Symbolically Understanding Christoffel Symbol and Riemann Curvature Tensor using EinsteinPy

import sympy

from einsteinpy.symbolic import MetricTensor, ChristoffelSymbols, RiemannCurvatureTensor



sympy.init_printing()  # enables the best printing available in an environment

Defining the metric tensor for 3d spherical coordinates

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)

# creating metric object

m_obj = MetricTensor(metric, syms)

m_obj.tensor()

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

Calculating the christoffel symbols

ch = ChristoffelSymbols.from_metric(m_obj)

ch.tensor()

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

ch.tensor()[1,1,0]

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

Calculating the Riemann Curvature tensor

# Calculating Riemann Tensor from Christoffel Symbols

rm1 = RiemannCurvatureTensor.from_christoffels(ch)

rm1.tensor()

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

# Calculating Riemann Tensor from Metric Tensor

rm2 = RiemannCurvatureTensor.from_metric(m_obj)

rm2.tensor()

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

Calculating the christoffel symbols for Schwarzschild Spacetime Metric

• The expressions are unsimplified

syms = sympy.symbols("t r theta phi")

G, M, c, a = sympy.symbols("G M c a")

# using metric values of schwarschild space-time

# a is schwarzschild radius

list2d = [[0 for i in range(4)] for i in range(4)]

list2d[0][0] = 1 - (a / syms[1])

list2d[1][1] = -1 / ((1 - (a / syms[1])) * (c ** 2))

list2d[2][2] = -1 * (syms[1] ** 2) / (c ** 2)

list2d[3][3] = -1 * (syms[1] ** 2) * (sympy.sin(syms[2]) ** 2) / (c ** 2)

sch = MetricTensor(list2d, syms)

sch.tensor()

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

# single substitution

subs1 = sch.subs(a,0)

subs1.tensor()

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

# multiple substitution

subs2 = sch.subs([(a,0), (c,1)])

subs2.tensor()

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

sch_ch = ChristoffelSymbols.from_metric(sch)

sch_ch.tensor()

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

Calculating the simplified expressions

simplified = sch_ch.simplify()

simplified

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