increase of symmetry, but the scheme is more complicated: the vacuum of the theory (in modern parlance) could be 5-dimensional Minkowski spacetime, invariant under the 5-dimensional Poincaré group but this symmetry is spontaneously broken (again in modern parlance) to the 4-dimensional Poincaré group times $U(1)$ due to the (completely arbitrary) choice of vacuum (4-dimensional Minkowski spacetime times a circle). General covariance implies that these symmetries are local in the resulting effective theory, a fact that can be formulated as the *Kaluza-Klein Principle*:

    > *Global invariances of the vacuum are local invariances of the theory*.

    An, originally unwanted, feature of the theory is that a new massless field is predicted: the *Kaluza-Klein scalar* (or *radion*) $k$. Its v.e.v., related to the radius of the internal circle, can also be fixed arbitrarily because there is no potential for this scalar. Fixing (*stabilizing*) the v.e.v. of scalars such as $k$ that determine the size and shape of part of the vacuum spacetime (generically known as *moduli*) is nowadays known as the *moduli problem*. Explaining why the vacuum should be 4-dimensional Minkowski spacetime times a circle of all the possible classical solutions of 5-dimensional General Relativity is the simplest version of the *vacuum selection 