Physics Notes | Kerr-like Schwarzschild Black Hole

Physics and Implementation Notes

Kerr-like Schwarzschild Black Hole Renderer

1. Scope

This renderer integrates light rays using a Schwarzschild geodesic model, then adds Kerr-like rotation effects as a controlled approximation for visual frame dragging and accretion-disk dynamics.

It is designed for plausible real-time visualization, not as a full GR solver for Kerr null geodesics.

2. Units and Coordinates

Code units normalize the Schwarzschild radius to r_s = 1.
Light speed is normalized (c = 1) in integration logic.
The black-hole horizon is at r = 1; photon-sphere behavior appears near r ≈ 1.5.
Rays are integrated in a polar orbital-plane parameterization using u = 1/r and angle phi.

3. Schwarzschild Geodesic Model

Null-ray dynamics are advanced with an ODE in the orbital plane:

d²u/dphi² + u = (3/2)u², where u = 1/r

Integration uses RK4 with adaptive angular step shrinkage near strong curvature, equatorial zones, and photon-sphere proximity to preserve image stability.

4. Kerr-like Approximation Layer (Rotation + Frame Dragging)

4.1 ISCO Shift

Disk inner radius is clamped against a prograde Kerr ISCO approximation r_in = max(ACCRETION_MIN_R, r_isco(a*)).

4.2 Frame-Dragging Angular Rate

omega_fd(r) ∝ a* / r^3

This drives three separate visual effects: geodesic-plane twisting, disk-flow angular velocity offset, and procedural pattern advection.

4.3 Important Disclaimer

The ray geodesic itself remains Schwarzschild. Spin and frame-dragging terms are added as a Kerr-like approximation for rendering behavior and disk appearance.

5. Accretion-Disk Emission and Color Model

5.1 Plane Intersection

The disk is a tilted plane defined by a normal and tangent basis. Ray segments crossing this plane accumulate emission (non-opaque additive compositing).

5.2 Temperature Profile

T(r) ∝ r^-0.8 * (1 - sqrt(r_in / r))^0.35

5.3 Relativistic Frequency/Intensity Shift

delta = 1 / (gamma * (1 - v·n))
g_gr = sqrt(1 - 1/r)
g = delta * g_gr
T_obs = g * T_em, I_obs ~ g^p * I_em (p is shader-tuned)

Color is sampled through a precomputed black-body lookup texture using the shifted temperature.

6. Procedural Disk Density (Point-Field Approximation)

The disk uses a dense rotating point-field with:

radial density taper (inner + outer),
mass-biased settling toward inner radii,
azimuthal/radial turbulence modulation,
high-gain narrow ring boosts for visually sharp density bands.

This is a stylized emissivity structure for visual realism, not a particle MHD simulation.

7. Rendering Pipeline Summary

Build camera ray per pixel (including optional aberration/beaming of observer motion).
Integrate ray geodesic in Schwarzschild space with adaptive RK4.
Apply Kerr-like frame-dragging twist to the local propagation basis.
Accumulate emissive disk intersections with Doppler/gravitational shifts.
If the ray escapes, sample lensed star/galaxy background.
Output final radiance with observer-motion intensity correction.

8. Limitations

No full Kerr metric geodesic integration (Schwarzschild core + Kerr-like overlays).
No self-consistent GRMHD disk evolution or radiative transfer.
Several constants are artistically tuned for stable, expressive real-time output.