Visualizing Event Horizon and Ergosphere (Singularities) of Kerr Metric or Black Hole

Importing required modules

In [1]:
import astropy.units as u

import numpy as np

import astropy.units as u

import matplotlib.pyplot as plt



from einsteinpy.coordinates import BoyerLindquistDifferential

from einsteinpy.metric import Kerr

Defining black holes and obtaining singularities

In [2]:
# Metric or Black Hole parameters - Mass, M and Spin Parameter, a

M = 4e30 * u.kg

a1 = 0.4 * u.one

a2 = 0.9 * u.one # Extremal Kerr Black Hole



# Coordinate object to initialize metric with

# Note that, for this example

# the coordinate values below are irrelevant

bl = BoyerLindquistDifferential(

    t=0. * u.s,

    r=1e3 * u.m,

    theta=np.pi / 2 * u.rad,

    phi=np.pi * u.rad,

    v_r=0. * u.m / u.s,

    v_th=0. * u.rad / u.s,

    v_p=0. * u.rad / u.s,

)



# Defining two Kerr Black Holes, one with a higher spin parameter

kerr1 = Kerr(coords=bl, M=M, a=a1)

kerr2 = Kerr(coords=bl, M=M, a=a2)



# Getting the list of singularities

sing_dict1 = kerr1.singularities()

sing_dict2 = kerr2.singularities()



# Let's check the contents of the dicts

# 'ergosphere' entries should be functions

print(sing_dict1, sing_dict2, sep="\n\n")

{'inner_ergosphere': <function BaseMetric.singularities.<locals>._in_ergo at 0x7f733c8b0f70>, 'inner_horizon': 247.98878315867296, 'outer_horizon': 5692.939432136259, 'outer_ergosphere': <function BaseMetric.singularities.<locals>._out_ergo at 0x7f730f324af0>}

{'inner_ergosphere': <function BaseMetric.singularities.<locals>._in_ergo at 0x7f733c062310>, 'inner_horizon': 1675.668821582463, 'outer_horizon': 4265.259393712469, 'outer_ergosphere': <function BaseMetric.singularities.<locals>._out_ergo at 0x7f733c0620d0>}

Preparing singularities for plotting

In [3]:
# Sampling Polar Angle for plotting in Polar Coordinates

theta = np.linspace(0, 2 * np.pi, 100)



# Ergospheres

# These are functions

Ei1, Eo1 = sing_dict1["inner_ergosphere"], sing_dict1["outer_ergosphere"]

Ei2, Eo2 = sing_dict2["inner_ergosphere"], sing_dict2["outer_ergosphere"]



# Creating lists of points on Ergospheres for different polar angles, for both black holes

Ei1_list, Eo1_list = Ei1(theta), Eo1(theta)

Ei2_list, Eo2_list = Ei2(theta), Eo2(theta)



# For Black Hole 1 (a = 0.4)

Xei1 = Ei1_list * np.sin(theta)

Yei1 = Ei1_list * np.cos(theta)



Xeo1 = Eo1_list * np.sin(theta)

Yeo1 = Eo1_list * np.cos(theta)



# For Black Hole 2 (a = 0.9)

Xei2 = Ei2_list * np.sin(theta)

Yei2 = Ei2_list * np.cos(theta)



Xeo2 = Eo2_list * np.sin(theta)

Yeo2 = Eo2_list * np.cos(theta)



# Event Horizons

Hi1, Ho1 = sing_dict1["inner_horizon"], sing_dict1["outer_horizon"]

Hi2, Ho2 = sing_dict2["inner_horizon"], sing_dict2["outer_horizon"]



# For Black Hole 1 (a = 0.4)

Xhi1 = Hi1 * np.sin(theta)

Yhi1 = Hi1 * np.cos(theta)



Xho1 = Ho1 * np.sin(theta)

Yho1 = Ho1 * np.cos(theta)



# For Black Hole 2 (a = 0.9)

Xhi2 = Hi2 * np.sin(theta)

Yhi2 = Hi2 * np.cos(theta)



Xho2 = Ho2 * np.sin(theta)

Yho2 = Ho2 * np.cos(theta)

Plotting both black holes

In [4]:
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15,7.5))



ax1.fill(Xei1, Yei1, 'b', Xeo1, Yeo1, 'r', Xhi1, Yhi1, 'b', Xho1, Yho1, 'r', alpha=0.3)

ax1.set_title(f"$M = {M}, a = {a1}$", fontsize=18)

ax1.set_xlabel("X", fontsize=18)

ax1.set_ylabel("Y", fontsize=18)

ax1.set_xlim([-6100, 6100])

ax1.set_ylim([-6100, 6100])



ax2.fill(Xei2, Yei2, 'b', Xeo2, Yeo2, 'r', Xhi2, Yhi2, 'b', Xho2, Yho2, 'r', alpha=0.3)

ax2.set_title(f"$M = {M}, a = {a2}$", fontsize=18)

ax2.set_xlabel("X", fontsize=18)

ax2.set_ylabel("Y", fontsize=18)

ax2.set_xlim([-6100, 6100])

ax2.set_ylim([-6100, 6100])
Out[4]:
(-6100.0, 6100.0)

  • The surfaces are clearly visible in the plots. Going radially inward, we have Outer Ergosphere, Outer Event Horizon, Inner Event Horizon and Inner Ergosphere. We can also observe the following:

    • As $a \to 1$ (its maximum attainable value), the individual singularities become prominent.

    • As $a \to 0$, some singularities appear to fade away, leaving us with a single surface, that is the Event Horizon of a Schwarzschild black hole.