Animating the shadow: a black hole spun up from rest

The static, side-by-side spin comparison in The Kerr shadow: a rotating black hole’s asymmetric silhouette already shows that a black hole’s shadow shrinks and its accretion disk creeps inward as the spin \(a\) increases – but a continuous sweep makes the mechanism far more vivid: watch the innermost stable circular orbit (ISCO) contract from \(6M\) (Schwarzschild) down toward \(M\) (extremal Kerr, prograde), dragging the disk’s bright inner edge inward with it, while the shadow itself flattens and shifts off-center from frame dragging.

\[r_{\text{ISCO}}(a) : \quad 6M \ \xrightarrow{a \to M} \ M \qquad \text{(prograde)},\]

each frame’s disk inner edge set to exactly this spin-dependent ISCO.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.relativity.chapters.kerr import KerrBlackHole
from physicskit.relativity.visualizers.shadow_render import animate_shadow_spin_sweep

Sweep the spin from Schwarzschild to near-extremal Kerr

Each frame is a full independent backward ray-trace at that frame’s spin, with the disk’s inner edge set to that spin’s ISCO and the disk shaded by its combined gravitational and Doppler redshift, so the Doppler-brightened approaching side becomes progressively more pronounced as the disk edges closer to the horizon.

anim = animate_shadow_spin_sweep(
    M=1.0,
    ny=140,
    nx=140,
    inclination=1.3,
    r_disk_outer=16.0,
    redshift=True,
)
plt.show()

To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:

anim.save("kerr_shadow_spin_sweep.gif", writer="pillow", fps=5)

Check

def isco_bpt(a, prograde=True):
    # Bardeen, Press and Teukolsky (1972), M = 1.
    z1 = 1 + (1 - a**2) ** (1 / 3) * ((1 + a) ** (1 / 3) + (1 - a) ** (1 / 3))
    z2 = np.sqrt(3 * a**2 + z1**2)
    return 3 + z2 - np.sqrt((3 - z1) * (3 + z1 + 2 * z2)) * (1 if prograde else -1)


# Each frame's disk starts at the spin's ISCO, contracting from 6M toward
# M as the spin grows (Bardeen-Press-Teukolsky).
spins = np.linspace(0.0, 0.98, 50)
isco = np.array([6.0] + [KerrBlackHole(M=1.0, a=a).isco_radius(prograde=True) for a in spins[1:]])
np.testing.assert_allclose(isco, isco_bpt(spins), rtol=1e-6)
assert np.all(np.diff(isco) < 0) and isco[-1] < 1.7