Exact Wald GiRaFFEfood Initial Data for GiRaFFE¶

Author: Zach Etienne & Patrick Nelson¶

Formatting improvements courtesy Brandon Clark¶

NRPy+ Source Code for this module: GiRaFFEfood_NRPy/GiRaFFEfood_NRPy_Exact_Wald.py¶

Notebook Status: Validated

Validation Notes: This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented below. The initial data has validated against the original GiRaFFE, as documented here.

Introduction:¶

With the GiRaFFE evolution thorn constructed, we now need to "feed" our giraffe with initial data to evolve. There are several different choices of initial data we can use here; here, we will only be implementing the "Exact Wald" initial data, given by Table 3 in the original paper:

$$\begin{align} A_{\phi} &= \frac{C_0}{2} r^2 \sin^2 \theta \\ E_{\phi} &= 2 M C_0 \left( 1+ \frac {2M}{r} \right)^{-1/2} \sin^2 \theta \\ \end{align}$$ (the unspecified components are set to 0). Here, $C_0$ is a constant that we will set to $1$ in our simulations. Now, to use this initial data scheme, we need to transform the above into the quantities actually tracked by GiRaFFE and HydroBase: $A_i$, $B^i$, $\tilde{S}_i$, $v^i$, and $\Phi$. Of these quantities, GiRaFFEfood will only set $A_i$, $v^i$, and $\Phi=0$, then call a separate function to calculate $\tilde{S}_i$; GiRaFFE itself will call a function to set $B^i$ before the time-evolution begins. This can be done with eqs. 16 and 18, here given in that same order: $$\begin{align} v^i &= \alpha \frac{\epsilon^{ijk} E_j B_k}{B^2} -\beta^i \\ B^i &= \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k \\ \end{align}$$ In the simulations, $B^i$ will be calculated numerically from $A_i$; however, it will be useful to analytically calculate $B^i$ to use calculating the initial $v^i$.

Table of Contents:¶

$$\label{toc}$$

This notebook is organized as follows

1. Step 1: Import core NRPy+ modules and set NRPy+ parameters
2. Step 2: Set the vector $A_i$
3. Step 3: Calculate $v^i$ from $A_i$ and $E_i$
4. Step 4: Code Validation against GiRaFFEfood_NRPy.GiRaFFEfood_NRPy NRPy+ Module
5. Step 5: Output this notebook to $\LaTeX$-formatted PDF file

Step 1: Import core NRPy+ modules and set NRPy+ parameters [Back to top]¶

$$\label{initializenrpy}$$

Here, we will import the NRPy+ core modules, set the reference metric to Cartesian, and set commonly used NRPy+ parameters. We will also set up a parameter to determine what initial data is set up, although it won't do much yet.

Step 2: Set the vector $A_i$ [Back to top]¶

$$\label{set_a_i}$$

We will first build the fundamental vectors $A_i$ and $E_i$ in spherical coordinates (see Table 3). Note that we use reference_metric.py to set $r$ and $\theta$ in terms of Cartesian coordinates; this will save us a step later when we convert to Cartesian coordinates. Since $C_0 = 1$,

$$A_{\phi} = \frac{1}{2} r^2 \sin^2 \theta.$$

Step 3: Calculate $v^i$ from $A_i$ and $E_i$ [Back to top]¶

$$\label{set_vi}$$

Next, we will code up the Valencia velocity, which will require us to first code the electric and magnetic fields. The magnetic field is simply

$$B^i = \frac{[ijk]}{\sqrt{\gamma}} \partial_j A_k.$$

The electric field is

$$E_{\phi} = 2 M \left( 1+ \frac {2M}{r} \right)^{-1/2} \sin^2 \theta,$$
Where the other components are $0$.

We can then calculate the the velocity as

$$v_{(n)}^i = \frac{\epsilon^{ijk} E_j B_k}{B^2}.$$

Step 4: Code Validation against GiRaFFEfood_NRPy.GiRaFFEfood_NRPy NRPy+ module [Back to top]¶

$$\label{code_validation1}$$

Here, as a code validation check, we verify agreement in the SymPy expressions for the GiRaFFE Exact Wald initial data equations we intend to use between

1. this tutorial and
2. the NRPy+ GiRaFFEfood_NRPy.GiRaFFEfood_NRPy_Exact_Wald module.

Step 5: 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-GiRaFFEfood_NRPy.pdf (Note that clicking on this link may not work; you may need to open the PDF file through another means.)