import numpy as np
from astropy import units as u
from einsteinpy import constant
from einsteinpy.metric import BaseMetric
from einsteinpy.utils import CoordinateError
_c = constant.c.value
[docs]class Schwarzschild(BaseMetric):
"""
Class for defining Schwarzschild Geometry
"""
@u.quantity_input(M=u.kg)
def __init__(self, coords, M):
"""
Constructor
Parameters
----------
coords : ~einsteinpy.coordinates.differential.*
Coordinate system, in which Metric is to be represented
M : ~astropy.units.quantity.Quantity
Mass of gravitating body, e.g. Black Hole
"""
super().__init__(
coords=coords,
M=M,
name="Schwarzschild Metric",
metric_cov=self.metric_covariant,
christoffels=self._christoffels,
f_vec=self._f_vec,
)
[docs] def metric_covariant(self, x_vec):
"""
Returns Covariant Schwarzschild Metric Tensor \
in chosen Coordinates
Parameters
----------
x_vec : array_like
Position 4-Vector
Returns
-------
~numpy.ndarray
Covariant Schwarzschild Metric Tensor in chosen Coordinates
Numpy array of shape (4,4)
"""
if self.coords.system == "Spherical":
return self._g_cov_s(x_vec)
raise CoordinateError(
"Schwarzschild Metric is available only in Spherical Polar Coordinates."
)
def _g_cov_s(self, x_vec):
"""
Returns Covariant Schwarzschild Metric Tensor \
in Schwarschild Coordinates
Parameters
----------
x_vec : array_like
Position 4-Vector
Returns
-------
~numpy.ndarray
Covariant Schwarzschild Metric Tensor
Numpy array of shape (4,4)
"""
r, th = x_vec[1], x_vec[2]
r_s = self.sch_rad
g_cov = np.zeros(shape=(4, 4), dtype=float)
tmp = 1.0 - (r_s / r)
g_cov[0, 0] = tmp * _c ** 2
g_cov[1, 1] = -1.0 / tmp
g_cov[2, 2] = -(r ** 2)
g_cov[3, 3] = -((r * np.sin(th)) ** 2)
return g_cov
def _christoffels(self, x_vec):
"""
Returns Christoffel Symbols for Schwarzschild Metric in chosen Coordinates
Parameters
----------
x_vec : array_like
Position 4-Vector
Returns
-------
~numpy.ndarray
Christoffel Symbols for Schwarzschild Metric \
in chosen Coordinates
Numpy array of shape (4,4,4)
Raises
------
CoordinateError
Raised, if the Christoffel Symbols are not \
available in the supplied Coordinate System
"""
if self.coords.system == "Spherical":
return self._ch_sym_s(x_vec)
raise CoordinateError(
"Christoffel Symbols for Schwarzschild Metric are available only in Spherical Polar Coordinates."
)
def _ch_sym_s(self, x_vec):
"""
Returns the Christoffel Symbols for Schwarzschild Metric
in Schwarzschild Coordinates
Parameters
----------
x_vec : array_like
Position 4-Vector
Returns
-------
~numpy.ndarray
Christoffel Symbols for Schwarzschild Metric \
in Schwarzschild Coordinates
Numpy array of shape (4,4,4)
"""
r, th = x_vec[1], x_vec[2]
r_s = self.sch_rad
chl = np.zeros(shape=(4, 4, 4), dtype=float)
chl[1, 0, 0] = 0.5 * r_s * (r - r_s) * (_c ** 2) / (r ** 3)
chl[1, 1, 1] = 0.5 * r_s / (r_s * r - r ** 2)
chl[1, 2, 2] = r_s - r
chl[1, 3, 3] = (r_s - r) * (np.sin(th) ** 2)
chl[0, 0, 1] = chl[0, 1, 0] = -chl[1, 1, 1]
chl[2, 2, 1] = chl[2, 1, 2] = chl[3, 3, 1] = chl[3, 1, 3] = 1 / r
chl[2, 3, 3] = -np.cos(th) * np.sin(th)
chl[3, 3, 2] = chl[3, 2, 3] = 1 / np.tan(th)
return chl
def _f_vec(self, lambda_, vec):
"""
Returns f_vec for Schwarzschild Metric in chosen coordinates
To be used for solving Geodesics ODE
Parameters
----------
lambda_ : float
Parameterizes current integration step
Used by ODE Solver
vec : array_like
Length-8 Vector, containing 4-Position & 4-Velocity
Returns
-------
~numpy.ndarray
f_vec for Schwarzschild Metric in chosen coordinates
Numpy array of shape (8)
Raises
------
CoordinateError
Raised, if ``f_vec`` is not available in \
the supplied Coordinate System
"""
if self.coords.system == "Spherical":
return self._f_vec_s(lambda_, vec)
raise CoordinateError(
"'f_vec' for Schwarzschild Metric is available only in Spherical Polar Coordinates."
)
def _f_vec_s(self, lambda_, vec):
"""
Returns f_vec for Schwarzschild Metric
To be used for solving Geodesics ODE
Parameters
----------
lambda_ : float
Parameterizes current integration step
Used by ODE Solver
vec : array_like
Length-8 Vector, containing 4-Position & 4-Velocity
Returns
-------
~numpy.ndarray
f_vec for Schwarzschild Metric
Numpy array of shape (8)
"""
chl = self.christoffels(vec[:4])
vals = np.zeros(shape=vec.shape, dtype=vec.dtype)
vals[:4] = vec[4:]
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
