# Weyl Tensor calculations using Symbolic module¶

[1]:

import sympy
from sympy import cos, sin, sinh
from einsteinpy.symbolic import MetricTensor, WeylTensor

sympy.init_printing()


## Defining the Anti-de Sitter spacetime Metric¶

[2]:

syms = sympy.symbols("t chi theta phi")
t, ch, th, ph = syms
m = sympy.diag(-1, cos(t) ** 2, cos(t) ** 2 * sinh(ch) ** 2, cos(t) ** 2 * sinh(ch) ** 2 * sin(th) ** 2).tolist()
metric = MetricTensor(m, syms)


## Calculating the Weyl Tensor (with all indices covariant)¶

[3]:

weyl = WeylTensor.from_metric(metric)
weyl.tensor()

[3]:

$$\left[\begin{matrix}\left[\begin{matrix}0 & 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] & \left[\begin{matrix}0 & 0 & 0 & 0\\\cos^{2}{\left (t \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\\cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\\sin^{2}{\left (\theta \right )} \cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0 & 0\end{matrix}\right]\\\left[\begin{matrix}0 & \cos^{2}{\left (t \right )} & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}- \cos^{2}{\left (t \right )} & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0\\0 & 0 & 0 & \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & - \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & - \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0\end{matrix}\right]\\\left[\begin{matrix}0 & 0 & \cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & - \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}- \cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0 & 0\\0 & \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{4}{\left (\chi \right )}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & - \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{4}{\left (\chi \right )} & 0\end{matrix}\right]\\\left[\begin{matrix}0 & 0 & 0 & \sin^{2}{\left (\theta \right )} \cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )}\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & - \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )}\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & - \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{4}{\left (\chi \right )}\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}- \sin^{2}{\left (\theta \right )} \cos^{2}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0 & 0\\0 & \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{2}{\left (\chi \right )} & 0 & 0\\0 & 0 & \sin^{2}{\left (\theta \right )} \cos^{4}{\left (t \right )} \sinh^{4}{\left (\chi \right )} & 0\\0 & 0 & 0 & 0\end{matrix}\right]\end{matrix}\right]$$
[4]:

weyl.config

[4]:

'llll'