The ground state metric, after an extra dimension of space is compactified (to a circle) in Einsteinian gravity, is the metric which corresponds to the R_4 × S_1 geometry of the separated dimensions. How does one arrive at this metric, from the cross product of 4 dimensional gravity and the extra dimension?
JM Overduin's cover of Kaluza-Klein Gravity , on page 21 mentions the ground state metric, as the "vacuum expectation value of the full metric", which "determines the topology of the compact space". It says that it is, topologically, R_4 × S_1.
In answer to a question of mine, Luboš Motl explained that the ground state metric, in practice, is the one for which the geometry is the Cartesian product R_4 × S_1.
So, how could one derive the ground state metric, knowing the topologies R_4 and S_1, and taking their cross product?