Klein's Curled-Up Dimension (1926): Why Extra Dimensions Hide

Oskar Klein’s 1926 paper explained how a fifth dimension could exist without anyone noticing it. Building on Theodor Kaluza’s five-dimensional theory of 1921, Klein proposed that the extra dimension is closed into a circle far too small to see, and he connected it to the new quantum mechanics. The idea of curling up dimensions, now called compactification, starts here.

What the paper found

• A closed fifth dimension. Instead of Kaluza’s bare assumption that nothing depends on the fifth coordinate, Klein took the fifth dimension to be periodic: travel far enough along it and you come back where you started, like walking around a garden hose.
• Quantum theory meets geometry. Klein combined Kaluza’s five-dimensional relativity with the wave mechanics that Schrödinger and others were creating in 1925 and 1926. In a closed dimension, a quantum wave must fit a whole number of times around the circle, so motion in that direction comes in discrete steps.
• Electric charge explained. Klein linked those discrete steps to electric charge. Because charge always comes in whole multiples of a basic unit, the size of the circle is fixed by the electron’s charge and the strength of gravity.
• A very small circle. Klein’s 1926 analysis put the period at about 0.8 × 10^-30 centimeters, a scale so tiny that no experiment of his day, or of ours, could resolve it directly.

How he did it

Like Kaluza’s, Klein’s work was theoretical. He wrote the equations of five-dimensional general relativity, required the fifth direction to be a circle, and applied the rule from quantum mechanics that momentum around a closed loop takes only whole-number values. The periodicity replaced Kaluza’s cylinder condition with something physical: fields can vary around the circle, but only in a set of discrete patterns, and all but the simplest pattern correspond to particles far too heavy to produce.

That last point is the origin of what physicists call the Kaluza-Klein tower. Every particle that can move in a curled-up dimension shows up in four dimensions as a ladder of heavier copies, spaced in mass by an amount set by the size of the circle. The smaller the circle, the heavier the copies.

Why it mattered

Klein gave extra dimensions a mechanism for hiding. Compactification became the standard answer to the question “if there are more dimensions, where are they?”, and it is used today in string theory, which requires extra dimensions, and in the large extra dimensions proposed in 1998.

The Kaluza-Klein tower also gave experimenters something to look for. Searches at the Large Hadron Collider for gravitons escaping into extra dimensions, such as the ATLAS monojet search of 2021, are hunting for exactly this ladder of states.

What we know now

No curled-up dimension has been detected. Klein’s estimate puts the circle far below the reach of any experiment, including particle colliders, which is why the original Kaluza-Klein idea is not testable as written. Modern versions are more vulnerable: a curled-up dimension large enough to change gravity at short range must, for a single extra dimension, be smaller than about 30 micrometers, according to the 2020 Eöt-Wash torsion-balance test.

For the established, four-dimensional side of the story, see Minkowski’s spacetime and our explainer on the fourth dimension. The pillar guide on cosmology covers the universe that these theories must describe.

Who wrote it

Oskar Klein, a Swedish theoretical physicist, wrote the paper alone. He also published a short companion letter, “The Atomicity of Electricity as a Quantum Theory Law”, published in Nature on 9 October 1926. Historians Lochlainn O’Raifeartaigh and Norbert Straumann reconstruct Klein’s argument in their 2000 review of the origins of gauge theory in Reviews of Modern Physics.

Read the original

Klein, O. (1926). “Quantentheorie und fünfdimensionale Relativitätstheorie.” Zeitschrift für Physik 37, 895-906. DOI: 10.1007/BF01397481 (subscription). The companion letter is Klein, O. (1926), Nature 118, 516, DOI: 10.1038/118516a0.

Questions people ask

What does it mean for a dimension to be curled up?

A curled-up, or compactified, dimension closes back on itself like a circle. Seen from far away, a garden hose looks like a one-dimensional line, but an ant on it can also walk around it. If a dimension’s circle is small enough, nothing we can measure would reveal it.

How small did Klein think the fifth dimension is?

Klein estimated the circumference at about 0.8 × 10^-30 centimeters, fixed by the charge of the electron and the strength of gravity. That is far below the reach of any experiment, so the original idea cannot be tested directly.

Why can’t we see extra dimensions?

If extra dimensions exist, they are either curled up very small, as Klein proposed, or our particles are confined to a four-dimensional surface while only gravity spreads into the extra space. Either way, experiments limit their size: tabletop tests of gravity rule out curled-up dimensions bigger than tens of micrometers that gravity could feel.