of a perfect fluid contains only the diagonal components. Perfect fluids are a
fluid solution used in
general relativity to model idealized distributions of
matter, such as the interior of a star or an isotropic universe. In the latter case, the symmetry of the cosmological principle and the
equation of state of the perfect fluid lead to
Friedmann equation for the
expansion of the universe.
Formulation In space-positive
metric signature tensor notation, the
stress–energy tensor of a perfect fluid can be written in the form : T^{\mu\nu} = \left( \rho_m + \frac{p}{c^2} \right) \, U^\mu U^\nu + p \, \eta^{\mu\nu} , where
U is the
4-velocity vector field of the fluid and where \eta_{\mu \nu} = \operatorname{diag}(-1,1,1,1) is the metric tensor of
Minkowski spacetime. The case where p=0 describes a
dust solution. When p=\rho_m c^2/3, it describes a
photon gas (radiation). In time-positive
metric signature tensor notation, the
stress–energy tensor of a perfect fluid can be written in the form : T^{\mu\nu} = \left( \rho_\text{m} + \frac{p}{c^2} \right) \, U^\mu U^\nu - p \, \eta^{\mu\nu} , where U is the 4-velocity of the fluid and where \eta_{\mu \nu} = \operatorname{diag}(1,-1,-1,-1) is the metric tensor of
Minkowski spacetime. This takes on a particularly simple form in the rest frame : \left[ \begin{matrix} \rho_e &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix} \right] where \rho_\text{e} = \rho_\text{m} c^2 is the
energy density and p is the
pressure of the fluid. Perfect fluids admit a
Lagrangian formulation, which allows the techniques used in
field theory, in particular,
quantization, to be applied to fluids.
Relativistic Euler equations read :\partial_\nu T^{\mu\nu}=0 in the non relativistic limit, these equations reduce to the usual Euler equations. == See also ==