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| r''' | |
| unit test describing the hyperbolic half-plane with the Poincare metric. This | |
| is a basic model of hyperbolic geometry on the (positive) half-space | |
| {(x,y) \in R^2 | y > 0} | |
| with the Riemannian metric | |
| ds^2 = (dx^2 + dy^2)/y^2 | |
| It has constant negative scalar curvature = -2 | |
| https://en.wikipedia.org/wiki/Poincare_half-plane_model | |
| ''' | |
| from sympy.matrices.dense import diag | |
| from sympy.diffgeom import (twoform_to_matrix, | |
| metric_to_Christoffel_1st, metric_to_Christoffel_2nd, | |
| metric_to_Riemann_components, metric_to_Ricci_components) | |
| import sympy.diffgeom.rn | |
| from sympy.tensor.array import ImmutableDenseNDimArray | |
| def test_H2(): | |
| TP = sympy.diffgeom.TensorProduct | |
| R2 = sympy.diffgeom.rn.R2 | |
| y = R2.y | |
| dy = R2.dy | |
| dx = R2.dx | |
| g = (TP(dx, dx) + TP(dy, dy))*y**(-2) | |
| automat = twoform_to_matrix(g) | |
| mat = diag(y**(-2), y**(-2)) | |
| assert mat == automat | |
| gamma1 = metric_to_Christoffel_1st(g) | |
| assert gamma1[0, 0, 0] == 0 | |
| assert gamma1[0, 0, 1] == -y**(-3) | |
| assert gamma1[0, 1, 0] == -y**(-3) | |
| assert gamma1[0, 1, 1] == 0 | |
| assert gamma1[1, 1, 1] == -y**(-3) | |
| assert gamma1[1, 1, 0] == 0 | |
| assert gamma1[1, 0, 1] == 0 | |
| assert gamma1[1, 0, 0] == y**(-3) | |
| gamma2 = metric_to_Christoffel_2nd(g) | |
| assert gamma2[0, 0, 0] == 0 | |
| assert gamma2[0, 0, 1] == -y**(-1) | |
| assert gamma2[0, 1, 0] == -y**(-1) | |
| assert gamma2[0, 1, 1] == 0 | |
| assert gamma2[1, 1, 1] == -y**(-1) | |
| assert gamma2[1, 1, 0] == 0 | |
| assert gamma2[1, 0, 1] == 0 | |
| assert gamma2[1, 0, 0] == y**(-1) | |
| Rm = metric_to_Riemann_components(g) | |
| assert Rm[0, 0, 0, 0] == 0 | |
| assert Rm[0, 0, 0, 1] == 0 | |
| assert Rm[0, 0, 1, 0] == 0 | |
| assert Rm[0, 0, 1, 1] == 0 | |
| assert Rm[0, 1, 0, 0] == 0 | |
| assert Rm[0, 1, 0, 1] == -y**(-2) | |
| assert Rm[0, 1, 1, 0] == y**(-2) | |
| assert Rm[0, 1, 1, 1] == 0 | |
| assert Rm[1, 0, 0, 0] == 0 | |
| assert Rm[1, 0, 0, 1] == y**(-2) | |
| assert Rm[1, 0, 1, 0] == -y**(-2) | |
| assert Rm[1, 0, 1, 1] == 0 | |
| assert Rm[1, 1, 0, 0] == 0 | |
| assert Rm[1, 1, 0, 1] == 0 | |
| assert Rm[1, 1, 1, 0] == 0 | |
| assert Rm[1, 1, 1, 1] == 0 | |
| Ric = metric_to_Ricci_components(g) | |
| assert Ric[0, 0] == -y**(-2) | |
| assert Ric[0, 1] == 0 | |
| assert Ric[1, 0] == 0 | |
| assert Ric[0, 0] == -y**(-2) | |
| assert Ric == ImmutableDenseNDimArray([-y**(-2), 0, 0, -y**(-2)], (2, 2)) | |
| ## scalar curvature is -2 | |
| #TODO - it would be nice to have index contraction built-in | |
| R = (Ric[0, 0] + Ric[1, 1])*y**2 | |
| assert R == -2 | |
| ## Gauss curvature is -1 | |
| assert R/2 == -1 | |