Cylinder Condition

Cylinder Condition

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Notation on this page

The cylinder condition is the technical assumption, originally imposed by Theodor Kaluza in 1921, that all physical fields are independent of the fifth coordinate:

$ \partial _{5}\,\phi =0\quad {\text{for every field }}\phi (x^{M}) $

Geometrically, this means physics is invariant under translations along x⁵. The 5D spacetime is a "cylinder" — extended infinitely in x⁵ but with all physics the same at every x⁵ position.

The cylinder condition is the simplifying assumption that allows the original Kaluza-Klein derivation to go through. Its modern interpretation, status, and alternatives are the subject of this page.

Kaluza's original use

Kaluza (1921) imposed the cylinder condition by hand to make his 5D unification finite:

• Without it: 5D fields depend on x⁵; the theory has 5D dynamics on top of the 4D dynamics; the result is not a 4D theory.
• With it: 5D fields reduce to 4D fields; integration over x⁵ gives a finite x⁵-volume (after compactification) and a clean 4D action.

Kaluza did not give a physical justification — the condition was taken as a postulate.

Klein's improvement

Klein (1926) gave the cylinder condition a physical foundation: the fifth dimension is compact (a small circle of radius L). Fields on a compact dimension can be Fourier-decomposed:

$ \phi (x^{\mu },x^{5})=\sum _{n}\phi _{n}(x^{\mu })\,e^{inx^{5}/L} $

The n = 0 mode is x⁵-independent. The n ≠ 0 modes (the Kaluza-Klein tower) have mass |n|/L. If L is small (Planck scale or near), these modes are extremely heavy and inaccessible at low energies.

The cylinder condition then becomes a low-energy effective truncation: below the energy ℏc/L, only the n = 0 mode is relevant. The cylinder condition is exact for the zero mode and a leading approximation for the rest.

This converted the cylinder condition from an ad-hoc assumption into a consequence of compactification.

Equivalent reformulations

The cylinder condition can be stated several equivalent ways:

1. ∂₅ φ = 0 for all fields φ — direct form.
2. Zero-mode truncation of the Kaluza-Klein Fourier expansion.
3. Translation symmetry along x⁵ — physics is invariant under x⁵ → x⁵ + a.
4. U(1) global symmetry — the translation symmetry, upon compactification, becomes the gauge symmetry that gets identified with electromagnetism.

The fourth interpretation is the deepest: U(1) gauge symmetry emerges as the residual diffeomorphism that survives the cylinder condition. This is the geometric origin of electromagnetic gauge symmetry in Kaluza-Klein theory.

Status in modern physics

• Original (Kaluza 1921): ad-hoc postulate.
• Klein 1926: consequence of compactification and low-energy effective theory.
• Modern Kaluza-Klein: allow the full KK tower; cylinder condition is just the zero-mode approximation.
• Wesson induced-matter theory: abandons the cylinder condition. Treats x⁵-dependence as physical — and interprets 4D matter as the geometric signature of 5D x⁵-dependence in vacuum.
• String theory: cylinder condition is replaced by Calabi-Yau compactification; fields can depend on the internal Calabi-Yau coordinates, generating realistic 4D field content.

Cylinder condition in the framework

In the framework's 5D action:

• The ψ field is taken as zero-mode in the standard derivation.
• This is the leading-order approximation, justified by compactification of x⁵ on a scale L below any current experimental probe.
• Beyond-zero-mode corrections to the framework would predict additional heavy ψ-like modes. These are not relevant to ordinary psi phenomena but could matter at very high-energy or near-Planck-scale physics.

Subtleties and critiques

• Why a cylinder, not a sphere or other manifold? — Kaluza's choice was the simplest. More elaborate Kaluza-Klein constructions use other compact manifolds and produce other gauge groups.
• Stability — the radius L can in principle change; this gives rise to the dilaton problem. See Dilaton.
• Cylinder condition vs. Wesson approach — both are mathematically consistent; they make different physical assumptions about whether x⁵-dependence is physical or unphysical.
• Why does the cylinder condition produce precisely electromagnetism, not some other gauge theory? — because the cylinder is one-dimensional (S¹), the residual gauge symmetry is U(1), the gauge group of electromagnetism. Higher-dimensional compact manifolds give larger gauge groups.

Sanity checks

• Strict cylinder condition imposed → recover Kaluza's 1921 result. ✓
• Zero-mode truncation of compactified theory → equivalent to cylinder condition at low energy. ✓
• Full KK tower → cylinder condition violated by massive KK modes; recovers 5D dynamics. ✓
• ψ → 0 (in framework) → standard cylinder-condition reduction. ✓ (Sanity_Check_Limits §2.)

See Also

• Kaluza-Klein_Unification
• Compactification_in_Kaluza-Klein
• Wesson_Induced_Matter_Theory
• Higher-Dimensional_Physics
• 5D_Action_Principle

References

• Kaluza, T. (1921). "Zum Unitätsproblem der Physik." Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 966–972.
• Klein, O. (1926). "Quantentheorie und fünfdimensionale Relativitätstheorie." Zeitschrift für Physik 37: 895–906.
• Overduin, J. M., Wesson, P. S. (1997). "Kaluza-Klein gravity." Physics Reports 283: 303–378.
• Wesson, P. S. (1999). Space-Time-Matter: Modern Kaluza-Klein Theory. World Scientific.