Kaluza-Klein Theory Fritz Reitz, Ph.D. (unfortunately it’s not in physics...) 6/2/05.
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Transcript of Kaluza-Klein Theory Fritz Reitz, Ph.D. (unfortunately it’s not in physics...) 6/2/05.
![Page 1: Kaluza-Klein Theory Fritz Reitz, Ph.D. (unfortunately it’s not in physics...) 6/2/05.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649f005503460f94c15b49/html5/thumbnails/1.jpg)
Kaluza-Klein Theory
Fritz Reitz, Ph.D. (unfortunately it’s not in physics...)
6/2/05
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They might be giants
• “If I have seen further than others, it is by standing upon the shoulders of giants.” -- Isaac Newton
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What did Kaluza & Klein see?
1. add a 4th spatial dimension to spacetime
2. leave everything else independent of the extra dimension
– equations describing the extra dimension now have the same form as electromagnetism
3. curl up the extra dimension tightly– dimension now imperceptible plus it
explains the quantization of charge
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Giant shoulder #1
• Einstein (who in turn stood on the shoulders of Lorentz, Michelson, Planck, etc.) produced a working theory of special relativity
• got people thinking in 4-D (even before he did, e.g. Minkowski)
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Giant shoulder #2
• Hans Thirring, pointed out near-perfect analogy between equations of general relativity and electromagnetism
• (geodesic eqn., metric for incoherent matter, and Christoffels Poynting vector, 4-potential, Maxwell’s eqns, respectively)
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Giant shoulder #3• H. Weyl, produced unified 4-D
theory of space, time, and electromagnetism
• thought length constancy with vector transport was a historical accident, that g might in fact be locally scaled by some scalar function , such that GR should be rederived for invariance with g g
• only ratios between elements would have physical meaning
• He and Einstein were deeply divided by this issue
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In 5-D before it was hip -
• Gunnar Nordström, working with his own 4-D version of gravitation, found that it could be elegantly added as a 5th dimension to electromagnetism...
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Nordström’s Gravity
• g = A2()η
is a scalar potential field, like that of Newton
• close enough to make Albert nervous, but no cigar
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What if (almost) everything is symmetrical?
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Or, in other words, if /w 0,
• Bz/y - By/z - (1/c2)Ex/t + 0 = kx/c•-Bz/x + Bx/z + (1/c2)Ey/t + 0 = ky/c• By/x - Bx/y - (1/c2)Ez/t + 0 = kz/c• (1/c2) Ex/x + Ey/y + Ez/z + 0 = kt/c• x/x + y/y + z/z - t/t = kw/c
•first 3 lines are B = 0J + 00E/t (Ampere’s +)
•next line is •E = (1/0) (Gauss’)
•last line is = - m
•[J = (1/c){kt, kx, ky, kz}, kt/c, m= - kw/c]
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Similarly, permuting through all 3-way combinations --
•first line is •B = 0 •next 3 are E = - B/t (Faraday’s)
•last six lines are = 0*
*(I took his word for that part)
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Nordström’s paper in sum
•an elegant 4-D version of gravity•pulls the whole of electromagnetism virtually out of a hat•unifies all then-known forces
what could go wrong?
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D’OH!• his relationship with
Einstein was “...less than cordial.”
• Einstein likely knew of Nordström’s work
• somehow it slipped his mind by the time Kaluza came along
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Theodor Kaluza• Privatdozent at U of
Königsberg (at the time...)
• Poor as dirt (wife would cry and gesture at empty cupboards)
• Nice guy (deferred scholarship to a widow)
• Boring lecturer
• Spoke at least 7 languages fluently & lectured from memory
• believer in theory (swimming)
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Kaluza’s paper
• had great new idea - unification of gravity with electromagnetism in 5-D...
• cited Thirring, Weyl
• hadn’t heard of Nordström
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Kaluza’s paper
• new idea worked by setting /w 0!
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Crash course in GR part 1:Christoffel symbols, “ Γ ”
• correction for changing basis
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Crash course in GR part 1:Christoffel symbols, “ Γ ”
• correction for changing basis
• basis can change due to simply remapping, as before, or due to actual curvature of space, which messes with the metric tensor directly...
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Crash course in GR part 1:Christoffel symbols, “ Γ ”
• Kaluza said wow, F and Γ look similar• hmm, F would need another index to match up
properly• why, that would only happen if there was - GASP
- another spatial dimension
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Crash course in GR part 1:Christoffel symbols, “ Γ ”
• Kaluza added the vector potential along the sides of the metric tensor essentially like so -
• figure after Kaku, “Hyperspace”
• Then, when you calculate Γ5 in the usual way, it turns out Γ5 F, and Γ55 = /
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Crash course in GR part 2:The Riemann Curvature Tensor
• The big punchline of GR is the “Einstein Equation”, basically equating spacetime curvature to the “Energy-Momentum” tensor T like so -
• when uncontracted, R is the “Riemann Curvature Tensor”; it distills curvature information from the Christoffel symbols like so -
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Crash course in GR part 1:Christoffel symbols, “ Γ ”
• after all this mess -
• it turns out that R5 F/x, and R55 = -
from Griffithsremember -
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and charge is velocity?
from Griffiths:
from Kaluza:
from Hartle (eqn. 22.51):
• 0u J
• J F/x
• F/x R5
• R5 T5
• T5 u5u 0u
• 0 u5
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Kaluza’s paper in sum
• extend GR to 5-D, with /w 0
• apologize for extra dimension
• EM falls out as wrinkles in 5th dimension
• charge is ~ velocity in this direction
• a few more calculations to verify that GR and EM haven’t been broken by the additions
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Einstein’s reaction(s)• 4/21/19: “I like your idea at first sight
very much... I will be pleased to present your paper...”
• 4/28/19: “...arguments brought forward do not appear convincing enough.”
• 5/29/19: “It is not easy for me to advise you whether to publish the idea as formulated...”
• 10/14/21: “I am having second thoughts about having restrained you from publishing... I shall present your paper to the academy after all...”
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Oskar Klein• a lot of alsos (Klein-Gordon
eqn. of relativistic waves, Jordan-Klein matrices & 2nd quantization, Klein-Nishina formula for high-energy photon scattering from electrons)
• a lot of almosts (would have beat Schrodinger to publication if he hadn’t gotten sick, proposed a Yang-Mills-esque strong interaction theory earlier)
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“Almost” #106 -
• Klein -“Yo, W-Dogg, look what happens when you extend GR to 5-D with /w 0!”
• Pauli -“You mean like in Kaluza’s paper that has already been published?”
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D’OH! #2• Klein was a tad
disappointed to hear about Kaluza’s paper
• decided there was enough new stuff in his own work to go ahead and publish
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His additions• he also found u5 and thus
momentum, and thus 1 / (de Broglie wavelength h/p)
• he imagined the extra dimension wrapped in a circle, with an integer number of standing waves
• quantum of charge thus specifies radius of extra dimension < 10-30 in.
• bunch of other stuff including repeated use of the word “simply”
after Greene, Fabric of the Cosmos, Fig. 12-7
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Klein’s paper in sum
• rederives Kaluza's results, and -
• charge is quantized by standing waves in small dimension
• dimension is wrapped around on itself and super small, that’s why we’ve never noticed it
• atomic physics is “simple”
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Their immediate legacy
• surely, the genius of these giants of unification would be lauded by their peers for decades!
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• actually their theory was totally eclipsed by quantum mechanics for 60 years or so
D’OH! #3
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But THEN their theory was much
celebrated
• figure after Kaku, Hyperspace
• theories such as Supergravity & String theory invoke yet more curled dimensions
• with 10 dimensions, you can fit everything!
• hopefully, you can even make them work!
• figure after Greene, Fabric of the Cosmos
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Further reading• Popularizations re: Kaluza-Klein theory, string theory:
Halpern’s The Great Beyond (much biographical history), Kaku’s Hyperspace, Greene’s The Elegant Universe
• Original papers: Appelquist et al. Modern Kaluza-Klein Theories (Nordstrom, Kaluza, Klein), De Sabbata & Schmutzer Unified Field Theories of More Than 4 Dimensions (Kaluza, Klein, Kaluza-Einstein correspondence), Lorentz et al. The Principle of Relativity (Weyl’s paper cited by Kaluza), Verbin & Nielsen (2005) Gen. Rel. Grav. 37:427 (translation of Thirring’s paper cited by Kaluza)
• Papers on Nordstrom’s scalar theory of gravity: Camenzind on “General Relativity” (http://www.lsw.uni-heidelberg.de/users/mcamenzi/GR_05.ps.gz), Calogero & Lee (2004) Comm. Math. Sci. 2(1):19
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Personal footnote
• “If I have seen less than others, it is because I as yet but cling to the buttocks of giants.” -- Fritz Reitz