Relativistic Euler equations

Definition

Work in Minkowski space $\R^{1+3}$ with coordinates $x^0 = t, x^1, x^2, x^3$, the metric $\eta = \operatorname{diag}(-1, 1, 1, 1)$ and units with $c = 1$. Indices are raised and lowered with $\eta$, and repeated indices are summed. A perfect fluid is described by its four-velocity $u^\mu$, with $u^\mu u_\mu = -1$ and $u^0 > 0$, and by three scalar fields. These are its rest-frame energy density $e$, its pressure $p$ and its particle number density $n$. Its energy–momentum tensor is $$T^{\mu\nu} = (e + p) \, u^\mu u^\nu + p \, \eta^{\mu\nu}.$$ The relativistic Euler equations are $$\partial_\mu T^{\mu\nu} = 0 \quad (\nu = 0, 1, 2, 3), \qquad \partial_\mu (n u^\mu) = 0,$$ closed by an equation of state that gives $p$ in terms of $e$ and $n$.