General Theory of Relativity in Python: Introduction

In this article, using Python, what is needed in general relativity ・ ** Calculation of various tensors including weighing $ g_ {\ mu \ nu} $ ** ・ ** Simple calculation of Einstein equation ** Introduce how to do.

** [Reference] ** For computing related to tensor analysis and algebra in general relativity, 　Heinicke, C., et.al.- Computer Algebra in Gravity 　Korolkova, A., et.al.- Tensor computations in computer algebra systems There are some affordable Review papers such as, so please refer to them as appropriate. Also, for a similar study using SageMath, 　Gourgoulhon, E., et.al. - Symbolic tensor calculus on manifolds Etc. are known.

Introducing the GraviPy module

GraviPy is a module for tensor calculation that runs on Python3.

By collaborating with SymPy, which specializes in algebraic symbol processing, a stress-free tensor calculation environment is realized on Python.

Download the GraviPy module including SymPy according to the following GraviPy, Tensor Calculus Package for General Relativity (Version 0.1.0) (2014) (Access date: November 26, 2019)

You can actually use pip,

python

$ pip install GraviPy

If there is no problem, it will be installed. SymPy should be installed at the same time if the Python environment is ver.3.7 or higher.

I calculated the Schwarzschild metric

Use GraviPy to derive the Schwarzschild solution from the Schwarzschild metric.

Introduction of Schwarzschild weighing

The sign of the metric is $ (+,-,-,-) $ according to MTW, and the line element is determined as follows. $ds^2=g_{\mu\nu}dx^\mu dx^\nu .$ For the gravitational constant and the speed of light, if $ G = c = 1 $, the Schwarzschild metric is written as follows.

g_{\mu\nu}=\left[\begin{array}{cccc}
\Big( 1- \frac{2M}{r} \Big) & 0 & 0 & 0	\\
0 & -\Big( 1- \frac{2M}{r} \Big)^{-1} & 0 & 0	\\
0 & 0 & -r^2 & 0	\\
0 & 0 & 0 & -r^2sin^2\theta
\end{array}\right]. 

Here, the four-dimensional coordinate $ (t, r, \ theta, \ phi) $ is taken as the coordinate system, and $ M $ is defined as the black hole mass. The Schwarzschild metric shows a singularity at $ r = 2M $ in the radial direction, which is known as the so-called "event horizon".

How to derive using GraviPy

Let's derive the Schwarzschild solution from the above $ g_ {\ mu \ nu} $. Use GraviPy to calculate according to the following process.

1. Definition of space-time (determination of metric): $ g_ {\ mu \ nu} $
2. Calculation of Christoffel symbol: $ \ Gamma {} ^ \ mu {} _ {\ nu \ rho} $
3. Ricci curvature calculation: $ R_ {\ mu \ nu} $
4. Einstein tensor calculation: $ G_ {\ mu \ nu} $

Definition of space-time

First, let's define space-time. Take $ x = x ^ \ mu = (t, r, \ theta, \ phi) $ as the four-dimensional space-time coordinates. In GraviPy, it is defined as follows.

GR.py

#!/usr/bin/env python3
from gravipy import *
from gravipy import tensorial as ten  
from sympy   import *
import inspect 

# Coordinates (\ chi is the four - vector of coordinates )
t, r, theta, phi, M = symbols('t , r , theta , phi , M ')
x = ten.Coordinates('\chi',[t, r, theta, phi])

Here, the first argument '\ chi' often.Coordinates ()specifies that the second argument is a four-vector. Also, the reason why it is set to ten. ~ Is that this Coordinates () does not work unless it explicitly refers to the tensorial.

After that, the tensors that appear will be referred to by ten. ~.

Definition of Schwarzschild metric

Next, let's define the metric $ g_ {\ mu \ nu} $. The Schwarzschild metric described above may be described as follows.

GR.py

#Continued
# Metric tensor
Metric = diag((1 -2* M / r ) , -1/(1 -2* M / r ) , -r **2 , -r **2* sin( theta ) **2)
g = ten.MetricTensor('g', x , Metric )

Here, the output result of g = ten.MetricTensor ('g', x, Metric) is

Metric
#Or
g(ten.All,ten.All)

It can be displayed with, and it becomes as follows.

g_{\mu\nu}=
\displaystyle \left[\begin{matrix}- 2M/r + 1 & 0 & 0 & 0\\0 & \displaystyle- \frac{1}{- 2M/r + 1} & 0 & 0\\0 & 0 & - r^{2} & 0\\0 & 0 & 0 & - r^{2} \sin^{2}{\left(\theta \right)}\end{matrix}\right] 

Although there are some differences in the expression, the previous definition is accurately reflected. For example, when referring to the radial component $ g_ {rr} = g_ {11} $

g(1,1)

Then, it can be obtained for each element as follows.

g_{11}=- \frac{1}{\displaystyle- \frac{2M}{r} + 1} 

Christoffel's calculation

Next, let's calculate the Christoffel symbol $ \ Gamma ^ \ mu {} _ {\ nu \ rho} $ based on $ g_ {\ mu \ nu} $.

GR.py

#Continued
# Christoffel symbol
Ga = ten.Christoffel('Ga', g )

Ga is the Christoffel symbol for $ g_ {\ mu \ nu} $. Christoffel is a tensor on the 3rd floor, so if you limit it to $ \ mu = 0 $, you will get the following tensor on the 2nd floor.

\Gamma^{ 0}{}_{\nu\rho}=
\left[
\begin{matrix}
0 & \frac{M}{r^{2}} & 0 & 0 \\
\frac{M}{r^{2}} & 0 & 0 & 0 \\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{matrix}\right]

To get all the ingredients

Ga(ten.All,ten.All,ten.All)

You can do it as

\displaystyle
\Gamma^\mu{}_{\nu\rho}=
 \left[\begin{matrix}\left[\begin{matrix}0 & \frac{M}{r^{2}} & 0 & 0\\\frac{M}{r^{2}} & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right] & \left[\begin{matrix}- \frac{M}{r^{2}} & 0 & 0 & 0\\0 & \frac{M}{\left(2 M - r\right)^{2}} & 0 & 0\\0 & 0 & r & 0\\0 & 0 & 0 & r \sin^{2}{\left(\theta \right)}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & - r & 0\\0 & - r & 0 & 0\\0 & 0 & 0 & \frac{r^{2} \sin{\left(2 \theta \right)}}{2}\end{matrix}\right] & \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & - r \sin^{2}{\left(\theta \right)}\\0 & 0 & 0 & - \frac{r^{2} \sin{\left(2 \theta \right)}}{2}\\0 & - r \sin^{2}{\left(\theta \right)} & - \frac{r^{2} \sin{\left(2 \theta \right)}}{2} & 0\end{matrix}\right]\end{matrix}\right]

Will be. Since it is a third-order tensor, it can be formally obtained as a "one-dimensional array of two-dimensional array" (= a form in which the elements of the one-dimensional array are two-dimensional arrays).

Ricci curvature calculation

Next, calculate the Ricci curvature. (This name comes from the mathematician Ricci.)

# Ricci tensor
Ri = ten.Ricci ('Ri ', g )
# Display all compon
Ri(ten.All,ten.All)

The Ricci tensor in the Schwarzschild metric drops all components to zero.

R_{\mu\nu}=
\displaystyle \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right]

Of course, this is a special thing, in the Schwarzschild metric formula. $r \longrightarrow r^{3.5}$ If you try to calculate the Ricci curvature by substituting with, etc., the values will be completely different, so you should experiment.

Einstein tensor calculation

Finally, let's output the Einstein tensor using the Ricci tensor.

# Einstein tensor
G = ten.Einstein ('G', Ri )
G(ten.All,ten.All)

Result is,

G_{\mu\nu}=
\displaystyle \left[\begin{matrix}0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\end{matrix}\right]

It has become. This result is the assumption of the Schwarzschild solution, ** vacuum condition ** $T_{\mu\nu}=0$ Also conforms to the Einstein equation $G_{\mu\nu}=8\pi T_{\mu\nu}$ Meet.

References

The GraviPy tutorial has been released, and I referred to the following.

https://github.com/wojciechczaja/GraviPy

Also, the SymPy support page is below.

[https://docs.sympy.org/dev/index.html] (https://docs.sympy.org/dev/index.html)