Gravity and Strings

11.2 KK dimensional reduction on a circle S 1 299
generally rescale and/or rotate the fields among themselves. In particular, they act on
scalars, and thus scalars naturally parametrize a σ -model. In this case k parametrizes a
σ -model with target space R + .Asweexplained before, these transformations should not
be understood as symmetries but as dualities relating different theories. 8
Observe that, in Section 11.1, the radius of the compact dimension Rz appeared explicitly
in the metric. In curved spacetime and at each point of the lower-dimensional spacetime we
can define a local radius of the compact dimension Rz(x),
2πℓ 2πℓ
2π Rz(x µ ) =
0
dz|ˆgzz| 1 2 =
0
kdz. (11.31)
Thus, we see that the KK scalar measures the local size of the internal dimension. We
should require that, asymptotically, our five-dimensional metric approaches that of the vacuum
Eq. (11.2). Then, we find the following relation among the modulus Rz, the fundamental
scale length ℓ,and the asymptotic value of the KK scalar k0:
Rz = ℓk0, k0 = lim
r→∞ k. (11.32)
Sometimes the word modulus is used for the full scalar k. However, only its value at
infinity, which we will see is not determined by the equations of motion and thus has to be
set by hand as a datum defining the theory, really deserves that name.
Since masses are measured at infinity and, in KK theory, we know that these depend on
the radius of the compact dimension through Eq. (11.9), we expect that the masses will
depend on the value at infinity of the radius of the compact dimension Rz (which is why
we have used the same symbol to denote them).
11.2.1 The Scherk–Schwarz formalism
Having determined the relations Eqs. (11.28) and (11.27) between the lower- and higherdimensional
fields, one can simply plug them into the equations of motion of the higherdimensional
fields (here just Einstein’s equations) and obtain equations for the lowerdimensional
ones. This procedure automatically ensures that any field configuration that
solves the lower-dimensional equations of motion also solves (when it is translated to
higher-dimensional fields) the higher-dimensional equations of motion.
In this way one can see that it is not correct to set the KK scalar to a constant as was usually
done in the very early KK literature. As was first realized in [886], the KK scalar has
a non-trivial equation of motion, which we will find later, and, if one sets it to a constant,
this equation of motion transforms into a constraint for the vector-field strength. This constraint
is not generically satisfied and, therefore, solutions with k = k0 that do not satisfy
this constraint are not solutions of the original theory.
8 Observe that ℓ is fixed. Dualities change Rz only.