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I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2.

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

  2. 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:

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

  2. Victor I. Piercey, Kaluza-Klein Gravity, Lecture notes for PHYS 569, University of Arizona, 2008. The pdf file is available here.

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3 Answers 3

up vote 7 down vote accepted

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.

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Why is $g_{55}$ necessarily a scalar field? What's the motivation to promote it from being a constant to a field? Is it just because a 5th dimension with variable size is more general? –  elfmotat Feb 17 '13 at 11:38
    
@elfmotat yes, as the whole metric is a field, it is more general and consistent to consider its components, such as the dilaton, to be fields too. However, the dilaton field should better not vary too much at scales smaler than cosmilogical ones, since in ST it determines not only the size of compactified dimensions but the value of the string coupling constant too. And this again determins the fundamental laws of nature which should be about constant in our universe ... –  Dilaton Feb 17 '13 at 12:05
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And it should be stated that originally, Kaluza-Klein was an attempt to unify gravity with electromagnetism. They didn't want the dilaton field, so they just assumed that it was nondynamical, and gave it a zero value. Of course, any one value of the metric can be set to one with a choice of coordinates. –  Jerry Schirmer Jul 16 at 16:27
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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.

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My sources are: weylmann.com/kaluza.pdf math.arizona.edu/~vpiercey/KaluzaKlein.pdf I'm not sure why you say applying the the action principle on the 5D Ricci scalar naturally implies an additional field. When considering the second form of the metric in my OP, isn't the Ricci scalar just: $\tilde{R}=R+\frac{k}{4}F^{\mu \nu}F_{\mu \nu }$ If $k$ is a constant I don't see how that implies a scalar field. –  elfmotat Feb 17 '13 at 11:24
    
Yes as you mentioned, there is no reason to put $k=const$, the first paper you mentioned is the original way it was done, including the scalar field just makes it more general, the approach that taken in the second paper. –  TMS Feb 17 '13 at 12:02
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I disagree with your last paragraph. Since to unify gravity and EM you need five dimensions, an additional parameter is needed in front of the new fifth component when writing proper time (or distance) by the five dimensional KK metric. And this parameter is exactly the additional field (which my have a fairly constant value) that appears naturally as 55 component in the KK metric, it IS needed. –  Dilaton Feb 17 '13 at 12:39
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The second form is the way in which the metric was written in the age of Kaluza and Klein. Why? out of embarrassment. If you keep the scalar field when considering the action you get an scalar. That was an undesired feature those times and that's why they hid it making it constant (actually they made it equal to 1).

Now, is there a reason why the first is preferred? well no, and yes. Ask yourself, what are we really doing when writing the metric tensor parameterized this way? What we really are doing is expressing a $SO(1,4)$ representation (the full metric tensor) using $SO(1,3)$ representations (that's how you know that $\phi$ is a scalar, $A_{\nu}$ a vector and $g_{\mu\nu}$ a tensor; and by the way this is what is fundamentally wrong with Kaluza's metric, his full metric tensor is not a real tensor).(and by the way, this is not the only way to write the metric using a scalar a vector and a tensor).

So, is there a reason to prefer one of the two expressions you gave? well (as far as I know, and correct me if I am mistaken) neither gives rise to inconsistencies so regarding that both are on equal footing.

Nonetheless you will admit that it is rather arbitrary and inelegant to set $\phi=k$.. Why not for example $A_{\nu}=k_{\nu}$? This is only a prejudice Kaluza had.

The full metric is, let's say, more natural.

Now, which is the meaning of $\phi$? This has been pointed by Dilaton, that the $\phi$ dilaton field is related to the size of the 5th dimension and you can see this really easily.

Imagine a situation where the electromagnetic potential is zero. Then from the form of the metric you right away see that in this situation, so to say, the 5th dimension decouples from the other 4 and that $\phi$ is the metric tensor of the fifth dimension. So, if you let $\phi$ be a function you are saying that the metric, and hence the geometry of the 5th dimension can change with the dilaton field.

So, to answer you question in a sentence. Does Kaluza Klein theory require an additional scalar field? Require Require (as far as I know) no. But hiding it under a constant is rather inelegant.

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