The number of dimensions a planet can orbit in

Counting what comes out states Gauss’s law as a counting rule: draw any closed surface, and the field crossing it depends only on the charge inside. The shape decides the falloff turns the rule into exponents. A point source’s field lines spread over spheres, whose area grows as $r^2$, so the field falls as $1/r^2$; a line source’s spread over cylinders, and the field falls as $1/r$; a plane’s do not spread at all, and the field is constant. The exponent, that essay says, belongs to the geometry of the source rather than to the physics of the force.

There is one more piece of geometry the exponent depends on, and it is hidden because it never changes: the number of dimensions of space. A point source in a space of $d$ dimensions spreads its field over surfaces whose size grows as $r^{d-1}$. The inverse square is what that becomes when $d = 3$. Paul Ehrenfest asked in 1917 what the world would be like if $d$ were something else, and his answer was that it would have no planets and no atoms. The drawings follow his argument, in a slightly more modern dress.

The falloff is the number of dimensions minus one #

Every line in the drawing is the same law. In one dimension a point source’s field lines have nowhere to spread — they run off in two directions along the line — and the field is the same at every distance. In two dimensions they spread round circles, whose circumference grows as $r$, and the field falls as $1/r$; that is also the field of an infinite line charge in three dimensions, which is a two-dimensional problem in disguise, and the logarithmic potential it gives is what makes a slow cylinder’s drag depend on the most distant wall. In three dimensions the inverse square; in four the inverse cube; in five the inverse fourth power.

The one-dimensional case is less abstract than it looks. Between two quarks, the field of the strong force does not spread out into space: the gluons that carry it attract one another, and the field lines are squeezed into a narrow tube running from one quark to the other. A field confined to a tube is a one-dimensional field — its lines have nowhere to spread — and like the one-dimensional line in the drawing it does not weaken with distance. The energy stored in the tube therefore grows in proportion to its length, by nearly a billion electronvolts for every femtometre, and pulling two quarks apart only lengthens the tube until it is cheaper to snap it by creating a new quark and antiquark from the stored energy. No quark has ever been isolated, and the reason is, in this sense, Gauss’s law in one dimension.

The law also runs the other way, which is how it is used. Measuring the exponent with which gravity falls off measures the number of dimensions gravity can spread into. The scale that may not be where it looks describes theories in which gravity, unlike the other forces, spreads into extra dimensions curled up too small to see, so that below that size it would fall off faster than the inverse square. Torsion-balance experiments have looked for that departure down to about fifty micrometres and found none: at every separation measured, gravity spreads over the surfaces of a three-dimensional space.

The well a circular orbit needs #

Whether an orbit is stable is decided by the same construction the orbit that cannot be made smaller uses for black holes. Fix the orbit’s angular momentum, and the radial motion is that of a particle in an effective potential: a centrifugal barrier, rising as $1/r^2$ towards the centre, plus the attraction’s potential. A circular orbit sits where the two balance. It is stable if that point is the bottom of a well, so that a body nudged inward or outward is pushed back.

For an attraction falling as $1/r^{d-1}$, the potential goes as $-1/r^{d-2}$. In three dimensions that is $-1/r$, which is shallower than the centrifugal barrier’s $1/r^2$ near the centre and deeper far out: the barrier wins close in, the attraction wins far out, and between them is a well. In two dimensions the potential is logarithmic, shallower still, and the well is wider. In four dimensions the potential goes as $-1/r^2$, exactly the same power as the barrier. The two cancel or add at every distance in the same proportion, the effective potential is a single power law with no well in it, and a circular orbit is balanced on a flat floor. In five dimensions the attraction beats the barrier close in, and the circular orbit sits on top of a hill.

The same nudge in four kinds of space #

The drawing gives the same small push to a circular orbit in each space and follows what happens. In three dimensions the orbit becomes an ellipse, reaching a quarter further out at its far point and returning exactly to its starting point each turn. In two dimensions it becomes a rosette — the radius oscillates, but the oscillation does not fit the orbit, so the closest approaches creep round — yet it stays within ten per cent of the circle for ever. In four and five dimensions the push is enough to send the body away for good, within two turns. A push inward instead would have sent it spiralling into the centre.

Any real planet is nudged constantly — by other planets, by passing stars, by the pressure of sunlight — and in a space of four or more dimensions every one of those nudges would grow. There would be no long-lived planetary systems, no stable binary stars, no galaxies held together by orbits. The inverse square is what lets orbits last.

Radial and orbital frequencies #

The stability condition can be written as a single number. A nearly circular orbit wobbles in and out at a radial frequency $\kappa$ while going round at an orbital frequency $\Omega$, and for a force falling as $1/r^{d-1}$ the two are related by

$$\frac{\kappa^2}{\Omega^2} = 4 - d.$$

Positive means a real frequency and a stable wobble; negative means an imaginary one, and exponential growth. The line crosses zero at four dimensions and nowhere else.

In three dimensions the ratio is exactly one: the orbit wobbles in and out once per turn, which is why the orbit closes into an ellipse. That coincidence is the one the orbit that does not come back to itself traces to Bertrand’s theorem — only the inverse square and the linear spring give closed orbits for every bound starting condition — and the orbit special relativity cannot close shows how easily it is spoiled. Three dimensions not only allows stable orbits; it makes them close, which is what lets a planet’s orbit be described by a fixed ellipse.

The rosettes of a galaxy #

Two dimensions is not only a thought experiment, and the rosette in the drawings has a real counterpart. A spiral galaxy’s stars orbit in a disc whose rotation speed, measured far from the centre, is nearly the same at every radius — a flat rotation curve, one of the pieces of evidence for dark matter. A constant orbital speed means a force falling as $1/r$, which is the force of a point mass in two dimensions, and the potential is logarithmic. The stars of such a galaxy therefore move in exactly the field of the two-dimensional drawing, and their slightly non-circular orbits are its rosettes: for a perfectly flat rotation curve the ratio of radial to orbital frequency is exactly $\sqrt{2}$.

The Sun’s orbit round the Milky Way is one of them. Measured from the motions of nearby stars, the ratio of its radial to its orbital frequency is about 1.35 — close to the flat-curve value — so the Sun swings in and out by a few hundred light-years roughly one and a third times per circuit, and its path through the Galaxy is a rosette that has not closed on itself in the twenty-odd circuits since the Solar System formed. The frequency ratio also sets the pattern speeds at which spiral arms can persist, and much of the theory of spiral structure is the theory of these rosettes. A galaxy is not two-dimensional, but its dark-matter halo makes it behave, for this purpose, as if gravity spread over circles.

The black hole’s version of four dimensions #

The failure of stability in four dimensions has a version that happens in three. General relativity adds to the effective potential of an orbit round a mass a term that falls as $-1/r^3$ — steeper than the centrifugal barrier’s $1/r^2$. Far out it is negligible and orbits are Newtonian. Close in it wins, exactly as the steeper attraction wins in five dimensions, and below a certain radius the well that holds a circular orbit turns into a hilltop. That radius is $6GM/c^2$, the innermost stable circular orbit, and matter spiralling into a black hole through a disc orbits stably down to it and then plunges.

The comparison is the same one throughout: whether the attraction’s potential, at small distances, is shallower or steeper than the centrifugal barrier. In three-dimensional Newtonian gravity it is shallower everywhere, and orbits are stable at every radius; add a steeper term, by adding dimensions or by adding relativity, and there is a radius inside which they are not.

An atom has a size only below four dimensions #

The same competition decides whether atoms exist. Why an atom is the size it is finds the atom’s size by balancing two costs: confining an electron to a region of size $a$ costs kinetic energy of order $\hbar^2/ma^2$, the motion that cannot be stopped, and the nucleus pays for the confinement with an attraction of order $-e^2/a$. The first falls faster as $a$ grows, the second faster as $a$ shrinks, and the total has a minimum at the Bohr radius. The atom has a size and a lowest energy because the two terms have different powers of $a$.

In $d$ dimensions the attraction goes as $-1/a^{d-2}$, and the kinetic cost still as $1/a^2$. In four dimensions the powers are equal. The total is a single term: if the attraction is weaker than the kinetic cost, the electron is not bound at any size, and if it is stronger, the energy falls without limit as the electron is squeezed towards the nucleus — the atom collapses. In five dimensions the attraction dominates at small sizes whatever its strength. Stable atoms, like stable orbits, need fewer than four dimensions.

The four-dimensional case has a famous relative in three. The mass no cold matter can hold up finds that a white dwarf’s electrons, once relativistic, have a kinetic cost that falls as $1/a$ — the same power as the gravitational attraction holding the star together. Two terms with the same power leave no minimum, and above the Chandrasekhar mass a white dwarf collapses. The star’s collapse and the four-dimensional atom’s are the same failure: a competition between two powers that happen to be equal.

One comparison, two problems #

The orbit and the atom are the same problem in two costumes. The centrifugal barrier of an orbit is the cost of angular momentum, $L^2/2mr^2$; the kinetic cost of confining a quantum particle is $\hbar^2/2ma^2$. Both fall as the inverse square of a size, because both are the energy of motion across a region of that size — one with the motion fixed by angular momentum, the other by the uncertainty principle. The attraction, in each case, is the potential Gauss’s law gives. So the question “is there a stable balance” reduces, for both, to whether the potential falls more slowly or more quickly than the inverse square — and it falls more slowly exactly when space has fewer than four dimensions.

That is also why the classical atom failed and the quantum one did not. Classically an electron orbiting a nucleus radiates and spirals in, as a charge that turns must glow, and there is no lowest orbit to stop at. Quantum mechanics replaces the orbit’s centrifugal barrier with the uncertainty principle’s, which cannot radiate away, and in three dimensions that barrier holds. In four it would not, whatever the radiation did.

What happens below three #

The other side of the argument is less tidy. In two dimensions orbits are stable and atoms would exist — the logarithmic potential binds a particle at any strength, with infinitely many bound states — but two-dimensional life faces other problems: a tube running through a body cuts it in two, and two-dimensional waves do not travel cleanly. A sound or light pulse in two dimensions leaves a trailing tail behind it, so a sharp signal arrives smeared; only in odd numbers of dimensions greater than one do waves carry sharp signals without distortion, a result that goes back to Jacques Hadamard. Three is the only number of dimensions in which planets orbit stably, atoms have sizes and signals arrive clean.

Arguments of this kind have been pushed further, to the number of time dimensions — with more than one, the equations of physics lose their ability to predict the future from the present — and they have been used, cautiously, in discussions of why the universe has the dimensions it has. What they show with certainty is narrower and still striking: the familiar exponents of physics — the inverse square, the Coulomb potential, the Bohr radius — encode the dimension of space, and would have to change together if it did.

Where the dimensional argument stops #

Unchanged laws. Every drawing assumes the laws of motion and of electromagnetism carry over unchanged to other dimensions, with only Gauss’s law’s surface changed. In a genuinely different number of dimensions the quantities themselves change character — the electromagnetic field in four space dimensions has more components, and charge has different units — and the argument treats only the one feature that decides stability.

Point sources. Orbits round an extended body in any dimension see its field only outside it, and the field outside cannot find the core; inside, the field of a uniform body grows linearly with distance in every number of dimensions, which is stable. The instability is of orbits round compact masses.

An estimate for the atom. The atom’s energy is the uncertainty-principle estimate, not a solution of Schrödinger’s equation. The exact treatment of the four-dimensional case is subtler — a $1/r^2$ potential sits on a knife edge where quantum mechanics has to be supplemented by a length scale from outside — but it confirms the conclusion that there is no ordinary ground state.

What the drawings cannot show #

None of the figures shows a space of four or five dimensions. They show one-dimensional quantities — a field against distance, an energy against size — computed with an exponent changed, and the orbits are drawn in a plane because an orbit in any number of dimensions lies in a plane. What a four-dimensional space would look like, and what else would be different in it, the drawings do not attempt; they isolate one consequence of one number.

Still open: whether there are more dimensions at short range #

The torsion-balance experiments that test the inverse square have reached separations of a few tens of micrometres; below that, the forces between neutral surfaces — the Casimir force and electrostatic patches — swamp gravity, and extra dimensions smaller than that are not excluded. Particle colliders set limits of their own, from the absence of the energy that would leak into extra dimensions in high-energy collisions. Whether space has more than three dimensions at scales far below what has been measured is open, and is a prediction of several theories that attempt to unite gravity with quantum mechanics. If it does, the inverse square is a large-distance approximation, and the stability of orbits and atoms a consequence of their being so much larger than the scale at which space changes.

The habit worth carrying away is to ask where an exponent comes from before treating it as fundamental. The inverse square is the dimension of space written as a power, and stable orbits and atoms both require the attraction to fall more slowly than the kinetic cost of being close — which it does only when space has fewer than four dimensions.