I'm trying to reproduce Kaluza & Klein's result of obtaining the electromagnetic field by introducing a fifth dimension. The basic idea is that the extra components of the five-dimensional metric will materialize in four dimensions as components of the electromagnetic vector potential. For instance, by postulating the appropriate five-dimensional metric and writing up the equation of motion for a particle in empty space, we should be able to recover the four dimensional equation of motion for a charged particle in an electromagnetic field.
Dealing with a single particle, that's a rather special case. Texts on Kaluza-Klein usually focus instead on the relativistic action, which would be applicable to all mechanical systems. My goal here, however, was simply to outline the approach and demonstrate through a simple case how it works, not to develop a comprehensive theory; that has been done by Kaluza over 80 years ago.
My first attempt was a naïve one: I thought I might be able to derive the desired result in flat space, without having to consider curvature with the associated computational complications. That is not so: as I now discovered, curvature, in particular the Christoffel-symbols, play an essential role in the theory, as it is due to the Christoffel-symbols that the electromagnetic field tensor will appear in the four dimensional equation of motion.
We start with empty 5-space. We use upper-case indices for 5-dimensional coordinates (0...4), while lower-case indices will be used in four dimensions (0...3). The electromagnetic field tensor, $F_{ab}$, is defined as $F_{ab}=\nabla_aA_b-\nabla_bA_a=\partial_aA_b-\partial_bA_a$, the contributions of the Christoffel-symbols canceling out each other due to their symmetry in the first two indices. The metric tensor of 5-space is assumed to take the following form (the reason for this peculiar choice will become evident later on):
\[G_{AB}=\begin{pmatrix}g_{ab}+g_{44}A_aA_b&g_{44}A_a\\g_{44}A_b&g_{44}\end{pmatrix},\]
where $A_a$ is an arbitrary 4-vector. Writing up the metric tensor in this form does not imply any loss of generality. The inverse of the metric tensor takes the following form:
\[G^{AB}=\begin{pmatrix}g^{ab}&-A_a\\-A_b&g_{44}^{-1}+A^2\end{pmatrix}.\]
The result can be verified through direct calculation, i.e., by computing $G_{AB}G^{BC}$. What next? Why, computing the Christoffel-symbols of course:
\[\Gamma_{AB}^C=G^{CD}\Gamma_{ABD}=\frac{1}{2}G^{CD}(\partial_AG_{BD}+\partial_BG_{AD}-\partial_DG_{AB}).\]
Wherever the notation might appear ambiguous, I use an upper left index (4) or (5) to distinguish between the four-dimensional and the five dimensional Christoffel-symbols.
Now is the time to make some assumptions about the 5-dimensional metric. First, we assume that the component $g_{44}$ remains constant everywhere. Second, we postulate that the fifth direction forms a so-called Killing field, meaning that the metric will not change with respect to the fifth coordinate: $\partial_4G_{AB}=0$. This is Kaluza's celebrated "cylinder condition". These identities imply that $\Gamma_{a44}=\Gamma_{4b4}=\Gamma_{44c}=0$. Now let's try some of the other Christoffel-symbols:
\begin{align}
{}^{(5)}\Gamma_{4b}^c&=G^{cD}\Gamma_{4bD}=G^{cd}\Gamma_{4bd}+G^{c4}\Gamma_{4b4}=\frac{1}{2}g^{cd}(\partial_4G_{bd}+\partial_bG_{4d}-\partial_dG_{4b})\\
&=\frac{1}{2}[\partial_b(g_{44}A_d)-\partial_d(g_{44}A_b)]=\frac{1}{2}g_{44}g^{cd}(\partial_bA_d-\partial_dA_b)=\frac{1}{2}g_{44}g^{cd}F_{bd}=\frac{1}{2}g_{44}F_b{}^c,\\
{}^{(5)}\Gamma_{a4}^c&=G^{cD}\Gamma_{a4D}=G^{cd}\Gamma_{a4d}+G^{c4}\Gamma_{a44}=\frac{1}{2}g^{cd}(\partial_aG_{4d}+\partial_4G_{ad}-\partial_dG_{a4})\\
&=\frac{1}{2}g^{cd}[\partial_a(g_{44}A_d)-\partial_d(g_{44}A_a)]=\frac{1}{2}g_{44}g^{cd}(\partial_aA_d-\partial_dA_a)=\frac{1}{2}g_{44}g^{cd}F_{ad}=\frac{1}{2}g_{44}F_a{}^c,\\
{}^{(5)}\Gamma_{44}^b&=G^{bD}\Gamma_{44D}=G^{bd}\Gamma_{44d}+G^{b4}\Gamma_{444}=0.
\end{align}
There are more, but these are all we're going to need. With the Christoffel-symbols at hand, we can begin to rewrite the five-dimensional equation of motion in the hope that we can extract something useful and interesting about motion in four dimensions. In explicit notation, the equation of motion takes the following form (geodesic equation):
\[\frac{d^2x^A}{d\tau^2}+\Gamma_{BC}^A\frac{dx^B}{d\tau}\frac{dx^C}{d\tau}=0.\]
But since we are trying to recover the equation of motion in four dimensions, we can just ignore the $A=4$ case:
\[\frac{d^2x^a}{d\tau^2}+\Gamma_{BC}^a\frac{dx^B}{d\tau}\frac{dx^C}{d\tau}=0.\]
Rewriting this in terms of Christoffel-symbols that we can evaluate, and making some dummy index substitutions, we get:
\begin{align}\frac{d^2x^a}{d\tau^2}+\Gamma_{BC}^a\frac{dx^B}{d\tau}\frac{dx^C}{d\tau}&=\frac{d^2x^a}{d\tau^2}+{}^{(5)}\Gamma_{bc}^a\frac{dx^b}{d\tau}\frac{dx^c}{d\tau}+\Gamma_{4c}^a\frac{dx^4}{d\tau}\frac{dx^c}{d\tau}+\Gamma_{b4}^a\frac{dx^b}{d\tau}\frac{dx^4}{d\tau}+\Gamma_{44}^a\frac{dx^4}{d\tau}\frac{dx^4}{d\tau}\\
&=\frac{d^2x^a}{d\tau^2}+{}^{(5)}\Gamma_{bc}^a\frac{dx^b}{d\tau}\frac{dx^c}{d\tau}+\frac{1}{2}g_{44}F_c{}^a\frac{dx^c}{d\tau}\frac{dx^4}{d\tau}+\frac{1}{2}g_{44}F_b{}^a\frac{dx^b}{d\tau}\frac{dx^4}{d\tau}\\
&=\frac{d^2x^a}{d\tau^2}+{}^{(5)}\Gamma_{bc}^a\frac{dx^b}{d\tau}\frac{dx^c}{d\tau}+g_{44}F_b{}^a\frac{dx^b}{d\tau}\frac{dx^4}{d\tau}=0,
\end{align}
i.e.,
\[\frac{d^2x^a}{d\tau^2}+{}^{(5)}\Gamma_{bc}^a\frac{dx^b}{d\tau}\frac{dx^c}{d\tau}=-g_{44}\frac{dx^4}{d\tau}F_b{}^a\frac{dx^b}{d\tau},\]
which is formally identical to the equation of motion in 4D spacetime in an electromagnetic field characterized by $F_b{}^a$, for a particle with a charge-mass ratio of $-g_{44}dx^4/d\tau$ (in other words, the momentum in the fifth direction will be proportional to the charge.) There is, of course, some sleight of hand involved in what I have done, namely that what we see on the left is the five-dimensional Christoffel-symbol in what is supposed to be a 4-dimensional equation, consequently hiding a term in the form $g_{44}A_CF_b{}^a(dx^b/d\tau)(dx^c/d\tau)$, but this derivation nevertheless should suffice to demonstrate the basic idea: starting with empty 5-dimensional space, we can recover an equation of motion in four dimensions that contains the electromagnetic field tensor. In any case, I believe the sleigh of hand is necessary, because the case of a "pure" electromagnetic field would be a nonphysical situation in general relativity: the electromagnetic field itself carries energy and will also influence the particle's motion gravitationally by introducing curvature, which is what I suspect is hidden behind the unwanted term that I eliminated by cheating.
By the way, all this is, by and large, the Kaluza part of the theory. Klein's contribution was with regards to the compactification of the fifth dimension. No, not for aesthetic reasons, though a compactified dimension certainly helped explaining why the fifth dimension couldn't be seen; no, the main reason was to account for the quantized electric charge. It was through compactification that Kaluza achieved a fifth dimension admitting only discrete solutions.
