Visualizing Frame Dragging in Kerr Spacetime¶

Importing required modules¶

[1]:

import numpy as np

from einsteinpy.geodesic import Nulllike
from einsteinpy.plotting import StaticGeodesicPlotter


Setting up the system¶

• Initial position & momentum of the test partcle

• Spin of the Kerr Black Hole

• Other solver parameters

Note that, we are working in M-Units ($$G = c = M = 1$$). Also, setting momentum’s $$\phi$$-component to negative, implies an initial retrograde trajectory.

[2]:

position = [2.5, np.pi / 2, 0.]
momentum = [0., 0., -2.]
a = 0.99
steps = 7440  # As close as we can get before the integration becomes highly unstable
delta = 0.0005
omega = 0.01
suppress_warnings = True


Here, omega, the coupling between the hamiltonian flows, needs to be decreased in order to decrease numerical errors and increase integration stability. Reference: https://arxiv.org/abs/2010.02237.

Also, suppress_warnings has been set to True, as the error would grow exponentially, very close to the black hole.

Calculating the geodesic¶

[5]:

geod = Nulllike(
metric="Kerr",
metric_params=(a,),
position=position,
momentum=momentum,
steps=steps,
delta=delta,
return_cartesian=True,
omega=omega,
suppress_warnings=suppress_warnings
)


Plotting the geodesic in 2D¶

[6]:

sgpl = StaticGeodesicPlotter(bh_colors=("red", "blue"))
sgpl.plot2D(geod, coordinates=(1, 2), figsize=(10, 10), color="indigo") # Plot X vs Y
sgpl.show()


As can be seen in the plot above, the photon’s trajectory is reversed, due to frame-dragging effects, so that, it moves in the direction of the black hole’s spin, before eventually falling into the black hole.

Also, the last few steps seem to have a larger delta, but that is simply because of huge numerical errors, as the particle has crossed the Event Horizon.