2.5.1 Equations of Relativistic Hydrodynamics in Conservation Form

The evolution of a relativistic fluid is described by a system of equations which are the expression of local conservation laws: the local conservation of baryon number density

$$\nabla_{\mu}J^{\mu} = 0$$ (41)

and the local conservation of energy-momentum

$$\nabla_{\mu}T^{\mu\nu} = 0$$ (42)

the current $J^{\mu}$ and the energy-momentum tensor $T^{\mu\nu}$ being

$$J^{\mu} = \rho u^{\mu}$$ (43)

$$T_{\mu\nu} = \rho h u_{\mu}u_{\nu} + p g_{\mu \nu}$$ (44)

(Greek and Latin indices run, respectively, from 0 to 3 and from 1 to 3. Units are used in which the speed of light is equal to one).

In the above equations $\rho$ is the rest-mass density, $p$ is the pressure and $h$ is the specific enthalpy, defined by $h = 1 + \varepsilon + p/\rho$, where $\varepsilon$ is the specific internal energy, $u^{\mu}$ is the four-velocity of the fluid and $g_{\mu \nu}$ defines the metric of a general space-time $\cal M$ where the fluid evolves, $\nabla_{\mu}$ stands for the covariant derivative.

In Minkowski space-time, $\cal M$, the above system of equations, (42) and (43), can be written in a more compact way as:

$$\frac {\partial {\bf F}^{\mu}({\bf w})}{\partial x^{\mu}} = 0$$ (45)

being the five-vector of unknowns

$${\bf w} = (\rho, v^{i}, \varepsilon)$$ (46)

where the components of the three-velocity, $v^i$, are defined according to

$$v^i= \frac{u^i}{u^0}$$

and the Lorentz factor, defined by $W \equiv u^0$, satisfies the familiar relation $W=(1-{\rm v}^{2})^{-1/2}$, where ${\rm v}^{2}= \delta_{ij} v^i v^j$.

The quantities ${\bf F}^{\alpha}({\bf w})$ are

$${\bf F}^{0}({\bf w}) = \left(\rho W, \rho h W^2 v^j, \rho h W^{2}-p-\rho W\right)$$ (47)

$${\bf F}^{i}({\bf w}) = \left(\rho W v^{i}, \rho h W^2 v^j v^i + p \delta^{ij}, \right.$$

$$\left. \rho h W^2 v^i -\rho W v^i \right)$$ (48)

The components of ${\bf F}^{0}({\bf w})$

$$D \equiv \rho W$$ (49)

$$S^{j} \equiv \rho h W^2 v^j$$ (50)

$$\tau \equiv \rho h W^{2}-p-\rho W$$ (51)

can be considered, respectively, the coordinate rest-mass density, the relativistic momentum density and the total energy density (substracted the rest-mass density).

An equation of state $p=p(\rho,\varepsilon)$ closes, as usual, the system. A very important quantity derived from the equation of state is the local sound velocity $c_{s}$:

$$h c_{s}^{2} = \chi + (p/\rho^{2})\kappa$$ (52)

with $\chi = \partial p / \partial \rho$ and $\kappa = \partial p / \partial \epsilon$. As it can be verified, for an ideal gas $p = (\Gamma -1) \rho \epsilon$, the maximum value of the local sound velocity satisfies the following relations:

$$c_s^{max} = \sqrt{\Gamma -1} = \sqrt{{\tilde{\kappa}}}$$ (53)

$\tilde{\kappa}$ being the quantity $\tilde{\kappa} \equiv \kappa/\rho$.

In the system of equations defined by (122) it is possible to carry out a theoretical analysis of the structure of the characteristic fields - see the next subsection - which will be crucial for numerical applications.