def Ernst(B=symbols("B"), M=symbols("M")):
"""
Black holes in a magnetic universe.
J. Math. Phys., 17:54–56, 1976.
Frederick J. Ernst.
Parameters
----------
M : ~sympy.core.basic.Basic or int or float
Mass of the black hole. Defaults to ``M``.
B : ~sympy.core.basic.Basic or int or float
The magnetic field strength
Defaults to ``B``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# Helper functions
lambd = 1 + ((B * r * sin(th))**2)
w = 1 - ((2 * M) / r)
# define the metric
metric = diag(
-1 * (lambd**2) * w,
(lambd**2) / w,
((r * lambd)**2),
(((r * sin(th)) / lambd)**2),
).tolist()
return MetricTensor(metric, coords, "ll")
def JanisNewmanWinicour(
c=constants.c, G=constants.G, gam=symbols("gam"), M=symbols("M")
):
"""
Reality of the Schwarzschild singularity.
Phys. Rev. Lett., 20:878–880, 1968.
A. I. Janis, E. T. Newman, and J. Winicour.
Parameters
----------
M : ~sympy.core.basic.Basic or int or float
Mass parameter, this is used for defining the schwarzschild metric. Defaults to ``M``.
gam : ~sympy.core.basic.Basic or int or float
Parameter for scaling Schwarzschild radius, for gamma=1 this will return the Schwarzschild metric
Defaults to ``gam``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# Helper functions
r_s = (2 * G * M) / (c ** 2)
alpha = 1 - (r_s / (gam * r))
# define the metric
metric = diag(
-1 * (alpha ** gam),
(alpha ** -gam) / (c ** 2),
(r ** 2) * (alpha ** (-gam + 1)),
(r ** 2) * (alpha ** (-gam + 1)) * (sin(th) ** 2),
).tolist()
return MetricTensor(metric, coords, "ll", name="JanisNewmanWinicourMetric")
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")
def SchwarzschildMetric(symbolstr="t r theta phi"):
"""
Returns Metric Tensor of symbols of Schwarzschild Metric.
Parameters
----------
symbolstr : string
symbols to be used to define schwarzschild space, defaults to 't r theta phi'
Returns
-------
~einsteinpy.symbolic.metric.MetricTensor
Metric Tensor for Schwarzschild space-time
"""
raise_warning(
PendingDeprecationWarning,
"SchwarzschildMetric class would be deprecated with v0.3.0 !",
)
list2d = [[0 for i in range(4)] for i in range(4)]
syms = sympy.symbols(symbolstr)
c, a = sympy.symbols("c a")
list2d[0][0] = 1 - (a / syms[1])
list2d[1][1] = -1 / ((1 - (a / syms[1])) * (c**2))
list2d[2][2] = -1 * (syms[1]**2) / (c**2)
list2d[3][3] = -1 * (syms[1]**2) * (sympy.sin(syms[2])**2) / (c**2)
return MetricTensor(list2d, syms)
def AntiDeSitterStatic():
"""
Static form of Anti-de Sitter space
Hawking and Ellis (5.9) p131
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
metric = diag(-cosh(r) ** 2, 1, sinh(r) ** 2, sinh(r) ** 2 * sin(th) ** 2).tolist()
return MetricTensor(metric, coords, "ll", name="AntiDeSitterStaticMetric")
def DeSitter():
"""
de Sitter space
Hawking and Ellis p125
"""
coords = symbols("t x y z")
t = coords[1]
al = symbols("alpha")
expr = exp(2 * t / al)
metric = diag(-1, expr, expr, expr).tolist()
return MetricTensor(metric, coords, "ll", name="DeSitterMetric")
def Godel():
"""
Godel metric
Rev. Mod. Phys., v21, p447, (1949)
Stephani (10.25) 122
"""
coords = symbols("t x y z")
om = symbols("omega")
t, x, y, z = coords
# define the metric
metric = diag(-1, 1, -Rational(1, 2) * exp(2 * sqrt(2) * om * x), 1)
metric[0, 2] = metric[2, 0] = -exp(sqrt(2) * om * x)
metric = metric.tolist()
return MetricTensor(metric, coords, "ll")
def Minkowski(c=constants.c):
"""
Minkowski(flat) space-time. Space-time without any curvature or matter.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to 'c'.
"""
coords = symbols("t x y z")
metric = diag(-1, 1 / (c**2), 1 / (c**2), 1 / (c**2)).tolist()
return MetricTensor(metric, coords, "ll")
def Davidson():
"""
Davidson's cylindrically symmetric radiation perfect fluid universe
Davidson, J. Math. Phys., v32, p1560, (1991)
"""
coords = symbols("t r z phi")
t, r, z, phi = coords
expr = (1 + r**2)**Rational(2, 5)
metric = diag(
-(expr**3),
t**Rational(4, 3) * expr,
t**Rational(-2, 3) / expr,
t**Rational(4, 3) * r**2 / expr,
).tolist()
return MetricTensor(metric, coords, "ll", name="DavidsonMetric")
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")
def CMetric():
"""
The C-metric
Stephani (Table 16.2) p188
"""
coords = symbols("t x y phi")
x, y = coords[1], coords[2]
f, h = Function("f")(x), Function("h")(y)
metric = diag(
-h / (x + y)**2,
1 / ((x + y)**2 * f),
1 / ((x + y)**2 * h),
f / (x + y)**2,
).tolist()
return MetricTensor(metric, coords, "ll")
def AntiDeSitter():
"""
Anti-de Sitter space
Hawking and Ellis (5.9) p131
"""
coords = symbols("t chi theta phi")
t, ch, th, ph = coords
metric = diag(
-1,
cos(t) ** 2,
cos(t) ** 2 * sinh(ch) ** 2,
cos(t) ** 2 * sinh(ch) ** 2 * sin(th) ** 2,
).tolist()
return MetricTensor(metric, coords, "ll", name="AntiDeSitterMetric")
def MinkowskiPolar(c=constants.c):
"""
Minkowski(flat) space-time in Polar coordinates. Space-time without any curvature or matter.
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to 'c'.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
c2 = c**2
metric = diag(-1, 1 / c2, (r**2) / c2, (r**2 * sin(th)**2) / c2).tolist()
return MetricTensor(metric, coords, "ll", name="MinkowskiMetricPolar")
def AlcubierreWarp(
x_s=symbols("x_s"), sigma=symbols("sigma"), R=symbols("R"), v=symbols("v")
):
"""
Alcubierre Warp Drive Metric (G = c = 1) [1]_
.. [1] Classical and Quantum Gravity,
"The warp drive: hyper-fast travel within general relativity",
Miguel Alcubierre, 1994, **11(5)**, pp. 73-77
arXiv: `<https://arxiv.org/abs/gr-qc/0009013>`_)
Journal: `<https://doi.org/10.1088%2F0264-9381%2F11%2F5%2F001>`_
Parameters
----------
x_s : ~sympy.core.basic.Basic or int or float
Coordinate (a function of time, `t`)
sigma : ~sympy.core.basic.Basic or int or float
Arbitrary Parameter (See paper [1]_)
R : ~sympy.core.basic.Basic or int or float
Arbitrary Parameter (See paper [1]_)
v : ~sympy.core.basic.Basic or int or float
Coordinate Velocity
Returns
-------
~einsteinpy.symbolic.metric.MetricTensor
Alcubierre Warp Drive Metric Tensor
Alcubierre Warp Drive Metric (G = c = 1) [1]_
"""
coords = symbols("t x y z")
t, x, y, z = coords
r_s = sqrt((x - x_s) ** 2 + y + z)
f = (tanh(sigma * (r_s + R)) - tanh(sigma * (r_s - R))) / (2 * tanh(sigma * R))
v2 = v ** 2
f2 = f ** 2
metric = diag(v2 * f2 - 1, 1, 1, 1).tolist()
metric[0][1] = metric[1][0] = -v * f
return MetricTensor(metric, coords, "ll", name="AlcubierreWarpMetric")
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), -1 * (r**2) / c2,
-1 * ((r * sin(theta))**2) / c2).tolist()
return MetricTensor(metric, coords, "ll")
def BarriolaVilekin(c=constants.c, k=symbols("k")):
"""
Barriola-Vilekin monopol metric
Phys. Rev. Lett. 63, 341
Manuel Barriola and Alexander Vilenkin
Published 24 July 1989
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to 'c'.
k : ~sympy.core.basic.Basic or int or float
The scaling factor responsible for the deficit/surplus angle
Defaults to ``k``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# define the metric
metric = diag(-1, 1 / (c**2), ((k * r)**2) / (c**2),
((k * r * sin(th))**2) / (c**2)).tolist()
return MetricTensor(metric, coords, "ll", name="BarriolaVilekinMetric")
def BesselGravitationalWave(C=symbols("C")):
"""
Exact gravitational wave solution without diffraction.
Class. Quantum Grav., 16:L75–78, 1999.
D. Kramer.
An exact solution describing an axisymmetric gravitational wave propagating in the
z-direction in closed form. This solution to Einstein’s vacuum field equations has the
remarkable property that the curvature invariants decrease monotonically with increasing radial
distance from the axis and vanish at infinity. The solution is regular at the symmetry axis.
Parameters
----------
C : ~sympy.core.basic.Basic or int or float
Constant for Bessel metric, the choice of the constant is not really relavent for details see the paper. Defaults to 'C'.
"""
coords = symbols("t rho phi z")
t, rho, ph, z = coords
# Useful helper functions, these wrap the Bessel functions that are used. C is some constant here
U = C * besselj(rho, 0) * cos(t)
K = (
(1 / 2)
* (C ** 2)
* rho
* (
(rho * ((besselj(rho, 0) ** 2) + (besselj(rho, 1) ** 2)))
- (2 * besselj(rho, 0) * besselj(rho, 1) * (cos(t) ** 2))
)
)
# define the metric
metric = diag(
-1 * exp(-2 * U) * exp(2 * K),
exp(-2 * U) * exp(2 * K),
exp(-2 * U) * (rho ** 2),
exp(2 * U),
).tolist()
return MetricTensor(metric, coords, "ll")
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")
def BertottiKasner(c=constants.c, k=symbols("k"), lambd=symbols("l")):
"""
Birkhoff’s theorem with Λ-term and Bertotti-Kasner space
Phys. Lett. A, 245:363–365, 1998
W. Rindler
Parameters
----------
c : ~sympy.core.basic.Basic or int or float
Any value to assign to speed of light. Defaults to 'c'.
lambd : ~sympy.core.basic.Basic or int or float
The cosmological constant, note it must be postive.
Defaults to ``l``.
"""
coords = symbols("t r theta phi")
t, r, th, ph = coords
# define the metric
metric = diag(
-1,
exp(2 * sqrt(lambd) * c * t) / (c**2),
1 / (lambd * (c**2)),
(sin(th)**2) / (lambd * (c**2)),
).tolist()
return MetricTensor(metric, coords, "ll", name="BertottiKasnerMetric")
