A less formal approach to kaluza—klein formalism

Research paper by M. A. McKiernan

Indexed on: 01 Jun '69Published on: 01 Jun '69Published in: Aequationes mathematicae

Abstract

The ‘action’ integrals (a)$$\lambda (\tau _1 ) = \int\limits_{\tau _0 }^{\tau _1 } {\surd g_{ij} \dot y^i \dot y^j d\tau }$$ and$$\lambda (\tau _1 ) = \int\limits_{\tau _0 }^{\tau _1 } {\{ \surd h_{ij} \dot x^i \dot x^j - B_i \dot y^i \} d\tau }$$, corresponding respectively to gravitational and gravitational-electromagnetic phenomena, are shown to be related under continuous groups of null translations. This relation motivates a modified Kaluza—Klein formalism for which the classical cylindrical metric preserving transformations (c)y5 = =x5 +f5(xj),yi =fi(xj) fori = 1, 2, 3, 4 are replaced by (d)y5 =x5,yi =fi(xj,x5). The cylindrical metric of V5 is nevertheless preserved under (d), since it is assumed thatV5 admits a metric of the form$$(\dot y^5 )^2 - g_{ij} (y^k )\dot y^i \dot y^j$$ (corresponding to (a)) and that (d) defines a continuous group of null translations in theV4 metric defined bygij whenx5 is considered the group parameter. Application of (d) leads to the cylindrical metric$$(\dot x^5 + B_i \dot x^i )^2 - h_{ij} \dot x^i \dot x^j$$ corresponding to (b). The resulting electromagnetic fieldsFij =Bi,j −Bj,i are then related to the curvatures of theV4 corresponding togij andhij; in particular it is shown that$$B_i B_j \mathop R\limits_g ^{ij} = - \tfrac{1}{4}F_{ij} F^{ij}$$ and$$F_{,j}^{ij} = B_j \mathop R\limits_g ^{ij}$$. When$$\mathop {R_{ij} }\limits_g = 0$$ it is shown thatFij is a null electromagnetic field which is generally non-trivial. Some physical and geometric interpretations of the mathematical results are also presented.