{ "cells": [ { "cell_type": "markdown", "id": "4d483730", "metadata": {}, "source": [ "# Example over time\n", "\n", "Examples so far have shown how to use aurel to automatically calculate relativistic terms for a given coordinate time. In practice you may be interested in seeing how these quantities evolve over time. For this the aurel package has the function `over_time` to do this, here we demonstrate how this is used.\n", "\n", "First we need a big dictionnary containing the spacetime/matter: " ] }, { "cell_type": "code", "execution_count": 1, "id": "5476a275", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import aurel\n", "from aurel.solutions import LCDM as sol\n", "\n", "import matplotlib.pyplot as plt\n", "import matplotlib.cm as cm\n", "import matplotlib.colors as mcolors" ] }, { "cell_type": "code", "execution_count": 2, "id": "74df8ce8", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4th order finite difference schemes are defined\n", "Data keys: ['t', 'gammadown3', 'Kdown3', 'rho0']\n" ] } ], "source": [ "# Define grid and finite difference class\n", "param = { 'Nx': 64, 'Ny': 64, 'Nz': 64,\n", " 'xmin': 0.0, 'ymin': 0.0, 'zmin': 0.0,\n", " 'dx': 20.0, 'dy': 20.0, 'dz': 20.0}\n", "fd = aurel.FiniteDifference(param)\n", "x, y, z = fd.cartesian_coords\n", "\n", "# Generate data for Lamnbda-CDM solution\n", "Nt = 20\n", "tarray = np.linspace(1, 10, Nt)\n", "data = {key:[] for key in ['t', 'gammadown3', 'Kdown3', 'rho0']}\n", "for t in tarray:\n", " data['t'] += [t]\n", " data['gammadown3'] += [sol.gammadown3(t, x, y, z)]\n", " data['Kdown3'] += [sol.Kdown3(t, x, y, z)]\n", " data['rho0'] += [sol.rho(t)*np.ones((fd.Nx, fd.Ny, fd.Nz))]\n", "print('Data keys:', list(data.keys()))" ] }, { "cell_type": "markdown", "id": "5a35bf7d", "metadata": {}, "source": [ "## Calculate any quantity from `aurel.descriptions.keys()`" ] }, { "cell_type": "code", "execution_count": 3, "id": "7c5d6d77", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Processing t = 1.0\n" ] }, { "data": { "text/latex": [ "Cosmological constant set to AurelCore.Lambda = 1.04e-07" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gammaup3: $\\gamma^{ij}$ Spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated s_Gamma_udd3: ${}^{(3)}{\\Gamma^{k}}_{ij}$ Christoffel symbols of spatial metric with mixed spatial indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated Ktrace: $K = \\gamma^{ij}K_{ij}$ Trace of extrinsic curvature" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated null_ray_exp_in: $\\Theta_{in}$ List of expansion of null rays radially going in" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Now processing remaining time steps sequentially\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "62123c83781f4b2ba51095ff483613b8", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/19 [00:00" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Calculate further terms with AurelCore iterated over time\n", "data = aurel.over_time(data, fd, vars=['null_ray_exp_in'], \n", " Lambda=sol.Lambda)\n", "# all aurel kwargs can be passed here\n", "print('Data keys:', list(data.keys()))\n", "\n", "# Plot Cosmological horizon over time\n", "Bcolors = cm.Blues(np.linspace(0.3, 1, Nt))\n", "for i in range(Nt):\n", " plt.plot(fd.xarray, data['null_ray_exp_in'][i][:,0,0], color=Bcolors[i])\n", "plt.grid()\n", "plt.ylim(-0.9, 0.9)\n", "plt.xlabel(r'x [Mpc]')\n", "plt.ylabel(r\"$|\\Theta_{in}|$\")\n", "plt.title(r'Cosmological horizon $\\Theta_{in} = 0$ growing over time')\n", "\n", "# Colorbar\n", "norm = mcolors.Normalize(vmin=min(data['t']), vmax=max(data['t']))\n", "sm = cm.ScalarMappable(cmap=cm.Blues, norm=norm)\n", "sm.set_array([])\n", "cbar = plt.colorbar(sm, ax=plt.gca(), extend='both')\n", "cbar.set_label(r\"t [Mpc]\")" ] }, { "cell_type": "markdown", "id": "6fd787fb", "metadata": {}, "source": [ "## Calculate your custom quantity over time" ] }, { "cell_type": "code", "execution_count": 4, "id": "32d53a84", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Validating variable function 'backreaction'...\n", "✓ Custom function 'backreaction' validated successfully\n", "Processing t = 1.0\n" ] }, { "data": { "text/latex": [ "Cosmological constant set to AurelCore.Lambda = 1.04e-07" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Calculating custom variable 'backreaction'...\n" ] }, { "data": { "text/latex": [ "Calculated gammadet: $\\gamma$ Determinant of spatial metric" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated alpha: $\\alpha$ Lapse. I assume $\\alpha=1$, if not then please define AurelCore.data['alpha'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betax: $\\beta^{x}$ x component of the shift vector with indices up. I assume $\\beta^{x}=0$, if not then please define AurelCore.data['betax'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betay: $\\beta^{y}$ y component of the shift vector with indices up. I assume $\\beta^{y}=0$, if not then please define AurelCore.data['betay'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betaz: $\\beta^{z}$ z component of the shift vector with indices up. I assume $\\beta^{z}=0$, if not then please define AurelCore.data['betaz'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betaup3: $\\beta^{i}$ Shift vector with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betadown3: $\\beta_{i}$ Shift vector with spatial indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gdown4: $g_{\\mu\\nu}$ Spacetime metric with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gup4: $g^{\\mu\\nu}$ Spacetime metric with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated w_lorentz: $W$ Lorentz factor. I assume $W=1$, if not then please define AurelCore.data['w_lorentz'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup0: $u^t$ Lagrangian fluid four velocity with time indice up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velx: $v^x$ x component of Eulerian fluid three velocity with indice up. I assume $v^x=0$, if not then please define AurelCore.data['velx'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated vely: $v^y$ y component of Eulerian fluid three velocity with indice up. I assume $v^y=0$, if not then please define AurelCore.data['vely'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velz: $v^z$ z component of Eulerian fluid three velocity with indice up. I assume $v^z=0$, if not then please define AurelCore.data['velz'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velup3: $v^i$ Eulerian fluid three velocity with spatial indices up." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup3: $u^i$ Lagrangian fluid four velocity with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup4: $u^\\mu$ Lagrangian fluid four velocity with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hup4: $h^{\\mu\\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated udown4: $u_\\mu$ Lagrangian fluid four velocity with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hdown4: $h_{\\mu\\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Cleaning up cache after 20 calculations..." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size before cleanup: 188.00 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'betadown3' used 13 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gdown4' used 2 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gup4' used 2 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'hup4' used 2 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removed 4 items" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size after cleanup: 86.00 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hmixed4: ${h^{\\mu}}_{\\nu}$ Spatial metric orthonomal to fluid flow with mixed spacetime indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "WARNING: dtconserved only works for constant press/rho" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_D: $D$ Conserved mass-energy density in Wilson's formalism" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated eps: $\\epsilon$ Specific internal energy. I assume $\\epsilon=0$, if not then please define AurelCore.data['eps'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_E: $E$ Conserved internal energy density in Wilson's formalism" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated press: $p$ Pressure. I assume $p=0$, if not then please define AurelCore.data['press'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated enthalpy: $h$ Specific enthalpy of the fluid" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_Sdown4: $S_{\\mu}$ Conserved energy flux (or momentum density) in Wilson's formalism with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_Sdown3: $S_{i}$ Conserved energy flux (or momentum density) in Wilson's formalism with spatial indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betadown3: $\\beta_{i}$ Shift vector with spatial indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gdown4: $g_{\\mu\\nu}$ Spacetime metric with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gup4: $g^{\\mu\\nu}$ Spacetime metric with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_Sup4: $S^{\\mu}$ Conserved energy flux (or momentum density) in Wilson's formalism with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_Sup3: $S^{i}$ Conserved energy flux (or momentum density) in Wilson's formalism with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gammaup3: $\\gamma^{ij}$ Spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated s_Gamma_udd3: ${}^{(3)}{\\Gamma^{k}}_{ij}$ Christoffel symbols of spatial metric with mixed spatial indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated Ktrace: $K = \\gamma^{ij}K_{ij}$ Trace of extrinsic curvature" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated Kup3: $K^{ij}$ Extrinsic curvature with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtgammaup3: $\\partial_t \\gamma^{ij}$ Coordinate time derivative of spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated 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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated udown3: $u_\\mu$ Lagrangian fluid four velocity with spatial indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Cleaning up cache after 40 calculations..." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size before cleanup: 324.01 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'betax' used 37 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'betay' used 36 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'betaz' used 35 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'uup0' used 29 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'velx' used 28 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'vely' used 27 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'velz' used 26 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'velup3' used 25 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'uup3' used 24 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'uup4' used 19 calculations ago (size: 8.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'hdown4' used 20 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'hmixed4' used 19 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'conserved_Sdown4' used 7 calculations ago (size: 8.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'betadown3' used 7 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gdown4' used 10 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gup4' used 9 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'conserved_Sup4' used 8 calculations ago (size: 8.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'Kup3' used 3 calculations ago (size: 18.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removed 18 items" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size after cleanup: 122.00 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velx: $v^x$ x component of Eulerian fluid three velocity with indice up. I assume $v^x=0$, if not then please define AurelCore.data['velx'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated vely: $v^y$ y component of Eulerian fluid three velocity with indice up. I assume $v^y=0$, if not then please define AurelCore.data['vely'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velz: $v^z$ z component of Eulerian fluid three velocity with indice up. I assume $v^z=0$, if not then please define AurelCore.data['velz'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated velup3: $v^i$ Eulerian fluid three velocity with spatial indices up." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup3: $u^i$ Lagrangian fluid four velocity with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup0: $u^t$ Lagrangian fluid four velocity with time indice up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtalpha: $\\partial_t \\alpha$ Coordinate time derivative of the lapse. I assume $\\partial_t \\alpha=0$, if not then please define AurelCore.data['dtalpha'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtbetax: $\\partial_t\\beta^{x}$ Coordinate time derivative of the x component of the shift vector with indices up. I assume $\\partial_t\\beta^{x}=0$, if not then please define AurelCore.data['dtbetax'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtbetay: $\\partial_t\\beta^{y}$ Coordinate time derivative of the y component of the shift vector with indices up. I assume $\\partial_t\\beta^{y}=0$, if not then please define AurelCore.data['dtbetay'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtbetaz: $\\partial_t\\beta^{z}$ Coordinate time derivative of the z component of the shift vector with indices up. I assume $\\partial_t\\beta^{z}=0$, if not then please define AurelCore.data['dtbetaz'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtbetaup3: $\\partial_t\\beta^{i}$ Coordinate time derivative of the shift vector with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated st_Gamma_udd4: ${}^{(4)}{\\Gamma^{\\alpha}}_{\\mu\\nu}$ Christoffel symbols of spacetime metric with mixed spacetime indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated st_covd_udown4: $\\nabla_{\\mu} u_{\\nu}$ Spacetime covariant derivative of Lagrangian fluid four velocity with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated 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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated thetadown4: $\\Theta_{\\mu\\nu}$ Fluid expansion tensor with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated theta: $\\Theta$ Fluid expansion scalar" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated betadown3: $\\beta_{i}$ Shift vector with spatial indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gdown4: $g_{\\mu\\nu}$ Spacetime metric with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gup4: $g^{\\mu\\nu}$ Spacetime metric with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated uup4: $u^\\mu$ Lagrangian fluid four velocity with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Cleaning up cache after 60 calculations..." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size before cleanup: 460.01 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gammadet' used 39 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'udown4' used 13 calculations ago (size: 8.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'conserved_D' used 20 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'eps' used 36 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'conserved_E' used 20 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'press' used 22 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'enthalpy' used 34 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gammaup3' used 9 calculations ago (size: 18.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'Ktrace' used 24 calculations ago (size: 2.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'dtgammaup3' used 20 calculations ago (size: 18.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'velup3' used 16 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'dtbetaup3' used 9 calculations ago (size: 6.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'st_Gamma_udd4' used 8 calculations ago (size: 128.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'st_covd_udown4' used 7 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 's_covd_udown4' used 6 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'thetadown4' used 5 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removing cached value for 'gdown4' used 2 calculations ago (size: 32.00 MB)." ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: Removed 17 items" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "CLEAN-UP: data size after cleanup: 134.00 MB" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hup4: $h^{\\mu\\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gdown4: $g_{\\mu\\nu}$ Spacetime metric with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated udown4: $u_\\mu$ Lagrangian fluid four velocity with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hdown4: $h_{\\mu\\nu}$ Spatial metric orthonomal to fluid flow with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated hmixed4: ${h^{\\mu}}_{\\nu}$ Spatial metric orthonomal to fluid flow with mixed spacetime indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gammadet: $\\gamma$ Determinant of spatial metric" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_D: $D$ Conserved mass-energy density in Wilson's formalism" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated eps: $\\epsilon$ Specific internal energy. I assume $\\epsilon=0$, if not then please define AurelCore.data['eps'] = ... " ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated conserved_E: $E$ Conserved internal energy density in Wilson's formalism" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gammaup3: $\\gamma^{ij}$ Spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated Kup3: $K^{ij}$ Extrinsic curvature with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtgammaup3: $\\partial_t \\gamma^{ij}$ Coordinate time derivative of spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated dtbetaup3: $\\partial_t\\beta^{i}$ Coordinate time derivative of the shift vector with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated st_Gamma_udd4: ${}^{(4)}{\\Gamma^{\\alpha}}_{\\mu\\nu}$ Christoffel symbols of spacetime metric with mixed spacetime indices" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated st_covd_udown4: $\\nabla_{\\mu} u_{\\nu}$ Spacetime covariant derivative of Lagrangian fluid four velocity with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated 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" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated thetadown4: $\\Theta_{\\mu\\nu}$ Fluid expansion tensor with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated sheardown4: $\\sigma_{\\mu\\nu}$ Fluid shear tensor with spacetime indices down" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated shear2: $\\sigma^2$ Magnitude of fluid shear" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Calculated and freezed variable 'backreaction' in AurelCore\n", "Now processing remaining time steps sequentially\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "76177009efe84cba912c8c1ffc4169c9", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/19 [00:00" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated gammaup3: $\\gamma^{ij}$ Spatial metric with spatial indices up" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "Calculated Ktrace: $K = \\gamma^{ij}K_{ij}$ Trace of extrinsic curvature" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Processing estimation item: max\n", "Now processing remaining time steps sequentially\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { "model_id": "d9fde7ba075c4c09b8b01976c9b258aa", "version_major": 2, "version_minor": 0 }, "text/plain": [ " 0%| | 0/19 [00:00