What is real space?

A bucket on a rope

At the end of Lesson 2 Newton had an empty, motionless box called absolute space, and a promise: nobody can see it, but spinning gives it away. The test is a bucket of water.

In the Scholium to the definitions in the Principia (1687; Newton’s laws are told in Course 4), Newton hangs a bucket from a long twisted rope, fills it with water, waits until the water is still, and lets go. The bucket whirls. At first the water’s surface stays “plain,” meaning flat, though the bucket is turning around it. Little by little the bucket passes its motion to the water, which “recedes by little and little from the middle” and climbs the sides, making a hollow. It climbs highest when it turns with the bucket, “relatively at rest in it” (Andrew Motte’s translation, 1729).

Compare the stages. At the start, bucket and water move relative to each other as much as they ever will, and the surface is flat. Later they do not move relative to each other at all, and the surface is hollow. So the hollow cannot come from motion relative to the bucket. Newton read it as the water’s “endeavour to recede from the axe of its motion,” a sign of true circular motion.

The size is easy to work out. Water going round a circle needs a push toward the middle, and the deeper water at the rim supplies it. The rim stands higher than the middle by the rim’s speed, squared, divided by twice the pull of gravity (9.81 meters per second per second), which is also how high a stone thrown up at that speed would climb. Take a bucket 30 centimeters across, turning once a second. The water at the rim moves 0.94 meters a second, and 0.94 × 0.94 ÷ 19.62 is 0.045: the surface dips about 4.5 centimeters from rim to middle. Double the speed and the dip is four times as deep. The same effect flattens the spinning Earth (Course 20).

The activity below has four stages, and only the first two are Newton’s. Stage 3, the bucket stopped while the water still turns and the surface still hollow, and stage 4, everything at rest and flat, are a later textbook addition. It is a fair thought experiment, but it is not in his text.

What did Newton think this proved? Not that we can see absolute space. He held that true motion is told from apparent motion by its effects, which are forces: the water climbing the bucket, or the tension in the cord between two globes whirled about their common center, even with nothing else around. Steady straight motion cannot be detected from inside (Corollary V, Lesson 2). What shows is turning. A car at steady speed feels like standing still, and on a sharp curve it throws you sideways.

Three critics

Three critics said the bucket proves less than Newton thought, and two defenders answered. All five were serious thinkers, and the quarrel is not over.

Berkeley: only relative motion

George Berkeley (1685–1753), an Irish philosopher about forty years younger than Newton, argued in 1710 (Principles of Human Knowledge, sections 111–117) that there is no motion except relative motion: a single body alone in the world could not move. He kept a role for force: a body truly moves when its distance from another changes and a force acts on it. In De Motu (1721), a prize essay for the Paris Academy that did not win, he proposed measuring against the fixed stars. That is an early version of Mach’s idea below, but only a choice of body to measure against, not yet a claim about forces.

Leibniz and Clarke: the shifted universe

In November 1715 Gottfried Wilhelm Leibniz wrote to Caroline, Princess of Wales (Lesson 2 told how it began), and set off a year of ten papers with Samuel Clarke, who answered for Newton, with Newton advising behind the scenes. Leibniz died in November 1716, before he could reply to the last. Clarke printed them in 1717.

Leibniz held that space is “an Order of Coexistences, as Time is an Order of Successions”: nothing but how things stand to one another. His best argument used what he called sufficient reason: every fact has a reason why it is so and not otherwise, which is why a perfectly balanced scale does not tip. If space were a real thing, the same everywhere, God could have put the whole universe somewhere else in it, or turned it about, “changing East into West,” with every relation unchanged. He would have had no reason to choose, so there is no such space, and the two situations are one.

Clarke replied that God’s will is reason enough, that two places exactly alike are still two, and that real motion is real even when it cannot be felt, as when a ship sails so smoothly that a man shut in its cabin cannot tell it moves. Space, he said, is not a substance but a property of an eternal being: “immensity is not God.” Leibniz granted in his last paper that a body’s true motion differs from a mere change of place relative to another body, but he would not accept an absolute space to explain it. Many modern commentators say the real fight was over rotation, the bucket, and not the shifted universe, since Newton’s own laws could not detect a shift anyway (Lesson 2).

Euler: where is the law of inertia true?

Leonhard Euler, a Swiss mathematician, answered the relationists, above all Leibniz’s German followers, in a memoir written in 1748 and printed in 1750. Newton’s first law says a body keeps at rest or in steady straight motion unless a force acts. At rest relative to what? Not to any visible body, Euler said, since even the “fixed” stars are probably not strictly fixed. Yet mechanics works, so the law needs a standard that is not a body: absolute space.

Mach: walls several leagues thick

Ernst Mach (1838–1916), an Austrian physicist, wrote in The Science of Mechanics (1883; English by T. J. McCormack, 1893) that Newton’s experiment shows only that turning relative to the bucket’s walls has no noticeable effect. What if the walls were “several leagues thick” (a league is about three miles)? Nobody can say, he wrote. He did not predict that the water would hollow if the stars turned round a still bucket. That prediction became “Mach’s principle” (马赫原理, Mǎhè yuánlǐ), a name Einstein gave it in 1918: inertia, a body’s resistance to changes of motion, comes from its relation to all the matter in the universe, so “absolute” spin would be spin relative to the distant stars. Mach was a main stimulus for Einstein, who later gave up the principle in its original form.

Does Einstein’s general relativity (1915) make Mach right? Only partly. It has frame dragging: a spinning body drags the space-time near it round with it. Gravity Probe B, a satellite launched in 2004, measured it with gyroscopes. The 2011 result was about 37 thousandths of an arcsecond a year, against 39 predicted, with an uncertainty of about 7. That confirms the effect but does not show that the theory is Machian. In Einstein’s own sense it is not: Mach’s principle says that without matter space itself should not exist, yet the equations still have solutions with empty space-time in them, such as gravitational waves and rotating black holes. (Kerr’s rotating black hole of 1963 and Gödel’s rotating universe of 1949 are cited as counterexamples, though not knock-down ones.)

Euclid’s fifth

While the critics argued about whether space is a thing, a quieter argument concerned its shape. Newton’s box came with Euclid’s geometry built in. Euclid’s Elements (written in Alexandria about 300 BCE) rests on five postulates.

1. A straight line can be drawn from any point to any point.
2. A finite straight line can be extended, as far as you like, in a straight line.
3. A circle can be drawn with any center and any radius.
4. All right angles are equal to one another.
5. If a line crossing two other lines makes the inside angles on one side add up to less than two right angles, then those two lines, extended far enough, meet on that side.

The first four are short and look obvious. The fifth is long, and it concerns what happens far away: picture two rails with a plank across them, leaning toward each other on one side. Euclid proves his first 28 propositions without it and first uses it in Proposition 29. The fact that a triangle’s angles add up to 180 degrees depends on it. Given the other axioms, it is equivalent to simpler statements: through a point off a line passes exactly one line that never meets it (Playfair’s form, 1795), and triangles of the same shape but different sizes exist (John Wallis, 1663).

It looked like a theorem, and for about two thousand years geometers tried to prove it. Proclus (410–485) took apart a false proof by Ptolemy and gave another false one. Ibn al-Haytham (about 965 to about 1040) and Omar Khayyam (1048–1131 by one reckoning) tried too, and Khayyam in effect reached results that belong to non-Euclidean geometry without knowing it. Girolamo Saccheri’s Euclid Freed of Every Flaw (1733) supposed the fifth false and hunted for a contradiction. He found one for one alternative, and for the other only an unconvincing appeal to a “point at infinity.” The theorems he derived along the way were in fact a new geometry. Adrien-Marie Legendre spent about forty years on it, and each of his proofs hid an equivalent assumption. In 1813 Gauss noted privately that the theory of parallels was “a shameful part of mathematics.”

A common retelling says the fifth is long-winded because the Greeks shrank from the infinite. Not quite. Definition 23 defines parallels as lines that, extended “to infinity” in both directions (the Greek word is apeiron; Heath and Joyce translate it “indefinitely”), never meet. What is true is that Euclid’s lines are segments that can be extended as far as needed, and Aristotle says mathematicians need no more (Physics III.7): a potential infinity, not a completed one.

Some readers, this course among them, take the first postulate to say that space has no holes and the fourth that space is the same everywhere. That is a gloss, not Euclid, who has no axiom of continuity; Hilbert closed that gap in 1899. The fifth does look like a statement about shape: one translator’s footnote calls it “the geometry of flat, rather than curved, space.”

Three kinds of triangle

In the early 1800s several men saw that the fifth could be denied without contradiction. Gauss, who had pondered it since his teens, published nothing; in 1832 he told Farkas Bolyai that praising his son’s work would be praising himself. The Russian Nikolai Lobachevsky (Kazan University) published first, in Russian, in 1829–30. The Hungarian János Bolyai’s appendix to his father’s book followed, about 1831–32; in 1823 he had written to his father of “a new, another world out of nothing” (translations vary).

On 10 June 1854 Bernhard Riemann gave a trial lecture at Göttingen. Gauss chose its topic from three Riemann offered, and afterward spoke of it with rare enthusiasm. It was published in 1868. Space, said Riemann, may have a measure of distance that changes from place to place, and we must distinguish “unboundedness” from infinite extent: a ball’s surface has no edge, yet it is finite. Also in 1868, Eugenio Beltrami showed that the new geometry is as sound as Euclid’s, and in 1871 Felix Klein named the three geometries parabolic (Euclid’s), elliptic and hyperbolic.

There are three simple cases. On a plane, a triangle’s angles add up to 180 degrees. On a sphere, a triangle’s sides are arcs of great circles and the angles add up to more. Take the North Pole and two points on the equator a quarter of the way round from each other: each side is about 10,000 kilometers (the metre was first defined, in 1791, as one ten-millionth of the way from pole to equator along the meridian through Paris), each angle is 90 degrees, and the sum is 270. The two sides from the pole are both at right angles to the equator, yet they meet; in flat space they never would (Euclid’s Propositions 27 and 28). Move the two points apart and the sum is 180 degrees plus the angle at the pole. The excess, in radians (180 degrees is about 3.14), equals the area divided by the square of the sphere’s radius. Check: this triangle is one eighth of the Earth, about 63.8 million square kilometers, and 63.8 million ÷ 6,371² is 1.57 radians, or 90 degrees. A sphere has no parallels: any two great circles meet. On a saddle-shaped surface the angles add up to less than 180 degrees, and through a point outside a line pass infinitely many lines that never meet it.

Circles tell the same story. In flat space, circumference ÷ diameter is π, 3.14159, for every circle. On a sphere, measured along the surface, it falls as the circle grows. Take the equator as a circle around the North Pole: its radius is a quarter of a great circle, its diameter half a great circle, its circumference a whole one, so the ratio is exactly 2. (Distances are measured along the ground, not through the Earth.) On a saddle the ratio is larger than π, 3.692 at one radius. Small regions look flat: a circle of radius 1,000 kilometers on the Earth has a ratio of 3.1287, only 0.4 percent below π, which is why city maps work.

Riemann’s theory covers curved spaces of any dimension, with the sphere as its simplest case. A saddle shows only a patch of hyperbolic space, so the activity draws it as a disk in which rulers shrink toward the edge, as in Poincaré’s world below. Lesson 2 told how Kant took Euclid’s geometry for the form of our experience of space. Now there were three consistent candidates, and logic could not choose among them.

Is space curved? Gauss, Poincaré, Einstein

Gauss’s triangle

A famous story says Gauss measured a triangle between three mountaintops to test whether space is Euclidean. The facts are plainer. From 1818 he surveyed the Kingdom of Hanover in what is now northern Germany, sighting with the heliotrope, his 1821 invention that flashes sunlight from a mirror to a distant station. Three stations, the Brocken in the Harz, the Inselsberg in the Thuringian Forest and the Hoher Hagen near Göttingen, none in the Alps, make a triangle with sides of about 69, 85 and 107 kilometers. Its angles add up to 180 degrees plus about 15 arcseconds. Is that curved space? Work it out. The area is about 2,930 square kilometers, and 2,930 divided by the Earth’s radius squared (6,371², about 40.6 million) is 0.000072 radian, or 15 arcseconds. The Earth’s own curvature accounts for all of it. A test of space would need a further excess, bigger than the measurement errors.

Did Gauss mean it as a test? The story comes from his first biographer, in 1856. In 1972 Arthur Miller called it a myth in Isis, and others replied in 1974. The best-supported statement is that his survey papers give no direct evidence of such an aim. What is solid is that he cared whether the fifth postulate could be proved, and wrote in 1830 that geometry should rank with mechanics, not arithmetic: it has an empirical side.

Poincaré’s hot world

Suppose a surveyor did find an extra excess. The French mathematician Henri Poincaré saw that it would show only that angles measured with light rays and rods add up to more than 180 degrees, not whether space is curved or the rays and rods are misbehaving. In Science and Hypothesis (1902; English by W. J. Greenstreet, 1905) he imagined a world inside a large sphere, hottest at the center and cooling toward the edge, where every object, rulers included, is larger or smaller in proportion to its temperature. The inhabitants never reach the edge: “to its inhabitants it will appear infinite.” Light there travels in curves. Doing geometry with shrinking rods and bent rays, they find hyperbolic geometry, while we, looking in, see an ordinary Euclidean ball with strange physics. Both descriptions fit every measurement. Poincaré’s moral was that the choice of a geometry is a convention, and that experiment tells us “not what is the truest, but what is the most convenient geometry.” Euclid’s, being the simplest, will stay the most convenient. This view is called conventionalism.

Einstein’s answer

Einstein replied in Berlin on 27 January 1921, in an address to the Prussian Academy of Sciences, Geometry and Experience (English translation in Sidelights on Relativity, 1922). He granted the logic: only geometry together with the laws of physics can say anything about real things. “Sub specie aeterni Poincaré, in my opinion, is right,” he wrote, meaning in the long view. But for now, he went on, rods and clocks must be used as independent ideas, because we cannot yet build them out of theory. A fair way to say why he did not keep Euclid and add forces: gravity acts in the same way on every body and on light, so the “distortion” cannot be peeled away from the bodies, and the simplest whole theory puts it in the geometry.

That theory, general relativity (completed in November 1915), says matter and energy curve space-time. Its first big test was starlight bending near the Sun at the eclipse of 29 May 1919. Eddington’s two expeditions reported on 6 November 1919: at the Sun’s edge Newtonian physics gives 0.87 arcseconds, Einstein 1.75, and Príncipe measured 1.61, give or take 0.30. The error bars were large, and sharper confirmation came from radio astronomy in the 1960s and 1970s. In a lecture at Leiden on 5 May 1920 Einstein went further: according to general relativity, he said, “space is endowed with physical qualities.”

Ripples and flatness

On 14 September 2015 both LIGO detectors recorded gravitational waves, ripples of space-time that carry energy, from two black holes of about 36 and 29 times the Sun’s mass merging about 1.3 billion light-years away (give or take a lot). The announcement came on 11 February 2016. The wave changed the difference in length of the two 4-kilometer arms by about 4 × 10⁻¹⁸ meters, roughly a four-hundredth of a proton’s width. Space-time, it seems, has dynamics of its own.

Gauss’s idea lives on at the largest scale. The microwave background (Course 7) holds a ruler whose length physics can compute, seen at about 0.6 degrees, a little wider than the full Moon, and its apparent size depends on the curvature of the space in between. The Planck satellite’s 2018 results, combined with galaxy surveys, give a curvature parameter (zero for flat) of 0.0007 plus or minus 0.0019, usually rounded to 0.001 plus or minus 0.002: space is flat to about 0.2 percent. The microwave background alone leans a little toward a closed, sphere-like space. Even at the upper end of the one-sigma range, my arithmetic gives a curvature radius of at least about 285 billion light-years, against about 46 billion for the radius of the observable universe. So what we can see may be a small, flat-looking patch of something larger, as Gauss’s triangle was a small patch of the Earth. This is space averaged over huge distances; near black holes space-time is strongly curved. (Course 19 meets the same number.)