# Symbolically Understanding Christoffel Symbol and Riemann Curvature Tensor using EinsteinPy


```python
import sympy
from einsteinpy.utils import christoffel
from pprint import pprint
```

### Defining the metric tensor for 3d spherical coordinates


```python
syms = sympy.symbols('r theta phi')
# define the metric for 3d spherical coordinates
metric = [[0 for i in range(3)] for i in range(3)]
metric[0][0] = 1
metric[1][1] = syms[0]**2
metric[2][2] = (syms[0]**2)*(sympy.sin(syms[1])**2)
```

### Calculating the christoffel symbols


```python
ch = christoffel.christoffels(metric, syms)
sympy.Matrix(ch)
```




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




```python
ch[1][1][0]
```




$\displaystyle \frac{1}{r}$



### Calculating the Riemann Curvature tensor


```python
rm = christoffel.riemann_curvature_tensor(metric, syms)
sympy.Matrix(rm)
```




$\displaystyle \left[\begin{matrix}\left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right]\\\left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right]\\\left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right] & \left[ \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right], \  \left[ 0, \  0, \  0\right]\right]\end{matrix}\right]$



### Calculating the christoffel symbols for Kerr Spacetime Metric
  - The expressions are unsimplified


```python
kerr_ch = christoffel.kerr_christoffels()
sympy.Matrix(kerr_ch[0][0])
```




$\displaystyle \left[\begin{matrix}0\\\frac{R a r \left(\frac{2 R a r^{2} \sin^{2}{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} - \frac{R a \sin^{2}{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right) \sin^{2}{\left(\theta \right)}}{2 \left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right) \left(- \frac{R^{2} a^{2} r^{2} \sin^{4}{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} + \frac{\left(\frac{R r}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}} - 1\right) \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{2}{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right)} + \frac{\left(- \frac{2 R r^{2}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} + \frac{R}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right) \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{2}{\left(\theta \right)}}{2 \left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right) \left(- \frac{R^{2} a^{2} r^{2} \sin^{4}{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} + \frac{\left(\frac{R r}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}} - 1\right) \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{2}{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right)}\\\frac{R a^{2} r \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{3}{\left(\theta \right)} \cos{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{3} \left(- \frac{R^{2} a^{2} r^{2} \sin^{4}{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} + \frac{\left(\frac{R r}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}} - 1\right) \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{2}{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right)} + \frac{R a r \left(- \frac{2 R a^{3} r \sin^{3}{\left(\theta \right)} \cos{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} - \frac{2 R a r \sin{\left(\theta \right)} \cos{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right) \sin^{2}{\left(\theta \right)}}{2 \left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right) \left(- \frac{R^{2} a^{2} r^{2} \sin^{4}{\left(\theta \right)}}{\left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2}} + \frac{\left(\frac{R r}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}} - 1\right) \left(- a^{2} \left(- R r + a^{2} + r^{2}\right) \sin^{2}{\left(\theta \right)} + \left(a^{2} + r^{2}\right)^{2}\right) \sin^{2}{\left(\theta \right)}}{a^{2} \cos^{2}{\left(\theta \right)} + r^{2}}\right)}\\0\end{matrix}\right]$



### Calculating the simplified expressions


```python
kerr_ch_simplified = christoffel.simplify_christoffels(kerr_ch)
sympy.Matrix(kerr_ch_simplified[0][0])
```




$\displaystyle \left[\begin{matrix}0\\\frac{R \left(- a^{4} \cos^{2}{\left(\theta \right)} - a^{2} r^{2} \cos^{2}{\left(\theta \right)} + a^{2} r^{2} + r^{4}\right)}{2 \left(a^{2} \cos^{2}{\left(\theta \right)} + r^{2}\right)^{2} \left(- R r + a^{2} + r^{2}\right)}\\- \frac{2 R a^{2} r \sin{\left(2 \theta \right)}}{\left(a^{2} \cos{\left(2 \theta \right)} + a^{2} + 2 r^{2}\right)^{2}}\\0\end{matrix}\right]$



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ipynb
Jupyter Notebook
docs/source/examples/Symbolically_Understanding_Christoffel_Symbol_and_Riemann_Curvature_Tensor_using_EinsteinPy.ipynb
Tushar-Tyagi/einsteinpy
ffb2f3a7d64ee6e5a442e620cc82f59c362f49e4
MIT
docs/source/examples/Symbolically_Understanding_Christoffel_Symbol_and_Riemann_Curvature_Tensor_using_EinsteinPy.ipynb
Tushar-Tyagi/einsteinpy
ffb2f3a7d64ee6e5a442e620cc82f59c362f49e4
MIT
docs/source/examples/Symbolically_Understanding_Christoffel_Symbol_and_Riemann_Curvature_Tensor_using_EinsteinPy.ipynb
Tushar-Tyagi/einsteinpy
ffb2f3a7d64ee6e5a442e620cc82f59c362f49e4
MIT
Qwen/Qwen-72B
1. YES
2. YES