The throw a uniform field bends into a chain

The push that does not point where the body goes established the rule that governs everything here: in relativity a force changes momentum at the rate it always did, $d\mathbf p/dt = \mathbf F$, but momentum is no longer mass times velocity. It is $\gamma m\mathbf v$, which can grow without limit while the speed stays below $c$. The speed that stops while the energy keeps coming followed one consequence in a straight line: an electron pushed along a field by a constant force gains energy at a constant rate while its speed creeps towards $c$ and stops.

That essay pushed along the motion. This one pushes across it, which is the oldest problem in mechanics. A stone thrown horizontally keeps its horizontal speed, gains vertical speed at a steady rate, and draws a parabola, and Galileo’s insight was that the two motions are independent: nothing horizontal changes because the force is vertical. A charge thrown across a uniform electric field, slowly, does exactly the same. The interesting question is what happens to that independence when the charge is fast, and the answer is that it fails in a precise and instructive way.

Momentum fixed, speed not #

Put the field along $y$ and launch the charge along $x$ with momentum $p_0$. The force has no $x$ component, so the momentum along $x$ never changes: $p_x = p_0$ for ever. The momentum along $y$ grows steadily, $p_y = qEt$. Those two statements are exactly Galileo’s, transcribed to momentum, and they are exact in relativity.

What is not exact is the step from momentum to speed. In Newton’s mechanics speed is momentum divided by a fixed mass, so a fixed $p_x$ means a fixed $v_x$. In relativity speed is momentum times $c^2$ divided by the energy,

$$v_x = \frac{p_0 c^2}{\varepsilon}, \qquad \varepsilon = \sqrt{\varepsilon_0^2 + (qEct)^2},$$

and the energy is not fixed. The field does work on the charge as it moves along $y$, and the energy rises without limit. So the sideways speed falls. There is no sideways force; the sideways momentum is unchanged; and the charge crosses the field more and more slowly.

The second figure plots the components. The speed along the field rises towards $c$; the speed across it falls from its launch value towards zero; the total approaches $c$ from below and never reaches it. Newton’s version would keep the sideways speed at its launch value and let the speed along the field grow without limit, so that the total passed the speed of light within a time $\sqrt{1-\beta_0^2}\,mc/qE$ — 0.8 units for a launch at $0.6c$. Relativity has to stop that somewhere, and where it stops it is not in the direction of the force. The field pushes only along $y$, and the speed limit is enforced partly by slowing $x$.

This is the plainest example of the result that force and acceleration point in different directions at high speed. The acceleration has an $x$ component, negative, while the force has none. The push that does not point where the body goes derived the general relation; here it shows up as a charge that decelerates in a direction in which nothing touches it.

A path written in hyperbolic functions #

The positions follow by integrating the speeds. With lengths in units of $mc^2/qE$ — the distance over which the field does work equal to the charge’s rest energy, 0.51 metres for an electron in a field of a megavolt per metre — and $\varepsilon_0$ the launch energy,

$$x = \frac{p_0 c}{qE}\,\operatorname{asinh}\frac{qEct}{\varepsilon_0}, \qquad y = \frac{\sqrt{\varepsilon_0^2 + (qEct)^2} - \varepsilon_0}{qE},$$

and eliminating the time gives the path:

$$y = \frac{\varepsilon_0}{qE}\left(\cosh\frac{qE\,x}{p_0 c} - 1\right).$$

Near the vertex the hyperbolic cosine is a parabola, $1 + u^2/2$, and the path reduces to $y = qE\,x^2/2\gamma_0 m v_0^2$: Newton’s parabola, but with the inertia $\gamma_0 m$ in place of $m$. That is the first figure’s first lesson. Close to the launch, a fast charge is deflected as though it were $\gamma_0$ times heavier — seven times, for a launch at $0.99c$ — and the old idea of a “transverse mass” $\gamma m$, which the push that does not point where the body goes explained as a sideways shadow of the energy, gets the initial curvature right.

Further from the vertex the parabola is overtaken. The hyperbolic cosine grows exponentially where the parabola grows quadratically, so every path eventually turns steeply along the field. For the slow launches the turn comes after the charge has become relativistic along the field, and it comes from the sideways slowing: the charge takes ever longer to cover each unit of $x$, and the field, pushing all that time, carries it further along $y$ than Newton’s law with its constant sideways speed would allow.

The sideways distance is a clock #

The formula for $x$ has a reading that the third figure makes concrete. A moving clock ticks at a rate $d\tau/dt = mc^2/\varepsilon$, so the rate of change of $x$ with the charge’s own proper time is

$$\frac{dx}{d\tau} = \frac{dx}{dt}\,\frac{dt}{d\tau} = \frac{p_0c^2}{\varepsilon}\cdot\frac{\varepsilon}{mc^2} = \frac{p_0}{m},$$

a constant. The charge’s sideways distance is its proper time multiplied by a fixed rate, the component of its four-velocity across the field. Every unit of sideways distance costs the same amount of the charge’s own time, whatever the laboratory clock says. The clock that does not feel the turn showed that a clock’s rate depends only on its speed; here the clock’s reading is laid out along the $x$ axis like the scale of a ruler.

The proper time of a body under a constant force grows only as the logarithm of the laboratory’s time, because the body’s growing energy slows its clock ever more. The wall of silence behind a rocket found the same logarithm for a rocket accelerating at a steady one gravity: twenty years of shipboard time take it to the centre of the galaxy while tens of thousands of years pass outside. A charge in a uniform field is that rocket pushed sideways at launch, and its sideways progress inherits the rocket’s logarithmic clock. At a laboratory time of twenty units it has travelled 2.6 units across, where Newton’s constant sideways speed would have taken it twelve, and at two hundred units it will have managed only about 4.3.

The chain that hangs the same way #

The hyperbolic cosine is the curve of a chain hanging under its own weight, and the fourth figure lays the paths over it. Measured in units of $p_0c/qE$ along both axes, every path is the chain’s curve stretched along the field by the factor $\varepsilon_0/p_0c = 1/\beta_0$. A charge launched at $0.3c$ draws a chain stretched more than threefold; one launched at $0.999c$ draws the chain itself to a part in a thousand. For a particle moving at the speed of light — a photon, which has no rest energy, so that $\varepsilon_0 = p_0c$ exactly — the path in a uniform field would be the catenary with no stretching at all, if a photon felt the field; a massless charged particle, were there one, would hang across a uniform field exactly like a chain.

The coincidence of shapes has a reason, and it is the same bookkeeping in both. A hanging chain is in balance because its horizontal tension is the same at every point — nothing pulls it sideways — while its vertical tension at any point is the weight of the chain between that point and the lowest point, which grows with the length of chain. The slope of the chain is the ratio of the two. A charge in a uniform field has a sideways momentum that is the same at every moment, while its momentum along the field grows with time, and the slope of its path is the ratio of those. In both cases the growing quantity is not measured along the horizontal: the chain’s weight grows with arc length, not with horizontal distance, and the charge’s momentum grows with time, which at high energy runs ahead of its sideways progress in exactly the way arc length runs ahead of horizontal distance on a steep chain. The chain that swings in Bessel’s notes found the same chain’s tension, varying along its length, setting the frequencies of its swings.

Galileo knew both curves and mistook one for the other: he proposed that a hanging chain was a parabola, which is the approximation near the vertex that the projectile’s parabola is to the charge’s path. The chain’s true shape was found in 1691, in answer to a challenge from Jakob Bernoulli, by Leibniz, Huygens and Johann Bernoulli, and it took relativity to make the projectile’s path correspond to it.

Why gravity would not do this #

The calculation has a twin in which every number changes, and the difference says something about the equivalence principle. Replace the electric field by a uniform gravitational field, and the force on the thrown body is no longer fixed. An electric force is $qE$ whatever the charge’s speed, because charge does not grow with energy. Gravity pulls on energy, so a body moving fast is pulled harder in proportion to its inertia, and the cancellation that makes every body fall alike at low speed continues at high speed. A fast stone in a uniform gravitational field is not deflected $\gamma$ times less than Newton says, as the fast charge is near its launch; it is deflected by an amount that does not depend on what it is made of, only on how fast it is going.

What it does depend on is the speed, and the dependence is the one the half of the bend a slow body never feels found for bodies passing the Sun: a slow body feels only the curvature of time, and a body near the speed of light feels the curvature of space as well, and is bent twice as much. In a uniform field the curvature of space is absent — the geometry is that of an accelerating frame, flat but described from a rocket — and the rocket in which light travels on circles found that a beam sent across such a frame follows a circular arc. A slow stone in the same frame draws a parabola, and a fast one a curve between the two. A charge in an electric field draws something else again, the stretched chain, because the electric field cannot see the charge’s energy. The three curves — parabola, circle and hyperbolic cosine — are the same Newtonian parabola corrected in three different ways for what the force acts on.

An electron beam across a deflecting field #

The last figure puts the result into an apparatus. An electron with a kinetic energy of one megaelectronvolt, moving at 94 per cent of the speed of light, enters a region of uniform transverse electric field of one megavolt per metre — a strong field, near what a well-made deflector can hold in vacuum. After a metre it has been deflected by 39 centimetres. Newton’s law with the electron’s rest mass, using $\tfrac12 mv^2$ for the kinetic energy, gets 25 centimetres, and gets it from a speed of nearly twice that of light, which is the kind of answer that announces its own failure. Newton’s law with the correct speed and the inertia $\gamma m$ gets 37 centimetres, close at first and falling short as the electron’s sideways speed drops in the field.

The curvature at the launch carries a quantity worth naming. The parabola with inertia $\gamma_0 m$ has curvature $qE/\gamma_0 m v_0^2$, which is $qE/p_0 v_0$: an electric field bends a charge according to the product of its momentum and its speed. A magnetic field, whose force is $qv_0B$ and whose path is a circle of radius $p_0/qB$, bends it according to its momentum alone. So the two kinds of field sort particles differently — an electrostatic analyser selects a value of $pv$, which for slow particles is twice the kinetic energy, and a magnetic one selects $p$ — and instruments that pass a beam through both, as the earliest measurements of the electron’s mass at high speed did, can solve for the speed and the momentum separately. Those measurements, by Walter Kaufmann from 1901 and Alfred Bucherer in 1908, were how the growth of inertia with speed was first seen, in the bending of beta rays across exactly the kind of field drawn here.

The practical lesson is the one accelerator designers learned early: electric fields are poor at steering fast particles. The deflection over a fixed distance goes as the field divided by the particle’s energy and its speed squared, and for a particle near the speed of light the energy is the whole of its inertia. A magnetic field does better because its force grows with the speed, $qvB$, and at the speed of light a magnetic field of one tesla exerts the same force as an electric field of three hundred megavolts per metre, far beyond what any surface can hold. The turns a muon makes at any energy and every circular accelerator rely on that. Electrostatic deflectors survive where the particles are slow, in the cathode-ray tubes and electron microscopes the voltage that bends an electron like glass described, and in the gentle steering of beams that do not need to be turned far.

Where the clean curve stops #

The hyperbolic cosine is exact for a point charge in a field that is uniform everywhere and unchanging in time, with nothing else in the universe. A real field is uniform only over a finite region, and a charge that leaves it carries the momentum it gained. A charge accelerated this hard also radiates: an electron gaining a megaelectronvolt per metre loses a negligible fraction of it to radiation, but in fields approaching those found near the surfaces of pulsars, or in the focus of the most intense lasers, the radiation reaction that the light that takes back an electron’s energy computed changes the path. And a uniform electric field of arbitrary extent is itself unstable in quantum theory: above a critical strength of about $10^{18}$ volts per metre it creates electron–positron pairs from the vacuum, and the field is discharged.

What a trace on a screen does not record #

The figures draw where the charge is, and a path is the least informative way to describe its motion. The same curve is traced by charges of different masses launched with the same ratio of energy to momentum, and nothing in the curve tells which clock the charge carried, which is the most interesting thing about it: the third figure needed the formula, not the path, to show that the sideways distance is a proper time. Nor does any figure show the field’s energy flowing into the charge, or the momentum the charge’s own field carries along with it, which is the subject of a separate and still subtle calculation for charges that are accelerating.

Still open: whether a uniform field can be held at the critical strength #

The path drawn here assumes the field stays uniform while it accelerates the charge, and in the strongest fields that assumption is the one in question. The critical field at which the vacuum begins to produce pairs, about $1.3 \times 10^{18}$ volts per metre, has not been reached in any laboratory; the most intense lasers reach it only in the frame of an ultrarelativistic electron colliding with the beam, where the field is boosted by the electron’s Lorentz factor. Experiments of that kind have seen the first signs of the pair production the theory predicts. Whether a field at the critical strength can be sustained, or is discharged into pairs as fast as it is built, and how a charge’s path looks in the field that remains, is a question for experiments now being prepared.

The habit worth carrying away is to keep track of which quantity a force conserves. A uniform field across a charge’s motion leaves its sideways momentum fixed and adds to its energy, so its sideways speed $p_0c^2/\varepsilon$ falls with no sideways force — and its path is not Newton’s parabola but $y = (\varepsilon_0/qE)(\cosh(qEx/p_0c) - 1)$, a hanging chain stretched by $1/\beta_0$, along which the sideways distance is the charge’s own proper time times $p_0/m$. Whenever a conserved momentum is shared with a growing energy, the corresponding speed falls, and the curve stops being a parabola.