Reducing Five to Four: One Worked Case

Chapter 2 of this arc said that Kaluza's five-dimensional vacuum gravity contains four-dimensional gravity and electromagnetism. Here is the statement in a form you can test, with one case checked by computer algebra.

Klein's metric

Take four-dimensional spacetime with metric g_μν and electromagnetic potential A_μ, and add a fifth coordinate w. In Klein's form (as written in Ravndal, arXiv:1309.4113) the five-dimensional metric and its inverse are:

Nothing depends on w. The identity that makes the theory work is that the five-dimensional curvature scalar splits as the four-dimensional one minus a multiple of the electromagnetic energy:

Varying the action built from R₍₅₎ then gives Einstein's equations with the Maxwell stress as source, and Maxwell's equations. In this chapter I state the identity as the standard result and have not rederived it in general here; the next paragraph tests it in one case.

One case, checked

Take flat four-dimensional space, so R₍₄₎ = 0, with a uniform magnetic field B along z, given by A = B x dy. Then F_xy = B and F_μνF^μν = 2B². The identity predicts R₍₅₎ = −κ²B²/2. I computed the Christoffel symbols and Ricci scalar of the five-dimensional metric symbolically (Python sympy, signature −+++ +, metric components g_tt = −1, g_xx = g_zz = g_ww = 1, g_yy = 1 + κ²B²x², g_yw = κBx). The result was R₍₅₎ = −κ²B²/2, matching the prediction exactly.

A contrast with the contact condition

This pure magnetic field has F ∧ F = 0, so by the earlier chapter its Kaluza–Klein space is not contact, yet it is a perfectly good solution of the reduction. Contact-ness is an extra condition, not a feature of every Kaluza–Klein space. To get it you need E and B together and not orthogonal.