def flat_sphere():
import find_metric as fm
import tensor as t
g, diff = find_metric.flat_sphere()
R = t.Riemann(g, dim=2, sys_title='flat_sphere', flat_diff=diff)
C = Christoffel_2nd(metric=R.metric)
return C
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 toroid(u=1, v=None, a=1):
import find_metric as fm
g, diff = fm.toroidal_coordinates()
if v is None:
g = g.subs('u', u)[:2, :2]
else:
g = g.subs('v', v)[1:, 1:]
g = g.subs('a', a)
import tensor as t
R = t.Riemann(g, dim=2, sys_title='toroid')
C = Christoffel_2nd(metric=R.metric)
return C
def egg_carton():
import tensor as t
import find_metric as fm
g, diff = find_metric.egg_carton_metric()
R = t.Riemann(g, dim=2, sys_title='egg_carton', flat_diff=diff)
"""
# this works :
from sympy.abc import u,v
u_,v_ = R.system.coord_functions()
du,dv = R.system.base_oneforms()
metric = R.metric.subs({u:u_,v:v_,'dv':dv,'du':du})
"""
C = Christoffel_2nd(metric=R.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
import find_metric as fm g = fm.kerr_metric() import tensor as t R = t.Riemann(g, 'kerr_metric', 4) RC = R.Christoffel_tensor() RR = R.Ricci_tensor() scalarRR = R.scalar_curvature() print "\nThe Kerr curve element has the following metric" print g print "\nThe Ricci tensor is given as" print RR print "\nand the scalar curvature is" print scalarRR