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= Cylinder Condition =

{{Audience_Sidebar
| difficulty = Advanced
| reading_time = 5 minutes
| prerequisites = [[Kaluza-Klein_Unification]]; [[Compactification_in_Kaluza-Klein]].
| if_too_advanced_see = [[Why_Does_Physics_Need_Extra_Dimensions]]
| if_you_want_the_math_see = This page
}}

{{Notation
| signature = Mostly-plus (−,+,+,+,+) in 5D.
| units = Geometrized; capital Latin indices M = 0..4; the fifth coordinate is x5.
}}

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

:<math>\partial_5\,\phi = 0\quad\text{for every field }\phi(x^M)</math>

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 [[Kaluza-Klein_Unification|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:

:<math>\phi(x^\mu, x^5) = \sum_n \phi_n(x^\mu)\,e^{i n x^5 / L}</math>

The n = 0 mode is x5-independent. The n ≠ 0 modes (the [[Compactification_in_Kaluza-Klein|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:

# '''∂5 φ = 0 for all fields φ''' — direct form.
# '''Zero-mode truncation''' of the Kaluza-Klein Fourier expansion.
# '''Translation symmetry along x5''' — physics is invariant under x5 → x5 + a.
# '''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|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 [[5D_Action_Principle|framework's 5D action]]:

* The [[Psi_Field|ψ field]] is taken as zero-mode in the standard derivation.
* This is the leading-order approximation, justified by [[Compactification_in_Kaluza-Klein|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.

[[Category:Psionics]]
[[Category:Kaluza-Klein]]
[[Category:Physics]]

Cylinder Condition - Revision history

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to ; lint-clean per tools/wiki_latex_lint.py

JonoThora: Phase N (01b): LaTeX restoration — promote Unicode display-math to $; lint-clean per tools/wiki_latex_lint.py$