!pip install nrpylatex==1.0.8 > /dev/null
!pip freeze | grep nrpylatex
%load_ext nrpylatex.extension
nrpylatex==1.0.8
from UnitTesting.assert_equal import assert_equal
import NRPy_param_funcs as par, reference_metric as rfm
import BSSN.BSSN_quantities as Bq
import BSSN.BSSN_RHSs as Brhs
import BSSN.BSSN_gauge_RHSs as gaugerhs
import BSSN.BSSN_constraints as bssncon
par.set_parval_from_str('reference_metric::CoordSystem', 'Cartesian')
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic', 'True')
rfm.reference_metric()
Brhs.BSSN_RHSs()
gaugerhs.BSSN_gauge_RHSs()
bssncon.BSSN_constraints()
par.set_parval_from_str('BSSN.BSSN_quantities::LeaveRicciSymbolic', 'False')
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
We formatted the notebook in the following pedagogical scheme, shown below, to encourage a learned approach to parsing tensorial LaTeX.
(1) attempt to parse equation
(2) if ParseError or TensorError
debug error and goto (1)
(3) change equation and goto (1)
YouTube Tutorial (Notebook Walkthrough)
%%parse_latex --reset --ignore-warning
\begin{align}
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
ParseError: \partial_t \bar{\gamma}_{i j} = [\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
^
cannot index undefined tensor 'gammabarDD' at position 29
First, attempt to parse the equation without any modification, and notice the ParseError for indexing an undefined tensor 'gammabarDD'.
gammabarDD := ˉγij=hij+ˆγij where hDD is the deviation from the flat 3D Cartesian metric gammahatDD := ˆγij=δij.
vardef: define a variable, including a derivative option and a (anti)symmetry option.
assign: assign a vardef option to an already existing variable in the namespace.
parse: parse an equation without rendering that equation in a .tex document or a Jupyter Notebook.
-metric can assign the symmetry sym01 and automatically generate the metric inverse and determinant.
deltaDD using the vardef macro.parse macro.-metric option to gammahatDD using the assign macro.% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
Remark: If a derivative option should also apply to the metric inverse and determinant, then that option should precede -metric. Moreover, you need not define a scalar quantity. However, should a scalar quantity require a vardef option, you may assign one using either the vardef or assign macro.
hDD using the vardef macro.parse macro.-metric option to gammabarDD using the assign macro.% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
%%parse_latex --reset --ignore-warning
\begin{align}
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
ParseError: \partial_t \bar{\gamma}_{i j} = [\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
^
cannot index undefined tensor 'betaU' at position 386
Next, replace every instance of betaU with vetU, a different variable name chosen for the SymPy output, using the srepl macro, and then define the variable vetU using the vardef macro. The srepl macro can preserve the original LaTeX representation while changing the variable name internally.
Remark: If you require a variable name longer than one Latin or Greek letter, you should surround that variable name in the \text command.
srepl: string replacement given the assumption that string A and string B are equal whenever they are equivalent syntactically
Therefore, for example, S_{a b} T^{b c} and S_{ab}T^{bc} are considered equal according to the srepl macro.
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
Remark: You can check "syntactic equivalence" for the srepl macro using the code snippet provided below.
lexer = Lexer()
lexer.initialize(A)
syntax_A = [(lexer.lexeme, token) for token in lexer.tokenize()]
lexer.initialize(B)
syntax_B = [(lexer.lexeme, token) for token in lexer.tokenize()]
assert syntax_A == syntax_B
-persist will apply the replacement rule "..." -> "..." to every subsequent input (internal or external) of the parse function.
Remark: The srepl macro will not replace any text inside of a future srepl or ignore macro.
%%parse_latex --reset --ignore-warning
\begin{align}
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
ParseError: \bar{A}^k_k - \bar{D}_k \text{vet}^k) - 2 \alpha \bar{A}_{i j} \\
^
cannot index undefined tensor 'AbarUD' at position 598
AbarDD with aDD using the srepl macro.aDD (with symmetry sym01) using the vardef macro.aUD by assigning the metric gammabar to aDD.% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
Remark: The default metric for a variable is the metric associated with the diacritic (or lack thereof) in the variable name. In the current situation, we need to explicitly associate that metric since we removed the bar diacritic from the variable AbarDD using the srepl macro.
%%parse_latex --reset --ignore-warning
\begin{align}
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
TensorError: unbalanced free index {'t'} in gammabarDD_dD
Next, replace the partial derivative ∂tˉγ on the LHS with the variable name h_rhs using the srepl macro.
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
Remark: We received a TensorError that resulted from interpreting t as an index in the partial derivative ∂tˉγ on the LHS. However, if the basis included the symbol t, then numeric indexing would have occurred, i.e. t -> 0 provided that basis = [t, x, y, z] for example.
%%parse_latex --reset --ignore-warning
\begin{align}
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
keydef: define a keyword, i.e. define a basis or coordinate system, or define an index range (for looping or summation).
% keydef basis [x, y, z]
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
try:
assert_equal(h_rhsDD, Brhs.h_rhsDD)
except AssertionError:
print('Assertion Failed!')
Assertion Failed!
Finally, we upwind every partial derivative in each instance of the pattern \beta^{...} \partial_{...}. To enforce upwinding inside of the bracketed term on the RHS where that pattern occurs, use the vphantom command to dynamically change the derivative type to upwinded.
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k ...
Remark: You could, alternatively, replace the bracketed term with the Lie derivative \mathcal{L}_\beta \bar{\gamma}_{i j}, allow automatic generation of that Lie derivative, and then use an srepl macro with the -persist option and (advanced) capture group syntax to upwind the aforementioned pattern.
% srepl -persist "\text{vet}^<1> \partial_<1>" -> "\text{vet}^<1> \vphantom{dupD} \partial_<1>"
To simplify the calculation of ˉDkβk, substitute a useful tensor identity for that contraction and apply the -persist option.
ˉDkβk=∂kβk+βk∂kˆγ2ˆγ
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\end{align}
assert_equal(h_rhsDD, Brhs.h_rhsDD)
Assertion Passed!
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
\partial_t \phi &= \left[\beta^k \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
\end{align}
TensorError: unbalanced free index {'t'} in phi_dD
First, replace every instance of phi with the conformal factor cf = W = e^{-2\phi}, including every derivative (partial or covariant) of phi.
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4\phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
Remark: We assigned the -persist option to the third srepl macro to replace the partial derivative inside of each covariant derivative of phi.
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
\partial_t \phi &= \left[\beta^k \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
\end{align}
try:
assert_equal(cf_rhs, Brhs.cf_rhs)
except AssertionError:
print('Assertion Failed!')
Assertion Failed!
Next, enforce upwinding on the contracted partial derivative inside of the bracketed term using the vphantom command.
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k ...
Finally, replace every instance of K with trK using the srepl macro.
% srepl "K" -> "\text{trK}"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j} \left(\alpha
\bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
\end{align}
assert_equal(cf_rhs, Brhs.cf_rhs)
Assertion Passed!
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
\partial_t K &= \left[\beta^k \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
\end{align}
TensorError: unbalanced free index {'t'} in trK_dD
First, replace the partial derivative ∂tK on the LHS with the variable name trK_rhs using the srepl macro.
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
\end{align}
try:
assert_equal(trK_rhs, Brhs.trK_rhs)
except AssertionError:
print('Assertion Failed!')
Assertion Failed!
Next, enforce upwinding on the contracted partial derivative inside of the bracketed term using the vphantom command.
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K ...
Finally, assign the property -diff_type=dD to cf, trK, and alpha using the assign macro.
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
\end{align}
assert_equal(trK_rhs, Brhs.trK_rhs)
Assertion Passed!
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
ParseError: \partial_t \bar{\Lambda}^i = [\text{vet}^k \partial_k \bar{\Lambda}^i - \partial_k \text{vet}^i \bar{\Lambda}^k ] +
^
cannot index undefined tensor 'LambdabarU' at position 2099
First, replace every instance of LambdabarU with lambdaU using the srepl macro, and then define the variable lambdaU using the vardef macro.
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
TensorError: unbalanced free index {'t'} in lambdaU_dD
parse macro to define DeltaUDD.DeltaUDD) using the parse macro to define DeltaU.% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij} \\
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
TensorError: unbalanced free index {'t'} in lambdaU_dD
Next, replace the partial derivative ∂tˉΛ on the LHS with the variable name Lambdabar_rhs using the srepl macro.
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
ParseError: \qquad - 2 \text{a}^{i j} (\partial_j \alpha - 6 \alpha \partial_j \text{cf} \frac{-1}{2 \text{cf}} ) + 2 \alpha
^
unsupported operator '\qquad' at position 2793
Next, internally remove the formatting command \qquad using the ignore macro.
ignore: remove a LaTeX command or substring; equivalent to srepl "..." -> "" (empty replacement).
% ignore "\qquad"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
ParseError: - 2 \text{a}^{i j} (\partial_j \alpha - 6 \alpha \partial_j \text{cf} \frac{-1}{2 \text{cf}} ) + 2 \alpha
^
unsupported operator '-' at position 2816
Next, append a percent symbol % to the end of the line break preceding \qquad, and then remove that custom line break \\% using the ignore macro.
% ignore "\\%"
Remark: You cannot split an equation across a line break, and hence we removed that line break internally using the ignore macro.
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
try:
assert_equal(Lambdabar_rhsU, Brhs.Lambdabar_rhsU)
except AssertionError:
print('Assertion Failed!')
Assertion Failed!
Finally, enforce upwinding on the contracted partial derivative inside of the bracketed term using the vphantom command.
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k ...
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
\end{align}
assert_equal(Lambdabar_rhsU, Brhs.Lambdabar_rhsU)
Assertion Passed!
Remark: We must manually expand the trace-free term {…}TF and the symmetric term ˉD(iαˉDj) since neither notation is currenlty supported.
ˉD(iαˉDj)=12(ˉDiαˉDj+ˉDjαˉDi)Xij=−2αˉDiˉDjϕ+4αˉDiϕˉDjϕ+2ˉDiαˉDjϕ+2ˉDjαˉDiϕ−ˉDiˉDjα+αˉRijˆXij=Xij−13ˉγijˉγklXkl∂tˉAij=[βk∂kˉAij+∂iβkˉAkj+∂jβkˉAik]−23ˉAijˉDkβk−2αˉAikˉAkj+αˉAijK+e−4ϕˆXij%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
X_{i j} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi + 2 \bar{D}_i \alpha
\bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
\hat{X}_{i j} &= X_{i j} - \frac{1}{3} \bar{\gamma}_{i j} \bar{\gamma}^{k l} X_{k l} \\
\partial_t \bar{A}_{i j} &= \left[\beta^k \partial_k \bar{A}_{i j} + \partial_i \beta^k \bar{A}_{k j} + \partial_j \beta^k
\bar{A}_{i k} \right] - \frac{2}{3} \bar{A}_{i j} \bar{D}_k \beta^k - 2 \alpha \bar{A}_{i k}
\bar{A}^k{}_j + \alpha \bar{A}_{i j} K + e^{-4 \phi} \hat{X}_{i j} \\
\end{align}
ParseError: \bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
^
cannot index undefined tensor 'RbarDD' at position 3255
First, define the variable RbarDD (with symmetry sym01) using the vardef macro.
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{i j} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi + 2 \bar{D}_i \alpha
\bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
\hat{X}_{i j} &= X_{i j} - \frac{1}{3} \bar{\gamma}_{i j} \bar{\gamma}^{k l} X_{k l} \\
\partial_t \bar{A}_{i j} &= \left[\beta^k \partial_k \bar{A}_{i j} + \partial_i \beta^k \bar{A}_{k j} + \partial_j \beta^k
\bar{A}_{i k} \right] - \frac{2}{3} \bar{A}_{i j} \bar{D}_k \beta^k - 2 \alpha \bar{A}_{i k}
\bar{A}^k{}_j + \alpha \bar{A}_{i j} K + e^{-4 \phi} \hat{X}_{i j} \\
\end{align}
TensorError: unbalanced free index {'t'} in aDD_dD
Next, replace the partial derivative ∂tˉAij on the LHS with the variable name a_rhs using the srepl macro.
% srepl "\partial_t \bar{A}_{i j}" -> "\text{a_rhs}"
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{i j} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi + 2 \bar{D}_i \alpha
\bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
\hat{X}_{i j} &= X_{i j} - \frac{1}{3} \bar{\gamma}_{i j} \bar{\gamma}^{k l} X_{k l} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{i j} &= \left[\beta^k \partial_k \bar{A}_{i j} + \partial_i \beta^k \bar{A}_{k j} + \partial_j \beta^k
\bar{A}_{i k} \right] - \frac{2}{3} \bar{A}_{i j} \bar{D}_k \beta^k - 2 \alpha \bar{A}_{i k}
\bar{A}^k{}_j + \alpha \bar{A}_{i j} K + e^{-4 \phi} \hat{X}_{i j} \\
\end{align}
try:
assert_equal(a_rhsDD, Brhs.a_rhsDD)
except AssertionError:
print('Assertion Failed!')
Assertion Failed!
Finally, enforce upwinding on the contracted partial derivative inside of the bracketed term using the vphantom command.
\partial_t \bar{A}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k ...
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{i j} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi + 2 \bar{D}_i \alpha
\bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
\hat{X}_{i j} &= X_{i j} - \frac{1}{3} \bar{\gamma}_{i j} \bar{\gamma}^{k l} X_{k l} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{A}_{i j} + \partial_i \beta^k \bar{A}_{k j} + \partial_j \beta^k
\bar{A}_{i k} \right] - \frac{2}{3} \bar{A}_{i j} \bar{D}_k \beta^k - 2 \alpha \bar{A}_{i k}
\bar{A}^k{}_j + \alpha \bar{A}_{i j} K + e^{-4 \phi} \hat{X}_{i j} \\
\end{align}
assert_equal(a_rhsDD, Brhs.a_rhsDD)
Assertion Passed!
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% srepl "\beta" -> "\text{vet}"
% vardef -diff_type=dD 'vetU'
%% replace '\bar{D}_k \beta^k' with contraction identity
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% srepl "\bar{A}" -> "\text{a}"
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% assign -metric='gammabar' 'aDD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\gamma}_{i j} + \partial_i \beta^k
\bar{\gamma}_{k j} + \partial_j \beta^k \bar{\gamma}_{i k} \right] + \frac{2}{3} \bar{\gamma}_{i j}
\left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right) - 2 \alpha \bar{A}_{i j} \\
% assign -diff_type=dD 'cf'
%% replace 'phi' with conformal factor cf = W = e^{-2\phi}
% srepl "e^{-4 \phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "K" -> "\text{trK}"
% assign -diff_type=dD 'trK'
\partial_t \phi &= \left[\beta^k \vphantom{dupD} \partial_k \phi \right] + \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% assign -diff_type=dD 'alpha'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \left[\beta^k \vphantom{dupD} \partial_k K \right] + \frac{1}{3} \alpha K^2 + \alpha \bar{A}_{i j}
\bar{A}^{i j} - e^{-4 \phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi \right) \\
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% vardef -diff_type=dD 'lambdaU'
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \left[\beta^k \vphantom{dupD} \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k \right] +
\bar{\gamma}^{j k} \hat{D}_j \hat{D}_k \beta^i + \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad - 2 \bar{A}^{i j} \left(\partial_j \alpha - 6 \alpha \partial_j \phi \right) + 2 \alpha
\bar{A}^{j k} \Delta^i_{j k} - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{i j} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi + 2 \bar{D}_i \alpha
\bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi - \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{i j} \\
\hat{X}_{i j} &= X_{i j} - \frac{1}{3} \bar{\gamma}_{i j} \bar{\gamma}^{k l} X_{k l} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{i j} &= \left[\beta^k \vphantom{dupD} \partial_k \bar{A}_{i j} + \partial_i \beta^k \bar{A}_{k j} + \partial_j \beta^k
\bar{A}_{i k} \right] - \frac{2}{3} \bar{A}_{i j} \bar{D}_k \beta^k - 2 \alpha \bar{A}_{i k}
\bar{A}^k{}_j + \alpha \bar{A}_{i j} K + e^{-4 \phi} \hat{X}_{i j} \\
% srepl "\partial_t \alpha" -> "\text{alpha_rhs}"
\partial_t \alpha &= \left[\beta^k \vphantom{dupD} \partial_k \alpha \right] - 2 \alpha K \\
% srepl "B" -> "\text{bet}"
% vardef -diff_type=dD 'betU'
% srepl "\partial_t \text{vet}" -> "\text{vet_rhs}"
\partial_t \beta^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j \beta^i \right] + B^i \\
% vardef -const 'eta'
% srepl "\partial_t \text{bet}" -> "\text{bet_rhs}"
\partial_t B^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j B^i \right] + \frac{3}{4} \left(\partial_t \bar{\Lambda}^i - \left[\beta^j \vphantom{dupD} \bar{D}_j \bar{\Lambda}^i \right] \right) - \eta B^i \\
\end{align}
assert_equal(alpha_rhs, gaugerhs.alpha_rhs)
Assertion Passed!
assert_equal(vet_rhsU, gaugerhs.vet_rhsU)
Assertion Passed!
assert_equal(bet_rhsU, gaugerhs.bet_rhsU)
Assertion Passed!
%%parse_latex --reset --ignore-warning
\begin{align}
% keydef basis [x, y, z]
% ignore "\\%", "\qquad"
% vardef -kron 'deltaDD'
% parse \hat{\gamma}_{ij} = \delta_{ij}
% assign -diff_type=symbolic -metric 'gammahatDD'
% vardef -diff_type=dD -symmetry=sym01 'hDD'
% parse \bar{\gamma}_{ij} = h_{ij} + \hat{\gamma}_{ij}
% assign -diff_type=dD -metric 'gammabarDD'
% vardef -diff_type=dD 'vetU'
% srepl "\beta" -> "\text{vet}"
%% upwind pattern inside Lie derivative expansion
% srepl -persist "\text{vet}^<1> \partial_<1>" -> "\text{vet}^<1> \vphantom{dupD} \partial_<1>"
%% substitute tensor identity (see appropriate BSSN notebook)
% srepl "\bar{D}_k \text{vet}^k" -> "(\partial_k \text{vet}^k + \frac{\partial_k \text{gammahatdet} \text{vet}^k}{2 \text{gammahatdet}})"
% vardef -diff_type=dD 'alpha'
% vardef -diff_type=dD -symmetry=sym01 'aDD'
% srepl "\bar{A}" -> "\text{a}"
% parse \bar{A}^i_j = \bar{\gamma}^{ik} \bar{A}_{kj}
% assign -diff_type=dD 'aUD'
% srepl "\partial_t \bar{\gamma}" -> "\text{h_rhs}"
\partial_t \bar{\gamma}_{ij} &= \mathcal{L}_\beta \bar{\gamma}_{ij}
+ \frac{2}{3} \bar{\gamma}_{ij} \left(\alpha \bar{A}^k{}_k - \bar{D}_k \beta^k\right)
- 2 \alpha \bar{A}_{ij} \\
% vardef -diff_type=dD 'cf', 'trK'
% srepl "K" -> "\text{trK}"
%% replace 'phi' with conformal factor cf = W = e^{{-2\phi}}
% srepl "e^{-4\phi}" -> "\text{cf}^2"
% srepl "\partial_t \phi = <1..> \\" -> "\text{cf_rhs} = -2 \text{cf} (<1..>) \\"
% srepl -persist "\partial_{<1>} \phi" -> "\partial_{<1>} \text{cf} \frac{-1}{2 \text{cf}}"
% srepl "\partial_<1> \phi" -> "\partial_<1> \text{cf} \frac{-1}{2 \text{cf}}"
\partial_t \phi &= \mathcal{L}_\beta \phi
+ \frac{1}{6} \left(\bar{D}_k \beta^k - \alpha K \right) \\
% parse \bar{A}^{ij} = \bar{\gamma}^{jk} \bar{A}^i_k
% assign -diff_type=dD 'aUU'
% srepl "\partial_t \text{trK}" -> "\text{trK_rhs}"
\partial_t K &= \mathcal{L}_\beta K
+ \frac{1}{3} \alpha K^2
+ \alpha \bar{A}_{ij} \bar{A}^{ij}
- e^{-4\phi} \left(\bar{D}_i \bar{D}^i \alpha + 2 \bar{D}^i \alpha \bar{D}_i \phi\right) \\
% vardef -diff_type=dD 'lambdaU'
% srepl "\bar{\Lambda}" -> "\text{lambda}"
% parse \Delta^k_{ij} = \bar{\Gamma}^k_{ij} - \hat{\Gamma}^k_{ij}
% parse \Delta_{ijk} = \bar{\gamma}_{il} \Delta^l_{jk}
% parse \Delta^k = \bar{\gamma}^{ij} \Delta^k_{ij}
% srepl "\partial_t \text{lambda}" -> "\text{Lambdabar_rhs}"
\partial_t \bar{\Lambda}^i &= \mathcal{L}_\beta \bar{\Lambda}^i + \bar{\gamma}^{jk} \hat{D}_j \hat{D}_k \beta^i
+ \frac{2}{3} \Delta^i \bar{D}_k \beta^k + \frac{1}{3} \bar{D}^i \bar{D}_k \beta^k \\%
&\qquad- 2 \bar{A}^{ij} \left(\partial_j \alpha - 6 \alpha \partial_j \phi\right)
+ 2 \alpha \bar{A}^{jk} \Delta^i_{jk} - \frac{4}{3} \alpha \bar{\gamma}^{ij} \partial_j K \\
% vardef -diff_type=dD -symmetry=sym01 'RbarDD'
X_{ij} &= -2 \alpha \bar{D}_i \bar{D}_j \phi + 4 \alpha \bar{D}_i \phi \bar{D}_j \phi
+ 2 \bar{D}_i \alpha \bar{D}_j \phi + 2 \bar{D}_j \alpha \bar{D}_i \phi
- \bar{D}_i \bar{D}_j \alpha + \alpha \bar{R}_{ij} \\
\hat{X}_{ij} &= X_{ij} - \frac{1}{3} \bar{\gamma}_{ij} \bar{\gamma}^{kl} X_{kl} \\
% srepl "\partial_t \text{a}" -> "\text{a_rhs}"
\partial_t \bar{A}_{ij} &= \mathcal{L}_\beta \bar{A}_{ij}
- \frac{2}{3} \bar{A}_{ij} \bar{D}_k \beta^k
- 2 \alpha \bar{A}_{ik} \bar{A}^k_j
+ \alpha \bar{A}_{ij} K
+ e^{-4\phi} \hat{X}_{ij} \\
% srepl "\partial_t \alpha" -> "\text{alpha_rhs}"
\partial_t \alpha &= \mathcal{L}_\beta \alpha - 2 \alpha K \\
% vardef -diff_type=dD 'betU'
% srepl "B" -> "\text{bet}"
% srepl "\partial_t \text{vet}" -> "\text{vet_rhs}"
\partial_t \beta^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j \beta^i\right] + B^i \\
% vardef -const 'eta'
% srepl "\partial_t \text{bet}" -> "\text{bet_rhs}"
\partial_t B^i &= \left[\beta^j \vphantom{dupD} \bar{D}_j B^i\right]
+ \frac{3}{4} \left(\partial_t \bar{\Lambda}^i - \left[\beta^j \vphantom{dupD} \bar{D}_j \bar{\Lambda}^i\right]\right)
- \eta B^i \\
% parse \bar{R} = \bar{\gamma}^{ij} \bar{R}_{ij}
% srepl "\bar{D}^2" -> "\bar{D}^i \bar{D}_i", "\mathcal{<1>}" -> "<1>"
\mathcal{H} &= \frac{2}{3} K^2 - \bar{A}_{ij} \bar{A}^{ij} + e^{-4\phi} \left(\bar{R} - 8 \bar{D}^i \phi
\bar{D}_i \phi - 8 \bar{D}^2 \phi \right) \\
\mathcal{M}^i &= e^{-4\phi} \left(\hat{D}_j \bar{A}^{ij} + 6 \bar{A}^{ij}\partial_j \phi - \frac{2}{3} \bar{\gamma}^{ij}
\partial_j K + \bar{A}^{jk} \Delta^i_{jk} + \bar{A}^{ik} \Delta^j_{jk} \right) \\
\bar{R}_{ij} &= -\frac{1}{2} \bar{\gamma}^{kl} \hat{D}_k \hat{D}_l \bar{\gamma}_{ij} + \frac{1}{2} \left(\bar{\gamma}_{ki}
\hat{D}_j \bar{\Lambda}^k + \bar{\gamma}_{kj} \hat{D}_i \bar{\Lambda}^k\right) + \frac{1}{2} \Delta^k \left(\Delta_{ijk} + \Delta_{jik}\right) \\%
&\qquad+ \bar{\gamma}^{kl} \left(\Delta^m_{ki} \Delta_{jml} + \Delta^m_{kj} \Delta_{iml} + \Delta^m_{ik} \Delta_{mjl}\right)
\end{align}
assert_equal(H, bssncon.H)
Assertion Passed!
assert_equal(MU, bssncon.MU)
Assertion Passed!
assert_equal(RbarDD, Bq.RbarDD)
Assertion Passed!
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-LaTeX_Interface_Example-BSSN_Cartesian.pdf. (Note that clicking on this link may not work; you may need to open the PDF file through another means.)
import cmdline_helper as cmd # NRPy+: Multi-platform Python command-line interface
cmd.output_Jupyter_notebook_to_LaTeXed_PDF("Tutorial-LaTeX_Interface_Example-BSSN_Cartesian")
Created Tutorial-LaTeX_Interface_Example-BSSN_Cartesian.tex, and compiled
LaTeX file to PDF file Tutorial-LaTeX_Interface_Example-
BSSN_Cartesian.pdf