Worldline / a relativistic instrument

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.