Physics Notes: Kaluza-Klein Theory

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, Fab, is defined as Fab = ∇aAb – ∇bAa = ∂aAb – ∂bAa, 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):

GAB =  ┌
│
│
└		  gab + g44AaAb
g44Ab

     g44Aa
   g44

 ┐
│
│
┘

where Aa 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:

GAB =  ┌
│
│
│
│
└		 gab   –Aa ┐
│
│
│
│
┘
–Ab
1
  + A2
g44

The result can be verified through direct calculation, i.e., by computing GABGBC. What next? Why, computing the Christoffel-symbols of course:

ΓABC = GCDΓABD = 
GCD
 (∂AGBD + ∂BGAD – ∂DGAB)
2

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 g44 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: ∂4GAB = 0. This is Kaluza's celebrated "cylinder condition". These identities imply that Γa44 = Γ4b4 = Γ44c = 0. Now let's try some of the other Christoffel-symbols:

(5)Γ4bc = GcDΓ4bD = GcdΓ4bd + Gc4Γ4b4 = 
gcd
 (∂4Gbd + ∂bG4d – ∂dG4b) =
2
 = 
gcd
 [∂b(g44Ad) – ∂d(g44Ab)] = 
g44gcd
  (∂bAd – ∂dAb) 
g44gcd
 Fbd = 
g44
 Fbc
2 2 2 2
(5)Γa4c = GcDΓa4D = GcdΓa4d + Gc4Γa44 = 
gcd
 (∂aG4d + ∂4Gad – ∂dGa4) =
2
 = 
gcd
 [∂a(g44Ad) – ∂d(g44Aa)] = 
g44gcd
  (∂aAd – ∂dAa)   = 
g44gcd
 Fad = 
g44
 Fac
2 2 2 2

(5)Γ44b = GbDΓ44D = GbdΓ44d + Gb4Γ444 = 0

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):

d2xA
  + ΓBCA 
dxB
  
dxC
  = 0
dτ2 dτ dτ

But since we are trying to recover the equation of motion in four dimensions, we can just ignore the A = 4 case:

d2xa
  + ΓBCa 
dxB
  
dxC
  = 0
dτ2 dτ dτ

Rewriting this in terms of Christoffel-symbols that we can evaluate, and making some dummy index substitutions, we get:

d2xa
  + ΓBCa 
dxB
  
dxC
  = 
d2xa
  + (5)Γbca 
dxb
  
dxc
  + Γ4ca 
dx4
  
dxc
  + Γb4a 
dxb
  
dx4
  + Γ44a 
dx4
  
dx4
  = 
dτ2 dτ dτ dτ2 dτ dτ dτ dτ dτ dτ dτ dτ
 = 
d2xa
  + (5)Γbca 
dxb
  
dxc
  + 
g44
 Fca
dxc
  
dx4
  + 
g44
 Fba
dxb
  
dx4
  = 
dτ2 dτ dτ 2 dτ dτ 2 dτ dτ
 = 
d2xa
  + (5)Γbca 
dxb
  
dxc
  + g44Fba
dxb
  
dx4
  = 0
dτ2 dτ dτ dτ dτ

i.e.,

d2xa
  + (5)Γbca 
dxb
  
dxc
  =   –g44
dx4
  Fba
dxb
dτ2 dτ dτ dτ dτ

which is formally identical to the equation of motion in 4D spacetime in an electromagnetic field characterized by Fba, for a particle with a charge-mass ratio of –g44dx4/dτ (in other words, the momentum in the fifth direction will be proportional to the charge.) There is, of course, some sleigh 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 g44AcFba(dxb/dτ)(dxc/dτ), 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.

References

Lovelock, David & Rund, Hanno, Tensors, Differential Forms, and Variational Principles, Dover Publications, 1989
Wald, Robert M., General Relativity, The University of Chicago Press, 1984
van Dongen, Jerome, Einstein and the Kaluza-Klein particle, arXiv:gr-qc/0009087
Overduin, J. M. & Wesson, P. S., Kaluza-Klein Gravity, arXiv:gr-qc/9805018
Derix, Martijn & van der Schaar, Jan Pieter, Stringy Black Holes, http://www-th.phys.rug.nl/~schaar/htmlreport/report.html
Hossenfelder, Sabine, Kaluza-Klein Theories, http://www.th.physik.uni-frankfurt.de/~lxd/intern/KK/