Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 357 / 2 / a / Solution

LetThe identity decomposes asApplying this decomposition to both indices of $T_{\mu \nu }$ givesThe four terms are, from the definitions, $S_{\alpha \beta }$, $j_{\alpha }{n}_{\beta }$, $n_{\alpha }{j}_{\beta }$, and $\rho n_{\alpha }{n}_{\beta }$. Hence the 3+1 decomposition of the stress-energy tensor isIt also makes explicit that