Kaluza's Fifth Dimension (1921): Uniting Gravity and Light

Kaluza’s 1921 paper is where the modern idea of extra dimensions begins. Theodor Kaluza showed that if Einstein’s general relativity is written in five dimensions instead of four, the extra pieces of the equations look exactly like Maxwell’s theory of electricity and magnetism. Gravity and light would then be two faces of one geometry.

What the paper found

• A fifth dimension of the world. Kaluza took Einstein’s description of gravity as the geometry of four-dimensional spacetime and extended it to five dimensions, adding one more coordinate to the four of Minkowski’s spacetime.
• Electromagnetism falls out of geometry. The five-dimensional metric, the object that encodes distances, has extra components. Kaluza showed that they behave like the electromagnetic potential, and that the five-dimensional equations reproduce Maxwell’s equations alongside Einstein’s.
• Charge as motion in the fifth direction. In his reading, the electric charge of matter is essentially the fifth component of its energy and momentum.
• The “cylinder condition”. To explain why nobody notices the extra dimension, Kaluza assumed that nothing in nature depends on the fifth coordinate. He called this the cylinder condition. It was an assumption, not a derivation, and he said so.

How he did it

The paper is pure theory: a short calculation in the mathematics of curved spaces, the same tools Einstein used for general relativity. Kaluza wrote down the curvature of a five-dimensional space, imposed the cylinder condition, and kept only the leading terms for weak fields. What remained were the familiar four-dimensional laws of gravity plus a set of equations with the structure of electromagnetism.

Einstein himself presented the paper to the Prussian Academy of Sciences in Berlin; the printed version notes it was submitted by him on 8 December 1921. Kaluza closes by thanking Einstein for his interest in the work.

Why it mattered

In 1921 physics had exactly two known long-range forces, gravity and electromagnetism, and the dream of unifying them was in the air. Kaluza’s result was striking because nothing was put in by hand: add one dimension and Maxwell’s equations appear for free.

The idea had two obvious problems. Why can’t we see or move in the fifth dimension? And how does it fit with quantum mechanics, then being born? Five years later Oskar Klein answered both by curling the fifth dimension into a tiny circle and linking its size to the quantum of electric charge. The combined picture is still called Kaluza-Klein theory.

The same mathematical machinery returned decades later in string theory, which needs extra dimensions to be consistent, and in models such as the large extra dimensions of 1998 and the warped extra dimension of 1999. The tower of heavy copies of particles that appears whenever a dimension is curled up is known as the Kaluza-Klein tower, and experiments at the Large Hadron Collider search for it.

What we know now

No experiment has found a fifth dimension. Kaluza’s specific theory did not survive as a complete description of nature: among other gaps, it says nothing about the strong and weak nuclear forces, which were understood only later. Its central insight, that forces can be geometry in hidden dimensions, is one of the most productive ideas in theoretical physics.

Modern limits are strict. Tabletop gravity tests show that any extra dimension that gravity can feel must be smaller than about 30 micrometers if it is curled up, and collider searches have found no sign of particles leaking into extra dimensions. See our explainer on the fourth dimension and extra dimensions for the full picture.

Who wrote it

Theodor Kaluza wrote the paper alone; the printed version gives his location as Königsberg. He presents it as a second route to the goal Hermann Weyl had pursued a few years earlier: a single geometric source for both gravity and electromagnetism. Weyl’s attempt added a new geometric quantity to four dimensions; Kaluza added a dimension instead.

Read the original

Kaluza, T. (1921). “Zum Unitätsproblem der Physik.” Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), pages 966-972. The record is on NASA ADS. A revised English translation, “On the Unification Problem in Physics”, is free on arXiv (1803.08616) and was published in the International Journal of Modern Physics D 27, 1870001 (2018). The original has no DOI.

Questions people ask

What is the fifth dimension in physics?

In physics, a fifth dimension means one more dimension beyond the three of space and one of time. Kaluza proposed it in 1921 to unify gravity and electromagnetism. It is a hypothesis: no experiment has detected one, and if it exists it must be hidden, for example curled up very small.

What is Kaluza-Klein theory?

Kaluza-Klein theory is the idea that general relativity in five dimensions, with the fifth dimension curled into a tiny circle, contains both gravity and electromagnetism. Kaluza found the classical version in 1921 and Oskar Klein added the curled-up dimension and the link to quantum theory in 1926.

Did Einstein support Kaluza’s idea?

Einstein presented Kaluza’s paper to the Prussian Academy in December 1921, and Kaluza’s paper ends by thanking him for his valuable interest in the work.