Visualizing Precession in Schwarzschild Spacetime

Importing required modules

import numpy as np

from einsteinpy.geodesic import Timelike
from einsteinpy.plotting.geodesic import GeodesicPlotter

Setting up the system

• Initial position & momentum of the test partcle

• Spin of the Schwarzschild Black Hole (= 0)

• Other solver parameters

Note that, we are working in M-Units (\(G = c = M = 1\)). Also, since the Schwarzschild spacetime has spherical symmetry, the values of the angular components do not affect the end result (We can always rotate our coordinate system to bring the geodesic in the equatorial plane). Hence, we set \(\theta = \pi / 2\) (equatorial plane), with initial \(p_\theta = 0\), which implies, that the geodesic should stay in the equatorial plane.

position = [40., np.pi / 2, 0.]
momentum = [0., 0., 3.83405]
a = 0.
steps = 5500
delta = 1.

Calculating the geodesic

geod = Timelike(
    metric="Schwarzschild",
    metric_params=(a,),
    position=position,
    momentum=momentum,
    steps=steps,
    delta=delta,
    return_cartesian=True
)

Plotting the geodesic

gpl = GeodesicPlotter()
gpl.plot(geod)
gpl.show()

Apsidal Precession is easily observed through the plot, above, and as expected, the geodesic is confined to the equatorial plane. We can visualize this better, with a few 2D plots.

gpl.clear()
gpl.plot2D(geod, coordinates=(1, 2)) # Plot X & Y
gpl.show()

gpl.clear()
gpl.plot2D(geod, coordinates=(1, 3)) # Plot X & Z
gpl.show()

gpl.clear()
gpl.plot2D(geod, coordinates=(2, 3)) # Plot Y & Z
gpl.show()

Let’s see, how the coordinates change, with \(\lambda\) (Affine Parameter).

gpl.clear()
gpl.parametric_plot(geod, colors=("cyan", "magenta", "lime")) # Plot X, Y, Z vs Lambda (Affine Parameter)
gpl.show()

The lag between \(X1\) (\(x\)-component) and \(X2\) (\(y\)-component), in the parametric plot, reaffirms the results from above. Also, \(X3 \approx 0\) (\(z\)-component) in this plot, which is expected, as the geodesic never leaves the equatorial plane.