n = e^ψ c₁ = e^−ψ ṙ = c₁ v ṽ = c₁(I − v vᵀ)∇ψ a = (c²/2)∇ψ Purist (Exact ψ)

Exact DFD equations; ψ = 2GM/r. No damping, no caps, no clamped core. Contours are precomputed for clarity.

ψ contour

photon

test mass

Computing ψ-grid & contours…

Initializing

GM=—

Purist = exact equations. This step ensures clean contours & stable bends. No heuristics.

No contours yet. Click “Recompute grid/contours”.

Model disclaimers. Exact DFD ray & matter laws (2D slice): ψ(x,y) = 2GM/r, c₁ = e^−ψ, rays: ṙ = c₁ v, ṽ = c₁(I−vvᵀ)∇ψ; matter: a = (c²/2)∇ψ. No GR curvature language, no two-way-speed hacks. Optional “Approximate” adds a soft core & UX cap for frame-rate only; it’s off by default.

DFD Controls

Purist mode is exact. Changing GM or resolution re-computes ψ-grid and contours.

Mode

Purist (exact ψ; no caps) Approximate (soft core + UX cap)

Field

GM (strength) Contours

show ψ-contours show ∇ψ vectors finite radius core (Approximate only) R_core

Integrator

Ray time step (dt) Matter time step (dt)

Respect causal cone (|v| < c₁)

Fast mode (relaxes adaptivity; not physics) β_max (Approx.)

Fast Mode & β_max ignored in Purist (exact) mode.

Launch

stream photons periodic masses

Resolution

Higher grid density → crisper contours (recompute required).

Grid step (px)