Predefined Metrics in Symbolic Module

Importing some of the predefined tensors. All the metrics are comprehensively listed in EinsteinPy documentation.

import sympy
from sympy import simplify
from einsteinpy.symbolic import RicciScalar
from einsteinpy.symbolic.predefined import Schwarzschild, DeSitter, AntiDeSitter, Minkowski, find

sympy.init_printing()  # for pretty printing

Printing the metrics for visualization

All the functions return instances of :py:class:~einsteinpy.symbolic.metric.MetricTensor

sch = Schwarzschild()
sch.tensor()

$\displaystyle \left[\begin{matrix}1 - \frac{r_{s}}{r} & 0 & 0 & 0\\0 & - \frac{1}{c^{2} \left(1 - \frac{r_{s}}{r}\right)} & 0 & 0\\0 & 0 & - \frac{r^{2}}{c^{2}} & 0\\0 & 0 & 0 & - \frac{r^{2} \sin^{2}{\left(\theta \right)}}{c^{2}}\end{matrix}\right]$

Minkowski(c=1).tensor()

$\displaystyle \left[\begin{matrix}-1 & 0 & 0 & 0\\0 & 1.0 & 0 & 0\\0 & 0 & 1.0 & 0\\0 & 0 & 0 & 1.0\end{matrix}\right]$

DeSitter().tensor()

$\displaystyle \left[\begin{matrix}-1 & 0 & 0 & 0\\0 & e^{\frac{2 x}{\alpha}} & 0 & 0\\0 & 0 & e^{\frac{2 x}{\alpha}} & 0\\0 & 0 & 0 & e^{\frac{2 x}{\alpha}}\end{matrix}\right]$

AntiDeSitter().tensor()

$\displaystyle \left[\begin{matrix}-1 & 0 & 0 & 0\\0 & \cos^{2}{\left(t \right)} & 0 & 0\\0 & 0 & \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)} & 0\\0 & 0 & 0 & \sin^{2}{\left(\theta \right)} \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}\end{matrix}\right]$

Calculating the scalar (Ricci) curavtures

They should be constant for De-Sitter and Anti-De-Sitter spacetimes.

scalar_curvature_de_sitter = RicciScalar.from_metric(DeSitter())
scalar_curvature_anti_de_sitter = RicciScalar.from_metric(AntiDeSitter())

scalar_curvature_de_sitter.expr

$\displaystyle - \frac{2 e^{- \frac{2 x}{\alpha}}}{\alpha^{2}}$

scalar_curvature_anti_de_sitter.expr

$\displaystyle -12$

On simplifying the expression we got above, we indeed obtain a constant

simplify(scalar_curvature_anti_de_sitter.expr)

$\displaystyle -12$

Searching for a predefined metric

find function returns a list of available functions

find("sitter")

['AntiDeSitter', 'AntiDeSitterStatic', 'DeSitter']