def _set_user_coordinates(self, sys_title, user_coord):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M", self.dim)
patch = Patch("P", manifold)
if self.dim == 4:
system = CoordSystem(sys_title, patch, [str(user_coord[0]),str(user_coord[1]),\
str(user_coord[2]),str(user_coord[3])])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim == 3:
system = CoordSystem(
sys_title, patch,
[str(user_coord[0]),
str(user_coord[1]),
str(user_coord[2])])
u, v, w = system.coord_functions()
self.w = w
if self.dim == 2:
system = CoordSystem(
sys_title, patch,
[str(user_coord[0]), str(user_coord[1])])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def _set_flat_coordinates(self, sys_title, flat_diff):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M", self.dim)
patch = Patch("P", manifold)
flat_diff = sympy.Matrix(flat_diff)
N = flat_diff.shape[0]
coords = []
for i in range(0, N):
n = str(flat_diff[i, i]).find('*')
coord_i = str(flat_diff[i, i])[1:n]
coords.append(coord_i)
if self.dim == 4:
system = CoordSystem(sys_title, patch, [str(coords[0]),str(coords[1]),\
str(coords[2]),str(coords[3])])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim == 3:
system = CoordSystem(sys_title, patch, [str(coords[0]),str(coords[1]),\
str(coords[2])])
u, v, w = system.coord_functions()
self.w = w
if self.dim == 2:
system = CoordSystem(
sys_title, patch,
[str(coords[0]), str(coords[1])])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def _set_flat_coordinates(self,sys_title,flat_diff):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M",self.dim)
patch = Patch("P",manifold)
flat_diff = sympy.Matrix(flat_diff)
N = flat_diff.shape[0]
coords = []
for i in range(0,N):
n = str(flat_diff[i,i]).find('*')
coord_i = str(flat_diff[i,i])[1:n]
coords.append(coord_i)
if self.dim==4:
system = CoordSystem(sys_title, patch, [str(coords[0]),str(coords[1]),\
str(coords[2]),str(coords[3])])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim==3:
system = CoordSystem(sys_title, patch, [str(coords[0]),str(coords[1]),\
str(coords[2])])
u, v, w = system.coord_functions()
self.w = w
if self.dim==2:
system = CoordSystem(sys_title, patch, [str(coords[0]),str(coords[1])])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
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_scalar():
M = Manifold('Reisner-Nordstrom', 4)
p = Patch('origin', M)
cs = CoordSystem('spherical', p, ['t', 'r', 'theta', 'phi'])
t, r, theta, phi = cs.coord_functions()
f = Function('f')
assert (ToArray(1, coordinate_system=cs).to_tensor() == 1)
assert (ToArray(f(r)).to_tensor() == f(r))
def test_expand_tensor():
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)
assert (expand_tensor(metric) == metric)
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 _set_coordinates(self,sys_title):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M",self.dim)
patch = Patch("P",manifold)
if self.dim==4:
system = CoordSystem(sys_title, patch, ["u", "v", "w","t"])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim==3:
system = CoordSystem(sys_title, patch, ["u", "v", "w"])
u, v, w = system.coord_functions()
self.w = w
if self.dim==2:
system = CoordSystem(sys_title, patch, ["u", "v"])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def _set_coordinates(self, sys_title):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M", self.dim)
patch = Patch("P", manifold)
if self.dim == 4:
system = CoordSystem(sys_title, patch, ["u", "v", "w", "t"])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim == 3:
system = CoordSystem(sys_title, patch, ["u", "v", "w"])
u, v, w = system.coord_functions()
self.w = w
if self.dim == 2:
system = CoordSystem(sys_title, patch, ["u", "v"])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def _set_user_coordinates(self,sys_title,user_coord):
from sympy.diffgeom import CoordSystem, Manifold, Patch
manifold = Manifold("M",self.dim)
patch = Patch("P",manifold)
if self.dim==4:
system = CoordSystem(sys_title, patch, [str(user_coord[0]),str(user_coord[1]),\
str(user_coord[2]),str(user_coord[3])])
u, v, w, t = system.coord_functions()
self.w = w
self.t = t
if self.dim==3:
system = CoordSystem(sys_title, patch, [str(user_coord[0]),str(user_coord[1]),
str(user_coord[2])])
u, v, w = system.coord_functions()
self.w = w
if self.dim==2:
system = CoordSystem(sys_title, patch, [str(user_coord[0]),str(user_coord[1])])
u, v = system.coord_functions()
self.u, self.v = u, v
self.system = system
def mobius_strip():
import find_metric as fm
import tensor as t
g,diff = fm.mobius_strip()
R = t.Riemann(g,dim=2,sys_title='mobius_strip',flat_diff = diff)
#metric=R.metric
from sympy.diffgeom import TensorProduct, Manifold, Patch, CoordSystem
manifold = Manifold("M",2)
patch = Patch("P",manifold)
system = CoordSystem('mobius_strip', patch, ["u", "v"])
u, v = system.coord_functions()
du,dv = system.base_oneforms()
from sympy import cos
metric = (cos(u/2)**2*v**2/4 + cos(u/2)*v + v**2/16 + 1)*TensorProduct(du, du) + 0.25*TensorProduct(dv, dv)
C = Christoffel_2nd(metric=metric)
return C
def mobius_strip():
import find_metric as fm
import tensor as t
g, diff = fm.mobius_strip()
R = t.Riemann(g, dim=2, sys_title='mobius_strip', flat_diff=diff)
#metric=R.metric
from sympy.diffgeom import TensorProduct, Manifold, Patch, CoordSystem
manifold = Manifold("M", 2)
patch = Patch("P", manifold)
system = CoordSystem('mobius_strip', patch, ["u", "v"])
u, v = system.coord_functions()
du, dv = system.base_oneforms()
from sympy import cos
metric = (cos(u / 2)**2 * v**2 / 4 + cos(u / 2) * v + v**2 / 16 +
1) * TensorProduct(du, du) + 0.25 * TensorProduct(dv, dv)
C = Christoffel_2nd(metric=metric)
return C
def flat_kerr(a=0,G=1,M=0.5):
import find_metric as fm
from sympy.diffgeom import CoordSystem, Manifold, Patch, TensorProduct
manifold = Manifold("M",3)
patch = Patch("P",manifold)
kerr = CoordSystem("kerr", patch, ["u","v","w"])
u,v,w = kerr.coord_functions()
du,dv,dw = kerr.base_oneforms()
g11 = (a**2*sym.cos(v) + u**2)/(-2*G*M*u + a**2 + u**2)
g22 = a**2*sym.cos(v) + u**2
g33 = -(1 - 2*G*M*u/(u**2 + a**2*sym.cos(v)))
# time independent : unphysical ?
#g33 = 2*G*M*a**2*sym.sin(v)**4*u/(a**2*sym.cos(v) + u**2)**2 + a**2*sym.sin(v)**2 + sym.sin(v)**2*u**2
metric = g11*TensorProduct(du, du) + g22*TensorProduct(dv, dv) + g33*TensorProduct(dw, dw)
C = Christoffel_2nd(metric=metric)
return C
def flat_kerr(a=0, G=1, M=0.5):
import find_metric as fm
from sympy.diffgeom import CoordSystem, Manifold, Patch, TensorProduct
manifold = Manifold("M", 3)
patch = Patch("P", manifold)
kerr = CoordSystem("kerr", patch, ["u", "v", "w"])
u, v, w = kerr.coord_functions()
du, dv, dw = kerr.base_oneforms()
g11 = (a**2 * sym.cos(v) + u**2) / (-2 * G * M * u + a**2 + u**2)
g22 = a**2 * sym.cos(v) + u**2
g33 = -(1 - 2 * G * M * u / (u**2 + a**2 * sym.cos(v)))
# time independent : unphysical ?
#g33 = 2*G*M*a**2*sym.sin(v)**4*u/(a**2*sym.cos(v) + u**2)**2 + a**2*sym.sin(v)**2 + sym.sin(v)**2*u**2
metric = g11 * TensorProduct(du, du) + g22 * TensorProduct(
dv, dv) + g33 * TensorProduct(dw, dw)
C = Christoffel_2nd(metric=metric)
return C
# Create a manifold.
M = Manifold('M', 4)
# Create a patch.
patch = Patch('P', M)
# Basic symbols
c, r_s = symbols('c r_s')
# Coordinate system
schwarzchild_coord = CoordSystem('schwarzchild', patch,
['t', 'r', 'theta', 'phi'])
# Get the coordinate functions
t, r, theta, phi = schwarzchild_coord.coord_functions()
# Get the base one forms.
dt, dr, dtheta, dphi = schwarzchild_coord.base_oneforms()
# 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 * (
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
from sympy import symbols, Function
m = Manifold('m', 2)
p = Patch('p', m)
cs = CoordSystem('cs', p, ['a', 'b'])
cs_noname = CoordSystem('cs', p)
x, y = symbols('x y')
f = Function('f')
s1, s2 = cs.coord_functions()
v1, v2 = cs.base_vectors()
f1, f2 = cs.base_oneforms()
def test_point():
point = Point(cs, [x, y])
assert point == point.func(*point.args)
assert point != Point(cs, [2, y])
#TODO assert point.subs(x, 2) == Point(cs, [2, y])
#TODO assert point.free_symbols == set([x, y])
def test_rebuild():
assert m == m.func(*m.args)
assert p == p.func(*p.args)
assert cs == cs.func(*cs.args)
assert cs_noname == cs_noname.func(*cs_noname.args)
assert s1 == s1.func(*s1.args)
assert v1 == v1.func(*v1.args)
assert f1 == f1.func(*f1.args)
from sympy.diffgeom import Manifold, Patch, CoordSystem, TensorProduct
from sympy import sin, trigsimp, simplify
from sympy.abc import r
from sympy.diffgeom import metric_to_Riemann_components
dim = 2
m = Manifold("M", dim)
patch = Patch("P", m)
cartesian = CoordSystem("cartesian", patch, ["x", "y"])
flat_sphere = CoordSystem("flat_sphere", patch, ["theta", "phi"])
x, y = cartesian.coord_functions()
theta, phi = flat_sphere.coord_functions()
dtheta, dphi = flat_sphere.base_oneforms()
metric_diff_form = r**2 * TensorProduct(
dtheta, dtheta) + r**2 * sin(theta)**2 * TensorProduct(dphi, dphi)
R = metric_to_Riemann_components(metric_diff_form)
def tuple_to_list(t):
return list(map(tuple_to_list, t)) if isinstance(t, (list, tuple)) else t
simpR = tuple_to_list(R)
for m in range(dim):
for i in range(dim):
for j in range(dim):
for k in range(dim):
expr = str(R[m][i][j][k])
from sympy.diffgeom import Manifold, Patch, CoordSystem, Point
from sympy import symbols, Function
m = Manifold('m', 2)
p = Patch('p', m)
cs = CoordSystem('cs', p, ['a', 'b'])
cs_noname = CoordSystem('cs', p)
x, y = symbols('x y')
f = Function('f')
s1, s2 = cs.coord_functions()
v1, v2 = cs.base_vectors()
f1, f2 = cs.base_oneforms()
def test_point():
point = Point(cs, [x, y])
assert point == point.func(*point.args)
assert point != Point(cs, [2, y])
#TODO assert point.subs(x, 2) == Point(cs, [2, y])
#TODO assert point.free_symbols == set([x, y])
def test_rebuild():
assert m == m.func(*m.args)
assert p == p.func(*p.args)
assert cs == cs.func(*cs.args)
assert cs_noname == cs_noname.func(*cs_noname.args)
assert s1 == s1.func(*s1.args)
assert v1 == v1.func(*v1.args)
assert f1 == f1.func(*f1.args)
def test_subs():
assert s1.subs(s1,s2) == s2
from sympy.diffgeom import Manifold, Patch, CoordSystem
from sympy.diffgeom import TensorProduct as TP
from sympy.diffgeom import metric_to_Riemann_components as Riemann
from sympy.diffgeom import metric_to_Christoffel_2nd as Christoffel
from sympy import Symbol, Function, latex, sin
n = 2
M = Manifold('M', n)
P = Patch('P', M)
coord = CoordSystem('coord', P, ['x%s'%i for i in range(n)])
x = coord.coord_functions()
dx = coord.base_oneforms()
g = [[1, 0], [0, sin(x[0])**2]]
metric = sum([g[i][j]*TP(dx[i], dx[j]) for i in range(n) for j in range(n)])
C = Christoffel(metric)
R = Riemann(metric)
