The eigensystem of the general relativistic hydrodynamics (GRHD) equations

The time-step calculation that Gkeyll performs as part of the finite-volume update step for the general relativistic hydrodynamics (GRHD) equations within the Moment app depends upon knowledge of the eigenvalues of the GRHD system, and, furthermore, the implementation of a full Roe-type approximate Riemann solver requires knowledge of its eigenvectors too. Thus, for the sake of convenience, we present here the complete eigenystem for the GRHD equations. This was first calculated by [Anile1989] and [Eulderink1995] for the specific case of the ideal gas equation of state, and later generalized by [Banyuls1997] to the case of arbitrary equations of state. In this short technical note, we will follow the notational and terminological conventions introduced within the description of the GRHD equations in Gkeyll. In particular, we assume a \({3 + 1}\) “ADM” decomposition of our 4-dimensional spacetime into 3-dimensional spacelike hypersurfaces with induced metric tensor \(\gamma_{i j}\), lapse function \(\alpha\), and shift vector \(\beta^i\). We also assume a perfect relativistic fluid with (rest) mass density \(\rho\), spatial velocity components (as perceived by normal observers) \(v^i\), pressure \(P\), specific enthalpy \(h\), and local sound speed \(c_s\). Einstein summation convention (in which repeated tensor indices are implicitly summed over) is assumed throughout, and the Latin indices \(i, j, k, l\) range over the spatial coordinate directions \(\left\lbrace 1, \dots 3 \right\rbrace\) only.

See also this note on Gkeyll’s handling of general black hole spacetimes and the Kerr-Schild coordinate system, and this note on Gkeyll’s “robustified” conservative to primitive variable reconstruction algorithm for both special and general relativity.

Eigenvalues

In each of the 3 spatial coordinate directions \(x^k\), we can compute a 5-dimensional Jacobian matrix \(\mathbf{B}^k\):

\[\begin{split}\mathbf{B}^k = \alpha \frac{\partial \begin{bmatrix} \left( \frac{\rho h}{1 - \gamma_{i j} v^i v^j} - P - \frac{\rho}{\sqrt{1 - \gamma_{i j} v^i v^j}} \right) \left( v^k - \frac{\beta^k}{\alpha} \right) + P v^k\\ \left( \frac{\rho h v_l}{1 - \gamma_{i j} v^i v^j} \right) \left( v^k - \frac{\beta^k}{\alpha} \right) + P \delta_{l}^{k}\\ \left( \frac{\rho}{\sqrt{1 - \gamma_{i j} v^i v^j}} \right) \left( v^k - \frac{\beta^k}{\alpha} \right) \end{bmatrix}}{\partial \begin{bmatrix} \frac{\rho h}{1 - \gamma_{i j} v^i v^j} - P - \frac{\rho}{\sqrt{1 - \gamma_{i j} v^i v^j}}\\ \frac{\rho h v_l}{1 - \gamma_{i j} v^i v^j}\\ \frac{\rho}{\sqrt{1 - \gamma_{i j} v^i v^j}} \end{bmatrix}}.\end{split}\]

The 5 eigenvalues of each such Jacobian matrix \(\mathbf{B}^k\) can then be categorized into those corresponding to the material wave-speeds:

\[\lambda_{0}^{k} = \alpha v^k - \beta^k,\]

which each have algebraic multiplicity 3, and those corresponding to the acoustic wave-speeds:

\[\begin{split}\lambda_{\pm}^{k} = \frac{\alpha}{1 - \gamma_{i j} v^i v^j c_{s}^{2}} \left[ v^k \left( 1 - c_{s}^{2} \right) \right.\\ \left. \pm c_s \sqrt{\left( 1 - \gamma_{i j} v^i v^j \right) \left[ \gamma^{k k} \left( 1 - \gamma_{i j} v^i v^j c_{s}^{2} \right) - v^k v^k \left( 1 - c_{s}^{2} \right) \right]} \right] - \beta^k,\end{split}\]

which each have algebraic multiplicity 1. These are used by Gkeyll in the approximation of the maximum wave-speed across the computational domain, so as to ensure numerical stability through explicit enforcement of the CFL condition.

Right eigenvectors

The right eigenvectors in the \(x^1\) spatial coordinate direction (i.e. the right eigenvectors of the \(\mathbf{B}^1\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{r}_{0, 1}^{1} = \begin{bmatrix} \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right)}\\ v_1\\ v_2\\ v_3\\ 1 - \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right)\vert_{\rho} - c_{s}^{2} \right)} \end{bmatrix},\end{split}\]

\[\begin{split}\mathbf{r}_{0, 2}^{1} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^2\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^2\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^2 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^2\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^2 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^2 \end{bmatrix},\end{split}\]

and:

\[\begin{split}\mathbf{r}_{0, 3}^{1} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^3\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^3\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^3\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^3 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^3 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^3 \end{bmatrix},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{1}\) eigenvalues), and:

\[\begin{split}\mathbf{r}_{\pm}^{1} = \begin{bmatrix} 1\\ h \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_1 - \frac{v^1 - \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right)}{\gamma^{1 1} - v^1 \left( \frac{\lambda_{\pm}^{1} +\beta^1}{\alpha} \right)} \right)\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_2\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_3\\ \frac{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \gamma^{1 1} - v^1 v^1 \right)}{\gamma^{1 1} - v^1 \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right)} - 1 \end{bmatrix},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{1}\) eigenvalues). The right eigenvectors in the \(x^2\) spatial coordinate direction (i.e. the right eigenvectors of the \(\mathbf{B}^2\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{r}_{0, 1}^{2} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^1\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^1 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^1\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^1\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^1 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^1 \end{bmatrix},\end{split}\]

\[\begin{split}\mathbf{r}_{0, 2}^{2} = \begin{bmatrix} \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right)}\\ v_1\\ v_2\\ v_3\\ 1 - \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right)} \end{bmatrix},\end{split}\]

and:

\[\begin{split}\mathbf{r}_{0, 3}^{2} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^3\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^3\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^3\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^3 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^3 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^3 \end{bmatrix},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{2}\) eigenvalues), and:

\[\begin{split}\mathbf{r}_{\pm}^{2} = \begin{bmatrix} 1\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_1\\ h \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_2 - \frac{v^2 - \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right)}{\gamma^{2 2} - v^2 \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right)} \right)\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_3\\ \frac{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \gamma^{2 2} - v^2 v^2 \right)}{\gamma^{2 2} - v^2 \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right)} - 1 \end{bmatrix},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{2}\) eigenvalues). Finally, the right eigenvectors in the \(x^3\) spatial coordinate direction (i.e. the right eigenvectors of the \(\mathbf{B}^3\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{r}_{0, 1}^{3} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^1\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^1 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^1\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^1\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^1 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^1 \end{bmatrix},\end{split}\]

\[\begin{split}\mathbf{r}_{0, 2}^{3} = \begin{bmatrix} \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^2\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_1 v^2\\ h \left( 1 + 2 \left( 1 - \gamma_{i j} v^i v^j \right) v_2 v^2 \right)\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v_3 v^2\\ 2 h \left( 1 - \gamma_{i j} v^i v^j \right) v^2 - \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v^2 \end{bmatrix},\end{split}\]

and:

\[\begin{split}\mathbf{r}_{0, 3}^{3} = \begin{bmatrix} \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right)}\\ v_1\\ v_2\\ v_3\\ 1 - \frac{\frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho}}{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right)} \end{bmatrix},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{3}\) eigenvalues), and:

\[\begin{split}\mathbf{r}_{\pm}^{3} = \begin{bmatrix} 1\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_1\\ h \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) v_2\\ h \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_3 - \frac{v^3 - \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right)}{\gamma^{3 3} - v^3 \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right)} \right)\\ \frac{h \sqrt{1 - \gamma_{i j} v^i v^j} \left( \gamma^{3 3} - v^3 v^3 \right)}{\gamma^{3 3} - v^3 \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right)} - 1 \end{bmatrix},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{3}\) eigenvalues). The corresponding left eigenvectors may now be determined in each çase simply by inverting the matrix whose columns are given by the right eigenvectors, and then extracting the corresponding rows.

Left eigenvectors

The left eigenvectors in the \(x^1\) spatial coordinate direction (i.e. the left eigenvectors of the \(\mathbf{B}^1\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{l}_{0, 1}^{1} = \begin{bmatrix} \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( h v_1 v^1 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_2 v^2 + v_3 v^3 \right) - h \right) \right)}{c_{s}^{2} \left( v_1 v^1 - 1 \right)}\\ \frac{\left( c_{s}^{2} - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \right) v^1 \left( 1 + \left( v_2 v^2 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_1 v^1 - 1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^2 \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^3 \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \left( v_2 v^2 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_1 v^1 - 1 \right)} \end{bmatrix}^{\intercal},\end{split}\]

\[\begin{split}\mathbf{l}_{0, 2}^{1} = \begin{bmatrix} - \frac{v_2}{h \left( 1 - v_1 v^1 \right)}\\ \frac{v_2 v^1}{h \left( 1 - v_1 v^1 \right)}\\ \frac{1}{h}\\ 0\\ - \frac{v_2}{h \left( 1 - v_1 v^1 \right)} \end{bmatrix}^{\intercal},\end{split}\]

and:

\[\begin{split}\mathbf{l}_{0, 3}^{1} = \begin{bmatrix} - \frac{v_3}{h \left( 1 - v_1 v^1 \right)}\\ \frac{v_3 v^1}{h \left( 1 - v_1 v^1 \right)}\\ 0\\ \frac{1}{h}\\ - \frac{v_3}{h \left( 1 - v_1 v^1 \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{1}\) eigenvalues), and:

\[\begin{split}\mathbf{l}_{\pm}^{1} = \begin{bmatrix} \frac{\left( \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right) v^1 - \gamma^{1 1} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) + c_{s}^{2} \gamma^{1 1} v_1 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^1 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) v_1 v^1 + h \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) \left( v_1 v^1 - 1 \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) \left( v_2 v^2 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{1} - \lambda_{\mp}^{1}}{\alpha} \right) \left( v_1 v^1 - 1 \right) \left( v^1 v^1 - \gamma^{1 1} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \gamma^{1 1} - \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right) v^1 \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) v^1 + c_{s}^{2} \left( \gamma^{1 1} - \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) v^1 \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) v^1 \left( v_2 v^2 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{1} - \lambda_{\mp}^{1}}{\alpha} \right) \left( v_1 v^1 - 1 \right) \left( v^1 v^1 - \gamma^{1 1} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) \left( \gamma^{1 1} - \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right) v^1 \right) v^2 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{1} - \lambda_{\mp}^{1}}{\alpha} \right) \left( \gamma^{1 1} - v^1 v^1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) - v^1 \right) \left( \gamma^{1 1} - \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right) v^1 \right) v^3 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{1} - \lambda_{\mp}^{1}}{\alpha} \right) \left( \gamma^{1 1} - v^1 v^1 \right)}\\ \frac{\left( \left( \frac{\lambda_{\pm}^{1} + \beta^1}{\alpha} \right) v^1 - \gamma^{1 1} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) + c_{s}^{2} \gamma^{1 1} v_1 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^1 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) v_1 v^1 + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{1} + \beta^1}{\alpha} \right) \right) - v^1 \right) \left( v_2 v^2 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{1} - \lambda_{\mp}^{1}}{\alpha} \right) \left( v_1 v^1 - 1 \right) \left( v^1 v^1 - \gamma^{1 1} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{1}\) eigenvalues). The left eigenvectors in the \(x^2\) spatial coordinate direction (i.e. the left eigenvectors of the \(\mathbf{B}^2\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{l}_{0, 1}^{2} = \begin{bmatrix} - \frac{v_1}{h \left( 1 - v_2 v^2 \right)}\\ \frac{1}{h}\\ \frac{v_1 v^2}{h \left( 1 - v_2 v^2 \right)}\\ 0\\ - \frac{v_1}{h \left( 1 - v_2 v^2 \right)} \end{bmatrix}^{\intercal},\end{split}\]

\[\begin{split}\mathbf{l}_{0, 2}^{2} = \begin{bmatrix} \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( h v_2 v^2 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_1 v^1 + v_3 v^3 \right) - h \right) \right)}{c_{s}^{2} \left( v_2 v^2 - 1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^1 \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( c_{s}^{2} - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \right) v^2 \left( 1 + \left( v_1 v^1 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_2 v^2 - 1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^3 \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \left( v_1 v^1 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_2 v^2 - 1 \right)} \end{bmatrix}^{\intercal},\end{split}\]

and:

\[\begin{split}\mathbf{l}_{0, 3}^{2} = \begin{bmatrix} - \frac{v_3}{h \left( 1 - v_2 v^2 \right)}\\ 0\\ \frac{v_3 v^2}{h \left( 1 - v_2 v^2 \right)}\\ \frac{1}{h}\\ - \frac{v_3}{h \left( 1 - v_2 v^2 \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{2}\) eigenvalues), and:

\[\begin{split}\mathbf{l}_{\pm}^{2} = \begin{bmatrix} \frac{\left( \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right) v^2 - \gamma^{2 2} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) + c_{s}^{2} \gamma^{2 2} v_2 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^2 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) v_2 v^2 + h \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) \left( v_2 v^2 - 1 \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) \left( v_1 v^1 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{2} - \lambda_{\mp}^{2}}{\alpha} \right) \left( v_2 v^2 - 1 \right) \left( v^2 v^2 - \gamma^{2 2} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) \left( \gamma^{2 2} - \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right) v^2 \right) v^1 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{2} - \lambda_{\mp}^{2}}{\alpha} \right) \left( \gamma^{2 2} - v^2 v^2 \right)}\\ \frac{\left( \gamma^{2 2} - \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right) v^2 \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) v^2 + c_{s}^{2} \left( \gamma^{2 2} - \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) v^2 \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) v^2 \left( v_1 v^1 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{2} - \lambda_{\mp}^{2}}{\alpha} \right) \left( v_2 v^2 - 1 \right) \left( v^2 v^2 - \gamma^{2 2} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) - v^2 \right) \left( \gamma^{2 2} - \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right) v^2 \right) v^3 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{2} - \lambda_{\mp}^{2}}{\alpha} \right) \left( \gamma^{2 2} - v^2 v^2 \right)}\\ \frac{\left( \left( \frac{\lambda_{\pm}^{2} + \beta^2}{\alpha} \right) v^2 - \gamma^{2 2} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) + c_{s}^{2} \gamma^{2 2} v_2 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^2 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) v_2 v^2 + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{2} + \beta^2}{\alpha} \right) \right) - v^2 \right) \left( v_1 v^1 + v_3 v^3 \right) \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{2} - \lambda_{\mp}^{2}}{\alpha} \right) \left( v_2 v^2 - 1 \right) \left( v^2 v^2 - \gamma^{2 2} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{2}\) eigenvalues). Finally, the left eigenvectors in the \(x^3\) spatial coordinate direction (i.e. the left eigenvectors of the \(\mathbf{B}^3\) Jacobian matrix) are given by:

\[\begin{split}\mathbf{l}_{0, 1}^{3} = \begin{bmatrix} - \frac{v_1}{h \left( 1 - v_3 v^3 \right)}\\ \frac{1}{h}\\ 0\\ \frac{v_1 v^3}{h \left( 1 - v_3 v^3 \right)}\\ - \frac{v_1}{h \left( 1 - v_3 v^3 \right)} \end{bmatrix}^{\intercal},\end{split}\]

\[\begin{split}\mathbf{l}_{0, 2}^{3} = \begin{bmatrix} - \frac{v_2}{h \left( 1 - v_3 v^3 \right)}\\ 0\\ \frac{1}{h}\\ \frac{v_2 v^3}{h \left( 1 - v_3 v^3 \right)}\\ - \frac{v_2}{h \left( 1 - v_3 v^3 \right)} \end{bmatrix}^{\intercal},\end{split}\]

and:

\[\begin{split}\mathbf{l}_{0, 3}^{3} = \begin{bmatrix} \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( h v_3 v^3 + \sqrt{1 - \gamma_{i j} v^i v^j} \left( v_1 v^1 + v_2 v^2 \right) - h \right) \right)}{c_{s}^{2} \left( v_3 v^3 - 1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^1 \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) v^2 \left( 1- \gamma_{i j} v^i v^j \right)}{c_{s}^{2}}\\ \frac{\left( c_{s}^{2} - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \right) v^3 \left( 1 + \left( v_1 v^1 + v_2 v^2 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_3 v^3 - 1 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( 1 + \left( v_1 v^1 + v_2 v^2 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} \left( v_3 v^3 - 1 \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 3 material waves (corresponding to the 3 \(\lambda_{0}^{3}\) eigenvalues), and:

\[\begin{split}\mathbf{l}_{\pm}^{3} = \begin{bmatrix} \frac{\left( \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right) v^3 - \gamma^{3 3} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) + c_{s}^{2} \gamma^{3 3} v_3 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^3 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) v_3 v^3 + h \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} - c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) \left( v_3 v^3 - 1 \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) \left( v_1 v^1 + v_2 v^2 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{3} - \lambda_{\mp}^{3}}{\alpha} \right) \left( v_3 v^3 - 1 \right) \left( v^3 v^3 - \gamma^{3 3} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) \left( \gamma^{3 3} - \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right) v^3 \right) v^1 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{3} - \lambda_{\mp}^{3}}{\alpha} \right) \left( \gamma^{3 3} - v^3 v^3 \right)}\\ \frac{\left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) \left( \gamma^{3 3} - \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right) v^3 \right) v^2 \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{3} - \lambda_{\mp}^{3}}{\alpha} \right) \left( \gamma^{3 3} - v^3 v^3 \right)}\\ \frac{\left( \gamma^{3 3} - \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right) v^3 \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) v^3 + c_{s}^{2} \left( \gamma^{3 3} - \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) v^3 \right) + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) - v^3 \right) v^3 \left( v_1 v^1 + v_2 v^2 \right) \left( 1 - \gamma_{i j} v^i v^j \right) \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{3} - \lambda_{\mp}^{3}}{\alpha} \right) \left( v_3 v^3 - 1 \right) \left( v^3 v^3 - \gamma^{3 3} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)}\\ \frac{\left( \left( \frac{\lambda_{\pm}^{3} + \beta^3}{\alpha} \right) v^3 - \gamma^{3 3} \right) \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) + c_{s}^{2} \gamma^{3 3} v_3 - \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} v^3 - c_{s}^{2} \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) v_3 v^3 + \left( \frac{1}{\rho} \left. \left( \frac{\partial P}{\partial \varepsilon} \right) \right\vert_{\rho} + c_{s}^{2} \right) \left( \left( \frac{\lambda_{\mp}^{3} + \beta^3}{\alpha} \right) \right) - v^3 \right) \left( v_1 v^1 + v_2 v^2 \right) \left( 1 - \gamma_{i j} v^i v^j \right)}{c_{s}^{2} h \left( \frac{\lambda_{\pm}^{3} - \lambda_{\mp}^{3}}{\alpha} \right) \left( v_3 v^3 - 1 \right) \left( v^3 v^3 - \gamma^{3 3} \right) \left( \sqrt{1 - \gamma_{i j} v^i v^j} \right)} \end{bmatrix}^{\intercal},\end{split}\]

for the 2 acoustic waves (corresponding to the 2 \(\lambda_{\pm}^{3}\) eigenvalues).

References

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A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics, Cambridge University Press. 1989.

[Eulderink1995]

F. Eulderink and G. Mellema, “General Relativistic Hydrodynamics with a Roe solver”, Astronomy and Astrophysics Supplement Series 110: 587-623. 1995.

[Banyuls1997]

F. Banyuls, J. A. Font, J. M. Ibáñez, J. M. Martí and J. A. Miralles, “Numerical {3 + 1} General Relativistic Hydrodynamics: A Local Characteristic Approach”, The Astrophysical Journal 476 (1): 221-231. 1997.