General Relativistic Hydrodynamics

Mathematical Model

The equations of general relativistic hydrodynamics under the xCFC approximation can be expressed as a system of hyperbolic conservation laws, representing the conservation of baryon number, momentum, and energy, all as measured by an Eulerian observer (as defined in the 3+1 decomposition).

The conserved variables are nonlinear functions of the primitive variables: the baryon mass density measured by a comoving observer, \(\rho\), the fluid three-velocity measured by an Eulerian observer, \(v^{i}\), and the internal energy density measured by a comoving observer, \(e\). Collecting the primitive variables into a column vector, \(\boldsymbol{V}\), we have

\[\boldsymbol{V} := \left( \rho \, , v^{i} \, , e \right)^{T} \, .\]

These are related to the conserved baryon mass density, \(D\), as

\[D := \rho \, W \, ,\]

where \(W\) is the Lorentz factor of the fluid measured by an Eulerian observer; the momentum density, \(S_{j}\), as

\[S_{j} := \rho \, h \, W^{2} \, v_{j} \, ,\]

where \(h\) is the specific enthalpy, defined as \(h := 1 + \left( e + p \right) / \rho\), with \(p\) the thermal pressure; and a conserved energy density, \(\tau\), defined as

\[\tau := \rho \, h \, W^{2} - p - \rho \, W \, .\]

We collect the conserved variables into a column vector, \(\boldsymbol{U}\),

\[\boldsymbol{U} := \left( D \, , S_{j} \, , \tau \right)^{T} \, .\]

With that, we can write the GRHD equations as

\[\frac{\partial \left( \psi^{6} \, \boldsymbol{U} \right)}{\partial t} + \frac{\psi^{6}}{\sqrt{\gamma}} \frac{\partial \left( \alpha \, \sqrt{\gamma} \, \boldsymbol{F}^{i}\left(\boldsymbol{U}\right) \right)} {\partial x^{i}} = \alpha \, \psi^{6} \, \boldsymbol{S}\left(\boldsymbol{U}\right) \, ,\]

where \(\alpha\) is the lapse function, \(\psi\) is the conformal factor, and \(\sqrt{\gamma}\) is the square root of the determinant of the spatial three-metric. (TODO: define the fluxes and sources)