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