# Predefined Metrics in Symbolic Module¶

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

[1]:

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

sympy.init_printing()  # for pretty printing


## Printing the metrics for visualization¶

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

[2]:

sch = Schwarzschild()
sch.tensor()

[2]:

$\displaystyle \left[\begin{matrix}1 - \frac{r_{s}}{r} & 0 & 0 & 0\\0 & - \frac{1}{c^{2} \left(1 - \frac{r_{s}}{r}\right)} & 0 & 0\\0 & 0 & - \frac{r^{2}}{c^{2}} & 0\\0 & 0 & 0 & - \frac{r^{2} \sin^{2}{\left(\theta \right)}}{c^{2}}\end{matrix}\right]$
[3]:

Minkowski(c=1).tensor()

[3]:

$\displaystyle \left[\begin{matrix}-1 & 0 & 0 & 0\\0 & 1.0 & 0 & 0\\0 & 0 & 1.0 & 0\\0 & 0 & 0 & 1.0\end{matrix}\right]$
[4]:

DeSitter().tensor()

[4]:

$\displaystyle \left[\begin{matrix}-1 & 0 & 0 & 0\\0 & e^{\frac{2 x}{\alpha}} & 0 & 0\\0 & 0 & e^{\frac{2 x}{\alpha}} & 0\\0 & 0 & 0 & e^{\frac{2 x}{\alpha}}\end{matrix}\right]$
[5]:

AntiDeSitter().tensor()

[5]:

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

## Calculating the scalar (Ricci) curavtures¶

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

[6]:

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

[7]:

scalar_curvature_de_sitter.expr

[7]:

$\displaystyle - \frac{2 e^{- \frac{2 x}{\alpha}}}{\alpha^{2}}$
[8]:

scalar_curvature_anti_de_sitter.expr

[8]:

$\displaystyle \frac{\left(\sin^{2}{\left(t \right)} - 1\right) \sinh^{2}{\left(\chi \right)} - 2 \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}}{\cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}} + \frac{\left(\sin^{2}{\left(t \right)} - 1\right) \sin^{2}{\left(\theta \right)} \sinh^{2}{\left(\chi \right)} - 2 \sin^{2}{\left(\theta \right)} \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}}{\sin^{2}{\left(\theta \right)} \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}} - 6$

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

[9]:

simplify(scalar_curvature_anti_de_sitter.expr)

[9]:

$\displaystyle -12$

## Searching for a predefined metric¶

find function returns a list of available functions

[10]:

find("sitter")

[10]:

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