4.6 General Relativistic Hydrodynamic Equations

References: Banyuls, Font, Ibáñez, Martí and Miralles, ApJ, 476 221 (1997)

Font, Miller, Suen and Tobias, Phys. Rev. D (1999) in press

gr-qc/9811015

• Equations of motion:
  local conservation laws $J^{\mu}_{;\mu} = 0$ and $T^{\mu\nu} _{;\nu} = 0$ where the current $J^{\mu}$ and the energy-momentum tensor $T^{\mu\nu}$ are
  $$\begin{eqnarray*} J^{\mu} = \rho u^{\mu} \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} T_{\mu\nu} = \rho h u_{\mu}u_{\nu} + p g_{\mu \nu} \end{eqnarray*}$$
  
  
  
  
  
  
  $\rho$ is the rest-mass density, $p$ is the pressure, $h$ is the specific enthalpy, $h = 1 + \epsilon + p/\rho$, $\epsilon$ is the specific internal energy and $g_{\mu \nu}$ denotes the space-time metric.

• Equation of state:
  $p = p(\rho,\epsilon)$ and the local sound velocity $c_{s}$:
  $$\begin{eqnarray*} h c_{s}^{2} = \chi + (p/\rho^{2})\kappa \end{eqnarray*}$$
  
  
  
  
  
  
  with $\chi = \partial p / \partial \rho$ and $\kappa = \partial p / \partial \epsilon$.

• $\{3+1\}$ formalism:
  the metric is split into the objects $\alpha$ (lapse), $\beta^{i}$ (shift) and $\gamma_{ij}$.
  $$\begin{eqnarray*} ds^{2} = -(\alpha^{2}-\beta_{i}\beta^{i}) dt^{2}+ 2 \beta_{i} dx^{i} dt + \gamma_{ij} dx^{i}dx^{j} \end{eqnarray*}$$
  
  
  
  
  
  

• Conserved Variables:
  rest-mass density ($D$), momentum density in the $j$-direction ($S_j$) and total energy density ($E$). In terms of the primitive variables ${\bf w} = (\rho, v_{i}, \varepsilon)$

  $$\begin{eqnarray*} D &=& \rho W \\ S_j & =& \rho h W^2 v_j \\ E & =& \rho h W^2 - p \\ \end{eqnarray*}$$
  
  
  
  
  
  

  where $v_i = \frac{{\bf u} \cdot \partial_i}{-({\bf u \cdot n})}$, $v^i= \frac{u^i}{\alpha u^t} + \frac{\beta^i}{\alpha}$ and $W \equiv -({\bf u \cdot n})= \alpha u^{t}$ (Lorentz factor) $W=(1-{v}^{2})^{-1/2}$ with ${v}^{2}= \gamma_{ij} v^i v^j$.

• Fundamental System (Hyperbolic):

  $$\begin{eqnarray*} \frac{1}{\sqrt{-g}} \left( \frac {\partial \sqrt{\gamma}{\bf... ... F}^{i}({\bf w})} {\partial x^{i}} \right) = {\bf s}({\bf w}) \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} {\bf F}^{0}({\bf w}) &=& (D, S_j, \tau)^T \\ \\ {\bf F}^... ...l x^{\mu}} - T^{\mu \nu} \Gamma^0_{\nu \mu} \right) \right)^T \end{eqnarray*}$$
  
  
  
  
  
  

  where $\tau \equiv E - D$, $g\equiv \det(g_{\mu\nu})$, $\sqrt{-g} = \alpha\sqrt{\gamma}$, $\gamma\equiv \det(\gamma_{ij})$.

• Numerical Integration:

  $$\begin{eqnarray*} \int_{\Omega} \frac{1}{\sqrt{-g}} \frac {\partial \sqrt{\gamma... ...F}^{i}}{\partial x^{i}} d\Omega = \int_{\Omega} {\bf s} d\Omega \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} & (\displaystyle{\overline{\bf F}^0} {\Delta V})_{t+\Delta t} ... ...nonumber %%& + & \displaystyle{\int_{\Omega}} {\bf s} d\Omega \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} \displaystyle{\overline{\bf F}^0} = \frac{1}{\Delta V} \int_{\Delta V} \sqrt{\gamma} {\bf F}^0 dx^1 dx^2 dx^3 \end{eqnarray*}$$
  
  
  
  
  
  
  $$\begin{eqnarray*} {\Delta V} = \int^{x^1+\Delta x^1}_{x^1} \int^{x^2+\Delta x^2}_{x^2} \int^{x^3+\Delta x^3}_{x^3} \sqrt{\gamma} dx^1 dx^2 dx^3 \end{eqnarray*}$$
  
  
  
  
  
  

• Characteristic fields ($x$ direction)

  Eigenvalues:

  $$\begin{eqnarray*} \lambda_0 &=& \alpha v^x - \beta^x \mbox{ (triple)} \\... ...{v}^{2}c_{s}^{2}) -v^x v^x (1-c_{s}^{2})]} \right\} - \beta^x \end{eqnarray*}$$
  
  
  
  
  
  

  Right-eigenvectors:

  
  

  $${\bf r}_{\pm} = \left[ \begin{array}{c} 1 \\ h W \left( ... ...displaystyle{\frac{{\cal K}}{h W}}} \\ \end{array} \right]$$
  
  

  
  

  $${\bf r_{0,2}} = \left[ \begin{array}{c} W v_y \\ h \left... ..._z }}\right) \\ W v_z (2hW - 1) \\ \end{array} \right]$$
  
  

  with $\tilde{\kappa} \equiv \kappa/\rho$, $\Lambda_{\pm}^i \equiv \tilde{\lambda}_{\pm} + \tilde{\beta}^i$, $\tilde{\lambda} \equiv \lambda/\alpha$ and $\tilde{\beta}^i \equiv \beta^i/\alpha$

In numerical relativity the GRH equations must be solved in conjunction with the gravitational field equations (Einstein equations)

ADM formulation ( $\gamma_{ij}, K_{ij}$)

• spacetime foliated into a set of non-intersecting spacelike hypersurfaces

• two kinematic variables describe the evolution between these surfaces: the lapse function $\alpha$, describing the rate of advance of time along a timelike unit vector $n^\mu$ normal to a surface, and the spacelike shift vector $\beta^i$ describing the motion of coordinates

• Metric:
  $$\begin{eqnarray*} ds^{2} = -(\alpha^{2}-\beta_{i}\beta^{i}) dt^{2}+ 2 \beta_{i} dx^{i} dt + \gamma_{ij} dx^{i}dx^{j} \end{eqnarray*}$$
  
  
  
  
  
  

• Evolution equations:
  $$\begin{eqnarray*} \partial_t \gamma_{ij} &=& - 2 \alpha K_{ij}+\nabla_i \beta_j+... ...abla_j \beta^m+K_{mj} \nabla_i \beta^m -\nabla_i \nabla_j \alpha \end{eqnarray*}$$
  
  
  
  
  
  
  where $\nabla_i$ denotes a covariant derivative with respect to the 3-metric $\gamma_{ij}$ and $R_{ij}$ is the Ricci curvature of the 3-metric

• Constraint equations:
  $$\begin{eqnarray*} {}^{(3)}R + K^2 - K_{ij} K^{ij} &=& 16 \pi {\rho}_{{}_{\verb+ADM+}} \\ \nabla_j K^{ij} - \gamma^{ij} \nabla_j K &=& 8 \pi j^i \end{eqnarray*}$$
  
  
  
  
  
  

  with

  $$\begin{eqnarray*} {\rho}_{{}_{\verb+ADM+}} &=& \rho h W^2 - P \\ j^i &=& \rho h... ...h W^2 v_i v_j + \gamma_{ij} P \\ S &=& \rho h W^2 v_i v^i + 3 P \end{eqnarray*}$$