Kaluza–Klein Reduction

A compact direction does more than hide a coordinate: its metric components become a lower-dimensional graviton, vector, and scalar, while momentum around the compact space becomes charge. This page performs the circle reduction with every normalization exposed, derives the gauge symmetry from a five-dimensional diffeomorphism, and identifies precisely when the four-dimensional description is reliable.

Required background. Gravitational Actions and Boundary Terms supplies the Einstein–Hilbert variational normalization and the boundary assumptions used in the reduction; Gauge Theory and String-Theory Preparation supplies the gauge transformations and compactification language used to interpret the off-diagonal metric.

Helpful background. Einstein Equations supplies the higher-dimensional vacuum equations, Formulations of General Relativity helps separate gauge conditions from physical fields, and Exact Spacetimes provides the product-background viewpoint used below.

A circle inside the metric

Use the site’s $(-,+,+,+)$ four-dimensional signature and take the fifth direction spacelike. Overduin and Wesson write the complementary $(+,-,-,-)$ package and an oppositely signed Einstein–Hilbert action; translating both the metric and action signs together gives the conventions below. Let $y\sim y+2\pi R_0$ and begin with

$$S_5=\frac{1}{16\pi G_5}\int d^4x\,dy\,\sqrt{-g_5}\,\mathcal R_5.$$

The Einstein-frame zero-mode ansatz is

$$ds_5^2=e^{\phi/\sqrt{3}}g_{\mu\nu}(x)dx^\mu dx^\nu +e^{-2\phi/\sqrt{3}}\bigl(dy+qA_\mu(x)dx^\mu\bigr)^2 .$$

Here $q$ has length dimension, $φ$ is dimensionless in this normalization, and all retained fields are independent of $y$. At $φ=0$ the circle has circumference $2πR_0$; generally its local physical radius is $R_{\rm phys}=R_0e^{-\phi/\sqrt{3}}$. The Weyl exponents are not decorative. They make the coefficient of the four-dimensional Ricci scalar independent of $φ$, so $g_{\mu\nu}$ is the Einstein-frame metric.

The components have distinct lower-dimensional roles. The ten components $g_{\mu\nu}$ contain the metric, the four components $g_{\mu y}$ contain $A_\mu$, and $g_{yy}$ contains the radion. These are field components, not physical polarizations. A massless graviton in five dimensions has $5(5-3)/2=5$ polarizations; its zero modes become the four-dimensional graviton’s two, the Maxwell field’s two, and the radion’s one. This is a decomposition of one five-dimensional metric, not the addition of three unrelated fields. The historical Jordan-frame form and its Einstein-frame rescaling are developed explicitly in Overduin and Wesson 1997, §§3.2–3.7, PDF.

Reduction and normalization

Substituting the ansatz and integrating once around the periodic circle gives the action below, up to a four-dimensional divergence. Periodicity removes internal boundary terms; the external divergence additionally requires compactly supported variations or matched boundary terms and data:

$$S_4=\frac{1}{16\pi G_4}\int d^4x\sqrt{-g}\left[ R-\frac12(\nabla\phi)^2-\frac{q^2}{4}e^{-\sqrt{3}\phi}F_{\mu\nu}F^{\mu\nu} \right], \qquad G_4=\frac{G_5}{2\pi R_0},$$

where $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$. The derivation has three useful checkpoints. First, $\sqrt{-g_5}=e^{\phi/\sqrt{3}}\sqrt{-g}$, while the part of $\mathcal R_5$ proportional to $R$ contributes $e^{-\phi/\sqrt{3}}R$; their factors cancel. Second, $[G_5]=L^3$ and $[G_4]=L^2$, so division by the circle length is dimensionally forced. Third, the radion kinetic sign is negative in the Lagrangian for the $(-,+,+,+)$ convention, which gives positive Hamiltonian energy.

For a normalization cross-check, Ortín 2004, §11.2.1, pp. 301–303, Eqs. (11.39)–(11.48) uses the circle scale $k$. Set $d=4$, $k=e^{-\phi/\sqrt3}$, and $A^{\mathrm{src}}=qA$, then translate his opposite-signature curvature convention. His scalar coefficient $\tfrac32(\partial\ln k)^2$ becomes our $-\tfrac12(\partial\phi)^2$, and his $k^3F^2$ becomes $q^2e^{-\sqrt3\phi}F^2$.

If the Maxwell term is instead written outside the gravitational prefactor as $-F^2/(4g_4^2)$, then

$$\frac{1}{g_4^2}=\frac{q^2}{16\pi G_4}$$

at $φ=0$. Thus a statement about “the gauge coupling” is meaningless until the coordinate period, the coefficient $q$, the vector normalization, and the scalar background have all been fixed. Rescaling $A_\mu\to cA_\mu$ while sending $q\to q/c$ leaves the five-dimensional metric unchanged but changes the symbol called $g_4$.

A diffeomorphism becomes gauge symmetry

Consider the coordinate change

$$y'=y-q\lambda(x),\qquad A'_\mu=A_\mu+\partial_\mu\lambda.$$

It leaves $dy+qA$ invariant, so the four-dimensional $U(1)$ transformation is exactly the remnant of a five-dimensional diffeomorphism. The field strength and the reduced action are invariant. This also explains charge geometrically. A periodic field admits

$$\Psi(x,y)=\sum_{n\in\mathbb Z}\psi_n(x)e^{iny/R_0}, \qquad m_{n,E}^2=e^{\sqrt{3}\phi}\frac{n^2}{R_0^2}.$$

Under the same coordinate change, $\psi_n\to e^{in q\lambda/R_0}\psi_n$. Consequently the covariant derivative is

$$D_\mu\psi_n=\left(\partial_\mu-i\frac{nq}{R_0}A_\mu\right)\psi_n,$$

and Fourier momentum is quantized $U(1)$ charge. The displayed mass is measured with the four-dimensional Einstein metric on a constant-$\phi$ background. The local inverse geometric radius is instead $R_{\rm phys}^{-1}=e^{\phi/\sqrt{3}}/R_0$; the extra conformal factor between those quantities is why the frame must be named. Klein’s periodic expansion and the mass gap $|n|/R_0$ at $\phi=0$ are stated in Overduin and Wesson 1997, §4.1, PDF. Non-Abelian gauge fields arise analogously from several internal Killing vectors, but their normalization depends on the internal metric and its volume.

Domain, checks, and failure limits

The zero-mode theory is a low-energy description for characteristic Einstein-frame energies and curvatures well below $m_{1,E}$ and $m_{1,E}^2$. Compactification itself retains the entire infinite tower; setting every $n\ne0$ mode to zero is an additional truncation. On a circle, translation invariance prevents zero modes from sourcing nonzero Fourier momentum, but that statement can fail for a different ansatz or for a finite selection containing interacting massive modes.

Two limits discriminate distinct claims. When $R_{\rm phys}\to0$ at fixed $G_4$, the tower decouples but the radion and vector remain unless stabilized or consistently set to special configurations. When $R_{\rm phys}\to\infty$, the tower becomes continuous: this is decompactification, not recovery of four-dimensional GR. Likewise, imposing constant $φ$ while keeping generic $F_{\mu\nu}F^{\mu\nu}\ne0$ violates the scalar equation in the simplest model; Overduin and Wesson 1997, §3.4, Eqs. (6)–(10), PDF gives the corresponding historical consistency check.

For current empirical limits on the radius or couplings, see Tests of Gravity. For the ultraviolet origin of the compact space, see Quantum Gravity. Both questions use the classical normalization derived here.

Common pitfalls

Confusing compactification with truncation. Periodicity reorganizes a higher-dimensional field into an infinite tower. Discarding the tower is a separate dynamical claim whose consistency and energy regime must be checked.

Freezing the radion by notation. A constant $g_{yy}$ is an equation-level restriction, not merely a convenient coordinate choice. Omitted scalar equations can impose conditions such as $F^2=0$.

Reading a coupling before canonical normalization. The coefficient multiplying $A_\mu$ in the metric can be moved by a field rescaling. Only a convention-complete comparison with the canonical kinetic term defines $g_4$.

Exercises

A neutral, massless five-dimensional scalar has modes $\psi_n e^{iny/R_0}$. Show directly that the mass and charge of its nonzero $n$th mode scale in the same way with $n$, and determine what happens to the spectrum under $R_0\to2R_0$ at fixed $φ$, $q$, and $G_5$.

Solution

After multiplying the five-dimensional mass-shell condition by the factor that normalizes the four-dimensional Einstein metric, the $y$-derivative gives $m_{n,E}=|n|e^{\sqrt{3}\phi/2}/R_0$ for a massless parent field. The gauge transformation gives charge $Q_n=nq/R_0$, hence $\lvert Q_n\rvert/m_{n,E}=|q|e^{-\sqrt{3}\phi/2}$, independent of $n$. Doubling $R_0$ halves both $m_{n,E}$ and $Q_n$ in this chosen vector normalization, while $G_4=G_5/(2\pi R_0)$ is also halved. Holding $G_4$ fixed instead would require changing $G_5$; that is a different limiting prescription.

References

• Ortín, T., Gravity and Strings, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2004). DOI.
• Overduin, J. M., and P. S. Wesson, “Kaluza–Klein Gravity,” Physics Reports 283 (1997) 303–378, DOI, Open PDF.