8.5 Kaluza–Klein example
One example of when relation (326) holds is the Kaluza–Klein model. In this model one starts
with a -dimensional theory of gravity, which is then compactified so that one has
compact directions, forming, for example, an -torus with characteristic size , and
4 non-compact directions, which form our 4-dimensional geometry. The higher-dimensional
Planck scale is considered fundamental in this model and plays the role of the UV cutoff,
while the 4-dimensional Planck scale (or 4-dimensional Newton’s constant) is derived,
Suppose that, in higher dimensions, there are only one particle - the massless graviton. From the
four-dimensional point of view one has, additionally to a single massless graviton, a theory of the tower of
spin-2 massive Kaluza–Klein (KK) particles. Truncating the tower at the cut-off , one finds that
is precisely the numebr of these Kaluza–Klein species. Thus, as was noted in [77, 79], the
relation (327) is a particular example of relation (326) in which should be understood as the number
of the KK species.
In the KK example, the entanglement entropy is equal to the Bekenstein–Hawking entropy as
demonstrated by Dvali and Solodukhin . Now consider a large black hole with horizon size .
Such a large black hole fills all compact directions so that, from the higher-dimensional point of
view, the black hole horizon is a product of a 2-sphere of radius and an -dimensional
torus of size . The Bekenstein–Hawking entropy in the -dimensional theory is
where is the area of -dimensional horizon. From the 4-dimensional point of view this
horizon is a 2-sphere of radius and the Bekenstein–Hawking entropy in the 4-dimensional theory is
We observe that these two entropies are equal so that the two pictures, the higher dimensional
and 4-dimensional one, are consistent. Let us now discuss the entanglement entropy. In the
-dimensional theory, there is only one field, the massless graviton. Its entropy is
where the cut-off is the higher dimensional Planck scale . On the other hand, in the 4-dimensional
theory one computes the entanglement entropy of KK fields
These two entropies are equal to each other so that the two ways to compute the entanglement entropy
agree. Moreover, the entanglement entropy (331) and (330) exactly reproduces the Bekenstein–Hawking
entropy (328) and (329).
However, discussing this result, we should note that the massless and massive gravitons are
non-minimally coupled particles. It remains to be understood how the problem of the non-minimal coupling
is overcome in this example.