nrpylatex 1.1.1

LaTeX Interface to SymPy (CAS) for General Relativity

NRPyLaTeX

NRPy+'s LaTeX Interface to SymPy (CAS) for General Relativity

• automatic expansion of
  • Einstein summation convention
  • Levi-Civita and Christoffel symbols
  • Lie and covariant derivatives
  • metric inverses and determinants
• automatic index manipulation
• arbitrary coordinate system (default)
• robust exception handling

§ Alternate CAS (Computer Algebra System)

If you are using Mathematica instead of SymPy,

from sympy import mathematica_code

namespace = parse_latex(...)
for var in namespace:
    exec(f'{var} = mathematica_code({var})')

If you are using a different CAS, reference the SymPy documentation to find the relevant printing function.

§ Installation

To install NRPyLaTeX using PyPI, run the following command

$ pip install nrpylatex

§ Interactive Tutorial (MyBinder)

Getting Started | Guided Example (Cartesian BSSN)

§ Documentation and Usage

Simple Example (Kretschmann Scalar)

Python REPL or Script (*.py)

>>> from nrpylatex import parse_latex
>>> parse_latex(r"""
...     \begin{align}
...         % attrib coord [t, r, \theta, \phi]
...         % vardef -zero gDD::4D
...         % vardef -const G, M
...         %% define Schwarzschild metric diagonal
...         g_{t t} &= -\left(1 - \frac{2GM}{r}\right) \\
...         g_{r r} &=  \left(1 - \frac{2GM}{r}\right)^{-1} \\
...         g_{\theta \theta} &= r^2 \\
...         g_{\phi \phi} &= r^2 \sin^2\theta \\
...         %% generate metric inverse gUU, determinant det(gDD), and connection GammaUDD
...         % assign -metric gDD
...         R^\alpha{}_{\beta \mu \nu} &= \partial_\mu \Gamma^\alpha_{\beta \nu} - \partial_\nu \Gamma^\alpha_{\beta \mu}
...             + \Gamma^\alpha_{\mu \gamma} \Gamma^\gamma_{\beta \nu} - \Gamma^\alpha_{\nu \sigma} \Gamma^\sigma_{\beta \mu} \\
...         K &= R^{\alpha \beta \mu \nu} R_{\alpha \beta \mu \nu}
...     \end{align}
... """, ignore_warning=True)
('G', 'GammaUDD', 'gDD', 'gUU', 'epsilonUUUU', 'RUDDD', 'K', 'RUUUU', 'M', 'r', 'theta', 'RDDDD', 'gdet')
>>> from sympy import simplify
>>> print(simplify(K))
48*G**2*M**2/r**6

IPython REPL or Jupyter Notebook

In [1]: %load_ext nrpylatex.extension
In [2]: %%parse_latex --ignore-warning
   ...: \begin{align}
   ...:     % attrib coord [t, r, \theta, \phi]
   ...:     % vardef -zero gDD::4D
   ...:     % vardef -const G, M
   ...:     %% define Schwarzschild metric diagonal
   ...:     g_{t t} &= -\left(1 - \frac{2GM}{r}\right) \\
   ...:     g_{r r} &=  \left(1 - \frac{2GM}{r}\right)^{-1} \\
   ...:     g_{\theta \theta} &= r^2 \\
   ...:     g_{\phi \phi} &= r^2 \sin^2\theta \\
   ...:     %% generate metric inverse gUU, determinant det(gDD), and connection GammaUDD
   ...:     % assign -metric gDD
   ...:     R^\alpha{}_{\beta \mu \nu} &= \partial_\mu \Gamma^\alpha_{\beta \nu} - \partial_\nu \Gamma^\alpha_{\beta \mu}
   ...:         + \Gamma^\alpha_{\mu \gamma} \Gamma^\gamma_{\beta \nu} - \Gamma^\alpha_{\nu \sigma} \Gamma^\sigma_{\beta \mu} \\
   ...:     K &= R^{\alpha \beta \mu \nu} R_{\alpha \beta \mu \nu}
   ...: \end{align}
Out[2]: ('G', 'GammaUDD', 'gDD', 'gUU', 'epsilonUUUU', 'RUDDD', 'K', 'RUUUU', 'M', 'r', 'theta', 'RDDDD', 'gdet')
In [3]: from sympy import simplify
In [4]: print(simplify(K))
Out[4]: 48*G**2*M**2/r**6

Einstein Summation Convention

If the same index appears exactly twice in any single term, assume summation over the range of that index.

M^{n k} v_k := \sum_k M^{n k} v_k = M^{n 0} v_0 + M^{n 1} v_1 + ...

Indexing Ambiguity

If v and vU are both in the namespace and you attempt to parse the expression v^2, the output from NRPyLaTeX will be vU[2] (the third component of vU). To resolve the indexing ambiguity between vU[2] and v**2 (v squared), we suggest using the notation v^{{2}} since v^2 and v^{{2}} are rendered identically in LaTeX. Similarly, if v and vD are both in the namespace and you are parsing the symbol v_2, we suggest replacing v_2 with \text{v_2} using the srepl macro to preserve the LaTeX rendering and build a compound symbol.

srepl "v_{<1..>}" -> "v<>{<1..>}", "v_<1>" -> "\text{v_<1>}", "v<>{<1..>}" -> "\text{v_<1..>}"

PARSE Command

parse - parses an equation without typesetting

USAGE
    parse EQUATION
EQUATION
    single LaTeX assignment + '\\'

IGNORE Command

ignore - ignores a LaTeX command or substring during parsing

USAGE
    ignore SUBSTRING, ...
SUBSTRING
    double-quoted string

VARDEF Command

vardef - defines a variable with (optional) attributes

USAGE
    vardef [OPERAND ...] [OPTION] VARIABLE [DIMENSION], ...
OPERAND
    metric=METRIC
        desc: assign metric for index manipulation
        default: metric associated with diacritic (or lack thereof)
    weight=NUMBER
        desc: assign weight for Lie derivative generation
        default: 0
    diff_type=DIFF_TYPE
        desc: assign derivative type {symbolic | dD | dupD (upwind)}
        default: symbolic
    symmetry=SYMMETRY
        desc: assign (anti)symmetry
        default: nosym
        example(s):  sym01 -> [i][j] = [j][i], anti01 -> [i][j] = -[j][i]
OPTION
    const
        label variable type: constant
    kron
        label variable type: delta function
    metric
        label variable type: metric
    zero
        assign zero to each component
VARIABLE
    alphabetic string
    example(s): vU, gDD, alpha
DIMENSION
    variable dimension (array length)
    default: 3D
    example(s): vU::4D

ATTRIB Command

attrib - overrides global properties of NRPyLaTeX

USAGE
    attrib OPERAND VARIABLE
OPERAND
    coord=COORD
        desc: define coord (or coordinate system)
        example(s): [x, y, z], [r, \phi], default
    index=RANGE
        desc: override index range
        example(s): i::4D, [a-z]::2D
VARIABLE
    alphabetic string
    example(s): vU, gDD, alpha

ASSIGN Command

assign - assigns attributes to already defined variables

USAGE
    assign [OPERAND ...] [OPTION] VARIABLE, ...
OPERAND
    metric=VARIABLE
        desc: assign metric for index manipulation
        default: metric associated with diacritic (or lack thereof)
    weight=NUMBER
        desc: assign weight for Lie derivative generation
        default: 0
    diff_type=DIFF_TYPE
        desc: assign derivative type {symbolic | dD | dupD (upwind)}
        default: symbolic
    symmetry=SYMMETRY
        desc: assign (anti)symmetry
        default: nosym
        example(s):  sym01 -> [i][j] = [j][i], anti01 -> [i][j] = -[j][i]
OPTION
    metric
        label variable type: metric
VARIABLE
    alphabetic string
    example(s): vU, gDD, alpha

SREPL COMMAND

srepl - performs string replacement with (lexeme) capture group support

USAGE
    srepl [OPTION] RULE, ...
OPTION
    persist
        apply rule(s) to every subsequent input of the parse() function (internal or external)
RULE
    syntax: "..." -> "..."
    remark (1): whitespace ignored
    remark (2): substring can include a lexeme capture group
        syntax: <i> (single), <i..> (continuous) where i = 0, 1, 2, ...