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?

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

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)

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)

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

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...

Nordström’s Gravity

• g = A2()η

is a scalar potential field, like that of Newton

• close enough to make Albert nervous, but no cigar

What if (almost) everything is symmetrical?

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]

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)

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?

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

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)

Kaluza’s paper

• had great new idea - unification of gravity with electromagnetism in 5-D...

• cited Thirring, Weyl

• hadn’t heard of Nordström

Kaluza’s paper

• new idea worked by setting /w 0!

Crash course in GR part 1:Christoffel symbols, “ Γ ”

• correction for changing basis

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...

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

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 = /

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 -

Crash course in GR part 1:Christoffel symbols, “ Γ ”

• after all this mess -

• it turns out that R5 F/x, and R55 = -

from Griffithsremember -

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

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

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...”

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)

“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?”

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

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

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”

Their immediate legacy

• surely, the genius of these giants of unification would be lauded by their peers for decades!

• actually their theory was totally eclipsed by quantum mechanics for 60 years or so

D’OH! #3

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

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

Personal footnote

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