Usage

RicciPy makes extensive usage of sympy for managing tensor products and contractions. To create a tensor, it is required to construct an array to represent the components of the tensor. This can be done by using most nested array types: a nested list, a sympy.Array instance, a sympy.Matrix instance, or a numpy array will work.

Before beginning any involved application, it is first necessary to define a Metric instance so that indices can be appropriately raised and lowered automatically.

In the following example, the electromagnetic tensor is defined in a Minkowski spacetime. To begin, we declare the components of the electromagnetic tensor using sympy for representing variables:

>>> from sympy import diag, symbols
>>> from riccipy import Tensor, Metric, indices, expand_array
>>> E1, E2, E3, B1, B2, B3 = symbols('E1:4 B1:4')
>>> em = [[0, -E1, -E2, -E3],
          [E1, 0, -B3, B2],
          [E2, B3, 0, -B1],
          [E3, -B2, B1, 0]]
>>> t, x, y, z = symbols('t x y z')

Next, the Minkowski metric is defined along with the Tensor object for the electromagnetic tensor. Here, the symmetry keyword is optional but is used to declare the electromagnetic tensor as being antisymmetric (refer to sympy’s documentation for the sympy.tensor.tensor module).

>>> eta = Metric('eta', [t, x, y, z], diag(1, -1, -1, -1))
>>> F = Tensor('F', em, eta, symmetry=[[2]])
>>> mu, nu = indices('mu nu', eta)

mu and nu are now variables that can be used to represent the indices of either the metric, eta, or the tensor F. Negative signs indicate that the particular index is a subscript (covariant) rather than a superscript (contravariant).

This first calculation demonstrates how contractions are handled: simply multiply two indexed tensors and matching indices will automatically apply the Einstein summation convention. Symbolically, indices that are to be contracted are denoted by the metric those indices belong to (in this case eta_0 and eta_1).

To convert a symbolic tensor expression into components, pass the expression to expand_array.

>>> expr = F(mu, nu) * F(-mu, -nu)
>>> expr
F(eta_0, eta_1)*F(-eta_0, -eta_1)
>>> expand_array(expr)
2*B_1**2 + 2*B_2**2 + 2*B_3**2 - 2*E_1**2 - 2*E_2**2 - 2*E_3**2

This next calculation demonstrates the consequence of having defined F as being antisymmetric:

>>> expr = F(mu, nu) + F(nu, mu)
>>> expand_array(expr)
0

>>> from riccipy.metrics import find
>>> find('sitter')
['anti-de sitter', 'de sitter', 'schwarzschild-de sitter']

>>> g, variables, functions = load_metric('g', 'anti-de sitter')
>>> g.as_array()
[[-1, 0, 0, 0],
[0, cos(t)**2, 0, 0],
[0, 0, cos(t)**2*sinh(chi)**2, 0],
[0, 0, 0, sin(theta)**2*cos(t)**2*sinh(chi)**2]]

Installation

To install RicciPy the following dependencies are required:

• Sympy (version >= 1.4)

• Numpy (version >= 1.15)

Installation is handled automatically by using

$ pip install riccipy

Contributing & Questions

RicciPy is in it’s early stages of development and thus contributions are very welcome, yet they will be handled on a person-to-person basis until sufficient interest accumulates in the project. Feel free to email the primary author at calvinjayross@gmail.com if you have any questions or interest in developing RicciPy.