Nothing is allowed to be rigid

Push one end of a rod and the other end moves. In every mechanics problem ever set, it moves at the same instant.

It does not. The push travels along the rod as a compression wave, at the material’s speed of sound, and the far end waits. For a three-metre steel rod that wait is six hundred microseconds — long enough to be measured with ordinary electronics, and long enough that any argument treating the rod as rigid has to justify itself.

The wait, and the floor beneath it #

The far end’s wait is the rod’s length over the speed at which the disturbance travels. Air gives 8.8 milliseconds over three metres, steel 600 microseconds, diamond 250.

The interesting number is the one at the bottom. Light crosses three metres in ten nanoseconds, and no disturbance in any material may cross faster. So the shortest possible wait is ten nanoseconds, and a rigid rod — whose far end moves at the same instant — would be a wait of zero.

The wait has a floor beneath it, and the floor is the interval. Two events on a worldline are separated by a proper time no observer disagrees about, and no signal can connect events separated by less of it than light needs. That is the whole constraint: a push applied at one end of a rod cannot be known at the other end sooner than light would carry the news, whatever the rod is made of.

That gap is the whole of the argument. Rigidity is not a large stiffness; it is an infinite signal speed, which is a different kind of quantity from any material property. Steel’s wait is sixty thousand times the light-crossing time, and the stiffest conceivable material would still wait the full ten nanoseconds. There is no sequence of harder materials that converges on rigidity.

The right-hand panel makes that geometry explicit. The delay falls as one over the speed, so it is a straight line on logarithmic axes, and the light cone is a horizontal floor that the line runs into and cannot cross.

Where the ceiling on stiffness comes from #

A material’s speed of sound is the square root of a stiffness over a density. Requiring the speed to stay below light’s therefore caps the stiffness — and it does so in a form that constrains an equation of state directly.

$$c_{\text{s}}^2 = \frac{\mathrm{d}P}{\mathrm{d}\rho} \le c^2$$

Drawn as a graph of pressure against energy density, the constraint is that the curve may nowhere be steeper than the diagonal. That is a strong statement about a substance and it has nothing to do with what the substance is made of; it follows from the requirement that pushing one part of it not tell a distant part faster than light.

The steepest slope each model in that figure reaches is measured off the drawn curve: 0.231 for an ideal gas of nucleons, 0.577 for relativistic degenerate matter — which is exactly $1/\sqrt3$ — and 0.962 for a deliberately stiff nuclear model built to approach the ceiling without crossing it.

The relativistic degenerate value is worth a note. A gas of ultra-relativistic particles has $P = \rho c^2/3$ exactly, so its sound speed is $c/\sqrt3$ and no more. That is the stiffest an ideal relativistic gas can be, and everything stiffer requires interactions.

What that caps #

A star supported by pressure is a competition between gravity and stiffness, so an upper bound on the stiffness is an upper bound on what can be supported.

What that caps, in the one place it is nearly reached, is the mass of a neutron star. Self-gravity has to be opposed by something, and for nuclear matter the opposition is pressure — but the causal ceiling limits how fast that pressure may rise with density. A stiffer equation of state would hold up more mass and is not available at any price, so the maximum mass is set by a speed limit rather than by the strength of anything.

The mass no cold matter can hold up is Chandrasekhar’s limit for electron degeneracy, and it is computed from a known equation of state. For neutron stars the equation of state is not known — it depends on nuclear physics at densities no laboratory reaches — so the corresponding limit cannot be computed the same way.

What can be done instead is to bound it. Assume nothing about nuclear matter except that it is causal, match a stiff equation of state onto the known low-density one at the highest density where the physics is understood, and integrate. The answer is a maximum mass a little under three solar masses, and it holds whatever nuclear physics turns out to be.

That is a remarkably strong result from a remarkably weak assumption, and it does real work. A compact object more massive than that bound and small enough to be compact cannot be held up by anything, so it is a black hole by elimination rather than by observation. When the gravitational-wave event GW190814 produced a companion of 2.6 solar masses, the argument about whether it was a very heavy neutron star or a very light black hole was conducted almost entirely in terms of how close to the causal bound a plausible equation of state could get.

Born’s rigid motion, and what survives of it #

Rigidity is forbidden as a property of a body. A weaker thing is possible and is worth distinguishing, because it is what “rigid” can honestly mean in relativity.

A body is in Born rigid motion if the distance between neighbouring points, measured in their own instantaneous rest frame, stays constant. That is achievable, and for straight-line acceleration it requires each point of the body to accelerate differently: the trailing end harder than the leading end, on hyperbolic worldlines that share a common asymptote.

Two things follow, and both are consequences rather than accidents.

A rod cannot be Born-rigidly accelerated if it is too long. The required proper acceleration diverges towards the trailing end, and at a distance $c^2/a$ behind the leading end it would be infinite. That distance is where the Rindler horizon sits, and no part of the rod may be beyond it.

And Born rigidity cannot be maintained through an arbitrary motion. Herglotz and Noether showed in 1910 that a Born-rigid body has only three degrees of freedom rather than six: once its motion is specified at one point for all time, the rest is fixed. A body cannot be rigidly rotated and accelerated independently, which means Born rigidity is not a substitute for the everyday notion.

The distinction matters for Bell’s spaceship problem, where two rockets accelerating identically is precisely not Born-rigid motion, and the string between them breaks for that reason.

What holds a body together, and how fast #

The reason a sound speed appears at all is worth spelling out, because it turns a relativistic prohibition into a statement about ordinary matter.

What holds a body together, at the level that decides its sound speed, is a chain of masses and springs. Push one and the spring between it and its neighbour begins to compress; the compression travels along the chain at a speed fixed by the spring constant and the mass. That speed is the material’s sound speed, and the whole of this essay is the observation that it cannot be infinite — because an infinitely stiff spring would transmit the push instantly.

A solid is atoms held apart by electromagnetic forces. Push one atom and it moves towards its neighbour; the neighbour feels a larger force and moves in turn. The disturbance is a wave in that chain, and its speed is the square root of the interatomic stiffness over the atomic mass.

Both of those are ordinary numbers. The stiffness is set by the curvature of the interatomic potential near its minimum, which for a covalent solid is a few electronvolts per square ångström; the mass is a few tens of nucleon masses. The ratio gives kilometres per second, and it is small compared with light’s for a reason that has nothing to do with relativity: the binding energy per atom is a few electronvolts and the rest energy per atom is a few tens of GeVs, so the sound speed is smaller than light’s by roughly the square root of that ratio, which is a factor of about $10^5$.

That is the deep reason ordinary matter is nowhere near the causal ceiling. It is not that materials could be stiffer and are not; it is that a substance held together by electromagnetism and made of nucleons has a fixed and enormous mismatch between the energy holding it together and the energy in its mass. Getting near the ceiling requires the two to be comparable, and the only place they are is where the binding is nuclear.

Where the assumption does the damage #

Rigid-body mechanics is a superb approximation and almost nothing in engineering needs correcting. The place it goes wrong is specific and is worth naming, because it is where the assumption is invariably reached for.

Every faster-than-light signalling proposal that involves a long stiff object is the same error. Push one end of a rod a light-year long, and the far end moves a year or two later rather than immediately, and no information has travelled faster than the wave.

The subtler cases are the ones where rigidity is assumed silently. A pole carried into a barn is a paradox only if the pole is rigid; once the compression wave is admitted, the two frames’ descriptions differ about when the pole’s parts are where, which is what the resolution consists of. A rigid body is one whose parts move together in some frame’s “now”, so assuming rigidity is assuming a slicing — and the arguments that use it are arguments about slicings.

There is a positive statement hidden in the negative one, and it is more useful than the prohibition. Because a body is not rigid, its parts have independent states of motion, and “the velocity of the object” is a summary rather than a fact. In relativity that summary has to be qualified by which parts and at what time in which frame — which is the same qualification a length needs, and for the same reason. Length contraction and the impossibility of rigidity are two readings of one thing: an extended object does not have properties at an instant unless an instant has been chosen.

A rod pushed at one end is genuinely compressed for as long as the wave takes to cross, and there is a frame in which its far end is still at rest while its near end is moving. Neither statement is available if the rod is treated as one object with one velocity.

Where the limit is nearly reached #

Nothing on Earth comes close to the causal ceiling, and it is worth knowing what does.

Where the limit is nearly reached, the constituents themselves have to be relativistic. A material’s sound speed is the speed of its collective excitations, and pushing that towards $c$ requires the particles carrying the excitation to be moving at a substantial fraction of it. That happens inside a neutron star and nowhere else within reach — which is why the ceiling is an astrophysical constraint rather than an engineering one.

Diamond’s sound speed is 12 kilometres per second, which is 4 parts in $10^5$ of light. Even the most exotic laboratory material differs from that by a factor of a few. Ordinary matter is nowhere near the ceiling, because its stiffness comes from electromagnetic binding at energies of electronvolts while its mass comes from nucleons at energies of GeVs, and that ratio of a billion is what the small sound speed is.

Inside a neutron star the two energies are comparable: the particles are relativistic and the binding is nuclear, so the sound speed reaches a substantial fraction of light’s. Modern equations of state fitted to the observed masses and radii want sound speeds above $c/\sqrt3$ in the core, which is above the ideal-gas value and taken as evidence that the matter there is strongly interacting rather than a gas of anything.

There is one more place worth mentioning, at the other extreme of density. The early universe’s radiation-dominated era had an equation of state $P = \rho c^2/3$ exactly, so its sound speed was $c/\sqrt3$ — and that number sets the size of the largest structures in the microwave background, because it is how far a pressure wave could travel before the plasma recombined. The acoustic peaks in that spectrum are a direct measurement of a relativistic sound speed, made on the largest object there is.

Whether it approaches $c$ itself is an open question with observational consequences, and it is one of the things gravitational-wave observations of merging neutron stars are expected to settle.

When a body may be called rigid #

The essay has said that rigid-body mechanics is excellent whenever the transit time is short compared with the timescale of the motion, which is true and vague. It can be made precise, and the precise version is more useful than the prohibition.

A rod of length $L$ in which disturbances travel at $c_s$ has a lowest longitudinal mode at $c_s/2L$ — the frequency at which one half-wavelength fits along it. For a three-metre steel rod that is about 830 hertz. Below that frequency the rod responds essentially as one object: every part moves nearly together, the internal deformation is a small correction, and rigid-body mechanics is right. Above it the rod has internal modes of its own, the far end can be moving one way while the near end moves the other, and the rigid description is simply the wrong description.

So rigidity is not a property of a material; it is a relation between two timescales, and the same object is rigid or not according to how fast it is being asked to move. Steel is rigid to a hand and not to a hammer blow; a diamond anvil is rigid to a press and not to a shock wave.

The engineering consequences are everywhere once the criterion is stated this way. A car’s body shell has its first bending mode somewhere between twenty and forty hertz, so it is a rigid body for a steering input at one hertz and a flexible structure for road noise at two hundred. A tall building’s first mode has a period of five to ten seconds, which is squarely inside the band an earthquake delivers, and treating it as rigid is the error the whole of seismic design exists to avoid.

The case where the criterion is most sharply enforced is spacecraft. A satellite’s solar arrays are large, light and floppy, with fundamental modes below one hertz, and its attitude-control loop is trying to point the whole thing. If the loop’s bandwidth reaches the array’s first mode, the controller and the structure drive each other and the vehicle tumbles. Control–structure interaction of exactly that kind has been responsible for lost missions, and the standard remedy is to filter the controller well below the lowest structural mode — which is to say, to force the spacecraft to be operated only in the regime where it is rigid.

That reframing also disposes of the objection that the relativistic prohibition is irrelevant to real engineering. It is relevant in the sense that the finiteness of the transit time is what every one of these designs is about; the light cone merely says that the transit time can never be made zero. What sets the practical limit is the enormously slower speed of sound in ordinary matter, and that limit is met daily.

The delay, measured and exploited #

The wait for the far end is not a thought experiment. It is measured routinely, and in one case it is the basis of an instrument.

The split-Hopkinson pressure bar is the standard way of measuring how a material behaves when it is deformed very fast. A striker hits one end of a long elastic bar, launching a compression pulse that travels along it at the bar’s own sound speed; the pulse crosses a specimen sandwiched between that bar and a second one; strain gauges on both bars record the incident, reflected and transmitted pulses as they pass.

The instrument works because the pulses are separated in time. The reflected pulse comes back past the gauge some hundreds of microseconds after the incident one went by, and it is that separation which lets the three be recorded independently and the specimen’s stress and strain inferred from them. A rigid bar would carry no distinguishable pulses and the technique would not exist.

At the other end of the size scale, the delay is an operational hazard. A freight train two kilometres long is braked by releasing pressure from a pipe running its length, and the release propagates at the speed of sound in the air in that pipe — a few hundred metres a second — so the rear brakes apply some seconds after the front ones. In the interval the rear of the train is still pushing, the slack in the couplings runs in, and the resulting longitudinal forces have derailed trains. The fix is not a stiffer coupling; it is to stop using a pressure wave as the signal, which is what distributed power and electrically controlled brakes do.

And the extreme case is a structure long enough that the delay is measured in hours. A space elevator cable would run tens of thousands of kilometres, and a longitudinal disturbance in a material stressed near its own limit travels at perhaps ten to twenty kilometres a second — so a tug at the bottom would reach the counterweight some hours later. Every proposal for controlling such a structure has to be designed around a control loop whose delay is longer than most of the disturbances it is supposed to reject, which is a much harder problem than holding up the mass.

What the pictures cannot show #

The sound speed is treated as one number. A real solid has at least two — longitudinal and transverse — and an anisotropic one has different speeds in different directions. The causal bound applies to the fastest of them.

The stiffness in the sound speed is not the stiffness in the equation of state. For a solid the relevant modulus involves shear as well as compression; the bound quoted here is the one for the compressional response of a fluid, which is what a neutron star is.

The neutron-star bound depends on where the known physics is matched on. Different choices of the matching density shift the maximum mass by a few tenths of a solar mass, which is why the number is quoted as “a little under three” rather than to three figures.

The rod is one-dimensional and the push is longitudinal. A real push at one end launches shear and surface waves too, and what arrives at the far end first is the fastest of them; the delay drawn is the earliest arrival rather than the full response.

And nothing here is a rigid rotation. A rotating disc raises a separate difficulty — the rim’s circumference and radius cannot both keep their rest-frame values — and that problem was what led Einstein towards curved spacetime in the first place. It is a different argument with a different resolution.

The ladder from here #

Later rungs on this anchor: the Herglotz–Noether theorem and the three degrees of freedom a Born-rigid body has; the relativistic elastic wave equation and the sense in which a stress tensor replaces a stiffness; the maximum-mass bound in detail, including how the matching density is chosen and what it costs; and the rotating disc, whose geometry is not flat and which is the historical entry point to general relativity.

The neighbouring ladders are the string that breaks between two rockets, which is what happens when a body is accelerated in a way that is not Born-rigid, the push that does not point where the body goes, which is the other way relativistic dynamics departs from the everyday, and the mass no cold matter can hold up, where a stiffness limit sets a stellar mass by a different route.