The tail that has to pull harder than the nose

The string that breaks between two rockets takes two identical rockets, starting at rest and programmed to accelerate exactly alike, with a thread tied between them. In the launch frame, the distance between them stays the same throughout, since their motions are the same. And yet the thread snaps. It breaks because a moving thread must be shorter than its rest length, and so it is being stretched in its own frame. That result surprises people, and it was John Bell’s point in telling the story. It leaves open what would have happened had the thread been kept whole. The rockets could not then have accelerated alike. How must they have accelerated? If they are to stay the same distance apart by their own rulers, what programme of thrust would achieve that?

Nothing is allowed to be rigid said that a rigid body in the sense of Newton does not exist in relativity. Push one end of a rod and the other cannot respond until a signal gets there. That is a statement about passive matter responding to a sudden push. It does not stop an active arrangement, with an engine at each point and a programme written beforehand, from keeping every distance fixed. Max Born studied such a motion in 1909, in one of the earliest attempts to work out what rigidity could mean in the new theory, and it is called Born rigid in his honour. The answer is surprising to anyone who assumes that the two ends of a rigid thing should move alike.

Worldlines that keep their distance #

The acceleration that gives constant speed in its own frame was established by the ship that never arrives at c: a hyperbola in spacetime, with $x^2 - c^2t^2$ constant. A point moving with fixed proper acceleration $a$ lies on the hyperbola at distance $c^2/a$ from a special point, the centre.

A family of such hyperbolas, all about the same centre, has a remarkable property. Any straight line through the centre cuts each of them at a right angle in the spacetime sense, and so it is a line of simultaneity for every one of them. All of the points on that line are at rest with respect to each other at that moment. And along that line, the distance between one hyperbola and the next is the same as it was on any other such line. This is just the invariance of the hyperbola under a boost: a boost rotates the lines through the centre and carries each hyperbola into itself.

This construction is the coordinate system due to Wolfgang Rindler: a position $\xi$, a distance from the centre fixed for each point of the body, and a shared time parameter that runs along the hyperbolas. In these coordinates, the body is at rest, permanently. The whole of this family is one motion, a boost that keeps acting, and the body it describes keeps its shape.

But hyperbolas closer to the centre are more sharply curved, and the curvature of a worldline is its proper acceleration. So a point at $\xi$ has

$$a(\xi) = \frac{c^2}{\xi}.$$

Rigidity forces different accelerations at different points. A nearer point must accelerate harder.

What people on the ship feel #

Put a ship’s tail at $\xi_0 = c^2/g$, so its tail accelerates at 1 g. At that acceleration the centre of the hyperbolas is $c^2/g = 0.97$ light-years behind the ship. A point a distance $L$ forward of the tail has proper acceleration $g/(1 + gL/c^2)$.

For any ship anyone might build, the difference is unmeasurable by any accelerometer of the usual kind. A hundred metres along, the fractional change is about a part in $10^{14}$. Yet it is exactly the same as what the clock that runs slow lower down deals with, and modern atomic clocks do see a difference over a height of a centimetre, which is a fractional shift of about $10^{-18}$. What seems unmeasurably abstract in the acceleration is at the edge of the measurable in the clock rate.

For a very long ship the difference is large. A passenger half a light-year ahead of the engines weighs a third less than one at the tail. The ship’s own frame has a “gravity” that falls with height. It falls as $1/\xi$. This is a precise statement of the floor that cannot be told from gravity, and it comes with an interesting qualification. Einstein’s elevator is uniform only over small regions; in a large accelerating frame, the effective gravity is not uniform at all. Nor is it the same as the gravity around a mass, which falls with a different law and has actual tidal forces, whereas this flat spacetime has none: there is no curvature here. An observer with a sensitive enough instrument would find that the gravity in the Rindler frame varies with height in the way that it would in a gravitational field, but that no pair of nearby free-falling particles ever draws together or apart. It is only a field in a manner of speaking.

A length no ship can have #

Now run the question the other way. Suppose the nose is to accelerate at 1 g. How much harder must the tail pull? The answer has a singularity in it.

The expression for the tail’s acceleration is $g/(1 - gL/c^2)$. At half a light-year it is 2.1 g, and it diverges as $L$ approaches $c^2/g$. At that distance behind the nose lies the Rindler horizon, the feature the wall of silence behind a rocket that never stops introduced. No signal from behind that surface can ever catch up with the nose, for as long as it keeps accelerating. A rigid tail cannot be behind the horizon, for then there would be no way of it being connected to the nose by anything at all. It cannot be on the horizon, since that would require infinite acceleration. So a rigid ship has a maximum length, a little under a light-year for a 1 g nose. Greater acceleration makes it shorter: $c^2/a$ for a nose accelerating at $a$. For a proton in a radio-frequency cavity, at about $10^{15}$ m/s², the same formula gives a limit of about 60 m, and for an electron in the fields of a laser wakefield plasma it comes to a few micrometres.

The formula has a familiar shape. A static body hanging in a gravitational field has a similar limit, in that a rope lowered towards a black hole’s horizon must carry a growing tension at its top and has a proper length that cannot reach the horizon without unlimited strength, a case discussed in the horizon that nothing marks. A Rindler observer sits in the near-horizon region of every black hole. The ship’s limit is in that sense the same problem as a person trying to lower a rope, at the same time as it is about rockets in flat space.

Clocks that disagree while at rest #

Another oddity follows directly. Each point on the body travels along its own hyperbola, and proper time passes at a rate proportional to the distance from the centre. Since a common line through the centre takes in the same range of hyperbolic angle for every point, a point at twice the $\xi$ accumulates twice the proper time between the same two instants of the ship.

This is a good thing to think about for anyone who still holds time dilation to be always a matter of relative motion. The clocks on the rigid ship never move relative to each other, by any measurement made on the ship. A radar pulse from the tail to the nose and back always takes the same time. They do tick at different rates, and over a long trip, the difference accumulates without limit. A light signal from the tail, sent once a second, arrives at the nose less than once a second as measured there. An observer at the nose reads this as a redshift, and an observer at the tail sees a blueshift of light from the nose. This is the clock that measures a height, computed without any gravity in the calculation at all. It was the first thing Einstein calculated, in 1907, when he noticed the equivalence, and he deduced the gravitational redshift from an accelerating box. So the very first derivation of gravity’s effect on time was the arithmetic of this figure at small $L$.

Part of the reason for the persistence of this effect is that the proper-time rate of the clock at $\xi$ has nothing to do with the acceleration it feels. A clock’s rate depends on the worldline as a whole, in the sense that a longer path through spacetime holds more time, and the clock that does not feel the turn has already made clear that an ideal clock is not directly affected by acceleration. The tail’s clock runs slower because of where it is in relation to the centre, not because it is feeling a stronger push.

The two programmes, side by side #

Set the rigid ship next to Bell’s pair of rockets, and the difference is clear.

So the launch frame’s account of the two is reversed from what a naive picture would suggest. In one, the launch frame sees nothing change in separation and the thread breaks. In the other, the launch frame sees the ship shrink as it accelerates, and its passengers feel nothing untoward. Since a contraction in the launch frame is required for a rigid body, the tail must catch up slowly with the nose, as seen from the launch frame. That is the sum in simple terms of why it needs to accelerate harder.

There is also the question of what “at the same time” means for the instructions. Bell’s rockets start at the same moment in the launch frame. A pair that were to be rigid would start with thrust ratios already set but at a common moment. After the first instant, a rigid body’s engines would have to follow programmes that are coordinated beforehand, since no signal could be exchanged in time for a very long ship. There is nothing wrong with that. It only means that a rigid ship must be planned, not merely pushed. To give a rigid response to a push at the tail is impossible in general; the nose hears about it a year later. This is the content of the older argument, and it is compatible with the rest.

What a uniform field would have to be #

The ship raises a question that has caught out a good many textbooks. Einstein’s elevator is said to be equivalent to a uniform gravitational field. So what is a uniform gravitational field? There are at least two possible meanings, and they turn out not to coincide. In one meaning, every freely falling test particle has the same acceleration at every height, as seen by the supporting structure. In the other, the proper acceleration needed to hold still is the same at every height. The first is what a Newtonian mind would mean, and it is roughly what the phrase suggests. The second is what Bell’s two rockets would have. Both fail to be the same thing when worked out relativistically.

The rigid ship provides the answer that is actually used. Its frame is static, and the metric in its coordinates does not depend on time. A body at rest on any deck stays at rest. In that frame, what supports an object at $\xi$ has to provide a proper acceleration of $c^2/\xi$, and this varies. So the field of a rigid frame cannot be uniform in the second sense. Yet spacetime is exactly flat there. No tidal stretching exists. The stretching that a Newtonian would attribute to a non-uniform field does not appear, because a pair of free particles released at different heights moves on perfectly straight parallel lines through spacetime, and the deck is what is doing the converging. The apparent variation of gravity with height is the result of a chosen way of measuring time, and not of curvature.

So a field that varies with height and has no tide is a thing that exists. That contradicts a very natural Newtonian reasoning, in which a varying field must produce a tidal stretching, and it is one of the places in which the term free fall cannot remove needs care. Its statement is that a true gravitational field has a piece that no choice of frame can remove. The Rindler frame has only the piece that can be removed. A variation of proper acceleration with height is therefore not by itself a sign of curvature. What is a sign of it is a change in the distance between neighbouring free particles, and those particles show nothing in this frame.

Lifting the same idea to everyday scales, a laboratory at the surface of the Earth is both. It has the Rindler part, because its floor is pushing it up, and a true tide. The gradient in the Earth’s gravity over a metre is about 3 parts in $10^{7}$ of $g$ per metre, and is far greater than the Rindler variation of $g^2/c^2$ per metre, of about $10^{-15}$ of $g$ per metre. A sensitive gravity gradiometer sees only the tidal part. A clock comparison over the same height reads the combination, and its leading term is of the first order in height and has nothing to do with tides. It is a result which would hold in a flat spacetime, as this essay’s ship does.

The ship’s mass is another curiosity. Calculating the total energy of the rigid rocket, as seen from the frame in which it is instantaneously at rest, requires that one take into account each point’s gravitational-like potential. Each deck is lifted to a different height in the Rindler potential, and this gives the rocket’s mass a correction, which contributes to what the box of light that weighs something would have called a weight. A rope hung in a shaft can be analysed the same way, and in both cases its stresses have to be included in order to get a consistent result.

What cannot be done with this picture #

The treatment assumes one dimension. The Rindler family extends to a full three-dimensional body, in a way that gets more peculiar: if the body rotates as it accelerates in general, it cannot be Born rigid. Gustav Herglotz and Fritz Noether showed in 1909–10 that Born-rigid motions have only three degrees of freedom. A rigid body in Newton’s mechanics can do six things. In relativity, a body that keeps its rigidity cannot be given an arbitrary motion, and in particular the disc that cannot be spun up from rest without any internal strain. So the mechanism here is far from universal. Only uniform acceleration works, and very few other choices.

The figures take no account of the materials or of how engines at each point would be powered, of how a ship could store the required energy, or of what an actual rod does. An elastic rod, pushed at one end, sends a sound wave along itself, moving much slower than light. The rod reaches Born rigid motion only in the limit of very slow changes to its acceleration. In a real rod a uniform acceleration from the tail would produce a compression that is approximately the classic result with corrections, and at length scales that matter here, it would have no strength to hold it anyway. A light-year-long structure at 1 g would be subject to a pressure far above that of any material. The thought experiment is valuable for what it reveals about spacetime, and it has no other use.

The rocket also starts from rest, and to get on the hyperbolas it is simply placed there. The exact family exists only if the body has always been on it, or starts abruptly; in the latter case, a discontinuous onset of a proper acceleration, which is not what any real body would do. The rigid motion is the idealised steady state, with no account of how to start it smoothly. Starting a rigid ship from rest over a finite time needs a further choice of programme, and it can be done with a slowly changing family of Fermi–Walker frames, but it no longer has quite such a neat diagram.

Still open: what a rocket-sized accelerometer could see #

The clock difference between nose and tail is real and can be compared with what modern clocks can do. Optical lattice clocks are now reading height differences at the millimetre level, which is a clock comparison between positions in a gravity field, with the difference of order $10^{-19}$. It is an interesting possibility that a laboratory-scale accelerated setup, such as a spinning centrifuge, could do the same with a known kinematic acceleration and test the rate as a function of position in a controlled way, with no gravity anywhere in it. The clocks that must all slow together had in mind whether all forms of clock agree in such a test. The accuracy has reached the point where a laboratory could vary the acceleration, not only the height. Rotating systems are the only way of getting large accelerations in small devices, and rotating clocks face the problem of their rotation changing their own frequencies, so that whether such a test improves on the ones already done with height is a question for the instrument builders.

A body that keeps its proper length while accelerating lies on hyperbolas with a common centre, with proper acceleration $c^2/\xi$ falling with distance from it: 0.66 g half a light-year ahead of a 1 g tail, a limit at $c^2/g$, 0.97 light-years, for a 1 g nose, and clock rates differing by $1 + aL/c^2$ with no relative motion at all. It is the exact opposite of the two rockets on a thread, whose separation was the same and whose thread broke. The habit worth carrying is to ask what is being held fixed, in which frame, because “the same” is a statement about a frame before it is a statement about a ship.