RelativisticPy

RelativisticPy - Genaral Relativity for physicists in a hurry. Simple equation looking User Interface. Write and Go! It is not the most complex, conprehensive or fast symbolic calculator, but it is a symbolic calculator.

PyPi Source

TL;DR - Features & Installation:

install python environment

pip install relativisticpy

(Black Hole Solution in < 10 lines)

Schild_solution.py file:

import relativisticpy as rel

wb = rel.Workbook()

wb.expr(
'''python
Coordinates := [t, r, theta, phi]

g_{mu}_{nu} := [ 
                 [-A(r),0,0,0], 
                 [0,B(r),0,0], 
                 [0,0,r**2,0], 
                 [0,0,0,r**2*sin(theta)**2]
               ]


# Now we have defined the metric above, we can call any individual component of the Ricci tensor itself (as it is metric dependent)
eq0 = Ric_{mu:0}_{nu:0}
eq1 = Ric_{mu:1}_{nu:1}
eq2 = Ric_{mu:2}_{nu:2}

eq5 = (eq0*B(r) + eq1*A(r))*(r*B(r))

B = RHS( dsolve(eq5, B(r)) )

eq6 = simplify( subs(eq2, B(r), B) )

A = RHS( dsolve(eq6, A(r)) )

g_{mu}_{nu} := [
                 [A,0,0,0], 
                 [0,1/A,0,0], 
                 [0,0,r**2,0], 
                 [0,0,0,r**2*sin(theta)**2]
               ]

# Step 5: We prove that C_1 and C_2 equations are in terms of c, G, M by comparing with Newton at large radius

a = C^{a:1}_{b:0 c:0}
solve( a*c**2 + G*M/r**2 ) # This shows us what 

              
g_{mu}_{nu} := [
                  [-(1 - (2 * G * M) / (c**2*r)), 0, 0, 0],
                  [0, 1 / (1 - (2 * G * M) / (c**2*r)), 0, 0],
                  [0, 0, r**2, 0],
                  [0, 0, 0, r**2 * sin(theta) ** 2]
              ]
              
Gamma^{a}_{c f} := (1/2)*g^{a b}*(d_{c}*g_{b f} + d_{f}*g_{b c} - d_{b}*g_{c f})

Riemann^{a}_{m b n} := d_{b}*Gamma^{a}_{n m} + Gamma^{a}_{b l}*Gamma^{l}_{n m} - d_{n}*Gamma^{a}_{b m} - Gamma^{a}_{n l}*Gamma^{l}_{b m}

Ricci_{m n} := Riemann^{a}_{m a n}

T1^{a f h i} := g^{i d}*(g^{h c}*(g^{f b}*Riemann^{a}_{b c d}))
T2_{a f h i} := g_{a n}*Riemann^{n}_{f h i}

S =  T1^{a f h i}*T2_{a f h i}
              
tsimplify( S )
''') # Will parse string and compute the equations

This Python package is designed to assist in performing mathematical operations, particularly in the field of General Relativity. It includes a variety of tools for working with symbolic expressions, including a workflow module that allows users to create linear mathematical workflows and solve tensor expressions.

Features

Some of the key features of this package include:

• Variable Assignment: Assign expressions to variables using equals sign, to use later on in your workflow.
• Integration: Integrate expressions with respect to specified variables.
• Differentiation: Differentiate expressions with respect to specified variables.
• Solve: Solve a algebraic equation, or set of algebraic equations with repect to set of variables.
• Simplify: Simplify a specified mathematical expression.
• Expand: Expand expression raised to some power.
• Substitute: Substitute a variable within an expression (if not already assigned.)
• Sum: Find the descrete sum of an expression.
• Series: Find the taylor series of some function/expr.
• Factor: Factor an expression if possible.
• Limit: Find the limit of an expression if possible.
• General Relativity Tensor Operations:
  • Define Metric and Basis using the assignment operation.
  • Write Tensor Expressions using indices to represent the summation of the components (Einstein summation convention.)
  • Get and Assign Specific Components from Tensors: Choice to assign indices to numbers, to return subset of components. And also assign those to other variables.

Package Philosophy

1. Ease of use. The main reason for this package's existance is to make Einstein tensor equations look exactly like they do in mathematical papers, but with a computational engine which can actually compute those symbolic tensor equations.

2. Keep it very lightweight. The packge is meant to have all the main functionality of a mathematical symbolic compotational tool, but yet not big and cluncky. We leave it to Sympy to do all the symbolic calculation work for us.

To put it simply, keep it simple for devs and users.

Directory Breakdown

Package Tree-Structure

TODO

Project Dependencies

• Sympy: This package has a large dependency in Sympy to perform all it's symbolic calculations and functions.

License

This package is licensed. See the LICENSE file for more information.

Contributing

If you would like to contribute to the development of this package, please feel free to fork the repository and submit a pull request. All contributions are welcome!