def test_cartesian_coords(self):
"""Test cartesian coordinate helper"""
cs = coords.cartesian_coords()
assert len(cs.base_symbols()) == 4 # 4 dim space
assert str(cs) == "cartesian"
cs = coords.cartesian_coords(dim=2)
assert len(cs.base_symbols()) == 2 # 4 dim space
assert str(cs) == "cartesian"
def friedmann_lemaitre_roberston_walker(curvature_constant: Symbol = symbols.k,
cartesian: bool = False):
"""Utility for constructing the FLRW metric in terms of a unit lapse and general
scale function `a`.
Args:
curvature_constant:
Symbol, default "k", the curvature parameter in reduced polar coordinates
cartesian:
bool, default False. If true create a cartesian FLRW and ignore curvature_constant argument
Returns:
Metric, the FLRW metric
References:
[1] S. Weinberg, Cosmology (Oxford University Press, Oxford ; New York, 2008).
"""
a = Function('a')(symbols.t)
if cartesian:
cs = coords.cartesian_coords()
dt, dx, dy, dz = cs.base_oneforms()
form = -c**2 * tpow(
dt, 2) + a**2 * (tpow(dx, 2) + tpow(dy, 2) + tpow(dz, 2))
else:
cs = coords.toroidal_coords()
_, r, theta, _ = cs.base_symbols()
dt, dr, dtheta, dphi = cs.base_oneforms()
form = -c**2 * tpow(dt, 2) + a**2 * (
1 / (1 - curvature_constant * r**2) * tpow(dr, 2) + r**2 *
(tpow(dtheta, 2) + sin(theta)**2 * tpow(dphi, 2)))
return Metric(twoform=form, components=(a, ))
def test_matrix_to_twoform(self):
"""Test matrix -> Expr"""
cs = coords.cartesian_coords(dim=2)
base_forms = cs.base_oneforms()
a, b = symbols('a b')
matrix = Matrix([[a**2, 0], [0, b**2]])
form = utilities.matrix_to_twoform(matrix, base_forms)
assert repr(
form) == "a**2*TensorProduct(dt, dt) + b**2*TensorProduct(dx, dx)"
def minkowski():
"""Utility for constructing the Minkowski metric
Returns:
Metric, the Minkowski metric for flat space
References:
[1] S. Weinberg, Cosmology (Oxford University Press, Oxford ; New York, 2008).
"""
cs = coords.cartesian_coords()
dt, dx, dy, dz = cs.base_oneforms()
form = -tpow(dt, 2) + tpow(dx, 2) + tpow(dy, 2) + tpow(dz, 2)
return Metric(twoform=form)
def test_simplify_deriv_notation(self):
"""Test simplify deriv notation"""
cs = coords.cartesian_coords()
t, x, *_ = cs.base_symbols()
dt, dx, *_ = cs.base_oneforms()
F = Function('F')(t)
G = Function('G')(t, x)
form = F ** 2 * tpow(dt, 2) + G ** 2 * tpow(dx, 2)
g = metric.Metric(twoform=form, components=(F, G))
# Check First-Order Terms
assert str(metric.simplify_deriv_notation(D(F, t), g, max_order=1)) == "F'(t)"
assert str(metric.simplify_deriv_notation(D(F, t), g, max_order=1, use_dots=True)) == r"\dot{F}(t)"
# Check Second-Order Pure Terms
assert str(metric.simplify_deriv_notation(D(D(F, t), t), g, max_order=2)) == "F''(t)"
assert str(metric.simplify_deriv_notation(D(D(F, t), t), g, max_order=2, use_dots=True)) == r"\ddot{F}(t)"
# Check Second-Order Mixed Terms
assert str(metric.simplify_deriv_notation(D(D(G, t), t), g, max_order=2)) == "G_{t t}(t, x)"
assert str(metric.simplify_deriv_notation(D(D(G, t), x), g, max_order=2)) == "G_{t x}(t, x)"