1 · The geodesic engine
For a photon in Schwarzschild spacetime the orbit equation in u = 1/r is the Binet equation with a single relativistic correction term. In Cartesian form — the form a fragment shader wants — it becomes a central force with one conserved quantity, and it is exact: not a post-Newtonian expansion, not a fit. The shader integrates it with fixed-step RK4, one ray per pixel per frame, traced backward from the camera.
Rays that end below the horizon paint the shadow. Rays that escape sample the background sky with their final direction — that is all gravitational lensing is. Rays with b just above bc wind around the photon sphere at r = 1.5 rs one or more times before leaving: they build the photon ring and the higher-order disk images, with no special-case code. On load, the page bisects its own integrator for the capture threshold and prints the result in the header strip next to the analytic value — if those two numbers ever disagree, do not trust the picture.
2 · The disk and the redshift factor
The disk is the Shakura-Sunyaev geometrically-thin, optically-thick disk, truncated at the ISCO (r = 6M = 3 rs), with matter on exact circular geodesics. Each disk pixel is shifted and beamed by the exact redshift factor for a circular-orbit emitter seen by a distant observer:
Three checks the formula passes by construction: seen face-on, g = √(1−3M/r) everywhere on a ring (pure gravitational + transverse-Doppler — at the ISCO exactly 0.7071, the same number as the orbiting clock in Entry XIV); the approaching side has g > 1 and is beamed brighter by g⁴, which is the asymmetry Luminet's 1979 photograph showed and the EHT ring shows; and with the Doppler switch off the disk becomes symmetric, isolating the purely gravitational redshift. The false-color g mode paints the disk by g directly — blue where the spectrum is blueshifted, red where redshifted — which is the single most useful teaching view on this page.
What the upper and lower arcs are: the disk's far side, whose light is bent over the top of (and beneath) the hole on its way to you. The thin ring just outside the shadow is the secondary image — light that crossed the disk plane after at least half an orbit of the photon sphere. Both were in Luminet's hand-plotted 1979 figure and both emerge here from the integrator alone.
3 · Configurations
Presets set camera, disk, and switches; everything stays live afterward. The Gargantua preset deliberately turns Doppler beaming off — the same choice the DNGR team made for Interstellar, because the true g⁴ asymmetry makes half the disk nearly invisible (James+ 2015, §4.2). Toggle it back on to see what Hollywood hid.
| Preset | camera | incl. | switches | what it teaches |
|---|---|---|---|---|
| Luminet 1979 | 30 r_s | 80° | blackbody · Doppler on | the original computed image: asymmetric brightness, doubled disk |
| Gargantua | 22 r_s | 85° | Doppler off | the Interstellar look, and why it is not what you would really see |
| EHT M87* | 60 r_s | 17° | false-color g | near-face-on ring; mild Doppler asymmetry; shadow at √27 M |
| Pure lensing | 30 r_s | 15° | disk off · grid bg | Einstein ring and sky distortion with no disk in the way |
| Photon ring | 110 r_s | 80° | narrow FOV 4° | shadow edge and the stacked higher-order images |
| No relativity | 30 r_s | 80° | straight rays | control case: flat-space camera, disk half-hidden, no ring, no shadow edge |
4 · Verification
The integrator in the shader is the same equation verified below. Rows marked live are computed by this page in JavaScript on load with the identical RK4 scheme; the offline rows were run at higher step counts during construction. Published anchors in the right column.
| Test | integrator result | analytic / published | status |
|---|---|---|---|
| Capture threshold b_c (live, bisection) | … | 3√3/2 = 2.59808 r_s | … |
| Deflection at b = 100 r_s (live) | … | 2 r_s/b = 2.000×10⁻² rad (+1.5% O(b⁻²) term) | … |
| Deflection at b = 500 r_s (offline) | 4.006×10⁻³ rad | 4.000×10⁻³ rad · ratio 1.0015 | ✓ converges to 4GM/c²b |
| Solar-limb deflection (analytic chain) | … | 1.75″ — Dyson, Eddington & Davidson 1920 | … |
| M87* shadow diameter | … | 42 ± 3 µas — EHT 2019 Paper VI | … |
| Sgr A* shadow diameter | … | 51.8 ± 2.3 µas — EHT 2022 Paper I | … |
| g at ISCO, face-on | 0.70711 (formula in §2) | √½ exactly — Bardeen, Press & Teukolsky 1972 | ✓ exact |
The M87*/Sgr A* rows convert √27 GM/c² to angular size with the EHT-adopted masses and distances (6.5×10⁹ M☉ at 16.8 Mpc; 4.297×10⁶ M☉ at 8.28 kpc). The Schwarzschild prediction lands inside both error bars; spin moves the predicted diameter by at most ~4% (Johannsen & Psaltis 2010), below EHT's current precision.
5 · Boundary of validity mandatory
What exact Schwarzschild ray tracing gives you — and what it doesn't
Model class: analytic (exact null geodesics + exact circular-emitter redshift) wrapped around one declared simplification (the Newtonian disk flux profile). The geometry of every photon path is exact; the disk's absolute brightness profile is not.
Lensing geometry exact
Shadow size, photon ring, Einstein rings, deflection angles, image multiplicity: all from the exact geodesic equation, verified against Darwin 1959 / Synge 1966 closed forms in §4 — including by the page itself at load time.
Redshift factor exact
g for circular-geodesic emitters is the exact Cunningham 1975 expression — gravitational, special-relativistic Doppler, and beaming combined with no expansion. Face-on ISCO value √½ ties this entry numerically to Entry XIV.
Schwarzschild, not Kerr
Astrophysical black holes spin. Kerr moves the ISCO (down to 1M prograde), drags the photon region into asymmetry, and displaces the shadow centroid. For the EHT sources the shadow-size shift is ≲4%, but disk appearance at high spin differs substantially — DNGR (James+ 2015) is the Kerr reference implementation.
Disk flux is Newtonian SS73
T ∝ r−3/4(1−√(rin/r))1/4 is the Shakura-Sunyaev profile; the fully relativistic Page-Thorne 1974 flux differs by tens of percent near the ISCO. Image geometry and g-factors are unaffected; relative brightness across the inner disk is approximate, and the display temperature scale is a rendering choice.
Thin disk ≠ EHT source
M87* and Sgr A* are optically-thin, geometrically-thick hot flows imaged at 230 GHz, not optically-thick thin disks. The valid comparison made here is shadow geometry only; reproducing the EHT ring brightness requires GRMHD + radiative transfer (EHT Paper V).
Static spacetime, static camera
No light-travel-time variability, no orbiting hot spots, no polarization (EHT 2021 Paper VII), and the camera is at rest — a camera plunging or orbiting would add aberration and its own Doppler field.
Bottom line. Every geometric statement this page makes — where light goes, what is hidden, what is doubled, what is shifted by how much — is exact Schwarzschild GR, and the page proves its own integrator at load. The disk's absolute photometry is declared Newtonian, and spin is the next entry's problem.
References
- Schwarzschild 1916. Über das Gravitationsfeld eines Massenpunktes. Sitzungsber. Preuss. Akad. Wiss. 189. ADS — the spacetime being traced.
- Darwin 1959. The gravity field of a particle. Proc. R. Soc. A 249, 180. RS — photon orbits, capture threshold 3√3 M, deflection closed forms.
- Synge 1966. The escape of photons from gravitationally intense stars. MNRAS 131, 463. ADS — the shadow angular-size formula used in the live panel.
- Bardeen 1973. Timelike and null geodesics in the Kerr metric. In Black Holes (Les Houches), 215. — the shadow ("escape cone") formalism, Kerr generalization.
- Cunningham 1975. The effects of redshifts and focusing on the spectrum of an accretion disk around a Kerr black hole. ApJ 202, 788. ADS — the transfer-function formalism; source of the g expression.
- Luminet 1979. Image of a spherical black hole with thin accretion disk. A&A 75, 228. ADS — the first computed image; the Luminet preset reproduces its geometry.
- Shakura & Sunyaev 1973. Black holes in binary systems. A&A 24, 337. ADS — the thin-disk temperature profile (declared simplification).
- Page & Thorne 1974. Disk-accretion onto a black hole. ApJ 191, 499. ADS — the relativistic flux this page declares it omits.
- Dyson, Eddington & Davidson 1920. A Determination of the Deflection of Light by the Sun's Gravitational Field. Phil. Trans. R. Soc. A 220, 291. RS — the 1.75″ measurement, verified in §4.
- James, von Tunzelmann, Franklin & Thorne 2015. Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar. CQG 32, 065001. IOP — DNGR; documents the Doppler-off rendering choice quoted in §3.
- EHT Collaboration 2019. First M87 EHT Results I & VI. ApJL 875, L1 & L6. IOP — 42 ± 3 µas ring; mass/distance used in §4.
- EHT Collaboration 2022. First Sgr A* EHT Results I. ApJL 930, L12. IOP — 51.8 ± 2.3 µas ring.
- Gralla, Holz & Wald 2019. Black hole shadows, photon rings, and lensing rings. Phys. Rev. D 100, 024018. APS — the modern decomposition of what this page renders into shadow / lensing ring / photon ring.
- Johannsen & Psaltis 2010. Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum II. ApJ 718, 446. ADS — shadow-size spin dependence ≲4%, quoted in §4–5.