from sympy import cos, diag, pi, sin, symbols
from einsteinpy.symbolic import constants
from einsteinpy.symbolic.metric import MetricTensor
[docs]def Schwarzschild(c=constants.c, sch=symbols("r_s")):
"""
Schwarzschild exterior metric in curvature coordinates
Schwarzschild, Sitz. Preuss. Akad. Wiss., p189, (1916)
Stephani (13.19) p157
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to ``c``.
sch : ~sympy.core.basic.Basic or int or float
Any value to assign to Schwarzschild Radius of the central object.
Defaults to ``r_s``.
"""
coords = symbols("t r theta phi")
t, r, theta, phi = coords
val1, c2 = 1 - sch / r, c ** 2
metric = diag(
val1, -1 / (val1 * c2), -(r ** 2) / c2, -((r * sin(theta)) ** 2) / c2
).tolist()
return MetricTensor(metric, coords, "ll", name="SchwarzschildMetric")
[docs]def Kerr(c=constants.c, sch=symbols("r_s"), a=symbols("a")):
"""
Kerr Metric in Boyer Lindquist coordinates.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to ``c``.
sch : ~sympy.core.basic.Basic or int or float
Any value to assign to Schwarzschild Radius of the central object.
Defaults to ``r_s``.
a : ~sympy.core.basic.Basic or int or float
Spin factor of the heavy body. Usually, given by ``J/(Mc)``,
where ``J`` is the angular momentum.
Defaults to ``a``.
"""
coords = symbols("t r theta phi")
t, r, theta, phi = coords
Sigma = r ** 2 + (a ** 2 * cos(theta) ** 2)
Delta = r ** 2 - sch * r + a ** 2
c2 = c ** 2
metric = diag(
1 - (sch * r / Sigma),
-Sigma / (Delta * c2),
-Sigma / c2,
-(
(r ** 2 + a ** 2 + (sch * r * (a ** 2) * (sin(theta) ** 2) / Sigma))
* (sin(theta) ** 2)
)
/ c2,
).tolist()
metric[0][3] = metric[3][0] = sch * r * a * (sin(theta) ** 2) / (Sigma * c)
return MetricTensor(metric, coords, "ll", name="KerrMetric")
[docs]def KerrNewman(
c=constants.c,
G=constants.G,
eps_0=constants.eps_0,
sch=symbols("r_s"),
a=symbols("a"),
Q=symbols("Q"),
):
"""
Kerr-Newman Metric in Boyer Lindquist coordinates.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to ``c``.
G : ~sympy.core.basic.Basic or int or float
Any value to assign to the Newton's (or gravitational) constant. Defaults to ``G``.
eps_0 : ~sympy.core.basic.Basic or int or float
Any value to assign to the electric constant or permittivity of free space. Defaults to ``eps_0``.
sch : ~sympy.core.basic.Basic or int or float
Any value to assign to Schwarzschild Radius of the central object.
Defaults to ``r_s``.
a : ~sympy.core.basic.Basic or int or float
Spin factor of the heavy body. Usually, given by ``J/(Mc)``,
where ``J`` is the angular momentum.
Defaults to ``a``.
Q: ~sympy.core.basic.Basic or int or float
Any value to assign to eletric charge of the central object.
Defaults to ``Q``.
"""
coords = symbols("t r theta phi")
t, r, theta, phi = coords
Sigma = r ** 2 + (a ** 2 * cos(theta) ** 2)
rQsq = ((Q ** 2) * G) / (4 * pi * eps_0 * (c ** 4))
Delta = r ** 2 - sch * r + a ** 2 + rQsq
c2 = c ** 2
metric = diag(
1 + ((rQsq - sch * r) / Sigma),
-Sigma / (Delta * c2),
-Sigma / c2,
(Delta * a ** 2 * sin(theta) ** 2 - (r ** 2 + a ** 2) ** 2)
* sin(theta) ** 2
/ (Sigma * c2),
).tolist()
metric[0][3] = metric[3][0] = (sch * r - rQsq) * a * (sin(theta) ** 2) / (Sigma * c)
return MetricTensor(metric, coords, "ll", name="KerrNewmanMetric")
[docs]def ReissnerNordstorm(
c=constants.c,
G=constants.G,
eps_0=constants.eps_0,
sch=symbols("r_s"),
Q=symbols("Q"),
):
"""
The Reissner–Nordström metric in spherical coordinates
A static solution to the Einstein–Maxwell field equations,
which corresponds to the gravitational field of a charged,
non-rotating, spherically symmetric body of mass M.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to ``c``.
G : ~sympy.core.basic.Basic or int or float
Any value to assign to the Newton's (or gravitational) constant. Defaults to ``G``.
eps_0 : ~sympy.core.basic.Basic or int or float
Any value to assign to the electric constant or permittivity of free space. Defaults to ``eps_0``.
sch : ~sympy.core.basic.Basic or int or float
Any value to assign to Schwarzschild Radius of the central object.
Defaults to ``r_s``.
Q: ~sympy.core.basic.Basic or int or float
Any value to assign to eletric charge of the central object.
Defaults to ``Q``.
"""
coords = symbols("t r theta phi")
t, r, theta, phi = coords
rQsq = ((Q ** 2) * G) / (4 * pi * eps_0 * (c ** 4))
Arn = 1 - sch / r + rQsq / r ** 2
c2 = c ** 2
metric = diag(
Arn, -1 / (Arn * c2), -(r ** 2) / c2, -(r ** 2) * sin(theta) ** 2 / c2
).tolist()
return MetricTensor(metric, coords, "ll", name="ReissnerNordstormMetric")
