# Integrating geodesics backwards in time

Is it possible to integrate a geodesic equation backwards in time in SageMath? I'm using the following code in order to ray trace null geodesics:

```
v0 = initial_vector(r0, b, al, ph0=0, inward=False)
geod = M.integrated_geodesic(g, (s, 50, 0), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.1, parameters_values={a: 0, q: 0},
verbose=True)
```

where 's' is my affine parameter.

From what i've understood, `solve_across_charts`

does use `scipy.odeint`

which supports backwards integration but when i try to run it
i get the following error:

```
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-58-13e5246d5cee> in <module>
14 geod = M.integrated_geodesic(g, (s, Integer(50), Integer(0)), v0, across_charts=True)
15 sol = geod.solve_across_charts(step=RealNumber('0.1'),parameters_values={a:Integer(0),q:Integer(0)},
---> 16 verbose=True)
17
18
/opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in solve_across_charts(self, charts, step, solution_key, parameters_values, verbose, **control_param)
1719 ics = initial_pt_coords + initial_tgt_vec_comps
1720 times = np.linspace(t_min, t_max, int((t_max - t_min) / step) + 1,
-> 1721 endpoint=True)
1722 nt = len(times)
1723
<__array_function__ internals> in linspace(*args, **kwargs)
/opt/sagemath-9.2/local/lib/python3.7/site-packages/numpy/core/function_base.py in linspace(start, stop, num, endpoint, retstep, dtype, axis)
113 num = operator.index(num)
114 if num < 0:
--> 115 raise ValueError("Number of samples, %s, must be non-negative." % num)
116 div = (num - 1) if endpoint else num
117
ValueError: Number of samples, -499, must be non-negative.
```

Which does make sense. Any workaround? I've been also trying to set my affine parameter as a negative but this also raises an error.

Thanks in advance.