At a glance
A time coordinate is not the moment the news of an event reaches you; it is a label a frame gives the event. This sim places events on one frame's grid, then adds a second frame moving past it, so you can compare two sets of labels for the same event.
What you'll learn
- What an event is, and what a frame adds to it.
- How a frame decides the time of an event far from its origin clock.
- How a second frame, moving at speed v, labels the same event, and the formulas that link the two sets of labels.
- Which quantity both frames still agree on.
Frame S (blue) with its clock at the origin and an event further along x. Frame S′ (amber) glides past at speed v.
What's on screen
- Blue grid: frame S, with x and y axes. A click on it places an event.
- Amber grid: frame S′, drawn when you switch it on. It moves along +x at speed v and carries its own clock at its origin.
- Event tags: each dot carries its label, E₁ (x, y, t) [S], plus an amber (x′, y′, t′) [S′] line when S′ is on.
- Top-right clocks: the S clock, t [S], and, with S′ on, a line for S′ and its speed v.
See itFrame S alone, with its clock running. Watch for: the clock in the top-right corner counting up, t [S].
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Click the grid | Places an event at the nearest whole-number (x, y) and tags it with its labels. | up to 10 events | |
| Relative speed v / c | How fast S′ moves along +x relative to S, in units of c. | 0–0.90 opens at 0.40 | |
| Lorentz factor γ | Read-only: γ = 1/√(1 − v²) for the current v. | 1 and up 1.091 at v = 0.40 | |
| ＋ Frame S′ | Draws the amber grid, the origin clock of S′, and an S′ line on every event tag. | on / off opens off | |
| ⏱ Reset clocks | Sets the clocks back to zero. Placed events stay. | button | |
| ✕ Clear events | Removes every event. The clocks keep running. | button | |
| Measure simultaneous events | Drops three events with one S time: two at x = 3 and one at x = 10. A second press clears them. | on / off | |
| ▶ Play / ⏸ Pause | Runs or freezes both clocks. Events can still be placed while paused. | top bar | |
| Speed | How fast the clocks run on screen. | 0.25×–4× opens at 1× | |
| ↻ Reset | Back to the start: clocks to zero, v = 0.40, S′ off, events cleared, paused. | top bar |
Fixed: c = 1. Lengths are in light-units, times in the matching time-units, and v is a fraction of c. S′ always moves along +x.
A frame is a lattice of clocks
- An event is one point of spacetime: a place and a moment, here (x, y, t). It is something that happens, not an object.
- A frame is a rule that gives events labels. Frame S is a lattice of rulers with a synchronised clock at every point. The sim draws only the clock at the origin; it stands in for all the others.
- An event's time is the reading of the clock beside it, at the moment it happens. The tag E₁ (x, y, t) [S] is that label, nothing more.
Frame S at the sim's opening settings. The event is out at x = 10.
The same event gets different labels in different frames.
Key idea. Coordinates belong to a frame. The event itself does not change when you change frame.
See itOne event placed at (4, 2) on frame S. Watch for: the tag beside the dot, E₁ (x=4, y=2, t=0.00) [S].
The time S gives a distant event
The time comes from the clock beside the event, so x and y do not enter it.
- An event at x = 10 gets the reading of the clock standing at x = 10. At the instant the origin clock reads 0, that is t = 0.
- With c = 1, the flash takes 10 time-units to reach the origin, so the origin first learns of it at t = 10. That is when the news arrives. The frame subtracts transit time; it never adds it.
- What you see is a picture built from light of different ages. What a frame reports is a map, with each event carrying its own label.
Spacetime diagram, c = 1: light runs at 45°. The event is at t = 0; t = 10 is when the origin finds out.
See itAn event placed at x = 10 while S's clock reads 0. Watch for: the tag reading t = 0.00 [S], not t = 10.
What S′ gives the same event
Frame S′ has its own rulers and its own synchronised clocks, moving at v along +x. It relabels events with the Lorentz transformation, which Lecture 10 derives; here it is the sim's rule.
The term −γvx is the whole difference at a shared S time.
v
These are the sim's own controls: the grid follows.
See itOne event at (6, 2), shown in both frames. Watch for: the amber t′ differing from the blue t.
See itRun both clocks with S′ drawn. Watch for: the S′ origin clock, t′, falling behind the S clock, t.
Simultaneity depends on the frame
- Two events with the same S time have Δt = 0, so S′ gives them Δt′ = −γvΔx. The offset grows with the separation along x, not with distance from any clock.
- Events simultaneous in S keep a common t′ only if they share x (or v = 0). The y values do not matter, since t′ has no y in it.
- The cause is that S and S′ synchronise their clocks differently. It is not signal delay.
Two events with one S time, labelled by both frames at v = 0.40.
If the clocks are running when you place events, all of them share whatever t the S clock shows. Every t′ moves up by γt together, so the differences are unchanged.
See itThree events with one S time: two at x = 3, one at x = 10, S′ on. Watch for: the two x = 3 tags sharing one t′, and the x = 10 tag reading lower.
The interval is the same in both frames
The two frames disagree on each label, but the relabelling keeps one combination fixed. For a pair of events, with c = 1:
Δs² is positive for a pair like this one, where space beats time. Here Δy = 0, so it drops out.
- Take events at x = 3 and x = 10 with the same S time and the same y. In S: Δx = 7, Δt = 0, so Δs² = 49.
- In S′ at v = 0.40: Δx′ = 7.64 and Δt′ = −3.06, so −3.06² + 7.64² ≈ 49.
- The sim does not print Δs². You can work it out from the tags.
Key idea. The labels belong to the frame. The interval belongs to the pair of events.
See itTwo events at x = 3 and x = 10, same y and same S time, with S′ on. Watch for: Δx′ coming out larger than 7 and Δt′ no longer zero, while Δx′² − Δt′² stays 49.
Where it matters
- Astronomy. Light from SN 1987A reached Earth in February 1987. The Large Magellanic Cloud is about 168,000 light-years away, so in the Galaxy's frame the star exploded roughly 168,000 years before. Astronomers subtract the transit time, just as frame S does.
- GPS. Each satellite broadcasts the time it sent its signal. The receiver subtracts that from its own clock reading. The difference, about 67 ms for a satellite about 20,000 km away, times c, gives the range.
- Setting clocks. Einstein's method: send a pulse from A at t₁ (A's clock), reflect it at B, receive it back at t₂. If light takes equal time each way, B's clock read (t₁ + t₂)/2 at the reflection. The delay is subtracted by construction.
Summary
In one line. An event's time is the reading of the clock beside it in a given frame. Another frame moving at v labels it t′ = γ(t − vx), and the interval is the same in both.
Key points
- A frame is rulers plus a synchronised clock at every point; distance never enters t.
- S′ labels events with t′ = γ(t − vx), x′ = γ(x − vt), y′ = y.
- The −γvx term splits S's single "now" by position: Δt′ = −γvΔx.
- Δs² = −Δt² + Δx² + Δy² is the same in both frames.
Common mix-ups
- "The recorded time should include the light-travel delay." That is when the flash arrives. The frame subtracts it, so the event keeps t = 0.
- "One clock at the origin does the timing." A frame has a synchronised clock at every point.
- "S′ gives the event t/γ." t/γ is the reading of the S′ clock at x′ = 0. An event at x ≠ 0 gets t′ = γ(t − vx).
Things to try on your own
- Turn on Frame S′, press Measure simultaneous events, and drag v from 0 up to 0.90. Read the three t′ tags at each setting.
- Place two events at the same x but different y, then compare their t′ tags.
- With Δy = 0, compute Δx′² − Δt′² from two tags at three values of v.