Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 1 / a / Solution

For a fixed scalar function $f$, the equation ${\square }_{g}f=0$ is coordinate invariant. A coordinate label $x^{\lambda }$, however, is not a geometrically specified scalar under an arbitrary change of chart: a nonlinear redefinition of harmonic coordinates need not remain harmonic. Thus ${\square }_g{x}^{\lambda }=0$ is a coordinate or gauge condition selecting a class of charts, rather than a tensor equation asserting the vanishing of a tensor field with index $\lambda$.

Solved by gpt-5.6-sol high.