def test_scalar_curvature_godel():
x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')
basis = [x0, x1, x2, x3]
e = sympy.exp(1)
a = sympy.Symbol('a')
matrix = (a**2) * sympy.Matrix([[1, 0, e**x1, 0], [0, -1, 0, 0],
[e**x1, 0,
(e**(2 * x1)) / 2, 0], [0, 0, 0, -1]])
metric = gr.get_tensor_from_matrix(matrix, basis)
cs = gr.get_chrisoffel_symbols_from_metric(metric)
R = gr.get_scalar_curvature(cs, metric)
R = R[(), ()]
assert R == 1.0 / a**2
def test_scalar_curvature_Schwarzschild():
t, r, theta, phi = sympy.symbols('t r \\theta \\phi')
Rs = sympy.Symbol('R_s')
basis = [t, r, theta, phi]
values = {
(0, 0): -(1 - Rs / r),
(1, 1): 1 / (1 - Rs / r),
(2, 2): r**2,
(3, 3): r**2 * sympy.sin(theta)**2
}
g = gr.Tensor(basis, (0, 2), values)
cs = gr.get_chrisoffel_symbols_from_metric(g)
R = gr.get_scalar_curvature(cs, g)
assert R[(), ()] == 0
def test_christoffel_symbols1_godel():
x0, x1, x2, x3 = sympy.symbols('x_0 x_1 x_2 x_3')
basis = [x0, x1, x2, x3]
e = sympy.exp(1)
matrix = sympy.Matrix([[1, 0, e**x1, 0], [0, -1, 0, 0],
[e**x1, 0, (e**(2 * x1)) / 2, 0], [0, 0, 0, -1]])
metric = gr.get_tensor_from_matrix(matrix, basis)
christoffel_symbols_1 = gr.get_chrisoffel_symbols_from_metric(metric)
christoffel_symbols_2 = {
((0, ), (0, 1)): 1,
((0, ), (1, 2)): (e**x1) / 2,
((1, ), (0, 2)): (e**x1) / 2,
((1, ), (2, 2)): (e**(2 * x1)) / 2,
((2, ), (0, 1)): -e**(-x1),
}
christoffel_symbols_2 = gr._dict_completer(christoffel_symbols_2, 1, 2, 4)
assert christoffel_symbols_1.get_all_values() == christoffel_symbols_2