Instructions
- Click to add events
- Drag events to move them
- Adjust β slider to change reference frame
- Scroll to zoom, drag to pan
- Show moving frame axes (x', ct')
Selected Event
Click an event to see details
Click to place events in spacetime, drag the β slider to change reference frames, and watch how simultaneity lines and light cones reveal the structure of special relativity
A Minkowski diagram is a powerful way to visualize spacetime and understand special relativity. Instead of thinking about space and time separately, we plot them together: the horizontal axis represents position in space (x), and the vertical axis represents time (ct, multiplied by the speed of light so both axes have the same units).
The gray axes represent the "rest frame" — the reference frame we usually think from. Vertical lines are objects at rest (moving through time but not space), while horizontal lines represent different moments in time occurring simultaneously.
When you adjust the β slider, blue axes appear. These represent a reference frame moving at velocity β·c relative to the rest frame. Notice how both axes tilt toward the light cone — this is the key to understanding relativity! The tilted ct' axis shows the path of an object at rest in the moving frame, while the tilted x' axis shows what's simultaneous in that frame.
These 45° lines represent light rays. Since we use units where c=1, light always travels at one unit of space per unit of time. Nothing can travel faster than light, so all possible worldlines (paths through spacetime) must stay within the light cone. Events inside your past light cone could have caused what you observe now; events inside your future light cone can be influenced by actions you take now.
These lines show events that are simultaneous in the moving frame. Here's where it gets wild: two events that happen "at the same time" in one frame happen at different times in another frame! This is the relativity of simultaneity, and it's not an illusion — there is no absolute "now" across distant locations.
The Relativity of Simultaneity: Place two events A and B at the same height (same ct value). They're simultaneous in the rest frame. Now drag the β slider — watch how their ct' coordinates become different! Enable simultaneity lines to see which events are simultaneous in the moving frame. Events that are simultaneous in one frame are not simultaneous in another.
Time Dilation: Place event A at (x=0, ct=0) and event B at (x=0, ct=4). These represent two ticks of a stationary clock. Now increase β and observe the ct' values. The moving observer measures a longer time interval (ct'_B - ct'_A > 4). Moving clocks run slow!
Length Contraction: Place events A and B at (ct=0, x=0) and (ct=0, x=4) — the endpoints of a 4-unit rod at rest. Now move the β slider. To measure the rod's length in the moving frame, you need to mark both endpoints simultaneously in that frame, meaning along a simultaneity line (blue dashed). The spatial distance (Δx') between where that line crosses the rod's worldlines is less than 4. The rod is contracted!
Causality and the Light Cone: Place event A at the origin. Notice that only events within the future light cone (above and within 45° lines) can be causally affected by A. Events outside the light cone are "spacelike separated" — you'd need to travel faster than light to connect them. Different observers may even disagree on their time ordering!
Velocity Addition: Create a sequence of events representing an object moving at some angle (worldline). Now boost to a moving frame. The object's velocity changes, but watch carefully — it never exceeds the speed of light (worldline never becomes shallower than 45°). Velocities don't add linearly like v₁+v₂; they combine relativistically.
The Invariant Interval: Place two events A and B anywhere. Select each and note their coordinates in both frames. Calculate s² = (Δct)² - (Δx)² in both frames. You'll find it's the same! The spacetime interval is invariant — all observers agree on it, even though they disagree on space and time separately.
Minkowski diagrams reveal that space and time are not separate, absolute backgrounds but are interwoven into a single spacetime continuum. What looks like "just space" to one observer looks like a mix of space and time to another. There is no universal "now," no absolute simultaneity, no preferred reference frame.
Yet despite this apparent chaos, there are invariants — quantities all observers agree on, like the spacetime interval s². These invariants are the true, objective features of reality. Special relativity teaches us that the universe doesn't care about our human notions of space and time as separate things. It speaks the language of spacetime.