Relativity Lab

Switch to full circle and the pointer can swing into all four quadrants. The right half is rightward motion (+x), the left half leftward (−x); both still point up, into the future.

The lower half is the interesting part. There the time component points down, into the past. A worldline running backward in time is, in the Feynman–Stückelberg reading, exactly an antiparticle running forward: a positron is an electron whose 4-velocity points into the past. The dial makes that literal. The arrow sweeping below the horizontal is the same particle, time-reversed, which 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. That single sign flip, plus instead of +, is the entire difference between Euclidean geometry and spacetime.

Because rapidities add where velocities do not, three 0.5c boosts in a row give tanh(3·atanh 0.5) ≈ 0.95c, 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 are the same 1/γ seen from two angles. The clock-rate and length curves here sit exactly on top of each other for that reason. 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.

One caveat the funnel can't show: this is only the space curvature. For a slowly orbiting planet the fall is dominated by time curvature instead, and this surface contributes almost nothing — the space part scales as (v/c)². It becomes a full half only for light, 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 time-dilation readout shows the same well slowing the orbiting clock.

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 exactly the statement that 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 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. For β→0 the prefactor is 1 and you recover the Newtonian 2M/b. Both superpose linearly because the potentials add, which is why dropping a second mass simply sums the bends.

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 0.99999c, 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. That jump is the entire resolution.

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, which dents the circle into the lopsided crescent the EHT actually sees.

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. There is a low altitude — about 3 200 km — where 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: factor 2. That single +1 is the 1919 eclipse result.

Concretely: the difference in clock rate between your head and your feet is about 10⁻¹⁶ — a part in ten quadrillion — yet spread over the 300 000 km of time you cross every second, that microscopic tilt is the entire 9.8 m/s² you feel right now.

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.

It is the cart that pulls aside when one wheel hits mud, the marching rank that wheels toward whoever shortens their stride, light bending into glass because one edge of the wavefront slows first (Huygens). In every case an extended thing crossing a gradient of speed turns toward the slow region.

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, and because the time difference is enormous measured against the distance light covers each second (see Why Things Fall), even a feather-light gradient bends the path by the full 9.8 m/s².

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 — the Newtonian limit, because for slow motion that is all the time-curvature leaves (see Why Things Fall) — sped up here so the drop visibly lands in 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); halve the radius and it quadruples. Escape speed is √(2GM/r), 11.2 km/s for Earth. The falling object feels none of it — free-fall is weightless; only something held off its geodesic (the surface, 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 that same effect appears as the cones tipping over. Redshift: crests emitted one tick apart deep down arrive more than a tick apart up high, by exactly rate(top)/rate(bottom).

The cheat is scale: honestly the faller climbs ~300 000 km up the time axis per second while sliding only metres sideways, so the horizontal axis is stretched enormously to make the lean visible. Flatten the mass to confirm — straight grid, straight rise, no fall.

Straighten is a coordinate change, 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 — 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 rushing up to meet you). This frame is only local: a faller released elsewhere would straighten in its own, slightly different shear, and that mismatch between neighbours is the tidal curvature 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 residual is real curvature, the part the equivalence principle cannot wave away. The 4-velocity dial echoes the Spacetime Dial — the needle tips from all-time toward space as the fall speeds up — and true scale briefly restores the honest axis ratio, collapsing the dramatic lean into the sliver of sideways drift it really is.

For a non-spinning hole the orbital factor dτ/dt = √(1 − 3M/r) is largest 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 — the whole of general relativity grows from this sealed box.

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 / 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 so the floor (the outer wall) pushes you inward-of-itself, i.e. "down" is 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 — which is why realistic habitats are large and the compact rings of most films would leave the crew 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 — that is how LIGO/Virgo weigh black holes a billion light-years off. Real strains are ~10⁻²¹, a thousandth of a proton 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.

For 1 g, ω = √(g/r): a 4 m centrifuge needs ~21 rpm (nauseating), a 100 m ring ~3 rpm, a 224 m ring just 2 rpm — the usual comfort ceiling. Coriolis deflection scales with ω, so the big slow ring also curves your dropped coffee far less. 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 catch. Reaching these speeds needs energy of order (γ−1)mc² per kilogram — for a round trip to Andromeda, 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 — that is what produces 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 drag the whole image sideways into the lopsided, flat-edged crescent a real Kerr hole shows. 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.