from sympy.diffgeom import *
TP = TensorProduct

m = Manifold('Schwarzschild', 4)
p = Patch('origin', m)
cs = CoordSystem('spherical', p, ['t', 'r', 'theta', 'phi'])

m, p, cs

t, r, theta, phi = cs.coord_functions()
t, r, theta, phi

dt, dr, dtheta, dphi = cs.base_oneforms()
dt, dr, dtheta, dphi

metric = exp(2*f(r))*TP(dt, dt) - exp(2*g(r))*TP(dr, dr) - r**2*TP(dtheta, dtheta) - r**2*sin(theta)**2*TP(dphi, dphi)
metric

twoform_to_matrix(metric)

ricci = metric_to_Ricci_components(metric)

ricci = [[simplify(ricci[i][j])
            for j in range(4)] for i in range(4)]


ricci[0][0]

ricci[1][1]

ricci[2][2]

ricci[3][3]

all(ricci[i][j]==0 for i in range(4) for j in range(4) if i!=j)

ch_2nd = metric_to_Christoffel_2nd(metric)
filt = [((i,j,k), simplify(ch_2nd[i][j][k]))
            for i in range(4) for j in range(4) for k in range(j,4)
            if ch_2nd[i][j][k]!=0]

filt[0:3]

filt[3:6]

filt[6:9]

all([ch_2nd[k][i][j] == ch_2nd[k][j][i] for k in range(4) for i in range(4) for j in range(4)])