pystein.curvature module

Symbolic Curvature Utilities

This module provides functions for computing components of various curvature-related tensors, such as Christoffel symbols, Riemann Tensor, and Ricci Tensor. While the sympy module does offer analogous functions in the sympy.diffgeom subpackage, there is an obnoxious bug in that derivatives must be take with respect to coordinate functions as opposed to the identical symbol. We remediate that shortcoming here, allowing for maximal interoperability with the other sympy utilies for solving differential equations and computing derivatives.

pystein.curvature.christoffel_symbol_component(lam: int, mu: int, nu: int, metric: pystein.metric.Metric) → sympy.core.expr.Expr

Compute a component of the Chrisoffel symbol for a particular metric corresponding to

G_mn^l = 1/2 g^ms (p_m g_ns + p_n g_sm - p_s g_mn)

Parameters

• lam – int, Upper index of Christoffel symbol

• mu – int, lower left index of Christoffel symbol

• nu – int, lower right index of Christoffel symbol

• metric – Metric

Returns

Expression of the G_mn^l

pystein.curvature.compute_components(metric: pystein.metric.Metric, non_trivial: bool = True)

pystein.curvature.display_components(cps)

pystein.curvature.einstein_tensor_component(mu: int, nu: int, metric: pystein.metric.Metric)

Compute a component of the Einstein Tensor for a particular metric corresponding to

G_mn = R_mn - (1/2) R g_mn

Parameters

• mu – int, lower left index of Einstein Tensor

• nu – int, lower right index of Einstein Tensor

• metric – Metric

Returns

Expression of the G_mn

pystein.curvature.ricci_scalar(metric: pystein.metric.Metric, simplify_intermediate: bool = False) → sympy.core.expr.Expr

Compute the Ricci Scalar for a particular metric corresponding to

R = R^l_l = g^mn R_mn

Parameters

metric – Metric

Returns

Expression of R

pystein.curvature.ricci_tensor_component(mu: int, nu: int, metric: pystein.metric.Metric, simplify_intermediate: bool = False)

Compute a component of the Ricci Tensor for a particular metric corresponding to

R_mn = R_m^l_ln

Parameters

• mu – int, lower left index of Ricci Tensor

• nu – int, lower right index of Ricci Tensor

• metric – Metric

Returns

Expression of the R_mn

pystein.curvature.riemann_tensor_component(rho: int, sig: int, mu: int, nu: int, metric: pystein.metric.Metric, simplify_intermediate: bool = False) → sympy.core.expr.Expr

Compute a component of the Riemann Tensor for a particular metric corresponding to

R^r_smn = p_m G_ns^r - p_n G_ms^r + G_ml^r G_ns^l - G_nl^r G_ms^l

Parameters

• rho – int, Upper index of the Riemann Tensor

• sig – int, lower left index of Riemann Tensor

• mu – int, lower middle index of Riemann Tensor

• nu – int, lower right index of Riemann Tensor

• metric – Metric

Returns

Expression of the R^r_smn