Mass and charge from higher dimensional geometry
The authors explore ways in which phenomenological physical quantities such as the rest mass and electric charge of a particle could be explained as properties of higher dimensional geometry. If the world is 5D in nature, one can write its line element \(dS^2 = g_{\alpha\beta}dx^\alpha dx^\beta -\Phi^2(d\ell + A_\alpha dx^\alpha)^2\) as a 4D part (general relativity) plus parts that involve a scalar field and electromagnetism. This split is of course motivated by the Kaluza-Klein ``miracle'', wherein the field equations in vacuum for this metric split naturally into Einstein's equation for \(g_{\alpha\beta}\), Maxwell's equation for \(A_\alpha\), and a scalar wave equation for \(\Phi\). There is a quantity \(B\equiv-\Phi^2(d\ell/dS+A_\alpha dx^\alpha/dS)\) associated with this metric which is not present in general relativity and depends, among other things, on the velocity in the extra dimension. It is a constant of motion if the 5D metric is independent of \(x^4 = \ell\), but has significance otherwise. Another quantity \(C\equiv g_{00}^{1/2}(1+ B^2/\Phi^2)^{1/2}/(1-\nu^2)^{1/2}+BA_0\) associated with the metric is present in general relativity in a simpler form, and depends on the velocity in ordinary 3D space. It is a constant of the motion if the 5D metric is independent of \(x^0 = t\). A study of dynamics based on the Lagrangian or the geodesic shows that \(B\) is related to charge and \(C\) is related to energy (including rest mass). Of these, the latter is more fundamental. In an appropriate coordinate system in the weak-field limit, the energy or mass of a neutral particle is \(E=\ell/(1-\nu^2)^{1/2}\). The rest mass of a charged particle is \(m_q=\ell(1+B^2/\Phi^2)^{1/2}\) and its charge is \(q = B\ell\). The ratio \(q/m_q\) which appears in the Lorentz law and figures prominently in particle physics is \(q/m_q=B/(1+B^2/\Phi^2)^{1/2}\) and is in general coordinate-dependent, as are \(m_q\) and \(q\). Modulo covariant versions of the 5D metric, the conclusions are that rest mass is related to the extra coordinate and electric charge is related to the extra momentum. In \(N\)-dimensional Riemannian field theory \(N\) is promiscuous, to be chosen in the most convenient way necessary to explain a given physical situation. As long as we follow the legacy of Einstein, the overriding principle is that all physical parameters have to be expressible in geometrical terms.
Kaluza-Klein and higher-dimensional theories
scientific article; zbMATH DE number 1098528