Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 6

Let $\xi$ denote geodesic distance from the charge; $R$ is the fixed curvature radius, not the radial variable. In round three-dimensional hyperbolic space, a sphere has areaUsing the convention that electric flux is $Q/{ϵ}_0$, Gauss's law gives the radial orthonormal electric fieldIts magnitude is $|Q|/A{ϵ}_0$, and a negative charge reverses the direction. In rationalized natural units set ${ϵ}_0=1$; Gaussian conventions instead use flux $4\pi Q$.

For $\xi \ll R$ this approaches $Q/(4\pi {ϵ}_0{\xi }^2)$. For $\xi \gg R$, $E\sim Q{e}^{-2\xi /R}/(\pi {ϵ}_0{R}^2)$. This hyperbolic-space Coulomb field falls rapidly because the sphere area grows exponentially; conserved electric flux is unchanged. A potential vanishing at infinity is $V=Q[\coth (\xi /R)-1]/(4\pi {ϵ}_{0}R)$.

The original PDF actually prints $\sin \theta$ rather than ${\sin }^2\theta$ in the angular metric tensor. The round angular factor used above is required by its description as hyperbolic space. If the printed angular metric tensor is taken literally, the sphere-area coefficient is insteadand a radial Gauss-law solution has $E=Q/[{ϵ}_0{\Omega }_{\ast }{R}^2{\sinh }^2(\xi /R)]$. That angular geometry is not a smooth round sphere, so the literal metric tensor does not describe the asserted homogeneous hyperbolic space. Both readings retain the same radial area-growth factor; the geometric repair is explicit.

Define $\Delta t=t-t^{\prime }$, $\mathbf{r}=\mathbf{x}-{\mathbf{x}}^{\prime }$ in the three noncompact spatial directions, and $\Delta w=w-w^{\prime }$. The compact source must be the periodic Dirac deltaFor the wave operator as explicitly defined in the question, a retarded circle-compactified massless propagator isThe poles $\omega =\pm \sqrt{{\mathbf{k}}^2+k_n^2}-i0$ are both below the real axis, making the integral vanish for $\Delta t<0$. Performing just the frequency integral gives an equivalent expressionThe first time derivative of $\Theta (t)\sin (\Omega t)/\Omega$ jumps by one at zero, so applying ${\partial }_t^2+{\Omega }^2$ yields $\delta (t)$. Together with the completeness relations for the Fourier series and Fourier transform this verifies the specified source and normalization. The symbol for the wave operator follows the question's explicit sign convention; it is the negative of $g^{AB}{\nabla }_A{\nabla }_B$ for its mostly-plus metric tensor.

Each compact Fourier mode obeys the noncompact equation with dispersion relation ${\omega }^2={\mathbf{k}}^2+m_n^2$, whereThus momentum in the extra dimension appears as rest mass to a four-dimensional observer. There is one massless zero mode and equally massive $n,-n$ modes. For a real scalar they can be organized into sine and cosine components rather than independent complex fields.

At equal compact positions all phase factors are one, and the propagator becomes the Kaluza-Klein towerThe factor $1/R$ follows from the normalized compact eigenfunctions. It cannot be discarded if the five-dimensional field and source normalizations are retained.

At distances and elapsed times much smaller than $R$, propagation probes the local covering space and resolves the compact direction. Formally $R^{-1}{\sum }_n$ becomes $\int d{k}_w/(2\pi )$ when many nearby compact momenta participate, giving the massless five-dimensional retarded Green function. Equivalently, the compact Green function is a sum of covering-space images at $\Delta w+jR$. Before a signal can travel to any nonzero winding image, only the direct image contributes. Short-scale physics is five-dimensional.

At low frequency and long wavelength, $|\omega |,|\mathbf{k}|\ll 2\pi /R$, massive modes cannot be excited as propagating particles, and the massless zero mode dominates at long noncompact range. Then $G\simeq R^{-1}{G}^{(4,0)}$, with higher-mode corrections and the corresponding effective source normalization. Low-energy long-scale physics is four-dimensional.

A precise static illustration avoids ambiguities of pointlike retarded signals. At equal compact positions and zero frequency, the Yukawa potential sum isThis static Green function across a circle compactification has limitsThey are the Coulomb kernels in four and three spatial dimensions respectively, with exponentially small massive corrections at large $r$.

For an unsmeared point-source retarded distribution, coordinate separations alone are not enough to impose a low-energy approximation. A high-frequency pulse can still excite massive modes and resolve winding signals at arbitrarily long travel distance/time; its image fronts satisfy $\Delta t^2=r^2+j^2{R}^2$. Thus large separation does not impose a Kaluza-Klein low-energy limit. The usual large-separation four-dimensional statement means low-energy sources or coarse-grained measurements, or a static long-distance limit. It is not a universal pointwise limit erasing all massive oscillations from a sharply resolved retarded pulse.