Relativity Lab

Switch to full circle and the dial shows all four quadrants. Your pointer sweeps only the upper (future) half — right of vertical for motion in +x, left for −x; its mirror image, the same worldline run backward in time, sweeps the lower half.

In the lower half, the time component points down, into the past. A worldline running backward in time is, in the Feynman–Stückelberg reading, an antiparticle running forward: a positron is an electron whose 4-velocity points into the past. On this dial that reading is literal — the arrow sweeping below the horizontal is the same particle, time-reversed. It is why pair creation and annihilation look like a single worldline bending back on itself.

Press photon to push β all the way to 1. The pointer swings flat onto the space axis: a massless particle spends all of its motion on space and none on time, so its arrow lies on the horizontal light line and proper time stops, dτ/dt = 0. There is no "rest frame" for light because there is no time component left to stand still in.

An ordinary rotation keeps x²+y² fixed and mixes axes with sin and cos. A Lorentz boost keeps x²−(ct)² fixed and mixes axes with sinh and cosh of the rapidity φ: ct′ = ct·coshφ − x·sinhφ, with tanh φ = β. A single sign separates Euclidean geometry from spacetime: minus where a rotation has plus.

Because rapidities add where velocities do not, three 0.5c boosts in a row give tanh(3·atanh 0.5) ≈ 0.93c, not 1.5c. Nothing crosses c no matter how many boosts you stack. The x′ axis tilts up by the same angle the ct′ axis tilts over, so they scissor symmetrically toward the 45° light line and never cross it.

γ = 1/√(1−β²) is gentle at first: at β=0.1 it is 1.005, a half-percent effect. It reaches only 1.15 at β=0.5. Then the square root starts to bite. At β=0.9 it is 2.3, at 0.99 it is 7.1, at 0.999 it is 22. The curve has a vertical asymptote at β=1, which is why no massive object reaches c: the energy γmc² needed diverges.

Time dilation and length contraction carry the same factor 1/γ, which is why the clock-rate and length curves here sit exactly on top of each other. They are not, however, one effect seen twice: they are two separate consequences of the Lorentz transformation, and length contraction needs a simultaneity convention on top of it — to have a length at all you must locate both ends at once, and which events count as "at once" depends on the frame. This is why stacking the two by hand goes wrong: apply it to a chasing light beam and you land on c/(1+β) rather than c. See Chasing Light.

The two Doppler curves are not 1/γ; they fold in the changing light travel-time as well, so the approach curve rises faster than γ and the recession curve falls toward zero.

The funnel is not the rubber-sheet cartoon where gravity is drawn as a ball denting a trampoline (that picture secretly uses gravity to explain gravity). It is Flamm's paraboloid, the true geometry of a spatial slice outside the mass. Distances measured along the curved surface are the real proper distances; the throat at r_s is where the surface turns vertical.

There is one thing the funnel cannot show: it is only the space curvature. For a slowly orbiting planet the fall is dominated by time curvature, and this surface contributes almost nothing; the space part scales as (v/c)². Only for light does it grow into a full half, which is why a photon's deflection is twice Newton's. The Why Things Fall tab isolates the time half that does the everyday work.

Newton's orbit closes into a fixed ellipse because the potential is exactly 1/r. The GR 3Mu² term breaks that, so the ellipse rotates a little each lap and traces a rosette. The closer the orbit to the mass (smaller p), the larger the per-orbit twist.

The two precession readouts disagree for a reason. The textbook figure, the one that gives Mercury its 43″ per century, is 6πM/p — and it is only the first term of an expansion in M/p. Mercury sits at p ≈ 3.8×10⁷ M, so for it that one term is the whole answer to many decimal places. This panel deliberately puts you at p = 16–24 M, where the orbit skims a few Schwarzschild radii and the higher-order terms are not small: at the default p = 18 M, e = 0.4 the exact advance is 114° per orbit against the formula's 60°. The exact row is measured from the integrated geodesic, perihelion to perihelion, and is the one to trust here; the weak-field row sits beside it so you can watch the textbook formula break down. The time-dilation readout shows how slow a clock hovering at the planet's current depth would run; the orbiting clock is also moving, so it ticks slower still — √(1−3M/r) on a circular orbit.

Every event sits at the tip of its own light cone. The future cone holds everything this event can still influence; the past cone holds everything that could have influenced it. The region outside both cones is the elsewhere: too far to reach even at light speed, so no cause and effect can pass either way.

A worldline tilted past 45° would mean travelling faster than light, which would let it exit its own future cone, reach the elsewhere, and in some frame arrive before it left. Keeping every worldline steeper than the cone wall is the same thing as saying causes precede effects for everyone. The simultaneity disk tilts by atan β, the mirror image of the worldline's tilt about the light line, so the faster you go the more your 'now' slices into what others call past and future.

Aberration moves stars: positions that were spread across the sky pull forward into a tight forward patch, so the bow fills with stars and the stern empties. Doppler recolours them: light ahead blueshifts (D>1), light behind redshifts (D<1). Beaming rebrightens them: because bolometric intensity goes as D⁴, the forward stars blaze and the rear ones fade almost to black. All three effects come from the same boost.

The forward half-angle is acos β: the whole rest-frame forward hemisphere squeezes into a cone of that opening. At β=0.99 that is 8.1°, so half the sky lives in a 16°-wide spot. The colour map here is illustrative; the position, Doppler factor, and D⁴ brightness are computed exactly from the formulas above.

Each ray is integrated with the weak-field deflection law dv̂/dl = −(1+β²)(∇Φ)⊥, with Φ = −Σ Mᵢ/rᵢ the summed Newtonian potential and (∇Φ)⊥ its component perpendicular to the ray. Only the perpendicular part bends the path, so speed is held fixed.

For light, β=1 and the prefactor is 2; integrating a distant flyby gives exactly ∫2(∇Φ)⊥ dl = 4M/b. Setting the prefactor to 1 — curved time only — gives 2M/b, the Newtonian prediction for a corpuscle crossing at c. Both superpose linearly because the potentials add, which is why dropping a second mass simply sums the bends.

Note that 4M/b is a small-angle result: it assumes the ray keeps to its original straight line while the field kicks it sideways. The rays drawn here are integrated in full, so a close pass gets dragged inward, feels a stronger field than the straight line would, and bends more than the formula says — by a fifth or so once the bend passes ~30°, and half again as much by ~70°. The readout therefore quotes the innermost ray that is still genuinely small-angle, and flags the comparison when a heavy enough mass pushes every ray out of that regime. The exact treatment, where a close enough ray is captured outright, is the Black Hole tab.

The β slider comes with a caveat: it dials only the (1+β²) space-curvature coupling while the beam still crosses at the same speed. It does not show the trajectory of a real particle slowed to β; a genuinely slow particle spends longer in the field and bends more, by 2M/bβ², the opposite trend. The slider answers a single question: how much of light's bend comes from curved space.

The grid is the same physics applied to a background lattice: each node is displaced by the deflection field α = Σ 4Mᵢ(x−xᵢ)/|x−xᵢ|², so straight coordinate lines appear pinched toward each mass, the visual signature of gravitational lensing. Softening near each mass keeps the weak-field picture valid; inside a few r_s the linear approximation breaks down and you would need the full Schwarzschild geodesics from the Gravity Well tab.

Each boost shifts rapidity by a fixed amount, so N identical boosts of β give rapidity N·atanh β and velocity tanh(N·atanh β). That tends to 1 but never arrives: ten 0.5c boosts give 0.99997c, not 5c. Because tanh saturates, c is an asymptote no finite stack of boosts can cross.

This is the same hyperbolic angle as the Minkowski boost: a velocity addition is a rotation through an imaginary angle, and rotation angles add. Velocities look awkward only because we read off tanh φ instead of φ itself.

Turn on simultaneity. On the outbound leg the traveller's lines of 'now' tilt one way; on the return they tilt the other. At the turnaround the traveller's notion of what is happening on Earth jumps forward across a whole band of Earth-time — the years that the naive "each sees the other's clock run slow" argument forgets. The whole resolution lives in that jump.

Nothing here needs acceleration math: the gap is geometric, set by the angle between the two simultaneity families, which is fixed by β.

Trace every captured ray backward and it came from inside an angular disk of radius set by b_crit = 3√3 M. No light from behind the hole can reach you through that disk, so it reads as a dark circle, the shadow, about 2.6 times wider than the horizon itself. This is the image the Event Horizon Telescope resolved for M87* in 2019.

The bright rim just outside is the photon ring: light that looped the photon sphere one or more times before escaping, piling up at the shadow's edge. Real black holes spin and are wrapped in glowing gas. Spin barely changes the shadow's size: seen edge-on, its vertical extent stays exactly 2√27 M at any spin. What spin does distort is the shape, and how much of that you see depends on where you stand. Edge-on at a = 0.99 the outline is 11 % narrower than it is tall, develops a straight edge on one side, and sits about 2.4M off the direction of the hole itself; push to extremal and it spans α ∈ [−2M, 7M], the classic Bardeen result. Viewed near face-on the distortion collapses: at M87*'s inclination of roughly 17° it is about 1 %, which is why that shadow reads as a circle and why the lopsided crescent is not the shape at all — it is Doppler beaming of gas orbiting at relativistic speed, the same D⁴ brightening as in the Starfield mode. The Gargantua tracer integrates Schwarzschild geodesics, so it draws the circle and never the flattening.

Causal structure is all about light cones, and this map keeps every cone at a rigid 45° everywhere on the page. So you can read off at a glance which events can signal which: just check whether you can get between them without ever tilting past 45°. Questions about infinity — does a ray escape, where does a worldline end — become questions about which edge you reach.

The same trick drawn for a black hole separates ℐ⁺ from the singularity by the horizon, which is how Penrose diagrams make causal traps like event horizons visually obvious.

A circular orbit fixes the speed: v = √(GM/r), so raising the orbit both slows the satellite (less SR slowdown) and lifts it higher (more GR speedup). Both pull the net positive as you climb. At one low altitude, about 3 200 km, the two effects cancel exactly and an orbiting clock keeps pace with the ground.

Constants used: GM⊕ = 3.986×10¹⁴ m³/s², R⊕ = 6 371 km, c = 299 792 458 m/s. Clocks on the ground also run slow from Earth's spin and equatorial bulge; those are smaller and left out here.

The weak-field line element is ds² = −(1+2Φ/c²)c²dt² + (1−2Φ/c²)dx². For a slow particle dx ≪ c·dt, so the dt² term dwarfs the dx² term: the geodesic equation keeps only d²x/dτ² = −∂Φ/∂x, which is Newton's law. Newtonian gravity is the time-curvature limit of general relativity.

The transverse pull on a particle crossing the field at speed β is a⊥ = −(1+β²)∂⊥Φ: the 1 is curved time (present for everything), the β² is curved space (only matters near light speed). Slow matter: factor 1. Light: β² = 1, so the space term contributes a second, equal share and the factor doubles; that doubling is the 1919 eclipse result. One caution about scope: this decomposition describes the transverse bend of something crossing the field. The tossed ball above falls radially, where the finite-β correction takes a different form, but at everyday speeds corrections of either kind are ~10⁻¹⁶, and "the fall is curved time" holds regardless of direction.

Concretely: the clock-rate gradient is g/c² ≈ 1.1×10⁻¹⁶ per metre of height, so between your head and your feet the difference is about 2×10⁻¹⁶ — a part in five quadrillion. And this gradient is measured, not inferred: Pound and Rebka read it up a 22.5 m tower in 1960 (gh/c² ≈ 2.5×10⁻¹⁵), and by 2022 optical lattice clocks resolved it across a single millimetre. Geodesic motion turns a rate-gradient dφ/dh into an acceleration c²·dφ/dh — feed in the measured gradient and 9.8 m/s² comes out. If you prefer it kinematically: a worldline crossing the gradient turns at g/c ≈ 3.3×10⁻⁸ radians per second, so after t seconds it has swung through g·t/c, and carrying that turn at c is a sideways speed of g·t — see Tethered Boats. Either way, the full 9.8 m/s² comes out of that microscopic tilt.

The two sliders in this app run in opposite directions, and it helps to know which is which. A demo β of 0.4 shows an 86 %/14 % time/space split: that is the time share dialled down from its real ≈100 %, so the space sliver becomes visible. Nothing in this panel is dialled up. The quantity the app does exaggerate is the clock-rate deficit, over in Curved Time, which labels the exaggeration on the panel.

By default this is Newtonian gravity, valid because every body moves far below c in a weak field — the regime where curved time reduces to F = −GMm/r² (see Why Things Fall). A lone two-body orbit is then a perfectly closed ellipse: it retraces the same path forever.

Turn on relativistic and each orbit gains the general-relativistic correction, an extra inward pull ≈ 3GM h²/(c²r⁴) (h = the body's angular momentum). The ellipse no longer closes — its near-point creeps forward a little each lap, tracing a slowly turning rosette. This is exactly perihelion precession — the anomaly in Mercury's orbit that first confirmed general relativity. The effect is exaggerated here so you can see it in a few orbits; Gravity Well shows the same precession from the exact Schwarzschild geometry.

Other notes: overlapping bodies merge, conserving momentum (accretion); the figure-8 is a real 1993 choreography of three equal masses on one looped path — drop a body on it and watch the chaos.

Left alone the orbits are Newtonian and, for an isolated pair, close into a fixed ellipse. Turn on relativistic and each orbit picks up the general-relativistic 3GM h²/(c²r⁴) correction: the ellipse precesses, sweeping out a rosette in 3D. It is the same perihelion precession that confirmed general relativity with Mercury, exaggerated here for visibility and shown exactly in Gravity Well.

The same thing happens to a cart when one wheel hits mud, to a marching rank when one end shortens its stride, and to light entering glass when one edge of the wavefront slows first (Huygens' principle). Whenever an extended thing crosses a gradient of speed, it turns toward the slow side.

In relativity the "speed" is the rate of proper time, which runs slower deeper in a gravitational well. A free object's worldline stays as straight as the geometry allows — a geodesic — so it veers toward slower time, toward the mass. That veer is gravity. The time difference is enormous when measured against the distance light covers each second (see Why Things Fall), so even this faint gradient bends the path by the full 9.8 m/s². The numbers work out exactly: the rod turns at g/c ≈ 3.3×10⁻⁸ radians per second, so after t seconds it has swung through g·t/c, and carrying that turn at c is a sideways speed of c·(g·t/c) = g·t — 9.8 m/s after one second, which is what falling means.

What this picture leaves out. A time gradient alone gives the 1 in a⊥ = −(1+β²)∂⊥Φ. For anything slow the β² term is negligible and the rod covers nearly everything. For light the β² term is just as large, and the beam bends twice as far as this analogy alone can explain. The missing half is curved space, the funnel in Gravity Well, and the doubling is what the 1919 eclipse measured. Curved Space shows the two halves side by side.

Clock rate is √(1 + 2Φ/c²), Φ = −GM/r; equal-time slices are t = τ/rate(x). The worldline is the geodesic d²x/dt² = −dΦ/dx, which is the Newtonian limit: for slow motion, that is all the time-curvature leaves (see Why Things Fall). It runs faster than life here so the drop lands inside the frame. The felt gravity readout is the surface gravity a static observer feels, g = GM/r², calibrated in Earth units: mass and radius of 1 give Earth's 9.8 m/s² (press ⊕ Earth), and halving the radius quadruples it. Escape speed is √(2GM/r), 11.2 km/s for Earth. The falling object feels none of this. Free fall is weightless; only something held off its geodesic, like the surface or a rocket, feels g.

Light cones use the coordinate light speed c·(1+2Φ/c²), which slows toward the mass, so the cones narrow and the future pinches inward. In free-fall coordinates the same effect appears as the cones tipping over. Redshift: crests emitted one tick apart deep down arrive more than a tick apart up high, spread by exactly rate(top)/rate(bottom).

The drawing cheats in three ways, and all of them are cosmetic. First, the scale: a real faller climbs about 300 000 km up the time axis every second while drifting only metres sideways, so the horizontal axis here is stretched enormously to make the lean visible at all. Flatten the mass to check: straight grid, straight rise, no fall. Second, the speed: the fall runs faster than life so it lands inside the frame. Third, the depth: the drawing's well uses a small speed of light (c² = 4 in sim units), which makes |2Φ/c²| ≈ 0.32 at the surface instead of Earth's real 1.4×10⁻⁹. That is the only reason the lean, the bowed slices and the picture-scale gauge are visible. The readouts skip all three cheats: clock rate, lag and speed are computed at true scale, with a uniform-density interior below the surface, and the launch slider carried over as its fraction of escape speed.

Straighten is a change of coordinates, not new physics. It shears the picture by the faller's own deflection, x → x − s·(x_geo(t) − x_geo(0)), landing in the comoving free-fall frame at s = 1. There the geodesic is straight and unaccelerated (this is the equivalence principle), so the spatial part of its 4-velocity arrow vanishes, and the held rocket, the mass and the surface are the ones that curve — the floor rushes up to meet you. The frame is only local, though: a faller released somewhere else would straighten under its own, slightly different shear, and the mismatch between neighbours is tidal curvature, which no single chart can shear away.

Turn on the test column to see that directly: a line of fallers released together stays parallel in flat space but converges toward the mass — geodesic deviation. Push straighten to 100% and the main path goes vertical, yet the column still squeezes. That residue is real curvature, the part the equivalence principle cannot remove. The spacetime dial echoes the Spacetime Dial mode: the needle tips from all-time toward space as the fall speeds up (picture scale, like the gauge). Truer scale squeezes the space axis 33-fold, and even that is nowhere near the real proportions. The default Earth drop takes about 1.9 hours, climbing two billion kilometres through time while drifting 19 000 km sideways — a lean of one part in 10⁵ that no screen could draw.

For a non-spinning hole the orbital factor dτ/dt = √(1 − 3M/r) is smallest at the innermost stable orbit, r = 6M: √(1−½) ≈ 0.71, so clocks run at worst ~1.4× slow. Inside 6M there are no stable circular orbits — anything there spirals through the horizon.

Spin changes the geometry. Frame dragging lets prograde stable orbits exist much closer in; as a → M the ISCO slides from 6M down toward M, into the region where dτ/dt → 0. Sitting just outside it gives an arbitrarily large factor. Kip Thorne fixed Gargantua's spin at a = 1 − 1.3×10⁻¹⁴ precisely so Miller's planet could orbit stably at the one-hour-per-seven-years rate the plot needs.

The same spin also drags space into a swirling Kerr "river" and warps light into the wrapped-disk image; this panel isolates just the clock.

Run the logic backward and you also get weightlessness: a freely-falling cabin cancels gravity exactly, which is why astronauts float. Einstein called realizing this "the happiest thought of my life." Promoting "you can't locally tell gravity from acceleration" to a law forces light to fall, clocks to run slow low down, and ultimately spacetime to curve. All of general relativity grows out of this sealed box.

The "locally" matters because a planet's field is not uniform: it points at a centre and weakens with height. Release two balls 3 m apart across the cabin and they converge, because both fall toward the same centre; release them 3 m apart vertically and they separate, because the lower one is pulled harder. The relative acceleration is 2GMd/R³, and over the fall the g cancels against t² = 2h/g, leaving a drift of exactly 18/R metres for a 3 m cabin: 2.8 µm on Earth, 10.4 µm on the Moon, since a smaller world curves its field more sharply. That is tiny, but an interferometer would see it. Under thrust the drift is exactly zero, so this one measurement does tell the two cabins apart, and it is why the principle is stated for a region small enough that tidal effects fall below your instruments. The tidal part survives every choice of frame; in Curved Time, the test column still squeezes with straighten at 100 %.

The two door-closing events are a distance L_barn apart and happen at the same barn-time. Lorentz-transform to the ladder frame: Δt′ = γβ L_barn / c = γ v L_barn / c² ≠ 0 — the doors shut that much apart in time, exactly enough that the ladder always pokes out one end. "Does the whole ladder fit at one instant?" depends on whose instant — there is no frame-independent answer, only the invariant events themselves.

All three shapes make gravity the same way: spin fast enough that the floor, the outer wall, keeps pushing you toward the axis, so "down" points outward. A wheel concentrates everything at one radius; an O'Neill cylinder adds length to live along; a dumbbell (two pods on a tether) is the cheapest to build but gives a strong head-to-foot gravity gradient if the pods are short. Comfort comes down to spin rate: under ~2 rpm the Coriolis force and the inner-ear conflict are unnoticeable, and reaching 1 g that slowly needs a radius of hundreds of metres. Realistic habitats are therefore large; the compact rings of most films would leave their crews queasy.

A flat universe expands as H(a) = H₀√(Ω_m/a³ + Ω_Λ); matter dilutes as it grows while dark energy stays constant, so the far future is exponential (de Sitter). Light stretched with space arrives redshifted by 1 + z = 1/a_emit. Recession v = H·d hits c at the Hubble radius c/H ≈ 14 Gly today; with Ω_Λ > 0 there is also a true cosmic event horizon — a comoving distance beyond which a photon emitted now never reaches — that the acceleration pulls steadily closer.

What is ΩΛ? Dark energy is modelled as a cosmological constant: a fixed energy density of space itself, about 0.68 of the total today. Because it does not dilute as the universe grows (matter thins as 1/a³, dark energy stays put), it must eventually win; and its negative pressure acts as repulsive gravity, so once it dominates the expansion accelerates and never stops, heading for a cold, empty de Sitter future. Set ΩΛ = 0 and the cosmos is matter-only: still expanding forever (it's flat), but ever more slowly, with no acceleration and no event horizon. Why Λ is so tiny, and why we live just as it takes over, are the cosmological-constant and coincidence problems.

The orbit loses energy to radiation ever faster as the separation shrinks, so the inspiral runs away: the frequency climbs as f ∝ (t_merge − t)^(−3/8) and the amplitude as f^(2/3). The chirp's shape encodes the masses (the "chirp mass") and the distance, which is how LIGO/Virgo weigh black holes a billion light-years away. Real strains are ~10⁻²¹ — about 1/400 of a proton's width across the 4 km arms; the breathing here is hugely exaggerated.

Relativistic velocity subtraction gives the speed of the beam in your frame as (c − v)/(1 − vc/c²) = c for any v < c; the c's cancel exactly. There is no frame in which light is slower (or faster) than c; that is what "invariant" means, and demanding it forces time dilation, length contraction, and the relativity of simultaneity all at once.

Why two of the three are not enough. It is tempting to say the beam stays at c because your clocks slow and your rulers shrink. Run that argument and it fails. In the ground frame the gap after time t is (1−β)ct; measure it with contracted rulers and you get γ(1−β)ct; divide by your dilated clock's t/γ and the answer is γ²(1−β)c = c/(1+β) — 0.625c at β = 0.6. Close, but not c, and not even constant.

What the argument left out is the −vx/c² in t′ = γ(t − vx/c²). "Where the beam is right now" is a statement about simultaneity, and you and the ground disagree about it, so you are not measuring the same pair of events. Put the term back: for the beam, x = ct, so x′ = γ(c−v)t and t′ = γ(1−β)t, and the ratio is exactly c. Dilation and contraction are consequences of the Lorentz transformation, not a substitute for it. The same lesson runs through the Twin Paradox and Ladder & Barn.

For 1 g, ω = √(g/r), where r is always the radius: a 2 m centrifuge needs ~21 rpm (nauseating), a 4 m one ~15 rpm, a 100 m ring ~3 rpm, a 224 m ring just 2 rpm — the usual comfort ceiling. A drop's deflection shrinks as 1/√R, so the 224 m ring curves your dropped coffee far less than the 2 m centrifuge. (At fixed radius, changing ω alone leaves the drop's curve unchanged, as the note above says: the ω in the Coriolis force cancels against the slower floor speed.) There is no relativity here at all — it is pure rotating-frame mechanics — but it is the only artificial gravity we actually know how to build, which is why it fills the hard-SF canon.

"Both engines start at the same instant" is a lab-frame statement. In the ships' moving frame the lead ship started earlier, so it is always a touch faster and pulls ahead. The proper separation grows as γL₀; the strain γ − 1 rises without bound, so for any real material the thread eventually snaps. It is the same relativity-of-simultaneity that resolves the ladder & barn — here it pulls a thread apart instead of fitting a pole.

With proper acceleration a and ship-time τ, the worldline is x = (c²/a)(cosh(aτ/c) − 1), ct = (c²/a) sinh(aτ/c), so β = tanh(aτ/c) and γ = cosh(aτ/c). The hyperbola asymptotes to the 45° light line: you approach c but never reach it, and light from events beyond that line can never catch you: a Rindler horizon trailing behind, the flat-space cousin of a black-hole horizon.

The kinematics is exact; the engineering is the hard part. Reaching these speeds needs energy of order (γ−1)mc² per kilogram; for a round trip to Andromeda that is far more fuel than the ship, even with perfect antimatter. The Expanse's drive is sub-relativistic, but its constant-thrust "flip and burn" is exactly this geometry at low β; push the thrust here and watch the same curve bend toward the light line.

There is a deeper limit the "1g to anywhere" table ignores: the universe is expanding, and the expansion is accelerating. Beyond a comoving distance of roughly 16–18 billion light-years, the cosmological event horizon, galaxies recede faster than any signal can close the gap, so no amount of ship-time ever reaches them; aim there and you brake into a cosmos that has already carried your target beyond reach. Only a few percent of the galaxies we can see are reachable even in principle. Nearby targets — stars, the Milky Way, the Local Group — are unaffected; expansion only bites at hundreds of millions of light-years and beyond.

For every pixel a null geodesic is integrated backward from the camera using the Schwarzschild orbit equation d²u/dφ² + u = 3M u² in the photon's own plane. The ray ends three ways: it falls past the horizon (black), it crosses the equatorial plane inside the disk's annulus (coloured by radius, hotter and brighter inward), or it escapes (sky). Rays that cross the plane outside the disk keep going, so they can still strike the disk on a later loop. Those later hits produce the wrapped halo and the secondary arc near the shadow.

The spin slider is approximate: it slides the disk's inner edge to the Kerr ISCO, so a faster-spinning hole lets the bright inner ring crowd in toward the shadow (and orbit faster, strengthening the beaming). The lensing here stays Schwarzschild, so the shadow remains a centred circle; true frame-dragging would also pull the image sideways into the lopsided, flat-edged crescent of a real Kerr hole. The film also turned off the Doppler beaming that makes one side far brighter, because Nolan wanted a symmetric disk — toggle Doppler beaming to put that asymmetry back.