Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 4 / b / Solution

SetDirect exterior differentiation givesSolving Cartan's first structure equation and imposing ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$ gives the independent mixed-index connection 1-formsThe remaining forms follow from Lorentz-signature antisymmetry: ${\omega }^1{}_0={\omega }^0{}_1$, ${\omega }^2{}_0={\omega }^0{}_2$, ${\omega }^3{}_0={\omega }^0{}_3$, and ${\omega }^j{}_i=-{\omega }^i{}_j$ for spatial indices.