metrictensor.py¶
Equation definitions and derivations using SymPy.
- class gme.core.metrictensor.MetricTensorMixin¶
Metric tensor equations supplement to equation definition class.
- define_g_eqns() → None¶
Define eqns for metric tensor \(g\) and its dual \(g^*\).
- gstar_varphi_pxpz_eqn¶
\(g^{*} = \left[\begin{matrix}\ \dfrac{2 p_{x}^{3} \varphi^{2}{\left(\mathbf{r} \right)}} {\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \ - \dfrac{3 p_{x}^{3} \left(p_{x}^{2} + \dfrac{3 p_{z}^{2}}{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} {\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \ + \dfrac{2 p_{x} \left(p_{x}^{2} + \dfrac{3 p_{z}^{2}}{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} {\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \ & \dfrac{3 p_{x}^{4} p_{z} \varphi^{2}{\left(\mathbf{r} \right)}}{2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \ - \dfrac{3 p_{x}^{2} p_{z} \varphi^{2}{\left(\mathbf{r} \right)}} {2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}}\\ \ \dfrac{3 p_{x}^{2} p_{z} \varphi^{2}{\left(\mathbf{r} \right)}} {\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \ - \dfrac{3 p_{x}^{2} p_{z} \left(p_{x}^{2} + \dfrac{3 p_{z}^{2}}{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} {\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \ & \dfrac{3 p_{x}^{3} p_{z}^{2} \varphi^{2} {\left(\mathbf{r} \right)}} {2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \ - \dfrac{p_{x}^{3} \varphi^{2}{\left(\mathbf{r} \right)}} {2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}}\ \end{matrix}\right]\)
- Type
- det_gstar_varphi_pxpz_eqn¶
\(\det\left(g^*\right) \ = \dfrac{p_{x}^{4} \left(- \dfrac{p_{x}^{2}}{2} \ + \dfrac{3 p_{z}^{2}}{4}\right) \varphi^{4}{\left(\mathbf{r} \right)}}{p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}}\)
- Type
- g_varphi_pxpz_eqn¶
\(g = \left[\begin{matrix}\ \dfrac{2 \left(p_{x}^{2} - 2 p_{z}^{2}\right) \sqrt{p_{x}^{2} + p_{z}^{2}}}{p_{x} \left(2 p_{x}^{2} - 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} \ & - \dfrac{6 p_{z}^{3} \sqrt{p_{x}^{2} + p_{z}^{2}}}{p_{x}^{2} \left(2 p_{x}^{2} - 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}}\\ \ - \dfrac{6 p_{z}^{3} \sqrt{p_{x}^{2} + p_{z}^{2}}}{p_{x}^{2} \left(2 p_{x}^{2} - 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} \ & - \dfrac{4 p_{x}^{6} + 14 p_{x}^{4} p_{z}^{2} + 22 p_{x}^{2} p_{z}^{4} + 12 p_{z}^{6}}{p_{x}^{3} \sqrt{p_{x}^{2} + p_{z}^{2}} \left(2 p_{x}^{2} - 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}}\ \end{matrix}\right]\)
- Type
- gstar_eigenvalues¶
\(\left[\begin{matrix}\ \dfrac{\varphi_0^{2} p_{x} x_{1}^{- 4 \mu} \left(\varepsilon x_{1}^{2 \mu} + \left(x_{1} - {r}^x\right)^{2 \mu}\right)^{2} \left(- 3 \left(p_{x}^{2} + p_{z}^{2}\right) \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + \left(p_{x}^{2} + 6 p_{z}^{2}\right) \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)\right)} {4 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}} \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)}\\\ \dfrac{\varphi_0^{2} p_{x} x_{1}^{- 4 \mu} \left(\varepsilon x_{1}^{2 \mu} + \left(x_{1} - {r}^x\right)^{2 \mu}\right)^{2} \left(3 \left(p_{x}^{2} + p_{z}^{2}\right) \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + \left(p_{x}^{2} + 6 p_{z}^{2}\right) \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)\right)} {4 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}} \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)}\ \end{matrix}\right]\)
- Type
- gstar_eigenvectors¶
\(\left[\ \begin{matrix}\dfrac{p_{x} p_{z}^{3} \left(p_{x}^{6} + 2 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + 2 p_{z}^{6} + \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}\right)}{p_{x}^{10} + 3 p_{x}^{8} p_{z}^{2} + 7 p_{x}^{6} p_{z}^{4} + 11 p_{x}^{4} p_{z}^{6} + p_{x}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2}p_{z}^{10}+4 p_{z}^{12}} + 10 p_{x}^{2} p_{z}^{8} + p_{x}^{2} p_{z}^{2} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} +20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + 4 p_{z}^{10} + 2 p_{z}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}}\\ \ 1 \ \end{matrix}\ \right] \\ \ \left[\ \begin{matrix}\dfrac{p_{x} p_{z}^{3} \left(p_{x}^{6} + 2 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + 2 p_{z}^{6} - \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}\right)}{p_{x}^{10} + 3 p_{x}^{8} p_{z}^{2} + 7 p_{x}^{6} p_{z}^{4} + 11 p_{x}^{4} p_{z}^{6} - p_{x}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + 10 p_{x}^{2} p_{z}^{8} - p_{x}^{2} p_{z}^{2} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + 4 p_{z}^{10} - 2 p_{z}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}}\\ \ 1 \ \end{matrix}\right]\)
- Type
Code¶
"""
Equation definitions and derivations using :mod:`SymPy <sympy>`.
---------------------------------------------------------------------
Requires Python packages/modules:
- :mod:`SymPy <sympy>`
- `GME`_
.. _GMPLib: https://github.com/geomorphysics/GMPLib
.. _GME: https://github.com/geomorphysics/GME
.. _Matrix:
https://docs.sympy.org/latest/modules/matrices/immutablematrices.html
.. _Equality: https://docs.sympy.org/latest/modules/core.html\
#sympy.core.relational.Equality
---------------------------------------------------------------------
"""
# Disable these pylint errors because it doesn't understand SymPy syntax
# - notably minus signs in equations flag an error
# pylint: disable=invalid-unary-operand-type, not-callable
# Library
import warnings
import logging
# from typing import Dict, Type, Optional # , Tuple, Eq, List
# SymPy
from sympy import Eq, simplify, Matrix, factor, diff
# GME
from gme.core.symbols import eta, gstar, det_gstar, g, varphi_r, rvec
warnings.filterwarnings("ignore")
__all__ = ["MetricTensorMixin"]
class MetricTensorMixin:
"""Metric tensor equations supplement to equation definition class."""
# Prerequisites
eta_: float
beta_type: str
rdot_vec_eqn: Eq
p_covec_eqn: Eq
varphi_rx_eqn: Eq
# Definitions
gstar_varphi_pxpz_eqn: Eq
det_gstar_varphi_pxpz_eqn: Eq
g_varphi_pxpz_eqn: Eq
gstar_eigen_varphi_pxpz: Eq
gstar_eigenvalues: Eq
gstar_eigenvectors: Eq
gstarhat_eqn: Eq # dummy
def define_g_eqns(self) -> None:
r"""
Define eqns for metric tensor :math:`g` and its dual :math:`g^*`.
Attributes:
gstar_varphi_pxpz_eqn (`Equality`_):
:math:`g^{*} = \left[\begin{matrix}\
\dfrac{2 p_{x}^{3} \varphi^{2}{\left(\mathbf{r} \right)}}
{\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \
- \dfrac{3 p_{x}^{3} \left(p_{x}^{2}
+ \dfrac{3 p_{z}^{2}}{2}\right)
\varphi^{2}{\left(\mathbf{r} \right)}}
{\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \
+ \dfrac{2 p_{x} \left(p_{x}^{2}
+ \dfrac{3 p_{z}^{2}}{2}\right)
\varphi^{2}{\left(\mathbf{r} \right)}}
{\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \
& \dfrac{3 p_{x}^{4} p_{z}
\varphi^{2}{\left(\mathbf{r} \right)}}{2 \left(p_{x}^{2}
+ p_{z}^{2}\right)^{\frac{5}{2}}} \
- \dfrac{3 p_{x}^{2} p_{z}
\varphi^{2}{\left(\mathbf{r} \right)}}
{2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}}\\ \
\dfrac{3 p_{x}^{2} p_{z} \varphi^{2}{\left(\mathbf{r} \right)}}
{\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}} \
- \dfrac{3 p_{x}^{2} p_{z} \left(p_{x}^{2}
+ \dfrac{3 p_{z}^{2}}{2}\right)
\varphi^{2}{\left(\mathbf{r} \right)}}
{\left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \
& \dfrac{3 p_{x}^{3} p_{z}^{2} \varphi^{2}
{\left(\mathbf{r} \right)}}
{2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{5}{2}}} \
- \dfrac{p_{x}^{3} \varphi^{2}{\left(\mathbf{r} \right)}}
{2 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}}\
\end{matrix}\right]`
det_gstar_varphi_pxpz_eqn (`Equality`_):
:math:`\det\left(g^*\right) \
= \dfrac{p_{x}^{4} \left(- \dfrac{p_{x}^{2}}{2} \
+ \dfrac{3 p_{z}^{2}}{4}\right) \varphi^{4}{\left(\mathbf{r}
\right)}}{p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2}
+ 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}}`
g_varphi_pxpz_eqn (`Equality`_):
:math:`g = \left[\begin{matrix}\
\dfrac{2 \left(p_{x}^{2} - 2 p_{z}^{2}\right) \sqrt{p_{x}^{2}
+ p_{z}^{2}}}{p_{x} \left(2 p_{x}^{2}
- 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r} \right)}} \
& - \dfrac{6 p_{z}^{3} \sqrt{p_{x}^{2} + p_{z}^{2}}}{p_{x}^{2}
\left(2 p_{x}^{2} - 3 p_{z}^{2}\right)
\varphi^{2}{\left(\mathbf{r} \right)}}\\ \
- \dfrac{6 p_{z}^{3} \sqrt{p_{x}^{2} + p_{z}^{2}}}{p_{x}^{2}
\left(2 p_{x}^{2} - 3 p_{z}^{2}\right)
\varphi^{2}{\left(\mathbf{r} \right)}} \
& - \dfrac{4 p_{x}^{6} + 14 p_{x}^{4} p_{z}^{2}
+ 22 p_{x}^{2} p_{z}^{4} + 12 p_{z}^{6}}{p_{x}^{3}
\sqrt{p_{x}^{2} + p_{z}^{2}} \left(2 p_{x}^{2}
- 3 p_{z}^{2}\right) \varphi^{2}{\left(\mathbf{r}
\right)}}\
\end{matrix}\right]`
gstar_eigen_varphi_pxpz (list of :class:`~sympy.core.expr.Expr`) :
eigenvalues and eigenvectors of :math:`g^{*}` in one object
gstar_eigenvalues (`Matrix`_):
:math:`\left[\begin{matrix}\
\dfrac{\varphi_0^{2} p_{x} x_{1}^{- 4 \mu} \left(\varepsilon
x_{1}^{2 \mu} + \left(x_{1} - {r}^x\right)^{2 \mu}\right)^{2}
\left(- 3 \left(p_{x}^{2} + p_{z}^{2}\right) \sqrt{p_{x}^{12}
+ 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4}
+ 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8}
+ 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + \left(p_{x}^{2}
+ 6 p_{z}^{2}\right) \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2}
+ 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)\right)}
{4 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}
\left(p_{x}^{6}
+ 3 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4}
+ p_{z}^{6}\right)}\\\
\dfrac{\varphi_0^{2} p_{x} x_{1}^{- 4 \mu} \left(\varepsilon
x_{1}^{2 \mu} + \left(x_{1} - {r}^x\right)^{2 \mu}\right)^{2}
\left(3 \left(p_{x}^{2} + p_{z}^{2}\right) \sqrt{p_{x}^{12}
+ 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20
p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2}
p_{z}^{10} + 4 p_{z}^{12}} + \left(p_{x}^{2}
+ 6 p_{z}^{2}\right) \left(p_{x}^{6} + 3 p_{x}^{4} p_{z}^{2}
+ 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)\right)}
{4 \left(p_{x}^{2} + p_{z}^{2}\right)^{\frac{3}{2}}
\left(p_{x}^{6} + 3 p_{x}^{4}
p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + p_{z}^{6}\right)}\
\end{matrix}\right]`
gstar_eigenvectors (list) :
:math:`\left[\
\begin{matrix}\dfrac{p_{x} p_{z}^{3} \left(p_{x}^{6}
+ 2 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + 2 p_{z}^{6}
+ \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8}
p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8}
+ 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}\right)}{p_{x}^{10}
+ 3 p_{x}^{8} p_{z}^{2} + 7 p_{x}^{6} p_{z}^{4} + 11 p_{x}^{4}
p_{z}^{6} + p_{x}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10}
p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6}
+ 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2}p_{z}^{10}+4 p_{z}^{12}}
+ 10 p_{x}^{2} p_{z}^{8} + p_{x}^{2} p_{z}^{2} \sqrt{p_{x}^{12}
+ 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} +20 p_{x}^{6}
p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10}
+ 4 p_{z}^{12}} + 4 p_{z}^{10} + 2 p_{z}^{4} \sqrt{p_{x}^{12}
+ 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8} p_{z}^{4} + 20p_{x}^{6}
p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10}
+ 4 p_{z}^{12}}}\\ \
1 \
\end{matrix}\
\right] \\ \
\left[\
\begin{matrix}\dfrac{p_{x} p_{z}^{3} \left(p_{x}^{6}
+ 2 p_{x}^{4} p_{z}^{2} + 3 p_{x}^{2} p_{z}^{4} + 2 p_{z}^{6}
- \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8}
p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8}
+ 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}\right)}{p_{x}^{10}
+ 3 p_{x}^{8} p_{z}^{2} + 7 p_{x}^{6} p_{z}^{4} + 11 p_{x}^{4}
p_{z}^{6} - p_{x}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2}
+ 10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6}
+ 25 p_{x}^{4} p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10}
+ 4 p_{z}^{12}} + 10 p_{x}^{2} p_{z}^{8} - p_{x}^{2} p_{z}^{2}
\sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} + 10 p_{x}^{8}
p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4} p_{z}^{8}
+ 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}} + 4 p_{z}^{10} -
2 p_{z}^{4} \sqrt{p_{x}^{12} + 4 p_{x}^{10} p_{z}^{2} +
10 p_{x}^{8} p_{z}^{4} + 20 p_{x}^{6} p_{z}^{6} + 25 p_{x}^{4}
p_{z}^{8} + 16 p_{x}^{2} p_{z}^{10} + 4 p_{z}^{12}}}\\ \
1 \
\end{matrix}\right]`
"""
logging.info("gme.core.metrictensor.define_g_eqns")
self.gstar_varphi_pxpz_eqn = None
self.det_gstar_varphi_pxpz_eqn = None
self.g_varphi_pxpz_eqn = None
self.gstar_eigen_varphi_pxpz = None
self.gstar_eigenvalues = None
self.gstar_eigenvectors = None
eta_sub = {eta: self.eta_}
self.gstar_varphi_pxpz_eqn = Eq(
gstar,
factor(
Matrix(
[
diff(self.rdot_vec_eqn.rhs, self.p_covec_eqn.rhs[0]).T,
diff(self.rdot_vec_eqn.rhs, self.p_covec_eqn.rhs[1]).T,
]
)
),
).subs(eta_sub)
self.det_gstar_varphi_pxpz_eqn = Eq(
det_gstar,
(simplify(self.gstar_varphi_pxpz_eqn.rhs.subs(eta_sub).det())),
)
if self.eta_ == 1 and self.beta_type == "sin":
print(
r"Cannot compute all metric tensor $g^{ij}$ equations "
+ r"for $\sin\beta$ model and $\eta=1$"
)
return
self.g_varphi_pxpz_eqn = Eq(
g, simplify(self.gstar_varphi_pxpz_eqn.rhs.subs(eta_sub).inverse())
)
self.gstar_eigen_varphi_pxpz = (
self.gstar_varphi_pxpz_eqn.rhs.eigenvects()
)
self.gstar_eigenvalues = simplify(
Matrix(
[
self.gstar_eigen_varphi_pxpz[0][0],
self.gstar_eigen_varphi_pxpz[1][0],
]
).subs({varphi_r(rvec): self.varphi_rx_eqn.rhs})
)
self.gstar_eigenvectors = [
simplify(
Matrix(self.gstar_eigen_varphi_pxpz[0][2][0]).subs(
{varphi_r(rvec): self.varphi_rx_eqn.rhs}
)
),
simplify(
Matrix(self.gstar_eigen_varphi_pxpz[1][2][0]).subs(
{varphi_r(rvec): self.varphi_rx_eqn.rhs}
)
),
]
#
