The rocket in which light travels on circles

The floor that cannot be told from gravity began with Einstein’s sealed laboratory, which cannot tell whether it is resting on a planet or being pushed through empty space by a rocket. One of the first consequences he drew from that, in 1907 and again in 1911, was that light must bend in a gravitational field. In a rocket accelerating upwards, a beam sent across the cabin travels in a straight line in the frame of the stars, while the cabin accelerates up past it; seen from inside, the beam curves down towards the floor. If the cabin cannot be told from a laboratory on a planet, light must curve down there too.

Einstein first stated the equivalence in a long review he wrote in 1907, at the Patent Office, almost as an afterthought to special relativity. He drew from it the slowing of clocks at lower heights and, a little more tentatively, the bending of light, and then set the subject aside for four years. When he returned to it in 1911 he computed the bending of starlight passing the Sun and proposed that it be looked for during an eclipse. The rocket — in 1907 simply a uniformly accelerated frame of reference, in his later popular account a chest hauled upward by a rope — was the whole of his theory of gravity for those four years, and it got the redshift exactly right.

The argument is usually stopped at the cabin wall, with a sag far too small to see. The wall of silence behind a rocket found that a rocket accelerating for ever has a horizon behind it, a surface that light from beyond can never cross to reach it, at a distance $c^2/a$ below. Following the bent beam all the way down to that horizon turns Einstein’s thought experiment into exact geometry, and the geometry is a surprise: every path light takes inside the rocket is a circle, and the circles obey a geometry in which parallel lines diverge.

Rays in the rocket’s coordinates #

The rocket’s natural coordinates measure height by proper distance from the horizon and time by the clock of a chosen observer in the rocket. A beam of light is a straight line in an inertial frame. To see what it does in the rocket, take each event on that straight line — each place the light is at each inertial time — and work out where and when it is in the rocket’s coordinates. Nothing else is needed; there is no physics in the transformation beyond the hyperbolic motion of the rocket’s points.

The result is clean. A ray launched straight across the rocket from a height $x_0$ above the horizon follows a quarter circle of radius $x_0$, centred on the point of the horizon directly below the source, and meets the horizon at right angles. A ray launched upward at an angle follows a larger arc, of radius $x_0/\cos\alpha$, rising before it falls. A ray launched downward follows the same circle in the other direction. A ray sent straight up is the one exception: it stays a straight line, the limiting circle of infinite radius. Every ray in the figure is fitted with a circle, and every circle’s centre lies on the horizon to a part in a thousand million.

That is the exact form of the sag. Near the source, where the ray has crossed only a small distance $w$ compared with its radius, a circle looks like a parabola, and the drop is $w^2/2x_0 = a w^2/2c^2$ — precisely the distance a stone would fall, under acceleration $a$, in the time $w/c$ that light takes to cross.

Light thrown upward comes back down #

The circles carry a second surprise. A ray launched upward at any angle short of vertical rises to a greatest height, $x_0/\cos\alpha$, and then falls back towards the horizon. In the rocket’s frame, light behaves like a thrown ball in uniform gravity that can never be thrown fast enough to escape: there is no angle below vertical at which it goes up for ever. Only the vertical ray rises without limit.

That sounds as though the rocket had an escape velocity larger than the speed of light, and in a sense it does. In the inertial frame nothing of the kind is happening; the ray is a straight line, and the rocket’s points, accelerating upward, overtake its horizontal progress so that it is left behind and below. Seen from the rocket, being left behind is falling. The analogue for a real planet is very different — light escapes from the Earth in every direction — because a planet’s gravity weakens with height and the rocket’s apparent gravity does not. The rocket’s field is uniform in the sense that every point of it feels a fixed acceleration for ever, and a field that never weakens holds everything that does not head straight out.

How small the sag is #

At the acceleration of the Earth’s gravity, $c^2/a$ is about a light-year, so a beam crossing a ten-metre cabin falls by $5.5\times10^{-15}$ metres, a few times the size of an atomic nucleus. Einstein knew this was hopeless to measure in a laboratory, which is why he turned to the Sun, where a beam grazing the edge travels through a field that acts over a million kilometres instead of ten metres. The figure follows the sag out to the scale where the circle becomes visible, which at one gravity is a light-year, and at a million gravities still billions of kilometres.

One factor for clocks and light #

Why circles? The cleanest answer comes from timing. In the rocket, clocks lower down run slow relative to clocks higher up — the gravitational redshift, which in the rocket is exact rather than approximate. A clock at height $x$ above the horizon runs at a rate proportional to $x$: at half the reference height it runs at half the rate, and at the horizon it stops.

This is not a separate fact from the bending; it is the same fact read with clocks instead of rulers. Alfred Schild turned it into an argument that gravity cannot be described in flat spacetime at all: two light signals sent up a tower, timed at the bottom and the top, do not close into a parallelogram when the clocks are compared. In the rocket the parallelogram fails to close for the same reason, and the rocket shows that the failure can happen in flat spacetime, provided the clocks are the rocket’s.

Light’s speed, measured with a local clock and a local ruler, is always $c$. But timed with the reference clock — which is what “the rocket’s time” means when a beam crosses many heights — light at height $x$ covers distance at $c\,x/x_0$, slower near the floor and faster near the ceiling, by exactly the factor that sets the clocks. That is the situation of light in a medium with a graded refractive index, $n = x_0/x$, rising towards the floor. In such a medium light bends without any surface to bend it, always towards the higher index, which is why a beam in the rocket bends down. And a medium whose index is inversely proportional to the distance from a plane is a well-known one in optics: its rays are exactly circles centred on that plane. The mirage over a hot road and the circles in the rocket are the same phenomenon, with the heat replaced by the slowing of clocks.

The same calculation gives the circles by Fermat’s principle: of all paths between two points, light takes the one of stationary travel time, and with the travel time measured by the reference clock, the stationary paths in a medium of index $x_0/x$ are semicircles standing on the horizon.

Geometry with the wrong angle sum #

The distances that light’s travel time defines — the time for light to go from one point to another, times $c$ — make the inside of the rocket into a geometry, and it is not Euclid’s. The paths light takes are its straight lines, and those are the semicircles. Two semicircles standing on the same line can fail to meet even if they start out heading towards each other; through a point off a given semicircle there are many semicircles that never meet it. This is the geometry Eugenio Beltrami and Henri Poincaré used to show that Euclid’s parallel postulate cannot be proved from his other axioms — the hyperbolic plane — and the upper half of a plane, with semicircles standing on its edge as straight lines, is Poincaré’s model of it.

A triangle drawn with light rays inside the rocket has angles summing to less than 180°. For the three points in the figure the sum is 156.6°, and the shortfall would grow for larger triangles. The ruler’s triangle through the same three points is perfectly Euclidean, because the space inside the rocket is ordinary flat space: rulers laid end to end obey Euclid exactly. What is non-Euclidean is the geometry of light-travel time, because the time is kept by clocks that run at different rates at different heights.

In a hyperbolic plane the shortfall is not arbitrary. It measures the triangle’s area: the amount by which the angles fall short of two right angles, in radians, equals the area divided by the square of the plane’s characteristic length — here $c^2/a$. The triangle in the figure falls short by 23.4°, or 0.41 radians, so its area in light-travel-time measure is 0.41 square units of $c^2/a$. A tiny triangle, a cabin’s width across, has a shortfall too small to measure, as the equivalence principle requires; the geometry is invisible at the scale where the rocket is a laboratory and dominant at the scale of its horizon.

The hyperbolic plane turns up in a second place in relativity, and it is not a coincidence. The space that speeds live in found that relativistic velocities, combined by the composition law, form a hyperbolic plane. The rocket’s points move with velocities that run through every value along a hyperbola, and the clocks that run slow are running slow because of those velocities; the hyperbolic geometry of the rays is the hyperbolic geometry of the speeds, seen from inside.

What the rocket gets right about the Sun, and what it misses #

Einstein’s 1911 calculation applied the rocket argument to the Sun: imagine a chain of small laboratories along the beam’s path, each falling or accelerating so as to be locally free of gravity, and add up the bending each one sees. He got 0.875 arcseconds for a beam grazing the Sun’s edge — exactly what a Newtonian calculation gives for a particle moving at the speed of light. In 1915, with the full theory, he doubled it. The bend Newton got half right explained the factor of two: half the deflection comes from clocks running at different rates in the Sun’s field, which the rocket captures, and half from the curvature of space around the Sun, which bends rulers as well as clocks and which no flat rocket contains.

That division is exactly what the rocket’s geometry shows. Inside the rocket, rulers obey Euclid and only the clocks are graded, so light bends by the clocks’ half alone. Near the Sun, rulers too are affected — the circumference of a circle round the Sun is not $2\pi$ times its radius — and light picks up the second half from that. The 1919 eclipse expeditions found a value near the full one; radio interferometry, measuring the deflection of quasars as the Sun passes near them, has since confirmed it to a few parts in ten thousand, and what a slow body would feel is the clocks’ half alone, because a slow body hardly samples space at all.

The rocket was right about light bending and right about the size of the effect it could see. What it could not do is reproduce a field that has curvature, since a uniformly accelerating frame is a portion of flat spacetime described in unusual coordinates. That is the precise sense in which the equivalence principle is local: it holds for regions small enough that the difference between gravity and acceleration — the tide — cannot be detected, and the half of the bending that comes from space curvature is a tidal effect accumulated along a path millions of kilometres long.

Where the circles end #

Every ray in the rocket that heads downward at all ends on the horizon, meeting it at right angles, and it takes an infinite amount of the reference clock’s time to get there: light approaching the horizon slows, in the rocket’s timing, without limit, just as a clock lowered towards it runs ever slower. From inside the rocket the light never arrives. In the inertial frame it crosses the horizon in a finite time and goes on, but by then it is in the region from which no signal can come back to the rocket. The circles stop at the horizon because the rocket’s coordinates stop there.

The horizon also gives the rocket a temperature. An accelerating observer sees the vacuum as warm, at a temperature proportional to the acceleration, and the same graded clocks that bend the rays redshift that warmth from the horizon upward — higher in the rocket the temperature is lower, in proportion to $1/x$, just as the index is. The rocket is optically a graded medium and thermally a graded oven, both because of one factor.

What the pictures cannot show #

The rays are drawn in two dimensions, height and one direction across. In three dimensions they are semicircles in vertical planes, and the geometry of light-travel time is the three-dimensional hyperbolic space, of which Poincaré’s half-space is the model. The figures also hold the rocket’s acceleration fixed for ever, so that its horizon exists. A real rocket accelerates for a while and stops; it has no permanent horizon, and its light rays are circles only during the acceleration and only in the region the formula describes.

The deflection figure compares a theory’s prediction for a beam grazing the Sun with a measurement, and the measurement is of radio waves from quasars, which pass at various distances from the Sun; the value quoted at the edge is what the measured coefficient implies there. The distinction between the clocks’ half and the space’s half is a statement about the theory in a particular coordinate system, as the factor of two always is.

The domain of the argument is flat spacetime seen from a frame of constant proper acceleration, below the height where the acceleration was set. Inside it, light’s paths are semicircles standing on the horizon and the geometry of light-travel time is hyperbolic. Near a real mass, it gets half the bending right.

Still open: what a rocket-borne optics experiment could see #

The sag of a beam across any real accelerating laboratory is far too small to measure directly, but its consequences for clocks are measured all the time: the gravitational redshift between two optical clocks a metre apart in height is now resolved routinely, and that redshift is the same graded factor that bends the rays. Whether the bending itself could be detected in a laboratory — in an interferometer whose arms are at different heights, where the phase difference picks up the clocks’ grading, or in experiments with atoms in free fall carrying light along with them — is a question of sensitivity that some proposed space missions aim at, and none has yet reached. What would be tested is not the bending, which follows from the redshift, but whether the two halves of light’s deflection behave as general relativity says when they can be separated in one apparatus.

The rocket’s geometry is settled. Inside a rocket with constant proper acceleration a, light timed by the rocket’s clocks moves at c·x/x₀, slow near the horizon and fast above it, so every ray is an exact semicircle standing on the horizon — the sag across a cabin is aw²/2c², 5.5 × 10⁻¹⁵ m over ten metres at 1 g — and triangles of rays have angles summing to less than 180°, although space in the rocket is flat. It is the clocks’ half of the bending of starlight by the Sun, and none of the other half.