Where Are The Other 7 Dimensions Hiding?

Summary

The video examines whether the universe could have more than the three spatial dimensions we observe. Roger Penrose argues that extra dimensions are mathematically compelling in some theories, but their physical existence remains unconfirmed and raises difficult questions about stability.

Why Three Spatial Dimensions Matter

• By Gauss’s law, a gravitational field spreads over the surface of a sphere. In three spatial dimensions, that surface grows with the square of distance, giving the familiar inverse-square law. In four spatial dimensions, gravity would instead fall off as an inverse cube.
• Penrose argues that this change would disrupt stable planetary orbits. He also describes a hydrogen atom in four spatial dimensions as lacking a lowest-energy state, undermining stable matter and chemistry.
• He invokes Huygens’ principle: in three dimensions, a sharp wave pulse can pass cleanly, whereas in four dimensions a pulse leaves a lingering “tail.” In the speaker’s account, that would make clear signals—and thus ordinary sight and hearing—impossible.
• These examples are presented as reasons three dimensions are closely tied to the conditions for a structured universe, not as proof that extra dimensions cannot exist.

How Extra Dimensions Entered Physics

• In 1919, Theodor Kaluza applied Einstein’s gravitational equations to five-dimensional spacetime. Under the assumption that physical quantities do not vary along the fifth dimension, parts of the five-dimensional geometry reproduce Maxwell’s equations for electromagnetism.
• The scheme also left an extra scalar quantity whose physical effects were problematic. Einstein initially supported publication, then delayed it while considering this issue; the paper was eventually published in 1921.
• In 1926, Oskar Klein proposed that the fifth dimension could be curled into a tiny circle. Motion around that circle would be quantized, offering a geometric account of quantized electric charge.
• The video traces modern dimension counts to mathematical consistency requirements in string theories: 26 spacetime dimensions for the original bosonic string, 10 for supersymmetric string theories, and 11 for M-theory. The 11-dimensional figure also matches the maximum dimension for supergravity described by Cremmer, Julia, and Scherk.
• To recover familiar four-dimensional physics, the additional dimensions must be compactified into special geometries: seven dimensions with (G_2) holonomy in the 11-dimensional account, or six dimensions in a Calabi–Yau manifold in the 10-dimensional version. Dominic Joyce constructed early compact (G_2) examples in 1996.

Penrose’s Objections and Proposed Responses

Penrose’s central critique is that extra spatial dimensions bring vastly more possible degrees of freedom for fields. He asks why the early universe would leave those degrees of freedom unexcited and why compact extra dimensions would remain stable. This is his argument, rather than an established result accepted by all physicists.

The standard response described in the talk is that tiny dimensions require enormous energies—around the Planck scale—to excite. Penrose counters that gravity can produce very high energies through compression and argues that the extra-dimensional geometry itself may be unstable.

He connects this concern to his work on trapped (or “captured”) surfaces: surfaces on which light is converging inward, a condition associated with gravitational collapse and singularities. He suggests that perturbations inside compact extra dimensions might create such conditions, causing them to collapse on timescales comparable to their tiny size. The talk also outlines responses from string theory:

• The theory is formulated using a two-dimensional string worldsheet, so Penrose’s field-counting argument may not apply directly.
• Fluxes may stabilize the size and shape parameters (“moduli”) of the compact dimensions.
• Holography, especially the AdS/CFT correspondence, may mean gravitational degrees of freedom are not counted in the bulk in the way his argument assumes.

The speaker notes that a recent technical paper challenged Penrose’s objections and presents the issue as unresolved.

Tests and Other Ideas

• Experiments by the Eöt-Wash group tested the inverse-square law of gravity down to roughly 50 microns; later work reached shorter distances. The talk reports no observed breakdown.
• The Large Hadron Collider has searched for signs such as missing energy from gravitons, microscopic black holes, and supersymmetric partners, but the video says no evidence was found.
• Penrose also discusses twistor theory. In his account, the apparent sky transforms under boosts by Möbius transformations of the Riemann sphere, linking the Lorentz symmetry of four-dimensional spacetime to complex geometry. He sees the theory’s special fit to four dimensions as potentially significant.
• The video highlights two major problems that extra dimensions do not, in Penrose’s view, solve:
  • The arrow of time: The laws of physics are largely time-symmetric, yet the early universe appears to have had extraordinarily low gravitational entropy. Penrose cites a degree of initial specialness of about one part in (10^{10^{123}}).
  • The measurement problem: The Schrödinger equation does not itself produce a single definite outcome. Penrose favors a possible gravity-related collapse mechanism, in which superpositions of different mass distributions become unstable; he says this idea is being tested.

Finally, Penrose considers a remote future containing only massless particles such as photons and gravitons. With no mass-based clocks or rulers, he argues, scale and duration may lose meaning while conformal structure—the pattern of light cones and causal relations—remains. He suggests that this may also illuminate the Big Bang and raises the possibility of a timeless geometric description of the universe.

Overall: The talk does not claim that extra dimensions have been disproved. It argues that their mathematical motivation is not yet empirical confirmation, and that they do not address some of the deepest puzzles in physics.

Researchers and Sources Mentioned

• Roger Penrose
• Paul Ehrenfest
• Theodor Kaluza
• Albert Einstein
• Oskar Klein
• James Clerk Maxwell
• Isaac Newton
• Carl Friedrich Gauss
• Christiaan Huygens
• Bernhard Riemann
• Georg Cantor
• Edward Witten
• Eugène Cremmer, Bernard Julia, and Joël Scherk
• Dominic Joyce
• Paul Dirac
• J. Robert Oppenheimer and Hartland Snyder
• Ivor Robinson
• Stephen Hawking
• Jacob Bekenstein
• M. C. Escher
• The Eöt-Wash experimental group
• The Large Hadron Collider (LHC)
• The AdS/CFT correspondence and an unnamed recent paper challenging Penrose’s objections