Kerr causality
What the diagram shows
The stacked tiles are the causal sectors of the maximally extended Kerr solution: exterior universes on top and bottom, between-horizon sectors in the middle, and the deep inner sheets around the ring singularity. The black lines are outer horizons, the orange lines are inner horizons, and the red dashed curves are the photon's radial turning surfaces. It is a topological map, not a metric-accurate embedding: distances between tiles do not reflect proper distance.
Light is allowed exactly where the radial potential is non-negative. The horizons are not walls; they are surfaces where which coordinate behaves like time changes.
The radial potential is the floor
Every legal null orbit lives in an allowed corridor: a connected interval of r where R(r) is non-negative. Turning points are real roots of R(r), where the radial momentum reverses. The cases tab ranks six scenarios by the width of that corridor. The figure-like ergoregion sits in a corridor that crosses both horizons. Switch to the positive-energy comparison and the upper turning point disappears: the photon can escape to infinity.
Three constants of motion
A null geodesic in Kerr carries three independent conserved quantities: energy E at infinity, axial angular momentum L, and the Carter constant Q. The Hamilton-Jacobi equation separates and the radial motion reduces to a single ordinary differential equation in r. The four sliders are exactly those three constants plus the spin a of the geometry the photon moves in.
The ergoregion makes E exotic
Outside the ergoregion a real photon must have positive E. Inside the ergoregion the time-translation Killing vector becomes spacelike, so E can be zero or even negative. The negative-energy case in this playground is the orbit fragment that, in the Penrose process, falls into the hole and reduces its mass while a sibling fragment escapes with more energy than the parent had. The borderline E = 0 case is what the original figure shows.
Mathematical, not astrophysical
This is the exact vacuum Kerr metric. Real rotating black holes are not expected to have such a clean traversable inner-horizon structure: the Cauchy horizon at r- is believed to be violently unstable through mass inflation. Read the diagram as the geometry the Kerr equations allow on paper, not as a tunnel through any real astrophysical hole.
Model changelog
- ported the demo SchematicPenrose to a black-and-lime SVG tile diagram with an animated photon dot and a snapshot ghost overlay.
- ported the radial-potential plot to a black-and-lime SVG with translucent allowed bands under R(r).
- added six named scenarios: figure-like ergoregion, positive-energy, negative-energy ergoregion, Schwarzschild limit, near-extremal, polar plunge.
- replaced the single-status readout with a four-tier regime classification (unbounded escape, captured outside, trapped across horizons, trapped inside r-).
- added the standard scientific panel suite: sweep across each of {a, E, L, Q}, sensitivity tornado on allowed corridor span, calibration table against closed-form and reader-assigned expected spans, ten assumption entries.
- snapshot comparison: save a configuration, change parameters, see the dashed-orange ghost overlay on both the causal diagram and the radial-potential plot.
- auto-play animation pattern: parameter changes restart the photon phase, with manual play/pause/replay and a scrub slider.