Special Relativity

Einstein’s key insight was not just that moving clocks run slow or moving rulers contract. It was that space and time have to be described together if the speed of light is the same for every inertial observer. A spacetime diagram gives us a way to see that geometry directly. Instead of tracking position alone, we track an event by its spatial coordinate $x$ and its time coordinate $ct$, where multiplying time by $c$ puts both axes in the same units.

Events, Worldlines, and Frames

An event is a single happening: a flash, a detector click, a spaceship passing a marker. On the diagram, each event is one point. A moving object leaves behind a worldline, the path of all its events through spacetime.

• A stationary object has a vertical worldline because its position stays fixed while time passes.
• A moving object has a slanted worldline because both position and time change.
• A light pulse travels on a $45^\circ$ line, since it obeys $x = \pm ct$.

The axes $(x, ct)$ describe one observer’s frame. A second observer moving at constant velocity relative to the first uses a different set of axes, $(x^{\prime}, ct^{\prime})$. The important change is not that one observer is “wrong” and the other is “right”; it is that they slice spacetime into “space” and “time” differently.

The Light Cone

The lines $x = ct$ and $x = -ct$ form the light cone. They mark the worldlines of light that passes through the origin. Different inertial observers may disagree about coordinates, but they agree on the causal structure:

• events inside the light cone can be connected by slower-than-light motion
• events on the cone can be connected by light
• events outside the cone cannot causally influence one another without exceeding $c$

As the axes of the moving frame tilt, they never cross outside the light cone. They lean toward it as $|\beta|$ increases, where

Simultaneity

In ordinary Newtonian intuition, “at the same time” feels universal. In relativity, it is not. For the unprimed observer, events on a horizontal line share the same value of $ct$, so they are simultaneous in that frame. But the moving observer uses tilted lines of constant $ct^{\prime}$ instead. That means two events can be simultaneous for one observer and not simultaneous for another.

This single geometric fact drives a lot of the rest of the theory:

• Time dilation emerges when one frame compares elapsed time along a moving clock’s worldline.
• Length contraction emerges because length must be measured using endpoints recorded at the same time in the measuring frame.

Invariant Interval

Although observers disagree about coordinates, they agree on certain combinations of them. The most important one in this diagram is the spacetime interval

That value is invariant under relativistic (Lorentz) transformations. So when you click an event in the simulation and compare its primed and unprimed coordinates, the numbers may change, but the interval does not.

Minkowski Diagrams

As you explore the diagram below, try this sequence:

1. Start with $\beta = 0$ and notice that the primed and unprimed axes coincide.
2. Increase $\beta$ and watch the $x^{\prime}$ and $ct^{\prime}$ axes tilt toward the light cone.
3. Place an event on the graph and compare its coordinates in both frames.
4. Select the event to see its projections onto both sets of axes.
5. Turn the light cone, grids, and ticks on and off to isolate the geometric idea you want to study.

What to Notice

As you work with the diagram, pay attention to three patterns:

• As the relative speed increases, the primed axes tilt closer to the light cone.
• Events that look simultaneous in one frame generally do not remain simultaneous in the other.
• The coordinates of an event change from frame to frame, but the invariant interval stays fixed.