle, and the figure above is that single line drawn four times.

Those two divisors were once sold as two masses. Longitudinal mass $\gamma^3 m$ and transverse mass $\gamma m$ appear in the literature of the 1900s, were used by Lorentz and by Einstein’s first paper, and were abandoned within a decade for the reason the figure makes plain: a property of a body that takes two different values depending on which way it is pushed is not a property of the body. The surviving quantity called mass is the invariant one, the same in every frame, and it is the length of the energy–momentum four-vector rather than a coefficient in a force law.

The constant force, followed exactly #

The cleanest case is a single body pushed by a constant force in a straight line — a rocket with an unvarying thrust, or a charged particle in a uniform field. Momentum rises linearly, because that is what a constant $d\mathbf{p}/dt$ means, and the whole of the peculiarity is in converting momentum to velocity.

This is where the usual verbal account goes wrong. Nothing becomes infinite. The momentum is finite at every finite time, the force is constant, the energy grows without bound but only linearly in the distance covered, and the reason the speed saturates is that $v = p/\sqrt{m^2 + p^2/c^2}$ is a saturating function of $p$. Speed is simply not a good coordinate for the motion.

Everything in relativistic dynamics that looks like a wall is one curve read from the wrong side. $\gamma$ rises without bound as $\beta$ approaches one, so a modest change in speed near the limit corresponds to an enormous change in energy and momentum, and the graph appears to show a barrier. Read the other way round — speed against $\gamma$ — it is a perfectly smooth function with nothing dramatic in it anywhere. Nothing resists; the coordinate is badly chosen.

The good coordinate is rapidity. Under a constant proper acceleration the rapidity increases at a constant rate, which is the natural statement of what a steady push does, and it is why boosts add in rapidity and speeds do not.

What the misalignment costs an engineer #

The rotation between force and acceleration is not a curiosity for anybody steering a relativistic beam. A quadrupole magnet focusing a beam applies a transverse force, and the resulting transverse acceleration is smaller than the Newtonian one by γ — so the focal length of a magnetic lens is proportional to the particle’s momentum and a beam line designed for one energy has to be rescaled entirely for another. That is why an accelerator’s optics are quoted in terms of a magnetic rigidity, Bρ, which is momentum divided by charge and has nothing about the mass in it.

Read as a curve rather than as a coordinate, the composition rule says the same thing: two speeds below $c$ compose to a speed below $c$, always, and the curve flattens against one rather than crossing it. That flattening is the whole of what the rapidity coordinate removes — plotted against rapidity the same relation is a straight line at 45°, because rapidities simply add.

The longitudinal case is worse and appears in a different place. Accelerating a bunch of particles along its own direction of motion divides the force by γ³, so a linear accelerator gains less speed per unit of accelerating field the further along it goes — and past a few tens of MeV for an electron it gains essentially none at all. The whole of a kilometre-long electron linac operates on particles travelling at 0.9999c or better, gaining energy and not speed, which is why the sections can all be identical: nothing about the timing has to change once the beam has stopped speeding up.

The rigidity that replaces the mass #

The remark about magnetic rigidity deserves unpacking, because it is the quantity an accelerator is actually designed around and it exists precisely because the mass has stopped being useful.

A charge in a magnetic field goes round a circle whose radius is set by balancing the magnetic force against the rate of change of momentum. Working it through, the radius is

$$\rho = \frac{p}{qB},$$

so the product $B\rho$ — with units of tesla-metres — is the momentum per unit charge and nothing else. No mass appears, no speed appears, and no Lorentz factor appears. A proton at one GeV/c and an electron at one GeV/c follow identical paths through the same magnet despite differing in mass by a factor of two thousand and in speed by a good deal.

The number is worth carrying: one GeV/c of momentum per unit charge is a rigidity of 3.34 tesla-metres. So a beam of that momentum bent through a right angle in a magnet of one tesla needs a radius of 3.34 metres, and everything about the size of a machine follows from that arithmetic and the field the magnets can reach.

Which explains a design decision that looks arbitrary from outside. A synchrotron keeps its beam on a fixed circle while accelerating it, which requires the radius to stay constant while the momentum rises — so the field must rise in exact proportion to the momentum, ramped through the acceleration cycle. A cyclotron, which keeps its field constant, has no choice but to let the radius grow, and its orbital frequency falls as $1/\gamma$ so its drive must be swept. Two machines, two things held fixed, and the same relation between them.

The absence of the mass from the rigidity is also why a detector reports momenta rather than velocities. What a curved track in a magnetic field measures is $B\rho$; converting that to a velocity requires knowing the particle’s identity, and identifying the particle is usually the point of the experiment rather than an input to it.

How the electron’s dynamics was settled #

The story of Kaufmann’s measurement deserves its ending, because the ending is about how a marginal experiment becomes a decisive one.

The situation in 1905 was that three theories of the electron predicted three slightly different dependences of the deflection on speed: Abraham’s, in which the electron is a rigid sphere; Lorentz’s, in which it contracts; and Bucherer’s, in which it contracts while keeping its volume. The differences between the three curves are a few per cent over the accessible range of speeds, and Kaufmann’s data — the best in the world — had scatter of about the same size.

He reported that his results favoured Abraham’s and excluded Lorentz’s, which would have excluded relativity too, since the Lorentz prediction is the relativistic one. Einstein’s response was that the question could not be regarded as settled with the data available, which was correct and unhelpful.

It was settled over the following decade by measurements with better velocity selection and better control of the fields — Bucherer’s own in 1908, and Neumann’s and others’ after him — and the answer was the relativistic one. The interesting part is why it took so long: the three predictions agree exactly at low speed and diverge only near the top of the range, where the beam was faintest and the systematics worst, so the whole discrimination rested on the least reliable end of every dataset.

That is a common shape and worth recognising. When rival predictions agree everywhere except in a corner, the experiment’s difficulty is concentrated in the corner, and an agreement over the easy range settles nothing at all.

What a constant force feels like on board #

One distinction has been used repeatedly above and is worth stating plainly, because the two quantities it separates are both called acceleration.

The proper acceleration is what an accelerometer on the body reads, and what a crew feels as weight. It is defined in the body’s own instantaneous rest frame, and a rocket with a steady thrust and a slowly changing mass has a nearly constant one.

The coordinate acceleration is $d^2x/dt^2$ in whatever frame is being used to describe the motion, and it is the quantity that goes to zero as the speed approaches $c$. For motion in a straight line the two are related by a factor of $\gamma^3$ — which is the same $\gamma^3$ that appeared in the longitudinal divisor, arriving now as a relation between two accelerations rather than between a force and one.

So the crew of a ship burning steadily at one $g$ feel one $g$ for ever, while an observer at the launch site sees their acceleration falling away toward nothing. Both descriptions are correct, neither is approximate, and the confusion between them is the source of most of the trouble people have with the speed limit — the ship never stops accelerating in the only sense anybody on board can measure.

The worldline that leaves something behind #

Draw the constant-proper-acceleration motion on a spacetime diagram and it is a hyperbola: $x^2 - c^2t^2 = (c^2/a)^2$. Its asymptotes are light rays through the origin, which is what “approaches $c$ and never reaches it” looks like when the picture is drawn properly.

The horizon is worth dwelling on because it is the same object as a black hole’s, arrived at without any gravity. Anything happening behind that asymptote can never reach the accelerating body while it keeps accelerating; the body is causally cut off from a region of spacetime by the fact of its own motion. Stop the engine and the horizon disappears, and everything that had been hidden arrives eventually. That is the sharpest difference between this horizon and a black hole’s, which cannot be undone by any decision of the observer’s — and the sharpest similarity is that neither is marked by anything local, so a body crossing one notices nothing.

The worldline of a body under constant proper acceleration is a hyperbola of constant spacetime interval from an event — one of the same curves that put a scale on a tilted axis. That is a coincidence with content in it: constant proper acceleration means constant curvature in spacetime, and the curve of constant interval is the curve of constant curvature. A body at rest in the accelerated frame therefore sits at a fixed interval from the origin, at every moment, for ever.

At one g the distance $c^2/a$ is 9.2 × 10¹⁵ metres — about a light year — so the horizon behind a spacecraft under Earth-like thrust is a year of light away. It is also why the front and rear of an accelerating rod must have different proper accelerations: they sit at different distances from the same horizon, and a rigid body in relativity is defined by that condition rather than by a fixed length.

Where it has actually been measured #

The corrections here are not exotic. Every particle accelerator built since the 1930s is an engineering test of them, and the first one to fail without them was the cyclotron.

The transverse case is the one accelerators live in and it is the cleaner of the two. A charge in a magnetic field feels a purely transverse force, so only the $\gamma m$ divisor appears and the radius of its circular path is exactly proportional to its momentum, with no approximation anywhere. The orbital frequency therefore falls as $1/\gamma$ as the particle is accelerated — a cyclotron with a fixed radio frequency drifts out of step with its own beam for that reason, and the synchrocyclotron, which sweeps the frequency to follow, was the first machine built to accommodate a relativistic correction.

The transverse case is also the one in which the old “transverse mass” survives usefully. A magnetic field does no work, so the speed is constant, so $\gamma$ is constant, so $F = \gamma m a$ holds with a genuinely constant coefficient — and every measurement of a particle’s momentum from the curvature of its track in a magnetic field rests on that. The quantity such a measurement returns is $p$, not $mv$, which is why detector papers quote momenta in units of energy over $c$ rather than quoting velocities.

An accelerator’s designers work along the energy axis rather than the speed axis, and the reason is the shape of the curve. Past a few tens of MeV an electron’s speed is within a fraction of a per cent of $c$ and stays there while the energy goes up by orders of magnitude: a 10 GeV electron and a 100 GeV electron differ in speed by two parts in $10^9$ and in every measurable dynamical property by a factor of ten. “How fast” stops being a useful question about a beam long before the machine is interesting.

The Newtonian limit, and how good it is #

It is worth putting a number on when any of this matters, because the answer is a reminder of how special the terrestrial regime is. The fractional error in $F = ma$ is of order $\beta^2$, so at a satellite’s orbital speed of 7.8 km/s it is 7 parts in 10¹⁰, at the speed of a rifle bullet 1 part in 10¹², and for a car 4 parts in 10¹⁶. Nothing built before the twentieth century could have detected it in a mechanics experiment.

The first place it showed up was not mechanics but the electron. Kaufmann in 1901 measured the deflection of beta particles in crossed electric and magnetic fields and found the charge-to-mass ratio falling with speed, which is exactly the $\gamma$ in the transverse divisor read as a mass. His data was good enough to see the effect and not good enough to distinguish the correct prediction from two rivals, and for several years it appeared to favour the wrong one — a reminder that an experiment agreeing with a theory is a weaker fact than it sounds when three theories are within the error bars.

Where the model stops #

The force is assumed to be given. In practice a force is produced by a field, and the way a field transforms between frames is a separate question with its own surprises — a purely electric force in one frame is electric and magnetic in another, and the two accounts of the resulting motion have to agree. That agreement is a strong check and it is where magnetism comes from.

Nothing here radiates. An accelerating charge loses energy to radiation, so a real charged particle under a constant force does not follow the curve drawn above; the correction is negligible in a linear accelerator and dominant in a circular one, which is why electron synchrotrons hit a ceiling that proton machines do not. The self-force that accounts for it is still not a settled subject.

The body is treated as a point with no internal structure. A real object being accelerated has parts, and the parts communicate at finite speed, so a push applied at one end takes time to be felt at the other and the object deforms while that is happening. The Newtonian idealisation of a rigid body is not available at all in relativity, and what replaces it is a family of definitions each of which keeps some of what rigidity meant and abandons the rest.

And “constant proper acceleration” is a statement about one worldline. An extended body cannot have a single proper acceleration without stretching or compressing, so the rigid rocket of the hyperbola figure is an idealisation that becomes internally inconsistent for any object long enough for the difference to matter. Its length is $c^2/a$, which is a light year at one g and a millimetre at the accelerations inside a particle beam.

What the pictures cannot show #

The hero figure draws two angles against each other and cannot show the thing being described, which is a body moving in a direction it is not being pushed. A vector diagram at one instant would show that, and would then have to be redrawn at the next instant because the speed has changed and $\gamma^2$ with it — so the phenomenon is a family of pictures rather than a picture.

Nor can the spacetime diagram show the horizon as a horizon. It is drawn as a dashed line, which looks like a boundary in space, and it is not: it is the locus of events from which a signal arrives exactly at infinity, and its position depends on the acceleration and on nothing local. An observer crossing it notices nothing, which is the property no drawing conveys.

Where this ladder goes next #

This is the first rung of a new anchor, and what it establishes is that the second law needs no repair, only reading correctly: the rate of change of momentum. Everything peculiar then follows from the relation between momentum and velocity rather than from a new dynamical principle.

The rungs above it are the ones this essay named and set aside. The four-force, which is what the law looks like when written so that it transforms properly, and which makes the $\gamma^3$ and $\gamma$ disappear into an index. The radiation reaction, where an accelerating charge’s own field acts back on it and the resulting equation has solutions that accelerate before being pushed. And collisions, where the conserved quantity is the total four-momentum and the threshold energy for producing a new particle turns out to be far above its rest energy — the reason colliders collide rather than striking fixed targets.

The habit worth carrying away is about the form a law is written in. A law that is stated in terms of a rate of change survives changes of framework that the same law solved for one variable does not. $F = dp/dt$ needed no amendment; $F = ma$ was the special case that had been mistaken for the law, and a century of physics was spent adding corrections to a rearrangement rather than to the thing itself.