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    • CommentRowNumber1.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 21st 2013

    Added a link to an expository talk I gave on “The geometry of force” giving an elementary explanation of the classical Kaluza-Klein mechanism (i.e. the idea that geodesics on the principal bundle project down to curved trajectories on base space apparently experiencing a “force”). Following the book of Bleecker, Gauge theory and variational principles.

    • CommentRowNumber2.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 21st 2013

    While preparing this talk, and especially after trying to understand the case of the frame bundle of S 2S^2 via this approach, I have the following question.

    Recall the main idea here (following the book of Bleecker). Let π:PM\pi : P\rightarrow M be a principal GG-bundle with connection ω\omega over a Riemannian manifold (M,g)(M, g), and let kk be an invariant inner product on the lie algebra of GG. Then there is an induced “pull-back” metric h=π *g+kωh = \pi^*g + k \omega on the bundle PP. The point is that geodesics γ\gamma on PP project down to curved trajectories γ¯:=πγ\bar{\gamma} := \pi \circ \gamma on MM which we think are “experiencing a force”. There is a precise equation for these projected geodesics which Bleecker writes down.

    Here’s my question. Don’t we also get, in this setup, a new connection α\alpha on MM itself? Namely, let σ\sigma be a curve in MM, and let FF be an orthonormal frame at x=σ(0)x = \sigma(0). Using the connection ω\omega on PP, we can lift σ\sigma to a curve σ˜\tilde{\sigma} in PP. But now (P,h)(P, h) is a Riemannian manifold as defined above, so it has a Levi-Civita connection, so we can parallel transport a frame in PP along σ˜\tilde{\sigma}. And a frame in PP is just a frame in MM plus a frame in the Lie algebra of GG, and we can choose some fixed one in the beginning. So we can parallel transport the frame in PP! Then we can project this down to MM. This should give a moving frame in MM.

    In other words, I have constructed a new connection α\alpha on MM. Is this correct, or am I making a geometric error?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 21st 2013
    • (edited Sep 21st 2013)

    Hi Bruce,

    thanks for adding this.

    I have expanded your line in the References-section

    An elementary exposition of the geometry behind the Kaluza-Klein mechanism (the idea that geodesics on the gauge bundle project to curved trajectories on the base manifold) can be found in this talk:

    to

    An elementary exposition of the geometry behind the Lorentz force in the Kaluza-Klein mechanism (the idea that geodesics on the gauge bundle project to curved trajectories on the base manifold) can be found in this talk:

    Okay?

    Then I have added to the section The mechanism the following paragraph:

    Then one finds that the KK-mechanics indeed not only reproduces gauge fields and their correct dynamics from pure gravity in higher dimensions, but also the forces which they excert on test particles. For instance the rajectory of a charged particle subject to the Lorentz force excerted by an electromagnetic field in dd-dimensional spacetime is in fact a geodesic in the field of pure gravity of the total space of the corresponding KK-un-compactified circle principal bundle. See (Bartlett 13) for a pedagogical discussion of this effect.

    • CommentRowNumber4.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 21st 2013

    Hi Urs, thanks yes that’s fine. I also added a reference to the book of Bleecker, and his name to the phrase “See (Bartlett13) for a pedagogical…”.

    • CommentRowNumber5.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 21st 2013

    I am still curious about my question. Let me make it more clear: I am suggesting that a connection on a GG-principal bundle over a Riemannian manifold (M,g)(M, g) gives rise to a new connection (different from the Levi-Civita connection) on MM.

    For instance, take M= 3M=\mathbb{R}^3 with the standard metric. And let PP be the trivial U(1)U(1)-bundle over . And let the connection on the U(1)\mathbb{R^U(1)-bundle PP be given by a magnetic field in the zz-direction (remember, this is a connection on PP, not on MM! A connection on MM is what I’m about to construct.) . This is a standard setup from 1st year physics. If we gave a particle in 3\mathbb{R}^3 an initial velocity in the xx direction, it travels in a circle in the xyxy-plane by the Lorentz force law, F=q(E+v×B)F = q(E + v \times B).

    I am claiming that we can interpret the magnetic field as giving rise to a new connection on 3\mathbb{R}^3, different from the standard flat connection. To give you this connection, I have to tell you how to parallel transport frames in 3\mathbb{R}^3 along a curve σ\sigma. Answer: each basis vector of the frame rotates as if experiencing the magnetic force appropriate to that basis vector being thought of as a velocity vector. So, if the curve σ\sigma goes along the zz-axis, nothing happens to the frame (no force). If σ\sigma is a circle in the xyxy-plane, the frame rotates around the zz-axis. For a general curve, a combination of these effects. I believe this connection has torsion, but I haven’t checked. I’m probably making a psychological error here.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2013
    • (edited Sep 22nd 2013)

    Bruce, for the sake of readability, could you dis-ambiguate your uses of “connection on” a bit? There are principal connections on a principal bundle in the game and at the same time affine connections on the tangent bundle over the total space of the principal bundle. When you say “connection on MM” is is clear that you mean a connection on TMMT M \to M, but when you say “connection on PP” then the reader has to start working out which of the two different meanings you have in mind (the actual principal connection on PMP \to M or the affine connection on TPPT P \to P).

    • CommentRowNumber7.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 22nd 2013

    I am using both those meanings, that’s the point. (Well… I always use connections on principal bundles, and never “affine” connections. But an affine connection on MM is the same thing as a connection on the principal frame bundle.) In particular, as you say, there is the connection on PP in the sense of PP being a principal bundle over MM (a Lie(G)Lie(G)-valued 1-form on PP), but there is also the Levi-Civita connection on the frame bundle of PP, since PP is equipped with the bundle metric h=π *g+kωh = \pi^* g + k \omega where kk is an inner product on Lie(G)Lie(G).

    When I say “I am suggesting that a connection on a G-principal bundle PP over a Riemannian manifold (M,g) gives rise to a new connection (different from the Levi-Civita connection) on M” I mean the following:

    “connection on a GG-principal bundle PP over a Riemannian manifold (M,g)(M,g)” means the usual notion of a connection on a principal bundle PP (an equivariant distribution of horizontal subspaces of PP).

    ” a new connection on MM” means a new connection on the frame bundle of MM, in the usual sense of a connection on a frame bundle (i.e. thinking of the frame bundle as a principal bundle).

    It can be a bit tricky, I agree.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2013
    • (edited Sep 22nd 2013)

    I don’t think it’s tricky, as we can all distinguish two things at a time. I just think that when you say “connection on PP” you are being ambiguous and making your readers work harder than necessary to figure out what you want to say.

    Another thing: I just noticed the edit of yours in verion 26 of the KK-entry, which is probably based on a misunderstanding of my citation convention. You changed the citation anchor “Bartlett13” to “Bartlett14” and introduced “Bleecker13”. But that number is supposed to be the year of publication. So I changed it (back) to “Bleecker81” and “Bartlett13”.

    • CommentRowNumber9.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 23rd 2013

    Yes, apologies for getting the conventions mixed up. But I guess there are no thoughts on my proposal?