Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

nas g = ^2 g^dx^x*, the expression of the Ricci curvature r g is the given bynr g = ^ tijdx i dx i with147' ~ 2 1 4^ U**&c' ^ 0*«0s> ^d«' d^dx'J(5) *.'=i+ quadratic terms in ~ox(in this formula, (gV) denotes the inverse matrix of (gy)).The key difFerence between the Riemannian and the Lorentzian settingsis of course that in the first case the system is elliptic, and in the second itis hyperbolic. On the space of metrics, the left hand side of the Einsteinfield equatìon has a variational nature. It is indeed the difTerential of the totalscalar curvature functional S. Since this fact was discovered by David Hilbert,S is often referred to as the Einstein-Hilbert Lagrangian.e) The originai Kaluza-Klein theoryIn 1919, Theodor Kaluza proposed a 5-dimensional version of GeneralRelativity. Although at first sight the theory presented some interesting features,Kaluza's paper (cf. [7]) was only accepted for publication by Einsteinin 1921.Kaluza considered IR 5 with coordinates (j 0 ,^1,!: 2 ,^3,^) as model for anextended space-time, which can be projected by a map p onto the ordinaryspace-time IR 4 with coordinates (x° 1 x 1 tx 2 }x 3 ). The main trick, the so-calledKaluza Ansatz, is to consider a Lorentzian metric £ on IR 5 of the form(6) / = * B (pV+-*(a + «)® (a+ #))where : IR 4 —* IR is a function, a 6 IR,^ is a Lorentzian metric on IR 4 and ais a 1-form on IR 4 .Provided one takes = 1, the key fact is thenKALUZA'S DISGOVERY. - The Einstein vacuum equations for i contain theMaxwell equations for the electromagnetic field u = da, and the Einsteinequations fori with right hand side T taken to be the interaction term comingfrom the electromagnetic field w.