Past exam of the relativity course of the University of Cambridge / 2021 / Section B / 4

For rest mass $m$ and electric charge $q$, the generally covariant Lorentz force law isHere $F_{ab}={\partial }_a{A}_b-{\partial }_b{A}_a$ is the electromagnetic field tensor for the chosen electromagnetic four-potential normalization, and the covariant derivative includes the spacetime Levi-Civita connection. Antisymmetry of $F_{ab}$ ensures $u_{a}D{u}^a/d\tau =0$, preserving the four-velocity norm and the particle's rest mass. An uncharged particle follows a geodesic.

Let a dot denote differentiation with respect to the five-dimensional affine parameter $\lambda =\tilde{u}$, and define $W={\dot{x}}^4+A_a{\dot{x}}^a$. Use the quadratic geodesic LagrangianThe assumed independence of $x^4$ makes it a cyclic coordinate, so the corresponding Euler-Lagrange equation isSince $\partial L/\partial {\dot{x}}^4=W$, this givesGeometrically, this is the conserved momentum associated with the fifth-direction Killing vector field ${\partial }_4$. It is the conserved fibre momentum underlying the Kaluza-Klein geodesic Lorentz force.

An affine geodesic has constant squared tangent norm, but that constant depends on the scale of its parameter. WriteUsing $W=\xi$ in the line element givesThe affine normalization in a Kaluza-Klein geodesic reduction matters: the specific formula requested in the paper uses $\mathcal{N}=1$, namely $\lambda$ equal to the positive five-dimensional arclength. With that normalization,A timelike four-dimensional projection has $d{u}^2=c^{2}d{\tau }^2>0$, requiring $|\xi |<1$ in this normalization. At $|\xi |=1$ the projection is null and has no proper-time four-velocity. For an arbitrary affine parameter the first boxed relation, with $\mathcal{N}$ retained, is the general result.

Differentiate the geodesic Lagrangian before substituting the conserved quantity. For a four-dimensional index $a$,Its Euler-Lagrange equation, together with $W=\xi$ and $\dot{W}=0$, isThus the sign of the projected Kaluza-Klein Lorentz force is fixed by the printed metric:The derivative order in the PDF's proposed equation is reversed. Its right-hand side is the negative of the result above for the stated $\xi$ and metric. This is not repaired by renaming the field tensor, because the two explicit derivative expressions have opposite signs. For example, $A_0=A_0(x^1)$, $A_1=0$ gives ${\ddot{x}}^1=-\xi A_0^{\prime }(x^1){\dot{x}}^0$ directly from the Lagrangian, whereas the printed expression gives the opposite. The corrected equation is used below.

Normalize the five-dimensional tangent as in part (c), and let $s=\sqrt{1-{\xi }^2}>0$. Since $u=c\tau$, one has $d\tau /d\lambda =s/c$, soSubstitution into the corrected equation in part (d) givesThis is the Lorentz force form in four-dimensional Minkowski spacetime. The effective charge-to-mass coefficient is proportional to the conserved fifth-direction momentum: in $c=1$ units and a potential normalization identifying $A$ with the electromagnetic four-potential, $q/m=\xi /\sqrt{1-{\xi }^2}$. The factor converting the geometrical $A$ to an SI potential fixes the physical charge units. Opposite signs of $\xi$ give opposite effective charges, and $\xi =0$ gives a neutral inertial projection. Antisymmetry gives $d({\eta }_{ab}{u}^a{u}^b)/d\tau =0$.

This is a Kaluza-Klein theory mechanism: a higher-dimensional geodesic appears as forced motion after projection. With the explicitly supplied $+---$ base signature, the positive fibre term gives a second positive direction in the five-dimensional metric. The derivation is a valid geometrical reduction of that metric; a conventional single-time spacetime would require a spacelike extra direction and corresponding sign changes. If one instead retains the sign printed in part (d), the inferred effective charge reverses, but that version is not the geodesic equation of the metric supplied here.