Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 58 / 5

Let a dot denote differentiation with respect to an affine geodesic parameter $s$. The five-dimensional geodesic Lagrangian isBecause $z$ is cyclic, its Euler-Lagrange equation gives the conserved quantityThis is the conserved momentum per unit five-dimensional mass associated with the Killing vector field ${\partial }_z$. For the other coordinates, differentiate $\partial L/\partial {\dot{x}}^i={\gamma }_{ij}{\dot{x}}^j+k{A}_i$. Using $\dot{k}=0$ givesThe first two terms combine into ${\gamma }_{ij}({\ddot{x}}^j+{\Gamma }^j{}_{lm}{\dot{x}}^l{\dot{x}}^m)$, with the Christoffel symbols of $\gamma$. Define the four-dimensional electromagnetic field tensor $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i$ and raise its index with $\gamma$. The complete reduced geodesic equations areThis is the Kaluza-Klein geodesic Lorentz force. The Christoffel symbols describe the gravitational motion in general relativity in the four-dimensional metric tensor, while the right-hand side is antisymmetric in its lowered indices and gives the relativistic Lorentz force. In particular,Thus the force preserves the four-dimensional velocity norm, as a Lorentz force must. A particle with $k=0$ is neutral and follows an ordinary four-dimensional geodesic. Nonzero compact momentum gives electric charge, with its sign determined by the direction of motion around the circle.

To identify the actual charge-to-mass ratio, suppose the five-dimensional particle has mass $M>0$ and $s$ is its five-dimensional proper time, so $g_5(\dot{X},\dot{X})=-1$. Then $\gamma (\dot{x},\dot{x})=-(1+k^2)$. Four-dimensional proper time obeys $d\tau /ds=\sqrt{1+k^2}$; since $k$ is constant, $u^i=d{x}^i/d\tau$ satisfiesHere $q$ is in the geometric normalization of the supplied potential $A$. The relation $m^2=M^2+p_z^2$ also follows directly from the five-dimensional mass shell, and continues to identify the effective mass when the higher-dimensional particle is massless. The distinction between the two proper times prevents mistaking $k$ itself for the physical four-dimensional charge-to-mass ratio. A rescaling of $A$ to a canonically normalized electromagnetic potential correspondingly rescales $q$.

For the wave calculation, take a scalar field of five-dimensional mass $M$ in units $\hslash =c=1$, obeying $({\square }_5-M^2)\Psi =0$. In the coordinate order $(x^i,z)$, direct inversion of the metric tensor givesThe determinant follows by the triangular change of coframe from $(d{x}^i,dz)$ to $(d{x}^i,dz+A_{i}d{x}^i)$, whose determinant is one. Write $J=\sqrt{|\det \gamma |}$. Since all coefficients are $z$-independent, the scalar covariant wave operator isExpanding the right-hand side verifies both cross terms $-2A^i{\partial }_i{\partial }_z\Psi$, the term $-J^{-1}{\partial }_i(J{A}^i){\partial }_z\Psi$, and the coefficient $1+A_i{A}^i$ of ${\partial }_z^2\Psi$.

Single-valuedness under the compact coordinate period requires the Fourier seriesFor the $n$th Kaluza-Klein mode, put $q_n=n/R$ and $D_i={\partial }_i-i{q}_n{A}_i$. The five-dimensional Klein-Gordon equation reduces toThis is the charged four-dimensional Klein-Gordon equation: the divergence expression includes both the spacetime connection and the gauge connection. Its interpretation as electric charge can also be checked directly. Under the fibre coordinate change $z^{\prime }=z-\chi (x)$, the metric retains its form with $A^{\prime }=A+d\chi$, and the scalar Fourier coefficient changes by ${\psi }_n^{\prime }=e^{i{q}_n\chi }{\psi }_n$. Hence $D_i^{\prime }{\psi }_n^{\prime }=e^{i{q}_n\chi }{D}_i{\psi }_n$, exactly the transformation law for a field of charge $q_n$.

The Kaluza-Klein charge quantization is thereforePositive and negative integers give opposite charges, and the zero mode is neutral. Restoring $\hslash$ gives compact momenta $p_z=n\hslash /R$, so their fundamental spacing is $\hslash /R$. These are charges in the metric's geometric normalization. If $A_i=\kappa {\mathcal{A}}_i$ for a physically normalized electromagnetic potential ${\mathcal{A}}_i$, the physical charge unit is $\kappa /R$ in natural units, or $\kappa \hslash /R$ with $\hslash$ restored. Determining $\kappa$ requires an electromagnetic action normalization not specified by the metric ansatz alone.