```python
import scipy
import sympy
import einsteinpy

from sympy import cos, sin, sinh
from einsteinpy.symbolic import MetricTensor, RicciTensor, RicciScalar, ChristoffelSymbols, RiemannCurvatureTensor, WeylTensor
from einsteinpy.symbolic.predefined import AntiDeSitter

sympy.init_printing()
```


```python
metric = AntiDeSitter()  # Define anti de Sitter metric
metric.tensor()
```




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




```python
Ric = RicciTensor.from_metric(metric) # Compute Ricci Tensor from AdS metric
Ric.tensor()
```




$\displaystyle \left[\begin{matrix}3 & 0 & 0 & 0\\0 & - 3 \cos^{2}{\left(t \right)} & 0 & 0\\0 & 0 & \left(\sin^{2}{\left(t \right)} - 1\right) \sinh^{2}{\left(\chi \right)} - 2 \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)} & 0\\0 & 0 & 0 & \left(\sin^{2}{\left(t \right)} - 1\right) \sin^{2}{\left(\theta \right)} \sinh^{2}{\left(\chi \right)} - 2 \sin^{2}{\left(\theta \right)} \cos^{2}{\left(t \right)} \sinh^{2}{\left(\chi \right)}\end{matrix}\right]$




```python
R = RicciScalar.from_riccitensor(Ric) # Compute Ricci Scalar from AdS metric
R.simplify()
R.expr
```


```python
ch = ChristoffelSymbols.from_metric(metric) # Compute Christoffel Symbols from AdS metric
ch.tensor()
```




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




```python
wt = WeylTensor.from_metric(metric) # Compute Weyl Tensor from AdS metric
wt.tensor()
```




$\displaystyle \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]$




```python

```

f5e18043cb1ab4f3ee7c17845ec956fec65cbd94
ipynb
Jupyter Notebook
Anti de Sitter Space.ipynb
SheepWaitForWolf/General-Relativity
e9eb0f8cc65be9368c6648c8afaa1f8e631516a8
MIT
2021-06-04T11:01:54.000Z
2021-06-04T11:01:54.000Z
Anti de Sitter Space.ipynb
SheepWaitForWolf/General-Relativity
e9eb0f8cc65be9368c6648c8afaa1f8e631516a8
MIT
Anti de Sitter Space.ipynb
SheepWaitForWolf/General-Relativity
e9eb0f8cc65be9368c6648c8afaa1f8e631516a8
MIT
Qwen/Qwen-72B
1. YES
2. YES