eminamitani / differential-geometry

several script for differential geometry using sympy

About this script

This is simple python script to calculate several tensors appears in differential geometry.

Ricci.py

class Ricci include modules to evaluate Christoffel Symbol, Riemann Tensor, Ricci Tensor, and Ricci Scalar for given metric. When create the instance of Ricci, metric and the index of local coordinate should be given as arguments. The metric should be NxN Matrix of Sympy and index of local coordinate should be a list of Symbol of Sympy.

Christoffel Symbol, Ricci Tensor and Ricci Scalar has the form of Array in Sympy. Here I comment on the order of index of Array and mathematical form.

A instance of Ricci is constructed under some metric g and coordinate.

Ric = Ricci(coordinate, g)

When Christoffel Symbol is defined by

CF=Ric.ChristoffelSymbol()

$Gamma^i_{jk}$ can be referred by

CF[i,j,k]

When Riemann Tensor is defined by

Rie=Ric.RiemannTensor()

$R^i_{jkl}$ can be referred by

Rie[i,j,k,l]

When Ricci Tensor is defined by

RT=Ric.RicciTensor()

$Ric_{ij}$ can be referred by

RT[i,j]

The __main__ is an example for Schwarzschild metric.