Gravity and Strings

292 The Kaluza–Klein black hole
Spacetime symmetries are associated with the graviton. It is, thus, natural to start by
studying the classical and quantum kinematics of a free massless particle representing a
graviton in flat five-dimensional spacetime with a compact fifth dimension of length equal
to 2π Rz and parametrized by the periodic coordinate x 4 = z which takes values in [0, 2πℓ],
z ∼ z + 2πℓ, (11.1)
that can be seen as the vacuum of the full KK theory just as Minkowski spacetime is the
vacuum of GR. ℓ is some fundamental length unit (the Planck length ℓPlanck, instring theory
the string length ℓs = √ α ′ etc.) Rz is a fundamental datum defining our KK vacuum
spacetime and is the simplest example of a modulus. The choice of vacuum in KK theory
is, however, arbitrary and one of the main objections to KK theories is that no dynamical
mechanisms explaining why one dimension is compact and has the size indicated by the
modulus are provided. This is generically known as the moduli problem.
The five-dimensional metric of this spacetime is, then, in these coordinates 2
d ˆs 2 = ηµνdx µ dx ν − (Rz/ℓ) 2 dz 2 . (11.2)
We can already see that the assumption that the fifth dimension is compact has an immediate
and important consequence: five-dimensional Poincaré invariance of the KK vacuum
is spontaneously broken,
ISO(1, 4) → ISO(1, 3) × U(1).
The five-dimensional Lorentz transformations that mix the compact and non-compact
dimensions are not symmetries of the metric (they leave it formally invariant if we set
ℓ = Rz but they change the periodicity properties of the coordinates). Amongst the fivedimensional
Poincaré transformations that do not mix compact and non-compact coordinates,
clearly Poincaré transformations in the four non-compact dimensions are a symmetry
of the theory and constant shifts in the internal coordinate z are also a U(1) symmetry of
the theory. These are the symmetries of the KK vacuum.
The rescalings of the compact coordinate rescale ℓ, butnot Rz, unless we choose to ignore
the rescaling of the period of z, which is the point of view that is usually adopted. In
this case, the rescalings are not a symmetry of the theory because they change the modulus
Rz which is part of our definition of the (vacuum of the) theory. This is a duality transformation
that takes us from one theory to another one (albeit of the same class).
We assume that the kinematics in the fifth dimension are the most straightforward generalization
of the four-dimensional ones. 3 Thus, we assume that a free, massless particle
2 Usually in the literature ℓ = Rz. Weprefer this parametrization which emphasizes the distinction between
Rz, which is a physical parameter, and ℓ, the range of z which is unphysical. One could also normalize
ℓ = 1/(2π) but coordinates have dimensions of length and it is useful to keep their dependence on ℓ. In
some cases it is easier to take ℓ = Rz and we will do so by indicating it explicitly.
3 It is always implicitly assumed that fundamental constants such as the speed of light c and Planck constant h
have the same value in the five-dimensional world and the extra dimension is always taken to be space-like.
These assumptions are completely ad hoc and should be taken as minimal assumptions, although it is known
that extra timelike dimensions give fields with kinetic terms with the wrong sign in lower dimensions and
this justifies the assumption.