Light Cone

Constructing the cone

Pick an event P in spacetime — for concreteness, "you snap your fingers right now at this exact spot." Imagine a flash of light goes off at P. After one nanosecond, that light has reached every point within 30 cm of P. After one second, every point within 300,000 km. After one year, a sphere of radius one light-year. Plot these in (ct, x, y, z) and the surface is a cone opening into the future — the future light cone.

Symmetrically, light that arrived AT P came from some shrinking sphere in the past. That past sphere traces a cone opening backward into the past — the past light cone.

In equations, the cone is the surface where the invariant interval from P vanishes:

c²(t − t_P)² − (x − x_P)² − (y − y_P)² − (z − z_P)² = 0

This is a 3D hypersurface in 4D spacetime. When we suppress one or two spatial dimensions (as we must for plotting), the cone becomes the familiar circular cone in 3D.

Four spacetime regions

Region	Defined by	Causal relation to P
Future light cone surface	c²Δt² = Δr², Δt > 0	Reachable by photon from P
Past light cone surface	c²Δt² = Δr², Δt < 0	Photon from event could reach P
Inside future cone (timelike future)	c²Δt² > Δr², Δt > 0	Reachable by <c signal; causal future
Inside past cone (timelike past)	c²Δt² > Δr², Δt < 0	Could have causally influenced P
Outside cones (spacelike "elsewhere")	c²Δt² < Δr²	No causal contact; time order frame-dependent
Cone apex (P itself)	Δt = Δr = 0	The event itself

Every event in the universe sits somewhere relative to P's cones, and that classification is invariant under Lorentz boosts. Boosting the diagram tilts the t and x axes hyperbolically, but the 45° cone surface is unchanged — light always moves at c for every observer (postulate 2).

Worked example — "Is the Andromeda Galaxy in our light cone?"

Take P = "you, right now." The Andromeda Galaxy is roughly 2.5 million light-years away.

• What you see right now. Light arriving at your eyes left Andromeda 2.5 Myr ago — that emission event is on your past light cone exactly.
• What's happening at Andromeda "right now." Spacelike-separated from you. Outside your light cone. You can't observe it, can't be influenced by it, and other inertial frames disagree about whether it's "before" or "after" your snap.
• What will be observed. Light that leaves Andromeda right now will reach you in 2.5 Myr — that future emission event is inside your future light cone after 2.5 Myr has passed.
• Tachyon hypothesis. If something travels faster than c, it could leave you "now" and arrive at Andromeda before its "now" in some frames. Some frames would see this as a signal arriving in their past — causal paradox. This is why physics rules out tachyons.

Lorentz boost preserves the cone

A boost transforms (ct, x) → (ct', x') = (γ(ct − βx), γ(x − β·ct)). Pick any point on the cone, say (ct, x) = (1, 1) (a future-pointing null ray). After boost:

ct' = γ(1 − β·1) = γ(1 − β)
x'  = γ(1 − β·1) = γ(1 − β)
⇒ ct' = x'    — still on the cone, in the new frame.

Same for x = −ct (the other null ray). The cone surface is mapped to itself. Geometrically, a boost is a hyperbolic rotation that pinches axes toward the cone; the cone itself is the asymptote.

Why this is the foundation of causality

Quantum field theory enforces "microcausality": for any two local operators φ(x) and ψ(y), the commutator [φ(x), ψ(y)] = 0 if x and y are spacelike-separated. This makes the operators commute, hence no signal between them, no detectable correlation that could be used to send information faster than light. The light cone defines what "spacelike" means here.

For a single particle in special relativity, the rule reduces to "world lines stay timelike or null." Massive particles have timelike world lines (inside the cone everywhere); photons have lightlike (on the cone); no particle has a spacelike world line.

JavaScript — classifying separations and boosts

const c = 299792458;  // m/s

// Classify the separation between two events
function classifySeparation(t1, r1, t2, r2) {
  const dt = t2 - t1;
  const dr = Math.sqrt((r2[0]-r1[0])**2 + (r2[1]-r1[1])**2 + (r2[2]-r1[2])**2);
  const s2 = c*c*dt*dt - dr*dr;
  if (s2 > 0) return dt > 0 ? 'future timelike' : 'past timelike';
  if (s2 < 0) return 'spacelike (elsewhere)';
  return dt > 0 ? 'future null (light)' : 'past null (light)';
}

// Andromeda example: you here-now vs Andromeda 2.5 Myr ago
const here_now = [0, [0, 0, 0]];
const andromeda_emission = [-2.5e6 * 365.25 * 86400, [2.5e6 * 9.461e15, 0, 0]];  // 2.5 Myr ago, 2.5 Mly away
console.log('Andromeda emission:', classifySeparation(here_now[0], here_now[1], andromeda_emission[0], andromeda_emission[1]));
// past null (light) — exactly on the past cone

// Andromeda 'now' — same spatial position, t = 0
const andromeda_now = [0, [2.5e6 * 9.461e15, 0, 0]];
console.log('Andromeda now:', classifySeparation(here_now[0], here_now[1], andromeda_now[0], andromeda_now[1]));
// spacelike (elsewhere) — not in our light cone

// Future arrival: light emitted at Andromeda now, will arrive here at t = 2.5 Myr
const future_arrival = [2.5e6 * 365.25 * 86400, [0, 0, 0]];
console.log('Future light arrival vs Andromeda-now:', classifySeparation(andromeda_now[0], andromeda_now[1], future_arrival[0], future_arrival[1]));
// future null

// Lorentz boost in x preserves the cone
function lorentzBoost(t, x, vx) {
  const beta = vx / c;
  const gamma = 1 / Math.sqrt(1 - beta * beta);
  return [gamma * (t - beta * x / c), gamma * (x - vx * t)];
}

// A point on the future light cone in some frame
const cone_point = [1, c];  // t=1, x=c·1; ct - x = 0
const boosted = lorentzBoost(cone_point[0], cone_point[1], 0.7 * c);
console.log('Boosted cone point:', boosted);
console.log('Still on cone? c*t - x =', (c*boosted[0] - boosted[1]).toExponential(3));  // ≈ 0

// Time order reversal for spacelike-separated pair
function timeOrderInBoostedFrame(t1, x1, t2, x2, vx) {
  const e1 = lorentzBoost(t1, x1, vx);
  const e2 = lorentzBoost(t2, x2, vx);
  return e2[0] - e1[0];  // > 0 means e1 before e2; < 0 means reversed
}

// Two spacelike-separated events: same time, different positions
const evtA = [0, 0];
const evtB = [0, 5e8];  // 500,000 km away simultaneously
console.log('At rest: t(B) - t(A) =', timeOrderInBoostedFrame(evtA[0], evtA[1], evtB[0], evtB[1], 0));
console.log('Boosted +0.5c: t(B) - t(A) =', timeOrderInBoostedFrame(evtA[0], evtA[1], evtB[0], evtB[1], 0.5*c));
console.log('Boosted -0.5c: t(B) - t(A) =', timeOrderInBoostedFrame(evtA[0], evtA[1], evtB[0], evtB[1], -0.5*c));
// One positive, one negative — time order reversed!

Where light cones matter

• Causality in QFT. Microcausality requires commutators of fields to vanish at spacelike separation — directly enforces the light cone structure.
• Cosmology. Particle horizons (limits of what we can see) and event horizons (limits of what can ever reach us) are tilted cones in cosmological spacetime.
• Black holes. Inside the event horizon, light cones tilt entirely inward — no future-directed paths lead out. The singularity is in your future, like Tuesday.
• Gravitational waves. Propagate at exactly c along null surfaces; detected when their wavefronts cross LIGO's mirrors.
• Numerical relativity. Causal-set and Cauchy-surface codes use the local cone structure to slice spacetime and integrate forward.
• Quantum information. No-signalling theorems and Bell tests rely on the experimenters being spacelike-separated, ensuring no causal link could explain the correlations.

Common misconceptions

• "Outside the cone = nowhere." The outside is the spacelike "elsewhere" — most of spacetime, in fact. It just has no causal contact with the apex.
• "Spacelike events happen at the same time." Only in one specific frame. In other frames the spacelike pair is time-ordered either way.
• Confusing the cone surface with its interior. The cone IS the null surface (light paths). The interior is the timelike region.
• "My light cone is a fixed object in space." Each event has its own cone. As you move through time, your cone shifts to a new apex.
• "Lorentz boost tilts the cone." It tilts the axes, not the cone. Light moves at c for everyone, so the 45° surface is invariant.
• "Tachyons just go fast." They would be SPACELIKE world lines, violating causality. No experiment has ever found one, and QFT excludes them as physical particles.