aurel.core ########## .. automodule:: aurel.core :noindex: .. _descriptions_list: descriptions ************ .. _required_quantities: Required quantities =================== **gxx**: $g_{xx}$ Metric with xx indices down (need to input) **gxy**: $g_{xy}$ Metric with xy indices down (need to input) **gxz**: $g_{xz}$ Metric with xz indices down (need to input) **gyy**: $g_{yy}$ Metric with yy indices down (need to input) **gyz**: $g_{yz}$ Metric with yz indices down (need to input) **gzz**: $g_{zz}$ Metric with zz indices down (need to input) **kxx**: $K_{xx}$ Extrinsic curvature with xx indices down (need to input) **kxy**: $K_{xy}$ Extrinsic curvature with xy indices down (need to input) **kxz**: $K_{xz}$ Extrinsic curvature with xz indices down (need to input) **kyy**: $K_{yy}$ Extrinsic curvature with yy indices down (need to input) **kyz**: $K_{yz}$ Extrinsic curvature with yz indices down (need to input) **kzz**: $K_{zz}$ Extrinsic curvature with zz indices down (need to input) **rho0**: $\rho_0$ Rest mass energy density (need to input) .. _assumed_quantities: Assumed quantities ================== $\Lambda = 0$, the Cosmological constant, to change this do **AurelCore.Lambda = ...** before running calculations **alpha**: $\alpha = 1$, the lapse, to change this do **AurelCore.data["alpha"] = ...** before running calculations **dtalpha**: $\partial_t \alpha = 0$, the time derivative of the lapse **betaup3**: $\beta^i = 0$, the shift vector with spatial indices up **dtbetaup3**: $\partial_t \beta^i = 0$, the time derivative of the shift vector with spatial indices up **press**: $p = 0$, the fluid pressure **eps**: $\epsilon = 0$, the fluid specific internal energy **w_lorentz**: $W = 1$, the Lorentz factor **velup3**: $v^i = 0$, the Eulerian fluid three velocity with spatial indices up Metric quantities ================= Spatial metric -------------- **gammadown3**: $\gamma_{ij}$ Spatial metric with spatial indices down **gammaup3**: $\gamma^{ij}$ Spatial metric with spatial indices up **dtgammaup3**: $\partial_t \gamma^{ij}$ Coordinate time derivative of spatial metric with spatial indices up **gammadet**: $\gamma$ Determinant of spatial metric **gammadown4**: $\gamma_{\mu\nu}$ Spatial metric with spacetime indices down **gammaup4**: $\gamma^{\mu\nu}$ Spatial metric with spacetime indices up Extrinsic curvature ------------------- **Kdown3**: $K_{ij}$ Extrinsic curvature with spatial indices down **Kup3**: $K^{ij}$ Extrinsic curvature with spatial indices up **Ktrace**: $K = \gamma^{ij}K_{ij}$ Trace of extrinsic curvature **Adown3**: $A_{ij}$ Traceless part of the extrinsic curvature with spatial indices down **Aup3**: $A^{ij}$ Traceless part of the extrinsic curvature with spatial indices up **A2**: $A^2$ Magnitude of traceless part of the extrinsic curvature Lapse ----- **alpha**: $\alpha$ Lapse (need to input or I assume =1) **dtalpha**: $\partial_t \alpha$ Coordinate time derivative of the lapse (need to input or I assume =0) Shift ----- **betaup3**: $\beta^{i}$ Shift vector with spatial indices up (need to input or I assume =0) **dtbetaup3**: $\partial_t\beta^{i}$ Coordinate time derivative of the shift vector with spatial indices up (need to input or I assume =0) **betadown3**: $\beta_{i}$ Shift vector with spatial indices down **betamag**: $\beta_{i}\beta^{i}$ Magnitude of shift vector Timeline normal vector ---------------------- **nup4**: $n^{\mu}$ Timelike vector normal to the spatial metric with spacetime indices up **ndown4**: $n_{\mu}$ Timelike vector normal to the spatial metric with spacetime indices down Spacetime metric ---------------- **gdown4**: $g_{\mu\nu}$ Spacetime metric with spacetime indices down **gup4**: $g^{\mu\nu}$ Spacetime metric with spacetime indices up **gdet**: $g$ Determinant of spacetime metric Matter quantities ================= Eulerian observer follows $n^\mu$ Lagrangian observer follows $u^\mu$ Lagrangian matter variables --------------------------- **press**: $p$ Pressure (need to input or I assume =0) **eps**: $\epsilon$ Specific internal energy (need to input or I assume =0) **rho**: $\rho$ Energy density **enthalpy**: $h$ Specific enthalpy of the fluid Fluid velocity -------------- **w_lorentz**: $W$ Lorentz factor (need to input or I assume =1) **velup3**: $v^i$ Eulerian fluid three velocity with spatial indices up (need to input or I assume =0) **uup0**: $u^t$ Lagrangian fluid four velocity with time indice up **uup3**: $u^i$ Lagrangian fluid four velocity with spatial indices up **uup4**: $u^\mu$ Lagrangian fluid four velocity with spacetime indices up **udown3**: $u_\mu$ Lagrangian fluid four velocity with spatial indices down **udown4**: $u_\mu$ Lagrangian fluid four velocity with spacetime indices down **hdown4**: $h_{\mu\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices down **hmixed4**: ${h^{\mu}}_{\nu}$ Spatial metric orthonomal to fluid flow with mixed spacetime indices **hup4**: $h^{\mu\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices up Energy-stress tensor -------------------- **Tdown4**: $T_{\mu\nu}$ Energy-stress tensor with spacetime indices down Eulerian matter variables ------------------------- **rho_n**: $\rho^{\{n\}}$ Energy density in the $n^\mu$ frame **fluxup3_n**: $S^{\{n\}i}$ Energy flux (or momentum density) in the $n^\mu$ frame with spatial indices up **fluxdown3_n**: $S^{\{n\}}_{i}$ Energy flux (or momentum density) in the $n^\mu$ frame with spatial indices down **angmomup3_n**: $J^{\{n\}i}$ Angular momentum density in the $n^\mu$ frame with spatial indices up **angmomdown3_n**: $J^{\{n\}}_{i}$ Angular momentum density in the $n^\mu$ frame with spatial indices down **Stressup3_n**: $S^{\{n\}ij}$ Stress tensor in the $n^\mu$ frame with spatial indices up **Stressdown3_n**: $S^{\{n\}}_{ij}$ Stress tensor in the $n^\mu$ frame with spatial indices down **Stresstrace_n**: $S^{\{n\}}$ Trace of Stress tensor in the $n^\mu$ frame **press_n**: $p^{\{n\}}$ Pressure in the $n^\mu$ frame **anisotropic_press_down3_n**: $\pi^{\{n\}_{ij}}$ Anisotropic pressure in the $n^\mu$ frame with spatial indices down **rho_n_fromHam**: $\rho^{\{n\}}$ Energy density in the $n^\mu$ frame computed from the Hamiltonian constraint **fluxup3_n_fromMom**: $S^{\{n\}i}$ Energy flux (or momentum density) in the $n^\mu$ frame with spatial indices up computed from the Momentum constraint Conserved variables ------------------- **conserved_D**: $D$ Conserved mass-energy density in Wilson's formalism **conserved_E**: $E$ Conserved internal energy density in Wilson's formalism **conserved_Sdown4**: $S_{\mu}$ Conserved energy flux (or momentum density) in Wilson's formalism with spacetime indices down **conserved_Sdown3**: $S_{i}$ Conserved energy flux (or momentum density) in Wilson's formalism with spatial indices down **conserved_Sup4**: $S^{\mu}$ Conserved energy flux (or momentum density) in Wilson's formalism with spacetime indices up **conserved_Sup3**: $S^{i}$ Conserved energy flux (or momentum density) in Wilson's formalism with spatial indices up **dtconserved**: $\partial_t D, \; \partial_t E, \partial_t S_{i}$ List of coordinate time derivatives of conserved rest mass-energy density, internal energy density and energy flux (or momentum density) with spatial indices down in Wilson's formalism Kinematic variables ------------------- **st_covd_udown4**: $\nabla_{\mu} u_{\nu}$ Spacetime covariant derivative of Lagrangian fluid four velocity with spacetime indices down **accelerationdown4**: $a_{\mu}$ Acceleration of the fluid with spacetime indices down **accelerationup4**: $a^{\mu}$ Acceleration of the fluid with spacetime indices up **s_covd_udown4**: $\mathcal{D}^{\{u\}}_{\mu} u_{\nu}$ Spatial covariant derivative of Lagrangian fluid four velocity with spacetime indices down, with respect to spatial hypersurfaces orthonormal to the fluid flow **thetadown4**: $\Theta_{\mu\nu}$ Fluid expansion tensor with spacetime indices down **theta**: $\Theta$ Fluid expansion scalar **sheardown4**: $\sigma_{\mu\nu}$ Fluid shear tensor with spacetime indices down **shear2**: $\sigma^2$ Magnitude of fluid shear **omegadown4**: $\omega_{\mu\nu}$ Fluid vorticity tensor with spacetime indices down **omega2**: $\omega^2$ Magnitude of fluid vorticity Curvature quantities ==================== Spatial curvature ----------------- **s_RicciS_u**: ${}^{(3)}R^{\{u\}}$ Ricci scalar of the spatial metric orthonormal to fluid flow **s_Gamma_udd3**: ${}^{(3)}{\Gamma^{k}}_{ij}$ Christoffel symbols of spatial metric with mixed spatial indices **s_Riemann_uddd3**: ${}^{(3)}{R^{i}}_{jkl}$ Riemann tensor of spatial metric with mixed spatial indices **s_Riemann_down3**: ${}^{(3)}R_{ijkl}$ Riemann tensor of spatial metric with all spatial indices down **s_Ricci_down3**: ${}^{(3)}R_{ij}$ Ricci tensor of spatial metric with spatial indices down **s_RicciS**: ${}^{(3)}R$ Ricci scalar of spatial metric Spacetime curvature ------------------- **st_Gamma_udd4**: ${}^{(4)}{\Gamma^{\alpha}}_{\mu\nu}$ Christoffel symbols of spacetime metric with mixed spacetime indices **st_Riemann_uddd4**: ${}^{(4)}{R^{\alpha}}_{\beta\mu\nu}$ Riemann tensor of spacetime metric with mixed spacetime indices **st_Riemann_down4**: ${}^{(4)}R_{\alpha\beta\mu\nu}$ Riemann tensor of spacetime metric with spacetime indices down **st_Riemann_uudd4**: ${}^{(4)}{R^{\alpha\beta}}_{\mu\nu}$ Riemann tensor of spacetime metric with mixed spacetime indices **st_Ricci_down4**: ${}^{(4)}R_{\alpha\beta}$ Ricci tensor of spacetime metric with spacetime indices down **st_Ricci_down3**: ${}^{(4)}R_{ij}$ Ricci tensor of spacetime metric with spatial indices down **st_RicciS**: ${}^{(4)}R$ Ricci scalar of spacetime metric **Kretschmann**: $K={R^{\alpha\beta}}_{\mu\nu}{R_{\alpha\beta}}^{\mu\nu}$ Kretschmann scalar Weyl decomposition ------------------ **st_Weyl_down4**: $C_{\alpha\beta\mu\nu}$ Weyl tensor of spacetime metric with spacetime indices down **Weyl_Psi**: $\Psi_0, \; \Psi_1, \; \Psi_2, \; \Psi_3, \; \Psi_4$ List of Weyl scalars for an null vector base defined with AurelCore.tetrad_to_use **Psi4_lm**: $\Psi_4^{l,m}$ Dictionary of spin weighted spherical harmonic decomposition of the 4th Weyl scalar, with AurelCore.Psi4_lm_radius and AurelCore.Psi4_lm_lmax. ``spinsfast`` is used for the decomposition. **Weyl_invariants**: $I, \; J, \; L, \; K, \; N$ Dictionary of Weyl invariants **eweyl_u_down4**: $E^{\{u\}}_{\alpha\beta}$ Electric part of the Weyl tensor on the hypersurface orthogonal to $u^{\mu}$ with spacetime indices down **eweyl_n_down3**: $E^{\{n\}}_{ij}$ Electric part of the Weyl tensor on the hypersurface orthogonal to $n^{\mu}$ with spatial indices down **bweyl_u_down4**: $B^{\{u\}}_{\alpha\beta}$ Magnetic part of the Weyl tensor on the hypersurface orthogonal to $u^{\mu}$ with spacetime indices down **bweyl_n_down3**: $B^{\{n\}}_{ij}$ Magnetic part of the Weyl tensor on the hypersurface orthogonal to $n^{\mu}$ with spatial indices down Null ray expansion ================== **null_ray_exp**: $\Theta_{out}, \; \Theta_{in}$ List of expansion of null rays radially going out and in respectively Constraints =========== **Hamiltonian**: $\mathcal{H}$ Hamilonian constraint **Hamiltonian_Escale**: [$\mathcal{H}$] Hamilonian constraint energy scale **Momentumup3**: $\mathcal{M}^i$ Momentum constraint with spatial indices up **Momentum_Escale**: [$\mathcal{M}$] Momentum constraint energy scale AurelCore ********* .. autoclass:: aurel.core.AurelCore :show-inheritance: :members: