Notebook

Tutorial: Setting up the energy-momentum tensor of a massless scalar field¶

Authors: Leonardo Werneck & Zach Etienne¶

This module documents the construction of the energy-momentum tensor of a massless scalar field.¶

Notebook Status: Validated

Validation Notes: The expressions generated by the NRPy+ module corresponding to this tutorial notebook are used to demonstrate that the initial data for a massless scalar field satisfy Einstein's equations as expected in this tutorial notebook.

Python module containing the final expressions constructed here: ScalarField/ScalarField_Tmunu.py¶

Table of Contents¶

$\label{toc}$

The module is organized as follows

Preliminaries: The energy momentum tensor of a massless scalar field
Step 1: Initialize core NRPy+ modules
Step 2: The 4-derivatives of the scalar field: $\partial^{\mu}\varphi$
Step 3: The energy momentum tensor: $T^{\mu\nu}$
Step 4: Validation against the ScalarField/ScalarField_Tmunu.py module
Step 5: Output this module to $\LaTeX$ -formatted PDF

Preliminaries: The energy momentum tensor of a massless scalar field [Back to top]¶

$\label{preliminaries}$

The energy-momentum tensor for a massless scalar field is given by eq. (5.232) of B&S, which we right here in contravariant form

$T^{\mu\nu} = \partial^{\mu}\varphi\partial^{\nu}\varphi - \frac{1}{2}g^{\mu\nu}\left(\partial^{\lambda}\varphi\partial_{\lambda}\varphi\right)\ .$

This is a key tensor in the problem of gravitational collapse of a massless scalar field, since it will be responsible for how the geometry changes in the presence of the scalar field. In this tutorial module we will be implementing this tensor.

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

$\label{initializenrpy}$

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

In [1]:

# Step 1.a: import all needed modules from NRPy+:
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
import reference_metric as rfm             # NRPy+: Reference metric support
import BSSN.BSSN_quantities as Bq          # NRPy+: BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as BtoA   # NRPy+: ADM quantities in terms of BSSN quantities
import BSSN.ADMBSSN_tofrom_4metric as ADMg # NRPy+: ADM 4-metric to/from ADM or BSSN quantities

# Step 1.b: Set the coordinate system for the numerical grid
coord_system = "Spherical"
par.set_parval_from_str("reference_metric::CoordSystem",coord_system)

# Step 1.c: Set spatial dimension (must be 3 for BSSN, as BSSN is
#           a 3+1-dimensional decomposition of the general
#           relativistic field equations)
DIM = 3
par.set_parval_from_str("grid::DIM",DIM)

# Step 1.d: Given the chosen coordinate system, set up
#           corresponding reference metric and needed
#           reference metric quantities
#    The following function call sets up the reference metric
#    and related quantities, including rescaling matrices ReDD,
#    ReU, and hatted quantities.
rfm.reference_metric()

# Step 1.e: Set the theta and phi axes to be the symmetry axes; i.e., axis "1" and "2",
#           corresponding to the i1 and i2 directions. This sets all spatial derivatives
#           in the theta and phi directions to zero (analytically).
par.set_parval_from_str("indexedexp::symmetry_axes","12")

# Step 1.e: Import all basic (unrescaled) BSSN scalars & tensors
Bq.BSSN_basic_tensors()
alpha = Bq.alpha
betaU = Bq.betaU

# Step 1.g: Define ADM quantities in terms of BSSN quantities
BtoA.ADM_in_terms_of_BSSN()
gammaDD = BtoA.gammaDD
gammaUU = BtoA.gammaUU

# Step 1.h: Define scalar field quantitites
sf_dD   = ixp.declarerank1("sf_dD")
Pi      = sp.Symbol("sfM",real=True)

Step 2: The 4-derivatives of the scalar field: $\partial^{\mu}\varphi$ [Back to top]¶

$\label{sf4d}$

Consider the ADM 4-metric (eq. 2.119 of B&S)

$g^{\mu\nu}=\begin{pmatrix} -\alpha^{-2} & \alpha^{-2}\beta^{i}\\ \alpha^{-2}\beta^{j} & \gamma^{ij} - \alpha^{-2}\beta^{i}\beta^{j} \end{pmatrix}\ ,$

and the definition of the scalar field's conjugate momentum, $\Pi$ , as given by eq. 2.522 of B&S

$\Pi\equiv-\frac{1}{\alpha}\left[\partial_{t}\varphi - \beta^{i}\partial_{i}\varphi\right]\ .$

Then we have

$\begin{align} \partial^{t}\varphi &= g^{tt}\partial_{t}\varphi + g^{ti}\partial_{i}\varphi\nonumber\\ &= -\alpha^{-2}\partial_{t}\varphi + \alpha^{-2}\beta^{i}\partial_{i}\varphi\nonumber\\ &= \alpha^{-1}\left[-\frac{1}{\alpha}\left(\partial_{t}\varphi - \beta^{i}\partial_{i}\varphi\right)\right]\nonumber\\ &= \frac{\Pi}{\alpha}\ . \end{align}$

In [2]:

# Step 2a: Set up \partial^{t}\varphi = Pi/alpha
sf4dU = ixp.zerorank1(DIM=4)
sf4dU[0] = Pi / alpha

Next, we look at

$\begin{align} \partial^{i}\varphi &= g^{it}\partial_{t}\varphi + g^{ij}\partial_{j}\varphi\nonumber\\ &=\alpha^{-2}\beta^{i}\partial_{t}\varphi + \gamma^{ij}\partial_{j}\varphi - \alpha^{-2}\beta^{i}\beta^{j}\partial_{j}\varphi\nonumber\\ &=-\alpha^{-1}\beta^{i}\left[-\frac{1}{\alpha}\left(\partial_{t}\varphi-\beta^{j}\partial_{j}\varphi\right)\right] + \gamma^{ij}\partial_{j}\varphi\nonumber\\ &=-\frac{\Pi}{\alpha}\beta^{i} + \gamma^{ij}\partial_{j}\varphi\ . \end{align}$

In [3]:

# Step 2b: Set up \partial^{i}\varphi = -Pi*beta^{i}/alpha + gamma^{ij}\partial_{j}\varphi
for i in range(DIM):
    sf4dU[i+1] = -Pi * betaU[i] / alpha
    for j in range(DIM):
        sf4dU[i+1] += gammaUU[i][j] * sf_dD[j]

The last step is to set up the contraction

$\begin{align} \partial^{\lambda}\varphi\partial_{\lambda}\varphi &= \partial^{t}\varphi\partial_{t}\varphi + \partial^{i}\varphi\partial_{i}\varphi\nonumber\\ &=\frac{\Pi}{\alpha}\partial_{t}\varphi - \frac{\Pi}{\alpha}\beta^{i}\partial_{i}\varphi + \gamma^{ij}\partial_{i}\varphi\partial_{j}\varphi\nonumber\\ &= -\Pi\left[-\frac{1}{\alpha}\left(\partial_{t}\varphi-\beta^{i}\partial_{i}\varphi\right)\right] + \gamma^{ij}\partial_{i}\varphi\partial_{j}\varphi\nonumber\\ &= -\Pi^2 + \gamma^{ij}\partial_{i}\varphi\partial_{j}\varphi\ . \end{align}$

In [4]:

# Step 2c: Set up \partial^{i}\varphi\partial_{i}\varphi = -Pi**2 + gamma^{ij}\partial_{i}\varphi\partial_{j}\varphi
sf4d2 = -Pi**2
for i in range(DIM):
    for j in range(DIM):
        sf4d2 += gammaUU[i][j] * sf_dD[i] * sf_dD[j]

Step 3: The energy momentum tensor: $T^{\mu\nu}$ [Back to top]¶

$\label{energy_momentum_tensor}$

We start by setting up the ADM 4-metric $g^{\mu\nu}$ , given by eq. (2.119) in B&S,

$g^{\mu\nu}=\begin{pmatrix} -\alpha^{-2} & \alpha^{-2}\beta^{i}\\ \alpha^{-2}\beta^{j} & \gamma^{ij} - \alpha^{-2}\beta^{i}\beta^{j} \end{pmatrix}\ .$

We do this be calling the BSSN.adm_four_metric_conversions.py module.

In [5]:

# Step 3a: Setting up g^{\mu\nu}
ADMg.g4UU_ito_BSSN_or_ADM("ADM",gammaDD=gammaDD,betaU=betaU,alpha=alpha, gammaUU=gammaUU)
g4UU = ADMg.g4UU

We then focus on the energy momentum tensor $T^{\mu\nu}$ for a massless scalar field, $\varphi$ (cf. eq. 5.232 of B&S with $V(\varphi)=0$ )

$T^{\mu\nu} = \partial^{\mu}\varphi\partial^{\nu}\varphi - \frac{1}{2}g^{\mu\nu}\left(\underbrace{\partial_{\lambda}\varphi\partial^{\lambda}\varphi}_{\equiv \rm sf4d2}\right)\ .$

In [6]:

# Step 3b: Setting up T^{\mu\nu} for a massless scalar field
T4UU = ixp.zerorank2(DIM=4)
for mu in range(4):
    for nu in range(4):
        T4UU[mu][nu] = sf4dU[mu] * sf4dU[nu] - sp.Rational(1,2) * g4UU[mu][nu] * sf4d2

Step 4: Validation against the ScalarField/ScalarField_Tmunu.py module [Back to top]¶

$\label{code_validation}$

Here we perform a code validation. We verify agreement in the SymPy expressions for the energy-momentum tensor of a scalar field between

this tutorial notebook and
the ScalarField/ScalarField_Tmunu.py NRPy+ module.

By default, we analyze the RHSs in Spherical coordinates, though other coordinate systems may be chosen.

In [7]:

import ScalarField.ScalarField_Tmunu as sfTmunu # NRPyCritCol: Scalar field energy-momentum tensor
sfTmunu.ScalarField_Tmunu()

print("Consistency check between this tutorial and the ScalarField.ScalarField_Tmunu.py module: ALL SHOULD BE ZERO\n")

for mu in range(4):
    for nu in range(4):
        print("T4UU["+str(mu)+"]["+str(nu)+"] - sfTmunu.T4UU["+str(mu)+"]["+str(nu)+"] = "+str(sp.simplify(T4UU[mu][nu] - sfTmunu.T4UU[mu][nu])))

Consistency check between this tutorial and the ScalarField.ScalarField_Tmunu.py module: ALL SHOULD BE ZERO

T4UU[0][0] - sfTmunu.T4UU[0][0] = 0
T4UU[0][1] - sfTmunu.T4UU[0][1] = 0
T4UU[0][2] - sfTmunu.T4UU[0][2] = 0
T4UU[0][3] - sfTmunu.T4UU[0][3] = 0
T4UU[1][0] - sfTmunu.T4UU[1][0] = 0
T4UU[1][1] - sfTmunu.T4UU[1][1] = 0
T4UU[1][2] - sfTmunu.T4UU[1][2] = 0
T4UU[1][3] - sfTmunu.T4UU[1][3] = 0
T4UU[2][0] - sfTmunu.T4UU[2][0] = 0
T4UU[2][1] - sfTmunu.T4UU[2][1] = 0
T4UU[2][2] - sfTmunu.T4UU[2][2] = 0
T4UU[2][3] - sfTmunu.T4UU[2][3] = 0
T4UU[3][0] - sfTmunu.T4UU[3][0] = 0
T4UU[3][1] - sfTmunu.T4UU[3][1] = 0
T4UU[3][2] - sfTmunu.T4UU[3][2] = 0
T4UU[3][3] - sfTmunu.T4UU[3][3] = 0

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

In [8]:

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

Created Tutorial-ScalarField_Tmunu.tex, and compiled LaTeX file to PDF file
    Tutorial-ScalarField_Tmunu.pdf