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Kaluza-Klein Theory

Kaluza-Klein Theory. Fritz Reitz, Ph.D. (unfortunately it’s not in physics...) 6/2/05. They might be giants. “If I have seen further than others, it is by standing upon the shoulders of giants.” -- Isaac Newton. What did Kaluza & Klein see?. add a 4th spatial dimension to spacetime

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Kaluza-Klein Theory

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  1. Kaluza-Klein Theory Fritz Reitz, Ph.D. (unfortunately it’s not in physics...) 6/2/05

  2. They might be giants • “If I have seen further than others, it is by standing upon the shoulders of giants.” -- Isaac Newton

  3. What did Kaluza & Klein see? • add a 4th spatial dimension to spacetime • leave everything else independent of the extra dimension • equations describing the extra dimension now have the same form as electromagnetism • curl up the extra dimension tightly • dimension now imperceptible plus it explains the quantization of charge

  4. 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)

  5. 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)

  6. 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

  7. 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...

  8. Nordström’s Gravity • g = A2()η •  is a scalar potential field, like that of Newton • close enough to make Albert nervous, but no cigar

  9. What if (almost) everything is symmetrical?

  10. 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 + 00E/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]

  11. 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)

  12. 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?

  13. 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

  14. 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)

  15. Kaluza’s paper • had great new idea - unification of gravity with electromagnetism in 5-D... • cited Thirring, Weyl • hadn’t heard of Nordström

  16. Kaluza’s paper • new idea worked by setting /w  0!

  17. Crash course in GR part 1:Christoffel symbols, “ Γ ” • correction for changing basis

  18. 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...

  19. 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

  20. Crash course in GR part 1:Christoffel symbols, “ Γ ” • Kaluza added the vector potential along the sides of the metric tensor essentially like so - • Then, when you calculate Γ5 in the usual way, it turns out Γ5  F, and Γ55 = /  • figure after Kaku, “Hyperspace”

  21. 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 -

  22. Crash course in GR part 1:Christoffel symbols, “ Γ ” • after all this mess - • it turns out that R5 F/x, and R55 = - from Griffiths remember -

  23. from Griffiths: and charge is velocity? • 0u J • J F/x • F/x  R5 • R5 T5 • T5 u5u  0u • 0 u5 from Kaluza: from Hartle (eqn. 22.51):

  24. 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

  25. 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...”

  26. 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)

  27. “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?”

  28. 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

  29. 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

  30. 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”

  31. Their immediate legacy • surely, the genius of these giants of unification would be lauded by their peers for decades!

  32. D’OH! #3 • actually their theory was totally eclipsed by quantum mechanics for 60 years or so

  33. But THEN their theory was much celebrated • theories such as Supergravity & String theory invoke yet more curled dimensions • figure after Greene, Fabric of the Cosmos • with 10 dimensions, you can fit everything! • hopefully, you can even make them work! • figure after Kaku, Hyperspace

  34. 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

  35. Personal footnote • “If I have seen less than others, it is because I as yet but cling to the buttocks of giants.” -- Fritz Reitz

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