Geodesics/tangent vector evaluation interpolation error

edit

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/refe... but it raises an error i can't find a solution to.

edit: Just to be precise and avoid taking up anyone's time. I am actually trying to create a poincare section of my orbit so i think i could use the numerical solution by calculating the velocity where the sol meets the criteria. Maybe there's an easier way (?). Couldnt think of anything else that can help me find the poincare section after and not during the integration.

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,r,th,ph> = M.chart(r't r:(2.0001,+inf) th:(0,pi):\theta ph:(0,2*pi):periodic\phi') 
X.coord_range()

var('m,b,a', domain='real')
g = M.metric()
m=1

g[0,0] = -(1-2*m*r/(r^2 + (a*cos(th))^2))
g[0,3] = -2*a*m*r*sin(th)^2/(r^2 + (a*cos(th))^2)
g[1,1] = (r^2 + (a*cos(th))^2)/(r^2 -2*m*r + a^2)
g[2,2] = (r^2 + (a*cos(th))^2)
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/(r^2 + (a*cos(th))^2))*sin(th)^2
g.display()

def g00(m,r,th,a):return g[0,0](m,r,th,a)
def g03(m,r,th,a):return g[0,3](m,r,th,a)
def g11(m,r,th,a):return g[1,1](m,r,th,a)
def g22(m,r,th,a):return g[2,2](m,r,th,a)
def g33(m,r,th,a):return g[3,3](m,r,th,a)
def D(m,r,th,a): return (g03(m,r,th,a))^2-g00(m,r,th,a)*g33(m,r,th,a)

E.<x,y,z> = EuclideanSpace()
phi = M.diff_map(E, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
phi.display()

def initial_vector(r0, b,al, ph0=0, E=1, inward=False):
    t0,th0=0,pi/2
    L = -b*E
    vt0=(E*g33(m,r0,th0,a)+L*g03(m,r0,th0,a))/(D(m,r0,th0,a))
    vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a)))/g11(m,r0,th0,a))
    if inward:
        vr0 = - vr0
    vth0 = al/(100*r0)
    vph0 = -(1/D(m,r0,th0,a))*(E*g03(m,r0,th0,a)+L*(g00(m,r0,th0,a)))
    p0 = M((t0, r0, th0, ph0), name='p_0')
    return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')

r0 = 100 
th0=pi/2
s = var('s')

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)

where using:

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-24-6fbce2379e9a> in <module>
----> 1 v = geod.tangent_vector_eval_at(Integer(200), verbose=True)

/opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
   2275             # partial test, in case future interpolation objects do not
   2276             # contain lists of instances of the Spline class
-> 2277             raise TypeError("unexpected type of interpolation object")
   2278 
   2279         interpolated_coordinates=[coordinate_curve_spline(t)

TypeError: unexpected type of interpolation object