Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 309 / 4 / a / Solution

Contract ${\nabla }_a{T}^{ab}=0$ for the perfect fluid in general relativity with $u_b$. Using $u_b{u}^b=-1$ and $u_b{\nabla }_a{u}^b=0$ gives the relativistic perfect-fluid energy equationthe local first law of thermodynamics. Project instead with $g^a{}_b+u^a{u}_b$ to obtain the relativistic Euler equation