import warnings
import astropy.units as u
import numpy as np
from einsteinpy import constant
from einsteinpy.coordinates import SphericalConversion
from einsteinpy.integrators import RK45
from einsteinpy.utils import schwarzschild_radius, schwarzschild_utils
_G = constant.G.value
_c = constant.c.value
[docs]class Schwarzschild:
"""
Class for defining a Schwarzschild Geometry methods
"""
@u.quantity_input(time=u.s, M=u.kg)
def __init__(self, sph_coords, M, time):
self.name = "Schwarzschild"
self.M = M
self.input_coords = sph_coords
self.time = time
pos_vec, vel_vec = (
self.input_coords.si_values()[:3],
self.input_coords.si_values()[3:],
)
time_vel = schwarzschild_utils.time_velocity(pos_vec, vel_vec, M)
self.initial_vec = np.hstack(
(self.time.value, pos_vec, time_vel.value, vel_vec)
)
self.scr = schwarzschild_radius(M)
[docs] @classmethod
@u.quantity_input(time=u.s, M=u.kg, a=u.m)
def from_coords(cls, coords, M, q=None, Q=None, time=0 * u.s, a=0 * u.m):
"""
Constructor
Parameters
----------
coords : ~einsteinpy.coordinates.velocity.CartesianDifferential
Object having both initial positions and velocities of particle in Cartesian Coordinates
M : ~astropy.units.quantity.Quantity
Mass of the body
time : ~astropy.units.quantity.Quantity
Time of start, defaults to 0 seconds.
"""
if coords.system == "Spherical":
return cls(coords, M, time)
sph_coords = coords.spherical_differential()
return cls(sph_coords, M, time)
def f_vec(self, ld, vec):
vals = np.zeros(shape=vec.shape, dtype=vec.dtype)
chl = schwarzschild_utils.christoffels(vec[1], vec[2], self.M.value)
vals[:4] = vec[4:8]
vals[4] = -2 * chl[0, 0, 1] * vec[4] * vec[5]
vals[5] = -1 * (
chl[1, 0, 0] * (vec[4] ** 2)
+ chl[1, 1, 1] * (vec[5] ** 2)
+ chl[1, 2, 2] * (vec[6] ** 2)
+ chl[1, 3, 3] * (vec[7] ** 2)
)
vals[6] = -2 * chl[2, 2, 1] * vec[6] * vec[5] - 1 * chl[2, 3, 3] * (vec[7] ** 2)
vals[7] = -2 * (chl[3, 3, 1] * vec[7] * vec[5] + chl[3, 3, 2] * vec[7] * vec[6])
return vals
[docs] def calculate_trajectory(
self,
start_lambda=0.0,
end_lambda=10.0,
stop_on_singularity=True,
OdeMethodKwargs={"stepsize": 1e-3},
return_cartesian=False,
):
"""
Calculate trajectory in manifold according to geodesic equation
Parameters
----------
start_lambda : float
Starting lambda(proper time), defaults to 0, (lambda ~= t)
end_lamdba : float
Lambda(proper time) where iteartions will stop, defaults to 100000
stop_on_singularity : bool
Whether to stop further computation on reaching singularity, defaults to True
OdeMethodKwargs : dict
Kwargs to be supplied to the ODESolver, defaults to {'stepsize': 1e-3}
Dictionary with key 'stepsize' along with an float value is expected.
return_cartesian : bool
True if coordinates and velocities are required in cartesian coordinates(SI units), defaults to False
Returns
-------
~numpy.ndarray
N-element array containing proper time.
~numpy.ndarray
(n,8) shape array of [t, x1, x2, x3, velocity_of_time, v1, v2, v3] for each proper time(lambda).
"""
ODE = RK45(
fun=self.f_vec,
t0=start_lambda,
y0=self.initial_vec,
t_bound=end_lambda,
**OdeMethodKwargs
)
vecs = list()
lambdas = list()
crossed_event_horizon = False
_scr = self.scr.value * 1.001
while ODE.t < end_lambda:
vecs.append(ODE.y)
lambdas.append(ODE.t)
ODE.step()
if (not crossed_event_horizon) and (ODE.y[1] <= _scr):
warnings.warn("particle reached Schwarzchild Radius. ", RuntimeWarning)
if stop_on_singularity:
break
else:
crossed_event_horizon = True
vecs, lambdas = np.array(vecs), np.array(lambdas)
if not return_cartesian:
return lambdas, vecs
else:
cart_vecs = list()
for v in vecs:
si_vals = SphericalConversion(
v[1], v[2], v[3], v[5], v[6], v[7]
).convert_cartesian()
cart_vecs.append(np.hstack((v[0], si_vals[:3], v[4], si_vals[3:])))
return lambdas, np.array(cart_vecs)
[docs] def calculate_trajectory_iterator(
self,
start_lambda=0.0,
stop_on_singularity=True,
OdeMethodKwargs={"stepsize": 1e-3},
return_cartesian=False,
):
"""
Calculate trajectory in manifold according to geodesic equation
Yields an iterator
Parameters
----------
start_lambda : float
Starting lambda, defaults to 0.0, (lambda ~= t)
stop_on_singularity : bool
Whether to stop further computation on reaching singularity, defaults to True
OdeMethodKwargs : dict
Kwargs to be supplied to the ODESolver, defaults to {'stepsize': 1e-3}
Dictionary with key 'stepsize' along with an float value is expected.
return_cartesian : bool
True if coordinates and velocities are required in cartesian coordinates(SI units), defaults to Falsed
Yields
------
float
proper time
~numpy.ndarray
array of [t, x1, x2, x3, velocity_of_time, v1, v2, v3] for each proper time(lambda).
"""
ODE = RK45(
fun=self.f_vec,
t0=start_lambda,
y0=self.initial_vec,
t_bound=1e300,
**OdeMethodKwargs
)
crossed_event_horizon = False
_scr = self.scr.value * 1.001
while True:
if not return_cartesian:
yield ODE.t, ODE.y
else:
v = ODE.y
si_vals = SphericalConversion(
v[1], v[2], v[3], v[5], v[6], v[7]
).convert_cartesian()
yield ODE.t, np.hstack((v[0], si_vals[:3], v[4], si_vals[3:]))
ODE.step()
if (not crossed_event_horizon) and (ODE.y[1] <= _scr):
warnings.warn("particle reached Schwarzchild Radius. ", RuntimeWarning)
if stop_on_singularity:
break
else:
crossed_event_horizon = True
