Compactification effects in Kaluza-K theory - Gauge away EM?

[center o bass]

I was first considering to post this in the GR section on the forum, but as I've understood it, compactification is essential in string theory, so I thought that perhaps you guys know the subject better also in Kaluza-Klein theory.

Compactification in Kaluza-Klein theory as I understand it is the statement that all fields become periodic with trespect to the fifth dimension and that the topology of $M^5$ goes to $M^4\times S^1$.

Klein introduced the idea in order to explain why we did not observe the extra dimension by making the circle 'small' - and since this also implies that the fields on the manifold are periodic, it provides a plausible explanation of the so called 'cylinder condition.. But what other consequences are there of this compactification?

In the following video lecture from Perimeter Institute (http://pirsa.org/displayFlash.php?id=13020069 at about 48 minutes ) the lecturer states that the compactification prevents us from 'gauging away' electromagnetism by a coordinate transformation due to the presence of a preferred frame. But why is this so?

As is noted here:http://staff.science.uva.nl/~jpschaar/report/node12.html
a general (infinitesimal) coordinate transformation $(x^\mu \to x^\mu + \xi^\mu)$ implies that the Kaluza-Klein vector potential changes to $A_\mu \to A_\mu + \delta A_\mu$ where

$$\delta A_\mu = A_\rho (\partial_\mu \xi^\rho) + \xi^\rho(\partial_\rho A_\mu) + \partial_\mu \xi^5$$

which implies a change in the maxwell field strength of $F_{\mu\nu} \to F_{\mu\nu} + \delta F_{\mu\nu}$ where

$$\delta F_{\mu\nu} = (\partial_\mu \xi^\rho) F_{\rho\nu} + (\partial_\nu \xi^\rho) F_{\mu\rho} +\xi^\rho \partial_\rho F_{\mu \nu} $$

Now what is stopping me from picking a $\xi^\mu$ that satisfies

$$F_{\mu\nu} + \delta F_{\mu \nu} = 0$$

where I would have found a frame with no electromagnetism? Is this somehow impossible due to compactification?
.

At the moment I'm also reading a paper on gravitational waves in a Kaluza-Klein space (found here: http://arxiv.org/abs/gr-qc/0411028) where one is perturbing a KK vacuum solution $j_{AB}$

$$g_{AB} = j_{AB} + h_{AB}$$

The author then states that

"When spontaneous compactification takes place, the universe acquires a kaluza-klein structure ($M^5 \to M^4 \times S^1$) and the 5D local Poincare group is spontaneousy broken into a 4d local Poincare group and a $U(1)$ local gauge group. The wave, originally a 5d object now feels the effect of compactification and it's components transform in a different way under 4d coordinate transformations"

Later the author also states that

"compactification implies that the general covariance is lost and 4d fields contained in the 5d perturbation acquire a different behaviour under 4d coordinate transformations becoming distinct 4-d dynamical fields."

Firs of all, could someone explain the first quote in terms that does not require a lot of group theory? Secondly, if I'm not entirely misunderstanding, it clearly seems like this author claims that the relevant fields get different transformation properties under compactification. This could possibly be related to my first question regarding the transformation properties of the field tensor, but the question is - how and why are these new transformation properties acquired? Do they somehow go from being non-physical to physical after compactification?

 

[fzero]

I was first considering to post this in the GR section on the forum, but as I've understood it, compactification is essential in string theory, so I thought that perhaps you guys know the subject better also in Kaluza-Klein theory.

Compactification in Kaluza-Klein theory as I understand it is the statement that all fields become periodic with trespect to the fifth dimension and that the topology of $M^5$ goes to $M^4\times S^1$.

Klein introduced the idea in order to explain why we did not observe the extra dimension by making the circle 'small' - and since this also implies that the fields on the manifold are periodic, it provides a plausible explanation of the so called 'cylinder condition.. But what other consequences are there of this compactification?

In the following video lecture from Perimeter Institute (http://pirsa.org/displayFlash.php?id=13020069 at about 48 minutes ) the lecturer states that the compactification prevents us from 'gauging away' electromagnetism by a coordinate transformation due to the presence of a preferred frame. But why is this so?

Both of the questions that you raise involve the way that singling out one of the 5d coordinates and making a periodic identification breaks the diffeomorphism and Lorentz symmetries of GR. Basically, once we demand the space is $M_4\times S^1$, we are no longer free to make transformations that mix the periodic coordinate $\sigma$ with the 4d coordinates $x^\mu$. In the example from the video, a 5d Lorentz tranformation is made that sends

$$ \sigma \rightarrow \sigma' = \sigma + \alpha x^1.~~~~(*)$$

Because $x^1$ is not periodic, $\sigma'$ is not periodic, so we don't preserve the topology of the circle. The tranformations that do preserve the topology are just the independent diffeomorphisms of $M_4$ and $S^1$.

To further elaborate, the original question is whether we can find some coordinate redefinition that let's us rewrite the 5d metric

$$ ds_5^2 = g_{\mu \nu} dx^\mu dx^\nu + e^{2\sigma} \left( d\sigma + A_\mu dx^\mu \right)^2$$

in such a way as to eliminate the cross-term:

$$ ds_5^2 = g'_{\mu' \nu'} dx^{\mu'} dx^{\nu'} + (d\sigma')^2. $$

To make the example simpler, let's just suppose that we have a metric

$$ ds^2 = dx^2 + \left( d\sigma + A dx \right)^2,$$

where $A$ is just a constant. The new coordinates

$$ \begin{split} & \sigma' = \sigma + A x \\
& x' = x, \end{split}$$

certainly removes the cross-term, since

$$ ds^2 = (dx')^2 + (d\sigma')^2,$$

but as in (*) above, the required coordinate reparameterization doesn't respect the topology.

As is noted here:http://staff.science.uva.nl/~jpschaar/report/node12.html
a general (infinitesimal) coordinate transformation $(x^\mu \to x^\mu + \xi^\mu)$ implies that the Kaluza-Klein vector potential changes to $A_\mu \to A_\mu + \delta A_\mu$ where

$$\delta A_\mu = A_\rho (\partial_\mu \xi^\rho) + \xi^\rho(\partial_\rho A_\mu) + \partial_\mu \xi^5$$

which implies a change in the maxwell field strength of $F_{\mu\nu} \to F_{\mu\nu} + \delta F_{\mu\nu}$ where

$$\delta F_{\mu\nu} = (\partial_\mu \xi^\rho) F_{\rho\nu} + (\partial_\nu \xi^\rho) F_{\mu\rho} +\xi^\rho \partial_\rho F_{\mu \nu} $$

Now what is stopping me from picking a $\xi^\mu$ that satisfies

$$F_{\mu\nu} + \delta F_{\mu \nu} = 0$$

where I would have found a frame with no electromagnetism? Is this somehow impossible due to compactification?
.

First, I'd note that the simple example above showed that in order to remove the cross term between $x^5$ and the 4d coordinates we would need to make a transformation that mixed $x^5$ with the 4d coordinates. But these were precisely the transformations that don't preserve the topology, so they are not allowed.

Second, I should point out that the transformation you wrote above is only the infinitesimal form of a general coordinate transformation that is of the form

$$ g'_{\hat{\mu}' \hat{\nu}' }(x') = g_{\hat{\mu} \hat{\nu}}(x) \frac{\partial x^{\hat{\mu}}}{\partial x^{\hat{\mu}'}} \frac{\partial x^{\hat{\nu}}}{\partial x^{\hat{\nu}'}} .$$

For a case where the gauge field is finite, the transformation has a more complicated form. To properly address the question, we'd need to work things out for the general case and then examine whether or not there are solutions with the properties that you suggest.

It's simpler to discuss the infinitesimal form, but we wouldn't expect an infinitesimal transformation to be able to generate a finite quantity that could cancel the gauge field. We can plod along for a specific example. If we have a point charge at the origin, we can consider a gauge potential of the form $A_0 = \alpha/r$. If you consider your transformation, there is a solution $\xi^r = r+r^2$, but this is no longer an infinitesimal transformation, so we can't consider the analysis valid.

In any case, I find the original argument about the cross-term and the transformation (*) to be compelling.

At the moment I'm also reading a paper on gravitational waves in a Kaluza-Klein space (found here: http://arxiv.org/abs/gr-qc/0411028) where one is perturbing a KK vacuum solution $j_{AB}$

$$g_{AB} = j_{AB} + h_{AB}$$

The author then states that

"When spontaneous compactification takes place, the universe acquires a kaluza-klein structure ($M^5 \to M^4 \times S^1$) and the 5D local Poincare group is spontaneousy broken into a 4d local Poincare group and a $U(1)$ local gauge group. The wave, originally a 5d object now feels the effect of compactification and it's components transform in a different way under 4d coordinate transformations"

Later the author also states that

"compactification implies that the general covariance is lost and 4d fields contained in the 5d perturbation acquire a different behaviour under 4d coordinate transformations becoming distinct 4-d dynamical fields."

Firs of all, could someone explain the first quote in terms that does not require a lot of group theory? Secondly, if I'm not entirely misunderstanding, it clearly seems like this author claims that the relevant fields get different transformation properties under compactification. This could possibly be related to my first question regarding the transformation properties of the field tensor, but the question is - how and why are these new transformation properties acquired? Do they somehow go from being non-physical to physical after compactification?

I think the authors are being somewhat confusing by using the terminology "transform in a different way." As I explained above, when we periodically identify the points in $x^5$ to form the circle, the symmetries of the original space that mixed $x^5$ with the 4d coordinates are broken. We could be more specific, which would lead to the group theory that the authors claim, but it isn't really necessary for us to get an idea of what has changed.

Rather than say "transform in a different way," I would say that the components only transform under the remaining symmetries. It's more accurate to say that the solution before compactification has additional symmetries that the solution in the compact space does not.

 

[center o bass]

First of all, thanks a lot for the reply fzero!

Both of the questions that you raise involve the way that singling out one of the 5d coordinates and making a periodic identification breaks the diffeomorphism and Lorentz symmetries of GR. Basically, once we demand the space is $M_4\times S^1$, we are no longer free to make transformations that mix the periodic coordinate $\sigma$ with the 4d coordinates $x^\mu$. In the example from the video, a 5d Lorentz tranformation is made that sends

$$ \sigma \rightarrow \sigma' = \sigma + \alpha x^1.~~~~(*)$$

Because $x^1$ is not periodic, $\sigma'$ is not periodic, so we don't preserve the topology of the circle. The tranformations that do preserve the topology are just the independent diffeomorphisms of $M_4$ and $S^1$.

To further elaborate, the original question is whether we can find some coordinate redefinition that let's us rewrite the 5d metric

$$ ds_5^2 = g_{\mu \nu} dx^\mu dx^\nu + e^{2\sigma} \left( d\sigma + A_\mu dx^\mu \right)^2$$

in such a way as to eliminate the cross-term:

$$ ds_5^2 = g'_{\mu' \nu'} dx^{\mu'} dx^{\nu'} + (d\sigma')^2. $$

To make the example simpler, let's just suppose that we have a metric

$$ ds^2 = dx^2 + \left( d\sigma + A dx \right)^2,$$

where $A$ is just a constant. The new coordinates

$$ \begin{split} & \sigma' = \sigma + A x \\
& x' = x, \end{split}$$

certainly removes the cross-term, since

$$ ds^2 = (dx')^2 + (d\sigma')^2,$$

but as in (*) above, the required coordinate reparameterization doesn't respect the topology.



First, I'd note that the simple example above showed that in order to remove the cross term between $x^5$ and the 4d coordinates we would need to make a transformation that mixed $x^5$ with the 4d coordinates. But these were precisely the transformations that don't preserve the topology, so they are not allowed.

I'm inclined to think that the topology of the circle is something that is independent of weather we chose to use one set of coordinates or another. I would agree that our analysis becomes harder if we transform away from the coordinates for which $x^5$ is periodic, but I do not see why we are not free to do so.

May my objection here be resolved in thinking about active diffeomorphisms instead of passive coordinate transformations? Are you talking about the first or the latter?




Since posting this thread I have given the subject some thought when thinking about it in terms of passive coordinate transformations: If one thinks about the fifth dimension as inaccessible small, then one can not resolve 'different' points in the fifth dimension, and thus in effect all points with coordinates $(x^\mu, x^5)$ (with varying $x^5$)will 'look' as just one point with coordinates $x^\mu$. If one then thinks about coordinate transformations as alternative ways of relabeling points in spacetime and takes into account that any point on the circle is from our perspective just one point, then the only meaningful(or accessible) transformation would be

$$x'^\mu = x'^\mu(x^\nu)$$
and from your analysis above it then also follows that we can not transform away electromagnetism.

To elaborate the transformation
$x'^\mu = x'^\mu(x^\nu, x^5)$

is not an accessible transformation from the macroscopic perspective since we do not know how to 'boost' into a dimension we can not observe and neither do we know how to relabel points in it. Thus the electromagnetic field appear to be something absolute (with the correct tensorial transformation law etc.) from a macroscopic point of view, but KK theory explains that it really is not if we only were able to reach a scale for which the extra degrees of freedom become directly observable. In contrast we can 'easily' put our self in an inertial frame of 4d gravitation by freely falling in the gravitational field and transform away gravity.

What do you think about this explanation?

Second, I should point out that the transformation you wrote above is only the infinitesimal form of a general coordinate transformation that is of the form

$$ g'_{\hat{\mu}' \hat{\nu}' }(x') = g_{\hat{\mu} \hat{\nu}}(x) \frac{\partial x^{\hat{\mu}}}{\partial x^{\hat{\mu}'}} \frac{\partial x^{\hat{\nu}}}{\partial x^{\hat{\nu}'}} .$$

For a case where the gauge field is finite, the transformation has a more complicated form. To properly address the question, we'd need to work things out for the general case and then examine whether or not there are solutions with the properties that you suggest.

It's simpler to discuss the infinitesimal form, but we wouldn't expect an infinitesimal transformation to be able to generate a finite quantity that could cancel the gauge field. We can plod along for a specific example. If we have a point charge at the origin, we can consider a gauge potential of the form $A_0 = \alpha/r$. If you consider your transformation, there is a solution $\xi^r = r+r^2$, but this is no longer an infinitesimal transformation, so we can't consider the analysis valid.


In any case, I find the original argument about the cross-term and the transformation (*) to be compelling.

I agree that apriori the argument does not appear to be valid, but if it's also true that some additional argument can not prove it's validity, there are many bad arguments out there in the literature. In the case of gauge freedom in linearized gravity, similar arguments seem to be standard. For example here

http://web.mit.edu/edbert/GR/gr6.pdf

at page 9 under the title "Gauge fixing: Transverse gauge". There is also an identical argument in Sean Carroll's "Spacetime and Geometry" in the gravitational wave section.

 

[center o bass]

The new coordinates

$$ \begin{split} & \sigma' = \sigma + A x \\
& x' = x, \end{split}$$

certainly removes the cross-term, since

$$ ds^2 = (dx')^2 + (d\sigma')^2,$$

but as in (*) above, the required coordinate reparameterization doesn't respect the topology.

Hi fzero! I've spent some time lately, studying topology. I've had a look at quotient spaces/(identification spaces) and I suspect that the compactification $M^5 \to M^4 \times S$ can be viewed as a 'quotient map' when $S^1$ seems obtainable from several different spaces through such a map. But then it seems that the metric tensor on the original five-dimensional manifold would be transformed in some way by this map. Would you happen to know anything about this?

Anyway.. Regarding your statement above (in quotes) you argued that one could not make a coordinate transformation which did not respect the topology. I agree that the new coordinates must somehow be periodic, but that does not mean that it has to be periodic in just one of the coordinates? Suppose for example that we had a cartesian coordinate system on a cylinder (x,y) with the y-coordinate periodic. If we now rotated this coordinate system into $(x',y')$ it seems that instead of just the y-coordinate being periodic, now the vector $\vec{x} = (x',y)^T$ pointing along the old $y$-axis would be periodic. This would also imply that both $x'$ and $y'$ would in general now be periodic. It thus does it not seems more correct to conclude that after compactification the allowable coordinate transformations are on the form

$$x'^a = f^a(x^\mu, x^5)$$

with the $x'^a$ periodic?