Converting between the 4-metric $g_{\mu\nu}$ and ADM variables $\left\{\gamma_{ij}, \alpha, \beta^i\right\}$ or BSSN variables $\left\{h_{ij}, {\rm cf}, \alpha, {\rm vet}^i\right\}$¶

Author: Zach Etienne¶

This notebook presents the NRPy+ Python module BSSN/ADMBSSN_tofrom_4metric.py, which facilitates conversion between the 4-metric and ADM or BSSN variables. The notebook also outlines the steps to write ADM/BSSN metric quantities in terms of 4-metric $g_{\mu\nu}$, while excluding extrinsic curvature $K_{ij}$, or the BSSN $\bar{A}_{ij}$, $K$.¶

Notebook Status: Self-validated, some additional tests performed

Validation Notes: This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented below. In addition, the construction of $g_{\mu\nu}$ and $g^{\mu\nu}$ from BSSN variables has passed the test $g^{\mu\nu}g_{\mu\nu}=4$ below. Additional validation tests may have been performed, but are as yet, undocumented. (TODO)

NRPy+ Source Code for this module: BSSN/ADMBSSN_tofrom_4metric.py¶

Table of Contents¶

$$\label{toc}$$

This notebook is organized as follows

1. Step 1: setup_ADM_quantities(inputvars): If inputvars="ADM" declare ADM quantities $\left\{\gamma_{ij},\beta^i,\alpha\right\}$; if inputvars="ADM" define ADM quantities in terms of BSSN quantities
2. Step 2: Write 4-metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ in terms of ADM or BSSN quantities
   1. Step 2.a: 4-metric $g_{\mu\nu}$ in terms of ADM or BSSN quantities
   2. Step 2.b: 4-metric inverse $g^{\mu\nu}$ in terms of ADM or BSSN quantities
   3. Step 2.c: Validation check: Confirm $g_{\mu\nu}g^{\mu\nu}=4$
3. Step 3: Write ADM/BSSN metric quantities in terms of 4-metric $g_{\mu\nu}$ (Excludes extrinsic curvature $K_{ij}$ or the BSSN $\bar{A}_{ij}$, $K$)
   1. Step 3.a: ADM in terms of 4-metric validation: Confirm $\gamma_{ij}\gamma^{ij}=3$
   2. Step 3.b: BSSN in terms of 4-metric validation: Confirm $\bar{\gamma}_{ij}\bar{\gamma}^{ij}=3$
4. Step 4: Code Validation against BSSN.ADMBSSN_tofrom_4metric NRPy+ module
5. Step 5: Output this notebook to $\LaTeX$-formatted PDF file

Step 1: setup_ADM_quantities(inputvars): If inputvars="ADM" declare ADM quantities $\left\{\gamma_{ij},\beta^i,\alpha\right\}$; if inputvars="ADM" define ADM quantities in terms of BSSN quantities [Back to top]¶

$$\label{setup_ADM_quantities}$$

Step 2: Write 4-metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$ in terms of ADM or BSSN variables [Back to top]¶

$$\label{admbssn_to_fourmetric}$$

Step 2.a: 4-metric $g_{\mu\nu}$ in terms of ADM or BSSN variables [Back to top]¶

$$\label{admbssn_to_fourmetric_lower}$$

Given ADM variables $\left\{\gamma_{ij},\beta^i,\alpha \right\}$, which themselves may be written in terms of the rescaled BSSN curvilinear variables $\left\{h_{ij},{\rm cf},\mathcal{V}^i,\alpha \right\}$ for our chosen reference metric via simple function calls to ADM_in_terms_of_BSSN() and BSSN_quantities.BSSN_basic_tensors(), we are to construct the 4-metric $g_{\mu\nu}$.

We accomplish this via Eq. 2.122 (which can be trivially derived from the ADM 3+1 line element) of Baumgarte & Shapiro's Numerical Relativity (henceforth B&S):

$$g_{\mu\nu} = \begin{pmatrix} -\alpha^2 + \beta^k \beta_k & \beta_i \\ \beta_j & \gamma_{ij} \end{pmatrix},$$ where the shift vector $\beta^i$ is lowered via (Eq. 2.121):

$$\beta_k = \gamma_{ik} \beta^i.$$

Step 2.b: Inverse 4-metric $g^{\mu\nu}$ in terms of ADM or BSSN variables [Back to top]¶

$$\label{admbssn_to_fourmetric_inv}$$

B&S also provide a convenient form for the inverse 4-metric (Eq. 2.119; also Eq. 4.49 in Gourgoulhon):

$$g^{\mu\nu} = \gamma^{\mu\nu} - n^\mu n^\nu = \begin{pmatrix} -\frac{1}{\alpha^2} & \frac{\beta^i}{\alpha^2} \\ \frac{\beta^i}{\alpha^2} & \gamma^{ij} - \frac{\beta^i\beta^j}{\alpha^2} \end{pmatrix},$$ where the unit normal vector to the hypersurface is given by $n^{\mu} = \left(\alpha^{-1},-\beta^i/\alpha\right)$.

Step 2.c: Validation check: Confirm $g_{\mu\nu}g^{\mu\nu}=4$ [Back to top]¶

$$\label{validationcontraction}$$

Next we compute $g^{\mu\nu} g_{\mu\nu}$ as a validation check. It should equal 4:

Step 3: Write ADM/BSSN metric quantities in terms of 4-metric $g_{\mu\nu}$ (Excludes extrinsic curvature $K_{ij}$, the BSSN $a_{ij}$, $K$, and $\lambda^i$) [Back to top]¶

$$\label{fourmetric_to_admbssn}$$

Given $g_{\mu\nu}$, we now compute ADM/BSSN metric quantities, excluding extrinsic curvature.

Let's start by computing the ADM quantities in terms of the 4-metric $g_{\mu\nu}$.

Recall that

$$g_{\mu\nu} = \begin{pmatrix} -\alpha^2 + \beta^k \beta_k & \beta_i \\ \beta_j & \gamma_{ij} \end{pmatrix}.$$

From this equation, we immediately obtain $\gamma_{ij}$. However we need $\beta^i$ and $\alpha$. After computing the inverse of $\gamma_{ij}$, $\gamma^{ij}$, we raise $\beta_j$ via $\beta^i=\gamma^{ij} \beta_j$ and then compute $\alpha$ via $\alpha = \sqrt{\beta^k \beta_k - g_{00}}$. To convert to BSSN variables $\left\{h_{ij},{\rm cf},\mathcal{V}^i,\alpha \right\}$, we need only convert from ADM via function calls to BSSN.BSSN_in_terms_of_ADM (tutorial).

Step 3.a: ADM in terms of 4-metric validation: Confirm $\gamma_{ij}\gamma^{ij}=3$ [Back to top]¶

$$\label{adm_ito_fourmetric_validate}$$

Next we compute $\gamma^{ij} \gamma_{ij}$ as a validation check. It should equal 3:

Step 3.b: BSSN in terms of 4-metric validation: Confirm $\bar{\gamma}_{ij}\bar{\gamma}^{ij}=3$ [Back to top]¶

$$\label{bssn_ito_fourmetric_validate}$$

Next we compute $\bar{\gamma}_{ij}\bar{\gamma}^{ij}$ as a validation check. It should equal 3:

Step 4: Code Validation against BSSN.ADMBSSN_tofrom_4metric NRPy+ module [Back to top]¶

$$\label{code_validation}$$

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

1. this tutorial and
2. the NRPy+ BSSN.ADMBSSN_tofrom_4metric module.

By default, we analyze these expressions in SinhCylindrical coordinates, though other coordinate systems may be chosen.

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-ADMBSSN_tofrom_4metric.pdf (Note that clicking on this link may not work; you may need to open the PDF file through another means.)