Kaluza Klein Theory

Kaluza first developed this method in 1919 in an atempt to unify Electromagnetism an General Relativity. His basic idea was to postulate an extra, fifth, dimension, but with all fields being independent of this extra dimension. The starting point would then be a 5-dimensional pure gravity in which because of the independence of the fifth coordinate, the fields can be expressed in 4-dimensional fields. We will see that in this way in 4 dimensions we end up with a metric, a Maxwell field and a scalar.

In 1926 Oskar Klein extended this idea. Instead of assuming total independence of the extra dimension, he assumed it to be compact. This means the fifth dimension would have the topoloy of a circle, with a radius of the order of the Planck length. Five dimensional space-time then has the topology $R^4\times S^1$, and the fifth coordinate y is periodic, $0\leq m y \leq 2\pi$, where $m$ is the inverse radius of the circle. In our normal perception of space-time we would never be able to see this extra dimension.

Because of the periodicy of the extra dimension we can make a Fourier expansion is this coordinate and we would end up with an infinite tower of fields in four dimensions. But we will focus on the first order terms of this expansion, which corresponds to the reduction as initialy introduced by Kaluza.

Let us first define our conventions: hatted quantities will be the five-dimensional ones and unhatted ones will be the four-dimensional fields. Five-dimensional indices: $\hat\mu = 0,1,2,3,5$ and ofcourse the four-dimensional indices: $\mu=0,1,2,3$ ( $x^{\hat\mu}=(x^\mu,y)$).

Kaluza made the following $4+1$ split of the five-dimensional metric

$$\hat g_{\hat\mu\hat\nu} = \left( \begin{array}{cc} g_{\mu\n... ...-\phi A_\mu \\ \\ -\sigma A_\nu & -\phi \end{array} \right)$$ (3.8)

The split is done in this way so the four-dimensional fields will have the proper transformation characteristics in four dimensions.

Consider an infinitesimal coordinate transformation in five dimensions:

$$x^{\hat\mu} \rightarrow x^{\hat\mu} + \epsilon \xi^{\hat\mu} (x^\mu)$$

where ofcourse the transformation is independent of the fifth coordinate. Under this coordinate transformation the five dimensional metric will transform in the following way

$$\delta \hat g_{\hat\mu\hat\nu} = \hat g_{\hat\mu\hat\rho} (\... ... \xi^{\hat\rho}(\partial _{\hat\rho} \hat g_{\hat\mu\hat\nu}).$$ (3.9)

For instance we can derive the transformation properties of the four-dimensional vector $A_\mu$:

$$\delta \hat g_{\mu 5} = -(\delta \phi) A_\mu - \phi (\delta A_\mu)$$

$$= \hat g_{\hat\rho 5} (\partial _\mu \xi^{\hat\rho}) + \xi^\rho(\partial _\rho \hat g_{\mu 5})$$

$$= -\phi A_\rho (\partial _\mu \xi^\rho) - \phi (\partial _\mu \xi^5 )- \xi^\rho (\partial _\rho\phi)D_\mu - \xi^\rho(\partial _\rho D_\mu)\phi$$

so

$$\delta A_\mu = A_\rho (\partial _\mu \xi^\rho) + \xi^\rho (\partial _\rho A_\mu) + \partial _\mu \xi^5.$$

This last term is a $U(1)$ gauge term and we see that indeed the vector field $A_\mu$ has right transformation properties in four dimensions. General coordinate invariance in five dimensions and independence of the fifth dimension results in the gauge symmetry of the four-dimensional vector. More complicated compactifications can result in more complicated gauge symmetries in four dimensions [16].

In the same way we can see that the four-dimensional metric and scalar have the correct transformation characteristics:

$$\delta g_{\mu\nu} = g_{\mu\rho} (\partial _{\nu} \xi^{\rho}) + g_{\rho\nu} (\partial _{\mu} \xi^{\rho}) + \xi^{\rho}(\partial _{\rho} g_{\mu\nu}),$$

$$\delta \phi = \xi^\rho \partial _\rho \phi$$

Note that we have set $\hat g_{55} = -\phi$ to ensure that while keeping the scalar field positive, the fifth coordinate stays space-like. This is merely a convenient choice.

From the relation $\hat g_{\hat\mu\hat\rho} \hat g^{\hat\rho\hat\nu}=\delta^{\hat\nu}_{\hat\mu}$ we can determine the inverse metric:

$$\hat g^{\hat\mu\hat\nu} = \left( \begin{array}{cc} g^{\mu\n... ... \\ \\ -A^\nu & -\frac{1}{\phi} + A^2 \end{array} \right).$$ (3.10)

Let us now continue with Kaluza's idea and start with a source free space-time (pure gravity) in five dimensions. The action descibing this system is:

$$S^{(5)}= - \int d^5x \sqrt{\hat g} \hat R$$ (3.11)

We now have express the five-dimensional quantities in four-dimensional ones. The metric determinant reduces in the simple manner:

$$\hat g = \mbox{det}(\hat g_{\hat\mu\hat\nu}) = -\mbox{det}(g_{\mu\nu})\phi=-g\phi$$ (3.12)

The reduction of the five-dimensional curvature scalar with much longer calculation and we will just give the result here:

$$\hat R = R + \frac{1}{2\phi^2}(\partial \phi)^2 - \frac{1}{\... ...x\phi + {\textstyle{1\over 4}}\phi F_{\mu\nu}(A)F^{\mu\nu}(A),$$ (3.13)

where ofcourse $F_{\mu\nu}=\partial _\mu A_\nu - \partial _\nu A_\mu$

If we put this back in the five-dimensional action and assume we can perform the integration over the fifth coordinate, $\int dx^5 = 1$, it becomes

$$S^{(4)} = \int d^4x \sqrt{-g\phi} \left\{ -R - \frac{1}{2\ph... ...1}{\phi}\Box\phi - {\textstyle{1\over 4}}\phi F(A)^2 \right\},$$ (3.14)

The two terms involving derivatives of $\phi$ can be written as a total derivative and thus do not contribute to the action. The action simplifies to

$$S^{(4)}=\int d^4x \sqrt{-g} \phi^{\frac{1}{2}} \left\{ -R - {\textstyle{1\over 4}}\phi F(A)^2 \right\}.$$ (3.15)

We want to write this action in the usual from involving an Einstein term plus some (exotic) matter terms. So we want to get rid of the scalar term in front of the curvature scalar. We can do this by performing a conformal rescaling of the metric,

$$g_{\mu\nu} \to g_{\mu\nu}' = \phi^{\frac{1}{2}} g_{\mu\nu}.$$

Under this tranformation the curvature scalar transforms in the following non-trivial manner:

$$R=\phi^{\frac{1}{2}} \left[ R' + {\textstyle{3\over 2}}( \na... ...la^\rho \phi ) - \frac{1}{4 \phi^2} (\nabla \phi)^2 ) \right].$$ (3.16)

The other terms in the action transform as follows

$$F^2 = \phi F'^2 ,$$

$$\sqrt{-g} = \phi^{-1} \sqrt{-g'}.$$

If we furthermore perform the scalar tranformation

$$\phi \to \phi'=\sqrt{3} \log \phi,$$

the four-dimensional action can be written in the conventional form (dropping the primes):

$$S = \int d^4x \sqrt{-g} \left\{ -R + {\textstyle{1\over 2}}\... ...e{1\over 4}}e^{-\sqrt{3} \phi} F_{\mu\nu} F^{\mu\nu} \right\}.$$ (3.17)

We see that Kaluza indeed succeeded in unifying Electromagnetism and Gravity, but the scalar that appears in the action was a bit of an embarrassment in the early 20's. Only when in the 80's and 90's people started to study higher dimensional effective actions, the Kaluza-Klein method experienced a big revival.