The ship that never arrives at c

A steadily accelerating rocket has a horizon behind it, a distance $c^2/a$ back, from beyond which no signal ever arrives. That is the first rung of this ladder and it is about what such a traveller cannot reach. This one is about what they can, and the answer is surprisingly cheerful before it becomes impossible.

The quantity that has no ceiling #

The speed of a body under constant proper acceleration is

$$\beta = \tanh\frac{a\tau}{c}$$

with $\tau$ its own elapsed time. The argument of that hyperbolic tangent is the rapidity, and it is the quantity that behaves the way a Newtonian velocity is supposed to.

Rapidity grows linearly with proper time, at $a/c$ per second, with no limit of any kind. It also adds: two successive boosts give a rapidity that is the sum of the two, which is why speeds refuse to add — the thing that adds is the rapidity and the velocity is its tangent, so combining velocities requires the addition formula rather than a sum.

So a ship at one gravity gains one unit of rapidity every 0.969 years of its own time, for ever, and the speed is whatever the tangent of the accumulated total happens to be. The tangent approaches one and never reaches it, which is the statement that nothing exceeds $c$; and the tangent’s approach is exponential, which is where all the interesting numbers come from.

Everything about the voyage follows from reading one graph with the vertical axis as the observable and the horizontal one as the thing the engine actually supplies. Rapidities of successive boosts add, exactly; velocities do not. So “hold one gravity for a year, then another year” is a statement about a sum in the coordinate the engine works in, and the speed that results is a hyperbolic tangent of that sum — which is why the speed saturates while the effort does not.

The three quantities #

Integrating the motion gives three functions of proper time, and they behave quite differently.

$$\beta = \tanh\eta, \qquad t = \frac{c}{a}\sinh\eta, \qquad x = \frac{c^2}{a}(\cosh\eta - 1)$$

with $\eta = a\tau/c$. The speed saturates. The coordinate time and the distance both grow exponentially in $\eta$ and therefore exponentially in the traveller’s own clock.

That combination is what makes the numbers come out as they do. After a year of shipboard time the ship is at 0.776 c and has covered 0.57 light-years. After ten it is at 0.99999 c and has covered a hundred and forty thousand.

For a journey with a turnover — accelerate for half, decelerate for half, so as to arrive at rest — the proper time to cross $D$ light-years is

$$\tau = \frac{2c}{a}\,\mathrm{arccosh}\!\left(\frac{aD}{2c^2}+1\right)$$

which grows as the logarithm of the distance once the ship is relativistic. Doubling the distance costs 0.67 years of shipboard time, not double.

The numbers, laid out #

It is worth having the table in front of the argument, because the pattern in it is the essay’s point.

At one gravity, with a turnover halfway and arrival at rest, the shipboard times are 3.5 years to Proxima Centauri at 4.24 light-years, 4.6 to Sirius at 8.6, 11.9 to the Pleiades at 444, 19.8 to the galactic centre at 26,000, and 28.6 to Andromeda at 2.5 million.

Read down that column and the distances rise by a factor of six hundred thousand while the shipboard times rise by a factor of eight. That is the logarithm doing its work, and it is why the interesting question about interstellar travel is never the distance.

The home-frame times are, to a very good approximation, the distance in years plus 1.9 — the extra being the time spent below relativistic speed at each end. So the round trip to the galactic centre takes 40 shipboard years and 52,000 home years, and the traveller returns to a world that has forgotten why they left. That asymmetry is the twin problem with the turnaround spread over a decade instead of an instant, and it comes out the same way for the same reason: the two worldlines between the same pair of events have different proper lengths.

One number in the table deserves special notice. Twenty-nine shipboard years reaches another galaxy. Nothing in relativity forbids a single crew from crossing intergalactic space; what forbids it is in the next section but one.

The Lorentz factor has an almost flat start and a vertical finish, and a voyage of this kind spends most of its shipboard time on the far right of it. There a small change in speed is an enormous change in everything else: in the energy that has to be supplied, in the fuel already spent, and in how much of the universe is reachable. The interesting part of the curve is the part where the speed has stopped being an informative number.

What the traveller sees #

It is worth being careful about the description from on board, because the popular account of it is usually wrong in a way that matters.

The traveller’s clock does not run slow. It is their clock; it ticks once a second by construction, and nothing about the ship’s interior is unusual at any point of the journey. Their scales read one gravity throughout, which is the whole point of choosing that acceleration.

What is different is the distance. In the ship’s instantaneous rest frame the galaxy is moving past at $\beta$, so it is length-contracted by $\gamma$ — which at turnover on the galactic-centre trip is about 27,000. The traveller’s honest description of the trip is not that time slowed but that the distance was, at the moment of maximum speed, about a light-year across.

Both descriptions are right and neither is a correction to the other. The home frame says the ship took 26,000 years and its clock ran slow; the ship says the distance was short. The two frames disagree about which pair of events is simultaneous and that disagreement is what reconciles them.

The fuel #

Everything above is kinematics and costs nothing. The dynamics is where it ends.

A rocket’s mass ratio is governed by the same logarithm as Tsiolkovsky’s, with rapidity in place of velocity. For a photon rocket — one converting mass entirely into a perfectly collimated beam, which is the theoretical best — the ratio for one leg is $e^{\eta}$.

Rapidity is what adds, so the exponents add. A one-way trip with a stop at the far end is two legs; a round trip is four. Proxima Centauri one way is 39 times the payload; there and back is 1,500. The galactic centre round trip is $10^{18}$.

Nothing improves that. A photon rocket is the endpoint of exhaust-velocity improvement, so there is no engine left to invent, and the difficulty is not that the number is large but that it is an exponential of a quantity that grows with distance. Multiplying the payload’s efficiency by a thousand moves the reachable distance by a few light-years.

The escapes are all outside the rocket equation: a beam-driven sail, which leaves the reaction mass at home; a ramjet scooping fuel on the way, whose drag turns out to exceed its thrust; or accepting far lower speeds and much longer trips. Every one of them is a way of not being a rocket.

The non-relativistic version of the same logarithm is the one every launch vehicle is designed against: the velocity gained goes as the log of the mass ratio, so reaching 9.4 km/s on a 4.44 km/s exhaust takes a mass ratio of about eight. The relativistic case replaces velocity with rapidity, which changes nothing about the shape of the difficulty and everything about how far the shape reaches — a rapidity of ten is a mass ratio of $e^{10}$, which is twenty-two thousand.

Why one gravity, and what happens if it is not #

The choice of one gravity is not arbitrary and changing it is instructive.

It is chosen because a crew has to live in it, and a human body is built for exactly that. Anything less gives the physiological problems of weightlessness spread over decades; anything much more is not survivable for long.

The scaling is simple. The characteristic length $c^2/a$ is 0.97 light-years at one gravity and the characteristic time $c/a$ is 0.97 years, and both scale as $1/a$. So doubling the acceleration halves the shipboard time to any given distant destination — asymptotically, since the arccosh’s argument doubles and its logarithm gains only $\ln 2$ — and halves the distance at which relativistic behaviour begins.

That last point is the one worth carrying. The whole character of the journey is set by whether the destination is much further than $c^2/a$. Below that distance the trip is Newtonian and the times are what school arithmetic gives; above it the times are logarithmic and everything in this essay applies. At one gravity the crossover is at about one light-year, which is why the solar system is a Newtonian problem and the nearest star is not.

And the fuel scales the wrong way. The rapidity needed for a given distance falls only logarithmically as the acceleration rises, so the mass ratio improves — but the proper time over which the fuel is consumed falls in proportion, so the power required rises. A ship at ten gravities needs less total fuel and must burn it ten times faster, and the power is what no design has.

The force that is not proportional to the acceleration #

One more asymmetry is worth recording because it is where the engine’s difficulty compounds.

A force applied to a fast body does not produce an acceleration along it, and the coefficient relating the two is different along the motion and across it — $\gamma^3$ one way and $\gamma$ the other. Holding proper acceleration constant, which is what a passenger feels and what this essay assumes, means the coordinate acceleration falls as $\gamma^{-3}$ and the thrust the engine must produce in the home frame rises accordingly.

That is not an extra effect on top of the fuel calculation; it is the same statement seen from the other frame, and the mass-ratio formula already contains it. It is worth naming because it explains why the ship’s own experience stays constant while the home frame’s description of what the engine is doing becomes extreme.

Read as a bill, the kinetic energy curve says the same thing in joules. The energy that has to be supplied rises without bound as the speed approaches $c$, while the speed it buys does not — and the mass ratio is the exchange rate between the two. Nothing about a better engine changes the shape of that curve; a better engine changes only the exhaust speed, which is the base of the logarithm.

The worldline of constant proper acceleration in the home frame is a hyperbola asymptotic to a light ray, and the constancy is of what a passenger feels. The coordinate acceleration falls away as the speed rises, so the ship is pushing just as hard and gaining less and less by the home frame’s reckoning — which is the same fact the mass ratio charges for, drawn as a curve instead of paid as fuel.

Where the acceleration itself becomes the subject #

There is a second reason to care about constant proper acceleration, and it has nothing to do with travel.

An accelerating observer’s frame has a horizon, and a horizon is a thermodynamic object. Unruh’s result is that such an observer finds the vacuum to be a thermal bath at temperature $\hbar a/2\pi c k$ — so the ship in this essay, at one gravity, is immersed in radiation at $4\times10^{-20}$ kelvin, which is far too small to detect and is not zero.

That temperature is the same expression as a horizon’s temperature with the surface gravity replaced by the proper acceleration, and the connection is not an analogy: the equivalence principle says a uniformly accelerating frame and a uniform gravitational field are locally indistinguishable, so if one has a temperature the other must.

The practical size of it is the problem. Reaching a kelvin needs $10^{20}$ gravities, which is beyond anything except the fields at the surface of a heavy nucleus or in the focus of an extreme laser. Both are proposed as experiments and neither has succeeded, and the effect remains one of the best-motivated undetected predictions in physics.

What the voyage figure supplies to that discussion is a sense of scale for the other side of the ledger: one gravity sustained for twenty years crosses a galaxy, and one gravity is twenty orders of magnitude short of a measurable Unruh temperature. The two applications of the same worldline live in completely different regimes.

The arithmetic is not a proposal #

It is worth saying plainly what this essay is and is not claiming, because the shipboard times are the sort of number that travels badly.

The kinematics is exact. A body under constant proper acceleration follows a hyperbola in spacetime, its rapidity grows linearly with its own clock, and the distances and times above are what that curve gives. Nothing in them is an approximation and none of it depends on anything unknown.

The dynamics is exact too, and it is what closes the subject. The mass ratio of an ideal rocket is fixed by momentum conservation and the ideal is a photon rocket; no physics permits better, and the numbers that result are not engineering targets but bounds.

What lies between the two is a large amount that is not physics at all — power densities, radiators, shielding, the behaviour of matter under a beam of relativistic protons — and none of it makes the bound easier. The honest summary is that relativity is unexpectedly generous about time and completely unforgiving about energy, and that the second is what decides.

That is a shape of answer worth recognising. The obvious obstacle, the speed limit, turns out not to be the obstacle; the constraint that binds is one that never appears in the popular statement of the problem, and it appears only when the bookkeeping is done properly. The same thing happens with an engine’s efficiency, where the temperature ratio and not the mechanism sets the ceiling.

The engine that carries no fuel, and why it is a brake #

The rocket equation charges for carrying the propellant, so the obvious escape is to pick it up on the way. Interstellar space contains about one hydrogen atom per cubic centimetre; a ship sweeping a large enough area at a large enough speed passes through a great deal of it.

That is Bussard’s ramjet, proposed in 1960, and the arithmetic that kills it is worth having because two separate things go wrong.

The first is the fuel itself. Fusing plain protons requires the first step of the chain that powers the Sun, and that step is governed by the weak interaction — which is why the Sun’s core, for all its reputation, produces a couple of hundred watts per cubic metre, less than a compost heap. A ramjet burning protons would need a reactor whose power density exceeds a star’s by an enormous factor, and no mechanism for that is known.

The second is worse, because it does not depend on the reactor at all. Collecting the interstellar medium means bringing it to rest relative to the ship, and that is a drag force: momentum has to be taken out of the incoming material before any can be given back. Detailed analyses in the 1970s and 1980s found that for any plausible scoop — necessarily magnetic, since a physical one thousands of kilometres across is not a proposal — the drag exceeds the thrust. The device slows the ship down.

Which turns out to be useful, in the one direction nobody was asking about. A magnetic scoop is an excellent decelerator, and braking against the interstellar medium on arrival is now a serious proposal for a mission that was launched by some other means. The idea survives with its sign reversed.

What the mass ratio costs in the currency of a star #

The exponents in the fuel figure are easier to feel with a payload attached to them.

Take a modest ship — a thousand tonnes, which is a few times the mass of a space station and generous for nothing but a crew and their supplies. The round trip to Proxima Centauri needs about fifteen hundred times that in a photon rocket, which is a million and a half tonnes of fuel converted entirely into a beam. The energy involved is around $10^{23}$ joules, or a few centuries of every kind of energy the human species currently uses, for the nearest star.

The galactic centre and back multiplies the mass ratio by $5\times10^{17}$. The same thousand-tonne ship would need something like a tenth of the Earth’s mass in fuel, and the energy released in converting it — about $10^{41}$ joules — is what the Sun radiates in several million years.

Andromeda is the one that ends the discussion. Four legs at that distance is a rapidity near sixty, so the mass ratio is $e^{60}$, of order $10^{25}$, and a thousand-tonne payload requires some twenty thousand solar masses of fuel.

The point of writing them out is the pattern rather than any of the numbers. The shipboard times went up by a factor of eight between Proxima and Andromeda; the fuel went up by twenty-two orders of magnitude. Everything comfortable about relativistic travel is logarithmic in the distance, and everything prohibitive is exponential in it — which is the whole essay in one sentence, and it is why the constraint that decides is not the one the popular version worries about.

What the picture cannot show #

The trip is drawn as a smooth hyperbola and would not be. Turnover requires reversing the thrust, which takes time; navigation requires corrections; and nothing in the calculation is about steering.

Nothing here is about what the journey would be like. At $\gamma$ in the thousands the interstellar medium arrives as a beam of relativistic protons, and a hydrogen atom per cubic centimetre at $\gamma = 10^4$ delivers a dose that would sterilise the ship many times over. Shielding it means carrying mass, which the mass ratio charges for again.

The photon rocket is a limit and not a design. Converting mass to a directed beam with no remainder has no known mechanism; the best actual proposal, matter–antimatter annihilation, produces neutral pions that decay to gamma rays in every direction and cannot be collimated by anything.

The horizon is still there and it is not a detail. The first rung of this ladder establishes that a permanently accelerating ship has a horizon $c^2/a$ behind it — 0.97 light-years at one gravity — and everything the ship left behind after a certain moment is beyond it. A message sent from Earth more than a couple of years after departure never catches up, for as long as the acceleration continues. The voyage is one-way in a stronger sense than the arithmetic of arrival times suggests.

And the destination has been treated as stationary. Over twenty-six thousand years of home time the galactic centre moves, the solar system moves, and Andromeda’s approach at 110 kilometres a second changes the distance by more than a light-year during a two-and-a-half-million-year crossing. Those are corrections to the arithmetic rather than to the physics, and they are larger than the arithmetic’s own precision.

The ladder from here #

Later rungs on this anchor: the Rindler coordinates in which the accelerating frame’s own description is written, and the horizon’s role in them; the Unruh effect, in which an accelerating observer finds the vacuum to be a thermal bath at a temperature proportional to the acceleration; the relativistic rocket equation derived properly, with the four-momentum ledger rather than by analogy; the beam-driven sail, where the reaction mass stays behind and the arithmetic changes character completely; and the twin problem posed with a smooth acceleration rather than an instantaneous turnaround, where the asymmetry between the two worldlines becomes a statement about proper length rather than about who turned round.

The neighbouring ladders are the rocket’s horizon, which is what this journey is running away from, and velocity addition, whose rapidity is the quantity everything here accumulates. The push that does not point where the body goes is what the engine has to overcome.