At a glance
A light pulse and a massive particle leave the same event and head for the same place, four light-seconds away. The sim draws both worldlines on a spacetime diagram and reads out the proper time each one accumulates, which is how relativity measures the length of a path through spacetime.
What you'll learn
- How the proper time along a worldline depends on the speed of whatever travels it.
- What the spacetime interval Δs² is, and what its sign tells you about two events.
- How energy is assigned to a massive particle and to a pulse of light.
- What limits how close a massive worldline can lean toward the 45° line.
Both worldlines start at E₁ and reach x = 4 light-seconds. The pulse arrives at E₂, the particle later, at E₂′.
What's on screen
- Pulse worldline (yellow): E₁ to E₂, with a glowing dot that moves along it. With Show energies on, it takes the pulse's wavelength colour.
- Particle worldline (cyan): E₁ to E₂′ at the v/c you set. Its dot carries a τ label, the reading of a clock travelling with it.
- Events E₁, E₂ and E₂′, a dashed line at ct = 4 where the pulse arrives, and a dashed vertical from E₂ to E₂′.
- Axes: x across and ct up, both in light-seconds on equal scales.
- Δτ wedge (violet): the shaded gap between the two worldlines, shown with Show comparison.
- Readout panel: Δτ for each worldline, plus the energies when they are on.
See itSend both out from E₁ together at your current v/c. Watch for: the yellow dot reaching E₂ while the cyan dot is still climbing toward E₂′.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| v/c (massive) | The particle's speed as a fraction of c. It tilts the cyan worldline and moves E₂′. | 0.05–0.995 opens at 0.500 | |
| m (eV/c²) | The particle's rest mass. It feeds the energy E = γmc². | 0.1–5 opens at 1.0 | |
| Speed presets | Jump v/c straight to a listed value. | 0.50, 0.80, 0.95, 0.99, 0.995 | |
| ▦ Show comparison | Shades the Δτ wedge between the two worldlines and labels each worldline's Δτ. | on / off opens off | |
| ⚡ Show energies | Adds the energy row to the Readout and brings in the ν slider. The pulse is then drawn in its wavelength colour. | on / off opens off | |
| ν (photon freq) | The pulse's frequency. It sets E = hν and the colour. Only visible with Show energies on. | 400–789 THz opens at 540 | |
| Readout | Δτ for the pulse and the particle, plus E for each when the energies are on. | panel opens folded | |
| ▶ Play / ⏸ Pause | Runs both worldlines out from E₁ in about 5 s at 1×, then stops at the end. Play again to rerun. | top bar | |
| Speed | How fast the run plays on screen. The worldlines and numbers stay the same. | 0.25×–4× opens at 1× | |
| ↻ Reset | Restarts the mode you are in from its beginning, with the opening settings. | top bar |
Fixed: Δx = 4 light-seconds, c = 1, so a time in seconds equals a ct value in light-seconds. Energies are in eV, with m in eV/c².
Proper time along a worldline
Between two events the sim uses the interval Δs² = −c²Δt² + Δx², with the time term negative. For the particle Δs² < 0, and a clock carried along its worldline reads the proper time Δτ = √(−Δs²)/c.
The spacetime interval.
The particle's Δt, Δs² and Δτ, with Δx = 4 light-seconds and c = 1.
v/c
m
These sliders move the sim too.
- Δt is the time on the frame's clocks. Δτ is the time on the clock the particle carries, always the smaller of the two.
- Mass does not enter. Δτ depends on Δx and v/c only. Slide m and Δτ stays put.
- Doubling v/c from 0.40 to 0.80 does not halve Δτ: it falls from 9.17 s to 3.00 s.
See itRun the finished worldline at m = 0.5, then at m = 4, with v/c unchanged. Watch for: the Δτ reading in the card staying the same while m changes.
The pulse's interval is zero
E₁ and E₂ are distinct events, four light-seconds apart in space and four seconds apart in time. Along the pulse Δx = cΔt, so the two terms of Δs² cancel exactly.
From the light signal to the proper time along it.
Three cases, set by the sign of Δs². The words are labels for those values.
The light cone at E₁. The 45° edge is where Δs² changes sign. The cyan line is the particle at v/c = 0.5.
- Zero separation between two distinct events cannot happen in Euclidean geometry. The minus sign in Δs² allows it.
- Every inertial observer measures light at c, so every one of them finds Δs² = 0 for this pair. The interval is invariant.
- A massive particle's worldline stays inside the cone. Only a signal at c lies on its edge.
See itDraw both finished worldlines from E₁. Watch for: the yellow line at exactly 45° from E₁ to E₂, with the cyan line steeper than it.
No inertial frame travels with the pulse
The limit v/c → 1 for a massive particle. It is approached and never reached.
- Δτ = Δx√(1−(v/c)²)/(v/c) is positive for every v/c < 1. At the slider's cap, 0.995, γ = 10.01 and Δτ = 0.40 s against Δt = 4.02 s.
- Reaching c would need γ → ∞, which is infinite energy. That is why the slider stops short of 1.
- The particle never closes in on the pulse. Velocities do not subtract the Newtonian way: for light, (c − v)/(1 − v/c) = c. Every inertial observer measures the pulse moving at c.
- So Δτ = 0 along the pulse is not a stopped clock. It is a property of the worldline: no inertial frame moves at c, so no clock can be carried along it.
| v/c | Δt (s) | Δτ (s) | γ |
|---|---|---|---|
| 0.50 | 8.00 | 6.93 | 1.15 |
| 0.90 | 4.44 | 1.94 | 2.29 |
| 0.99 | 4.04 | 0.57 | 7.09 |
| 0.995 | 4.02 | 0.40 | 10.01 |
Δx = 4 light-seconds throughout.
See itStep v/c through 0.50, 0.90, 0.99 and 0.995, with the wedge on. Watch for: the cyan line crowding the 45° line and the Δτ reading falling, without reaching zero.
Energy without mass
The massless limit. m = 0 does not make E or |p| zero. It ties them together.
- The pulse has E = |p|c = hν, set by its frequency alone. At the default ν = 540 THz, λ = c/ν = 555 nm (green) and E = 2.23 eV. The ν slider covers 400–789 THz, 1.65–3.26 eV.
- The particle has E = γmc², set by v/c and m. At v/c = 0.5 and m = 1 eV/c², E = 1.15 eV.
- E = γmc² does not apply to the pulse. γ is undefined at v = c, and m = 0 would make the product 0 × ∞. The Readout's energy row uses E = hν for it.
- Massless is not momentum-less: |p| = hν/c. The sim shows E only.
See itSwitch the energies on at your current settings. Watch for: the card listing E for the pulse and for the particle, and the pulse line drawn in its wavelength colour instead of plain yellow.
Where it matters
- Cosmic-ray muons. A muon formed about 15 km up at v/c = 0.998 needs 50 µs by ground clocks to reach the surface. Its own clock reads 3.2 µs, against a rest-frame mean lifetime of 2.2 µs. This is the sim's Δτ = Δt/γ at work.
- Sunlight. Light crosses 1 AU in 499 s (8.3 min) by Sun–Earth clocks. Its path has Δs² = 0, so there is no proper time to assign along it.
- Solar sails. Light carries momentum |p| = E/c. Full absorption of sunlight at Earth's distance gives a pressure of about 4.5 µPa.
Go deeper: the pulse as a limit
- Let m → 0 and v/c → 1 together, keeping E = γmc² fixed. Then |p|c/E = v/c → 1 and Δτ = Δx/((v/c)γ) → 0, which are the pulse's values.
- This makes the pulse the limit of a family of massive worldlines. It does not give it a rest frame.
- Along a null worldline τ cannot label the points, so the course uses an affine parameter λ instead.
Summary
In one line. Proper time is the length of a worldline, √(−Δs²)/c: positive and shrinking as v/c → 1 for the particle, zero for the pulse, and no inertial frame travels at c.
Key points
- For the particle, Δτ = Δx√(1−(v/c)²)/(v/c), independent of its mass.
- For the pulse Δx = cΔt, so Δs² = 0 in every inertial frame.
- Δτ = 0 describes the pulse's worldline. It does not describe a viewpoint.
- The pulse carries E = |p|c = hν, while the particle carries E = γmc².
Common mix-ups
- "Run fast enough and you see the photon at rest." The pulse recedes at c from every inertial observer, and no massive body reaches c.
- "The photon's clock is just stopped." A clock reading belongs to a frame, and none travels at c. Δτ = 0 is a fact about the path.
- "Massless means no energy and no momentum." E = hν and |p| = hν/c. It is the rest mass that is zero.
Things to try on your own
- Set v/c to 0.80 and note Δτ. Then jump to 0.99 and compare the two readings.
- Turn the energies on, move ν across its range, and record the pulse's E at both ends.
- With the energies on, set m = 5 and v/c = 0.995, then check the particle's E against γmc².