Contract $\nabla_aT^{ab}=0$ for the <perfect fluid in general relativity> with $u_b$. Using $u_bu^b=-1$ and $u_b\nabla_au^b=0$ gives the <relativistic perfect-fluid energy equation>
$$
\boxed{u^a\nabla_a\rho+(\rho+p)\nabla_au^a=0},
$$
the local <first law of thermodynamics>. Project instead with $g^a{}_b+u^au_b$ to obtain the <relativistic Euler equation>
$$
\boxed{(\rho+p)u^b\nabla_bu^a
=-(g^{ab}+u^au^b)\nabla_bp}.
$$

Solution

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

The spatial projection of stress-energy conservation is $(\rho+p)u^b\nabla_bu^a=-(g^{ab}+u^au^b)\nabla_bp$.


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 Incoming links: Relativistic Euler equation