Notebook

Brill-Lindquist Initial data¶

Author: Zach Etienne¶

Formatting improvements courtesy Brandon Clark¶

This module has been validated to exhibit convergence to zero of the Hamiltonian constraint violation at the expected order to the exact solution (see plot at bottom of the exact initial data validation start-to-finish tutorial notebook; momentum constraint is zero), and all quantities have been validated against the original SENR code.

NRPy+ Source Code for this module: BrillLindquist.py ¶

Introduction:¶

This module sets up initial data for a merging black hole system in Cartesian coordinates.

Table of Contents¶

$\label{toc}$

This notebook is organized as follows

Step 1: Initialize core Python/NRPy+ modules
Step 2: Setting up Brill-Lindquist initial data
Step 3: Code Validation against BSSN/BrillLindquist NRPy+ module
Step 4: Output this notebook to $\LaTeX$ -formatted PDF file

Step 1: Initialize core Python/NRPy+ modules [Back to top]¶

$\label{initializenrpy}$

Let's start by importing all the needed modules from Python/NRPy+:

In [1]:

# Step 1: Initialize core Python/NRPy+ modules
import sympy as sp             # SymPy: The Python computer algebra package upon which NRPy+ depends
import NRPy_param_funcs as par # NRPy+: Parameter interface
import indexedexp as ixp       # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support

Step 2: Setting up Brill-Lindquist initial data [Back to top]¶

$\label{initialdata}$

Here we set up Brill-Lindquist initial data (Brill & Lindquist, Phys. Rev. 131, 471, 1963; see also Eq. 1 of Brandt & Brügmann, arXiv:gr-qc/9711015v1):

The conformal factor $\psi$ for Brill-Lindquist initial data is given by

$\psi = e^{\phi} = 1 + \sum_{i=1}^N \frac{m_{(i)}}{2 \left|\vec{r}_{(i)} - \vec{r}\right|};\quad K_{ij}=0,$

where $\psi$ is written in terms of the 3-metric $\gamma_{ij}$ as

$\gamma_{ij} = \psi^4 \delta_{ij}.$

The extrinsic curvature is zero:

$K_{ij} = 0$

These data consist of $N$ nonspinning black holes initially at rest. This module restricts to the case of two such black holes, positioned anywhere. Here, we implement $N=2$ .

Inputs for $\psi$ :

The position and (bare) mass of black hole 1: $\left(x_{(1)},y_{(1)},z_{(1)}\right)$ and $m_{(1)}$ , respectively
The position and (bare) mass of black hole 2: $\left(x_{(2)},y_{(2)},z_{(2)}\right)$ and $m_{(2)}$ , respectively

Additional variables needed for spacetime evolution:

Desired coordinate system
Desired initial lapse $\alpha$ and shift $\beta^i$ . We will choose our gauge conditions as $\alpha=\psi^{-2}$ and $\beta^i=B^i=0$ .

In [2]:

# Step 2: Setting up Brill-Lindquist initial data

thismodule = "Brill-Lindquist"
BH1_posn_x,BH1_posn_y,BH1_posn_z = par.Cparameters("REAL", thismodule,
                                                   ["BH1_posn_x","BH1_posn_y","BH1_posn_z"],
                                                   [         0.0,         0.0,        +0.5])
BH1_mass = par.Cparameters("REAL", thismodule, ["BH1_mass"],1.0)
BH2_posn_x,BH2_posn_y,BH2_posn_z = par.Cparameters("REAL", thismodule,
                                                   ["BH2_posn_x","BH2_posn_y","BH2_posn_z"],
                                                   [         0.0,         0.0,        -0.5])
BH2_mass = par.Cparameters("REAL", thismodule, ["BH2_mass"],1.0)

# Step 2.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM",DIM)

Cartxyz = ixp.declarerank1("Cartxyz")

# Step 2.b: Set psi, the conformal factor:
psi = sp.sympify(1)
psi += BH1_mass / ( 2 * sp.sqrt((Cartxyz[0]-BH1_posn_x)**2 + (Cartxyz[1]-BH1_posn_y)**2 + (Cartxyz[2]-BH1_posn_z)**2) )
psi += BH2_mass / ( 2 * sp.sqrt((Cartxyz[0]-BH2_posn_x)**2 + (Cartxyz[1]-BH2_posn_y)**2 + (Cartxyz[2]-BH2_posn_z)**2) )

# Step 2.c: Set all needed ADM variables in Cartesian coordinates
gammaDD = ixp.zerorank2()
KDD     = ixp.zerorank2() # K_{ij} = 0 for these initial data
for i in range(DIM):
    gammaDD[i][i] = psi**4

alpha = 1/psi**2
betaU = ixp.zerorank1() # We generally choose \beta^i = 0 for these initial data
BU    = ixp.zerorank1() # We generally choose B^i = 0 for these initial data

Step 3: Code Validation against BSSN/BrillLindquist NRPy+ module [Back to top]¶

$\label{code_validation}$

Here, as a code validation check, we verify agreement in the SymPy expressions for Brill-Lindquist initial data between

this tutorial and
the NRPy+ BSSN/BrillLindquist.py module.

In [3]:

# Step 3: Code Validation against BSSN.BrillLindquist NRPy+ module

import BSSN.BrillLindquist as bl
bl.BrillLindquist()

def compare(q1, q2, q1name, q2name):
    if sp.simplify(q1 - q2) != 0:
        print("Error: "+q1name+" - "+q2name+" = "+str(sp.simplify(q1 - q2)))
        sys.exit(1) 

print("Consistency check between Brill-Lindquist tutorial and NRPy+ BSSN.BrillLindquist module.")
compare(alpha, bl.alpha, "alpha", "bl.alpha")
for i in range(DIM):
    compare(betaU[i], bl.betaU[i], "betaU"+str(i), "bl.betaU"+str(i))
    compare(BU[i], bl.BU[i], "BU"+str(i), "bl.BU"+str(i))
    for j in range(DIM):
        compare(gammaDD[i][j], bl.gammaDD[i][j], "gammaDD"+str(i)+str(j), "bl.gammaDD"+str(i)+str(j))
        compare(KDD[i][j], bl.KDD[i][j], "KDD"+str(i)+str(j), "bl.KDD"+str(i)+str(j))
print("ALL TESTS PASS")

Consistency check between Brill-Lindquist tutorial and NRPy+ BSSN.BrillLindquist module.
ALL TESTS PASS

Step 4: Output this notebook to $\LaTeX$ -formatted PDF file [Back to top]¶

$\label{latex_pdf_output}$

The following code cell converts this Jupyter notebook into a proper, clickable $\LaTeX$ -formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename Tutorial-ADM_Initial_Data-Brill-Lindquist (Note that clicking on this link may not work; you may need to open the PDF file through another means.)

In [4]:

import cmdline_helper as cmd    # NRPy+: Multi-platform Python command-line interface
cmd.output_Jupyter_notebook_to_LaTeXed_PDF("Tutorial-ADM_Initial_Data-Brill-Lindquist")

Created Tutorial-ADM_Initial_Data-Brill-Lindquist.tex, and compiled LaTeX
    file to PDF file Tutorial-ADM_Initial_Data-Brill-Lindquist.pdf