Does Kaluza-Klein Theory Require an Additional Scalar Field? I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. 


*

*In one, there is a constant as well as an additional scalar field introduced:
$$\tilde{g}_{AB}=\begin{pmatrix}
g_{\mu \nu}+k_1^2\phi^2 A_\mu A_\nu & k_1\phi^2 A_\mu \\ 
k_1\phi^2 A_\nu & \phi^2
\end{pmatrix}.$$

*In the other, only a constant is introduced:
$$\tilde{g}_{AB}=\begin{pmatrix}
g_{\mu \nu}+k_2A_\mu A_\nu & k_2A_\mu \\ 
k_2A_\nu & k_2
\end{pmatrix}.$$
Doesn't the second take care of any problems associated with an unobserved scalar field? Or is there some reason why the first is preferred?
References:


*

*William O. Straub,  Kaluza-Klein Theory, Lecture notes, Pasadena, California, 2008. The pdf file is available here.

*Victor I. Piercey, Kaluza-Klein Gravity, Lecture notes for PHYS 569, University of Arizona, 2008. The pdf file is available here.
 A: When you write the five dimensional Kaluza-Klein metric tensor as 
$$ g_{mn} = \left(
  \begin{array}{cc}
    g_{\mu\nu} & g_{\mu 5} \\
    g_{5\nu} &  g_{55}\\
  \end{array}
\right) $$
where $g_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $ g_{\mu 5}$ is the ordinary four dimensional vector potetial, $g_{55}$ appears as an additional scalar  field. This new scalar field, called a dilaton field, IS physically meaningful, since it defines the size of the 5th additional dimension in Kaluza-Klein theory. They are natural in every theory that hase compactified dimensions. Even though such fields have up to now not been experimentally confirmed it is wrong to call such a field "unphysical". 
"Unphysical" are in some cases fields introduced to rewrite the transformation determinant in calculations of certain generating functionals, or the additional fields needed to make an action local, which may have conversely to such dilaton field, no well defined physical meaning.
A: It will be better if you were mentioned the sources.
However, as I remember, that in some old works on this theory, they used to assume that $\phi=const$ because the main purpose of the theory was looking for a geometrical unification of Gravity and electromagnetism, and no physical scalar fields was known at that time
Also it will be not very accurate to say "Requires additional" scalar field, because this field in addition to the Electromagnetic field raises naturally (almost, considering cylindrical condition) in the theory after applying the least action principle on five dimensional scalar curvature, and this "natural way" is the whole point of the theory.
