Gravity and Strings

294 The Kaluza–Klein black hole
Table 11.1. In this table the decomposition of the five-dimensional graviton in
four-dimensional fields and the physical spectrum are displayed. As explained
in the main text, the three four-dimensional fields g (n)
µν , A (n)
µ , and k (n) for each
n combine via the Higgs mechanism and represent a massive spin-2 particle
(massive graviton) with mass m =|n|/Rz which has five degrees of freedom
(DOF). There are no massive scalars or vectors in the spectrum.
n ˆd = 5 DOF d = 4fields DOF Physical spectrum
0 ˆg (0)
ˆµˆν 5 gµν
Aµ
2
2
Graviton m = 0
Vector m = 0
k 1 Scalar m = 0
n = 0 ˆg (n)
ˆµˆν 5 g (n)
µν
A
2
(n)
µ
k
2 Graviton m =|n|/Rz
(n) 1
On substituting into the above equation, we see that each Fourier mode satisfies the Klein–
Gordon equation for massive fields ( = 1),
✷ − (n/Rz) 2 ϕ (n) (x) = 0, (11.12)
and, therefore, each Fourier mode corresponds to a scalar KK mode. Dimensional reduction
amounts to taking the zero mode alone. If ˆϕ is to be interpreted as a “relativistic wave
function,” this is all we need to know. However, if we want to do field theory, we are
interested in the Green function for the Klein–Gordon equation. For instance, for timeindependent
sources we are interested in the Laplace equation
(4) ˆϕ = δ (4) (x4), x4 = (x 1 ,...,x 4 ), (11.13)
and we want to know which kind of equations it implies for each KK mode and what its
solution is. That is, we want to know the harmonic function H R×S 1 in R 3 × S 1 and its
relation to harmonic functions in R 3 .Wewill deal with this problem in Appendix G.
The same analysis cannot be naively applied to the five-dimensional metric field ˆg ˆµˆν.
The Fourier modes of a five-dimensional scalar field can be interpreted as scalar fields in
four dimensions, but the Fourier modes of the five-dimensional metric cannot be interpreted
as four-dimensional metrics because they are 5 × 5 matrices. The same applies to vector
or spinor fields. We have to decompose the fields with respect to the four-dimensional
Poincaré group.
For the graviton, the result is represented schematically in Table 11.1. Let us first focus on
the Fourier zero mode, which is a 5 × 5 symmetric matrix. It can be decomposed (in several
ways) into a 4 × 4 symmetric matrix that can be interpreted as the four-dimensional metric
(graviton), a four-dimensional vector, and a scalar. We will see in detail in Section 11.2
how this four-dimensional massless mode of the five-dimensional graviton ˆg (0)
ˆµˆν (five helicity
states) can be decomposed into one massless graviton gµν (two helicity states), one massless
vector Aµ (two helicity states), which we will call a KK vector, and one massless scalar k