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from enum import Enum, auto
import numpy as np
from scipy.interpolate import RectBivariateSpline, interp1d
from scipy.integrate import solve_ivp
def _photon_dXdt(t, coord, sign, gXX, gtt, gXt):
"""
Null curve equation RHS (not a geodesic - not affinely parametrized).
gXX dX^2 + 2 gXt dX dt - gtt dt^2 = 0
=> dx / dt = (-gXt ± √(gXt^2 - gXX gtt)) / gXX
"""
gXt_val = gXt(t, coord)
gXX_val = gXX(t, coord)
gtt_val = gtt(t, coord)
return ((-gXt_val + sign * np.sqrt((gXt_val ** 2) - gtt_val * gXX_val)) / gXX_val).flatten()[0]
# terminate integration on reaching the outer boundaries
def _events_bnd_null_curves(spatial_coords):
def event_x_bound_upper(t, x, sign, *args):
return x[0] - spatial_coords[-1]
event_x_bound_upper.terminal = True
event_x_bound_upper.direction = 1.0
def event_x_bound_lower(t, x, sign, *args):
return x[0] - spatial_coords[0]
event_x_bound_lower.terminal = True
event_x_bound_lower.direction = -1.0
return [event_x_bound_upper, event_x_bound_lower]
def _calc_null_kernel_time(times, origin, sign, gXX, gXt, gtt, events):
idx = np.where(times == origin)[0][0]
t_fwd = times[idx:]
t_back = times[:idx + 1][::-1]
ray_fwd = [0.0]
if len(t_fwd) > 1:
sol = solve_ivp(_photon_dXdt, (t_fwd[0], t_fwd[-1]), (0,),
method = 'RK45', t_eval = t_fwd,
args = (sign, gXX, gtt, gXt),
dense_output = True, events = events,
rtol = 1e-6, atol = 1e-8)
ray_fwd = sol.y[0]
ray_fwd = np.concatenate((ray_fwd, [ray_fwd[-1]] * (len(t_fwd) - len(ray_fwd))))
ray_back = []
if len(t_back) > 1:
sol = solve_ivp(_photon_dXdt, (t_back[0], t_back[-1]), (0,),
method = 'RK45', t_eval = t_back,
args = (sign, gXX, gtt, gXt),
dense_output = True, events = events,
rtol = 1e-6, atol = 1e-8)
ray_back = sol.y[0]
ray_back = np.concatenate((ray_back, [ray_back[-1]] * (len(t_back) - len(ray_back))))
return np.concatenate((ray_back[::-1][:-1], ray_fwd))
def _calc_null_kernel_space(times, origin, sign, gXX, gXt, gtt, events):
sol = solve_ivp(_photon_dXdt, (times[0], times[-1]), (origin,),
method = 'RK45', t_eval = times, args = (sign, gXX, gtt, gXt),
dense_output = True, events = events, rtol = 1e-6, atol = 1e-8)
t, x = sol.t, sol.y[0]
if len(t) < len(times):
x_ext = np.empty_like(times)
x_ext[:x.shape[0]] = x
x_ext[x.shape[0]:] = x[-1] + sign * (times[x.shape[0]:] - t[-1])
x = x_ext
return x
class Curves(Enum):
SPATIAL_FORWARD = auto()
"photons are traced forward in time from (t=times[0], X=spatial_coords[i])"
SPATIAL_BACK = auto()
"photons are traced backward in time from (t=times[-1], X=spatial_coords[i])"
TEMPORAL = auto()
"""
photons are traced both forward and backward in time from
(t=times[i], X=0); the forward and backward segment are combined
to form the resulting curve
"""
def calc_null_curves(times, spatial_coords, gXX, gXt, gtt,
kind = Curves.SPATIAL_FORWARD):
"""
Compute null curves along a given axis.
Shoot a null ray from each point in spatial_coords, in the positive and
negative spatial direction and compute its trajectory.
:param array_like times: 1D array of coordinate times at which the spacetime
curvature is provided
:param array_like spatial_coords: 1D array of spatial coordinates
:param array_like gXX: 2D array containing the values of the XX component
of the spacetime metric, where X is the spatial
coordinate along which the rays are traced.
gXX[i, j] is the value at spacetime point
(t=times[i], X=spatial_coords[j]).
:param array_like gXt: same as gXX, but for the Xt component of the metric
:param array_like gtt: same as gXX, but for the tt component of the metric
:param Curves kind: specifies where to integrate the curves from
:return: Tuple of (ray_times, rays_pos, rays_neg). rays_*[i, j] contains the
X-coordinate of the ray shot from (t=ray_times[0],
X=spatial_coords[i]) at time t=ray_times[j].
"""
gXX_interp = RectBivariateSpline(times, spatial_coords, gXX)
gXt_interp = RectBivariateSpline(times, spatial_coords, gXt)
gtt_interp = RectBivariateSpline(times, spatial_coords, gtt)
if kind == Curves.TEMPORAL:
kernel = _calc_null_kernel_time
origins = times
else:
kernel = _calc_null_kernel_space
origins = spatial_coords
ray_times = times[::-1] if kind == Curves.SPATIAL_BACK else times
events = _events_bnd_null_curves(spatial_coords)
rays_pos = np.empty((origins.shape[0], times.shape[0]))
rays_neg = np.empty_like(rays_pos)
for tgt, sign in ((rays_pos, 1.0), (rays_neg, -1.0)):
for i, origin in enumerate(origins):
tgt[i] = kernel(ray_times, origin, sign,
gXX_interp, gXt_interp, gtt_interp,
events)
return (ray_times, rays_pos, rays_neg)
def calc_null_coordinates(times, spatial_coords, u_rays, v_rays,
gXX, gXt, gtt, kind = Curves.SPATIAL_FORWARD):
"""
Compute double-null coordinates (u, v) as functions of
position and time.
:param array_like times: 1D array of coordinate times at which the spacetime
curvature is provided
:param array_like spatial_coords: 1D array of spatial coordinates
:param array_like u_rays: 1D array assigning the values of u on the initial
time slice. u_rays[i] is the value of u at
X=spatial_coords[i].
:param array_like v_rays: same as u_rays, but for v.
:param array_like gXX: 2D array containing the values of the XX component
of the spacetime metric, where X is the spatial
coordinate along which the rays are traced.
gXX[i, j] is the value at spacetime point
(t=times[i], X=spatial_coords[j]).
:param array_like gXt: same as gXX, but for the Xt component of the metric
:param array_like gtt: same as gXX, but for the tt component of the metric
:return: tuple containing two 2D arrays with, respectively, values of u and
v as functions of t and X. u/v[i, j] is the value of u/v at
t=times[i], X=spatial_coords[j].
"""
_, X_of_ut, X_of_vt = calc_null_curves(times, spatial_coords, gXX, gXt, gtt, kind = kind)
u_of_tx = np.empty((times.shape[0], spatial_coords.shape[0]))
v_of_tx = np.empty_like(u_of_tx)
for i, t in enumerate(times):
Xu = X_of_ut[:, i]
Xv = X_of_vt[:, i]
u_of_tx[i] = interp1d(Xu, u_rays, fill_value = 'extrapolate')(spatial_coords)
v_of_tx[i] = interp1d(Xv, v_rays, fill_value = 'extrapolate')(spatial_coords)
return (u_of_tx, v_of_tx)
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