def test_curvature():
#For testing curvature conventions
M = Manifold('Reisner-Nordstrom', 4)
p = Patch('origin', M)
cs = CoordSystem('spherical', p, ['t', 'r', 'theta', 'phi'])
t, r, theta, phi = cs.coord_functions()
dt, dr, dtheta, dphi = cs.base_oneforms()
f = Function('f')
metric = f(r)**2 * TP(dt, dt) - f(r)**(-2) * TP(dr, dr) - r**2 * TP(
dtheta, dtheta) - r**2 * sin(theta)**2 * TP(dphi, dphi)
ch_2nd = metric_to_Christoffel_2nd(metric)
G = ToArray(metric)
rm = TensorArray(components=metric_to_Riemann_components(metric),
variance=[-1, 1, 1, 1],
coordinate_system=cs)
v = [Function(f) for f in ['v0', 'v1', 'v2', 'v3']]
V = TensorArray(components=[f(t, r, theta, phi) for f in v],
variance=[-1],
coordinate_system=cs)
dV = V.covD(ch_2nd)
ddV = dV.covD(ch_2nd)
#Commuted covariant derivative:
D2V = ddV - ddV.braid(0, 1)
rm = TensorArray(components=metric_to_Riemann_components(metric),
variance=[-1, 1, 1, 1],
coordinate_system=cs)
rmv = rm.TensorProduct(V).contract(1, 4).braid(0, 1).braid(1, 2)
rmvtensor = [(I, simplify(rmv.tensor[I])) for I in rmv.indices]
rmvtensor = [(I, coeff) for (I, coeff) in rmvtensor if coeff != 0]
rmvtensor.sort()
D2Vtensor = [(I, simplify(D2V.tensor[I])) for I in D2V.indices]
D2Vtensor = [(I, v) for (I, v) in D2Vtensor if v != 0]
D2Vtensor.sort()
for (a, b) in zip(rmvtensor, D2Vtensor):
assert (a == b)
def test_covar_deriv():
ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
cvd = BaseCovarDerivativeOp(R2_r, 0, ch)
assert cvd(R2.x) == 1
assert cvd(R2.x * R2.e_x) == R2.e_x
cvd = CovarDerivativeOp(R2.x * R2.e_x, ch)
assert cvd(R2.x) == R2.x
assert cvd(R2.x * R2.e_x) == R2.x * R2.e_x
def test_covar_deriv():
ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
cvd = BaseCovarDerivativeOp(R2_r, 0, ch)
assert cvd(R2.x) == 1
assert cvd(R2.x*R2.e_x) == R2.e_x
cvd = CovarDerivativeOp(R2.x*R2.e_x, ch)
assert cvd(R2.x) == R2.x
assert cvd(R2.x*R2.e_x) == R2.x*R2.e_x
def test_covar_deriv():
ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
cvd = BaseCovarDerivativeOp(R2_r, 0, ch)
assert cvd(R2.x) == 1
# This line fails if the cache is disabled:
assert cvd(R2.x * R2.e_x) == R2.e_x
cvd = CovarDerivativeOp(R2.x * R2.e_x, ch)
assert cvd(R2.x) == R2.x
assert cvd(R2.x * R2.e_x) == R2.x * R2.e_x
def Christoffel_2nd(g=None,metric=None): # either g is supplied as arugment or the two-form
from sympy.abc import u,v
from sympy.diffgeom import metric_to_Christoffel_2nd
from sympy import asin, acos, atan, cos, log, ln, exp, cosh, sin, sinh, sqrt, tan, tanh
if metric is None: # if metric is not specified as two_form
import tensor as t
R = t.Riemann(g,g.shape[0])
metric = R.metric
C = sym.Matrix(eval(str(metric_to_Christoffel_2nd(metric))))
return C
def test_covD():
M = Manifold('Reisner-Nordstrom', 4)
p = Patch('origin', M)
cs = CoordSystem('spherical', p, ['t', 'r', 'theta', 'phi'])
t, r, theta, phi = cs.coord_functions()
dt, dr, dtheta, dphi = cs.base_oneforms()
f = Function('f')
metric = f(r)**2 * TP(dt, dt) - f(r)**(-2) * TP(dr, dr) - r**2 * TP(
dtheta, dtheta) - r**2 * sin(theta)**2 * TP(dphi, dphi)
ch_2nd = metric_to_Christoffel_2nd(metric)
G = ToArray(metric)
assert (G.covD(ch_2nd).to_tensor() == 0)
def Christoffel_2nd(
g=None,
metric=None): # either g is supplied as arugment or the two-form
from sympy.abc import u, v
from sympy.diffgeom import metric_to_Christoffel_2nd
from sympy import asin, acos, atan, cos, log, ln, exp, cosh, sin, sinh, sqrt, tan, tanh
if metric is None: # if metric is not specified as two_form
import tensor as t
R = t.Riemann(g, g.shape[0])
metric = R.metric
C = sym.Matrix(eval(str(metric_to_Christoffel_2nd(metric))))
return C
def test_helpers_and_coordinate_dependent():
one_form = R2.dr + R2.dx
two_form = Differential(R2.x * R2.dr + R2.r * R2.dx)
three_form = Differential(R2.y * two_form) + Differential(
R2.x * Differential(R2.r * R2.dr))
metric = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dy, R2.dy)
metric_ambig = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dr, R2.dr)
misform_a = TensorProduct(R2.dr, R2.dr) + R2.dr
misform_b = R2.dr**4
misform_c = R2.dx * R2.dy
twoform_not_sym = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dx, R2.dy)
twoform_not_TP = WedgeProduct(R2.dx, R2.dy)
one_vector = R2.e_x + R2.e_y
two_vector = TensorProduct(R2.e_x, R2.e_y)
three_vector = TensorProduct(R2.e_x, R2.e_y, R2.e_x)
two_wp = WedgeProduct(R2.e_x, R2.e_y)
assert covariant_order(one_form) == 1
assert covariant_order(two_form) == 2
assert covariant_order(three_form) == 3
assert covariant_order(two_form + metric) == 2
assert covariant_order(two_form + metric_ambig) == 2
assert covariant_order(two_form + twoform_not_sym) == 2
assert covariant_order(two_form + twoform_not_TP) == 2
assert contravariant_order(one_vector) == 1
assert contravariant_order(two_vector) == 2
assert contravariant_order(three_vector) == 3
assert contravariant_order(two_vector + two_wp) == 2
raises(ValueError, lambda: covariant_order(misform_a))
raises(ValueError, lambda: covariant_order(misform_b))
raises(ValueError, lambda: covariant_order(misform_c))
assert twoform_to_matrix(metric) == Matrix([[1, 0], [0, 1]])
assert twoform_to_matrix(twoform_not_sym) == Matrix([[1, 0], [1, 0]])
assert twoform_to_matrix(twoform_not_TP) == Matrix([[0, -1], [1, 0]])
raises(ValueError, lambda: twoform_to_matrix(one_form))
raises(ValueError, lambda: twoform_to_matrix(three_form))
raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
raises(ValueError, lambda: metric_to_Christoffel_1st(twoform_not_sym))
raises(ValueError, lambda: metric_to_Christoffel_2nd(twoform_not_sym))
raises(ValueError, lambda: metric_to_Riemann_components(twoform_not_sym))
raises(ValueError, lambda: metric_to_Ricci_components(twoform_not_sym))
def test_helpers_and_coordinate_dependent():
one_form = R2.dr + R2.dx
two_form = Differential(R2.x*R2.dr + R2.r*R2.dx)
three_form = Differential(
R2.y*two_form) + Differential(R2.x*Differential(R2.r*R2.dr))
metric = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dy, R2.dy)
metric_ambig = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dr, R2.dr)
misform_a = TensorProduct(R2.dr, R2.dr) + R2.dr
misform_b = R2.dr**4
misform_c = R2.dx*R2.dy
twoform_not_sym = TensorProduct(R2.dx, R2.dx) + TensorProduct(R2.dx, R2.dy)
twoform_not_TP = WedgeProduct(R2.dx, R2.dy)
one_vector = R2.e_x + R2.e_y
two_vector = TensorProduct(R2.e_x, R2.e_y)
three_vector = TensorProduct(R2.e_x, R2.e_y, R2.e_x)
two_wp = WedgeProduct(R2.e_x,R2.e_y)
assert covariant_order(one_form) == 1
assert covariant_order(two_form) == 2
assert covariant_order(three_form) == 3
assert covariant_order(two_form + metric) == 2
assert covariant_order(two_form + metric_ambig) == 2
assert covariant_order(two_form + twoform_not_sym) == 2
assert covariant_order(two_form + twoform_not_TP) == 2
assert contravariant_order(one_vector) == 1
assert contravariant_order(two_vector) == 2
assert contravariant_order(three_vector) == 3
assert contravariant_order(two_vector + two_wp) == 2
raises(ValueError, lambda: covariant_order(misform_a))
raises(ValueError, lambda: covariant_order(misform_b))
raises(ValueError, lambda: covariant_order(misform_c))
assert twoform_to_matrix(metric) == Matrix([[1, 0], [0, 1]])
assert twoform_to_matrix(twoform_not_sym) == Matrix([[1, 0], [1, 0]])
assert twoform_to_matrix(twoform_not_TP) == Matrix([[0, -1], [1, 0]])
raises(ValueError, lambda: twoform_to_matrix(one_form))
raises(ValueError, lambda: twoform_to_matrix(three_form))
raises(ValueError, lambda: twoform_to_matrix(metric_ambig))
raises(ValueError, lambda: metric_to_Christoffel_1st(twoform_not_sym))
raises(ValueError, lambda: metric_to_Christoffel_2nd(twoform_not_sym))
raises(ValueError, lambda: metric_to_Riemann_components(twoform_not_sym))
raises(ValueError, lambda: metric_to_Ricci_components(twoform_not_sym))
# Auxiliar terms for the metric.
dt_2 = TP(dt, dt)
dr_2 = TP(dr, dr)
dtheta_2 = TP(dtheta, dtheta)
dphi_2 = TP(dphi, dphi)
factor = (1 - r_s / r)
# Build the metric
metric = factor * c**2 * dt_2 - 1 / factor * dr_2 - r**2 * (
dtheta_2 + sin(theta)**2 * dphi_2)
metric = factor * c**2 * dt_2 - 1 / factor * dr_2 - r**2 * (
dtheta_2 + sin(theta)**2 * dphi_2)
metric = metric / c**2
# Get the Christoffel symbols of the second kind.
christoffel = metric_to_Christoffel_2nd(metric)
# Let's print this in an elegant way ;)
# for i in range(4):
# for j in range(4):
# for k in range(4):
# if christoffel[i, j, k] != 0:
# display(Math('\Gamma^{}_{{{},{}}} = '.format(i, j, k) + latex(christoffel[i, j, k])))
## Specify c and r_s
christoffel_ = christoffel.subs({c: 2, r_s: 1})
def F(t, y):
u = y[0:4]
v = y[4:8]
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)
## 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
def metric_to_Christoffel_2nd(self):
from sympy.diffgeom import metric_to_Christoffel_2nd
return metric_to_Christoffel_2nd(self.metric)
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
def metric_to_Christoffel_2nd(self):
from sympy.diffgeom import metric_to_Christoffel_2nd
return metric_to_Christoffel_2nd(self.metric)
def diffgeometryEx2(request):
typesym = request.GET.get('typesym')
metric_input = request.GET.get('metric')
if not check_args(typesym, metric_input):
return {'is_valid': False}
def get_unique(lst):
lst = set(lst)
lst = list(lst)
lst.sort()
return lst
def replace_in_metric(m, lst1, lst2):
for i in range(len(lst1)):
m = m.replace(lst1[i], lst2[i])
return m
TP = TensorProduct
R = [R2.x, R2.y]
dR2 = [TP(R2.dx, R2.dx), TP(R2.dy, R2.dy)]
dR = (TP(R2.dx, R2.dy) + TP(R2.dy, R2.dx)) / 2
metric = re.sub(r'\^', '**', str(metric_input))
metric = re.sub(r'd\*s\*\*2=', '', metric)
# get vars from string-metric
variables = re.findall(r'd\*(\w+)', metric)
variables = get_unique(variables)
# get differentials from string-metric and replace 'd*x' to 'dx'
zamena_diff = re.findall(r'd\*\w+', metric)
zamena_diff = get_unique(zamena_diff)
zamena_diff_new = [el.replace('*', '') for el in zamena_diff]
metric = replace_in_metric(metric, zamena_diff, zamena_diff_new)
# get differentials at the 2 power
diffs2 = re.findall(r'd\w+\*\*2', metric)
diffs2 = get_unique(diffs2)
# get differentials 'dx*dy' and replace to 'dxdy'
diffs_diff = re.findall(r'd\w+\*d\w+', metric)
diffs_diff = get_unique(diffs_diff)
diffs_diff_new = [el.replace('*', '') for el in diffs_diff]
metric = replace_in_metric(metric, diffs_diff, diffs_diff_new)
# get sympy-eval from string-metric
transformations = standard_transformations + (function_exponentiation, implicit_application,)
try:
metric_sym = parse_expr(metric, transformations=transformations)
except:
return {'is_valid': False}
metric_input = latex(metric_sym)
# replace variables to 'R2.x' and "R2.y'
metric_sym = replace_in_metric(metric_sym, variables, R)
# replace differentials at the 2 power to 'TP(dx, dx)' and 'TP(dy, dy)'
metric_sym = replace_in_metric(metric_sym, diffs2, dR2)
# replace 'dxdy' to 'TP(dx, dy)'
for el in diffs_diff_new:
metric_sym = metric_sym.replace(el, dR)
metric_sym = expand(metric_sym)
#generate metric-matrix
metric_matrix = [latex(metric_sym.coeff(TP(a, b))) for a in [R2.dx, R2.dy] for b in [R2.dx, R2.dy]]
if typesym == '1':
try:
Christoffel = simplify(metric_to_Christoffel_1st(metric_sym))
except ValueError:
return {'is_valid': False}
elif typesym == '2':
try:
Christoffel = simplify(metric_to_Christoffel_2nd(metric_sym))
except ValueError:
return {'is_valid': False}
else:
return {'is_valid': False}
Christoffel = list(Christoffel)
# replace 'R2.x,y' to variables
index = list(map(''.join, product('12', repeat=3)))
answer = [('_{%s,%s%s}=' if typesym == '1' else '^%s_{ \ %s%s}=') % (index[i][0], index[i][1], index[i][2]) +
latex(el.subs([(R[i], variables[i]) for i in range(len(variables))], simultaneous=True))
for i, el in enumerate(Christoffel)]
return {'answer': answer, 'metric': metric_input, 'typesym': typesym, 'metric_matrix': metric_matrix, 'is_valid': True}
from sympy.diffgeom.rn import R2 LieDerivative(R2.e_x, R2.y) LieDerivative(R2.e_x, R2.x) LieDerivative(R2.e_x, R2.e_x) # The Lie derivative of a tensor field by another tensor field is equal to their commutator: LieDerivative(R2.e_x, R2.e_r) LieDerivative(R2.e_x + R2.e_y, R2.x) tp = TensorProduct(R2.dx, R2.dy) LieDerivative(R2.e_x, tp) LieDerivative(R2.e_x, tp).doit() #BaseCovarDerivativeOp from sympy.diffgeom.rn import R2, R2_r from sympy.diffgeom import BaseCovarDerivativeOp from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct TP = TensorProduct ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) ch cvd = BaseCovarDerivativeOp(R2_r, 0, ch) cvd(R2.x) cvd(R2.x*R2.e_x) #CovarDerivativeOp from sympy.diffgeom.rn import R2 from sympy.diffgeom import CovarDerivativeOp from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct TP = TensorProduct ch = metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy)) ch cvd = CovarDerivativeOp(R2.x*R2.e_x, ch) cvd(R2.x) cvd(R2.x*R2.e_x) #intcurve_series