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A less formal approach to kaluza-klein formalism

Authors:	M A McKiernan

Institution:	(1) University of Waterloo, Waterloo, Ont., Canada

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)y ⁵ = =x ⁵ +f ⁵(x ^j),y ⁱ =f ⁱ(x ^j) fori = 1, 2, 3, 4 are replaced by (d)y ⁵ =x ⁵,y ⁱ =f ⁱ(x ^j,x ⁵). The cylindrical metric of V⁵ is nevertheless preserved under (d), since it is assumed thatV ⁵ 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 theV ⁴ metric defined byg _ij whenx ⁵ 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 fieldsF _ij =B _i,j –B _j,i are then related to the curvatures of theV ⁴ corresponding tog _ij andh _ij; 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 thatF _ij is a null electromagnetic field which is generally non-trivial. Some physical and geometric interpretations of the mathematical results are also presented.Dedicated to Professor A. Ostrowski on the occasion of his 75th birthday

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