8.5 Kaluza–Klein example

One example of when relation (326View Equation) holds is the Kaluza–Klein model. In this model one starts with a (4 + n)-dimensional theory of gravity, which is then compactified so that one has n compact directions, forming, for example, an n-torus with characteristic size R, and 4 non-compact directions, which form our 4-dimensional geometry. The higher-dimensional Planck scale Λ is considered fundamental in this model and plays the role of the UV cutoff, while the 4-dimensional Planck scale MPL (or 4-dimensional Newton’s constant) is derived,
M 2PL = Λ2(R Λ )n . (327 )
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 Λ, one finds that N = (R Λ)n is precisely the numebr of these Kaluza–Klein species. Thus, as was noted in [77, 79], the relation (327View Equation) is a particular example of relation (326View Equation) in which N 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 [80]. Now consider a large black hole with horizon size rg ≫ R. 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 rg and an n-dimensional torus of size R. The Bekenstein–Hawking entropy in the (4 + n)-dimensional theory is

(4+n) n+2 2 n SBH = 4πΛ rgR , (328 )
where 4πr2Rn g is the area of (4 + n)-dimensional horizon. From the 4-dimensional point of view this horizon is a 2-sphere of radius rg and the Bekenstein–Hawking entropy in the 4-dimensional theory is
(4) SBH = 4πM 2PLr2g . (329 )
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 (4 + n)-dimensional theory, there is only one field, the massless graviton. Its entropy is
S(e4n+tn)= 4πr2gRn Λn+2 , (330 )
where the cut-off is the higher dimensional Planck scale Λ. On the other hand, in the 4-dimensional theory one computes the entanglement entropy of N KK fields
(4) 2 2 2 2 n Sent = N (4 πrgΛ ) = 4πr gΛ (R Λ) . (331 )
These two entropies are equal to each other so that the two ways to compute the entanglement entropy agree. Moreover, the entanglement entropy (331View Equation) and (330View Equation) exactly reproduces the Bekenstein–Hawking entropy (328View Equation) and (329View Equation).

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.

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