A5-3. Special Relativity: Spacetime and SimultaneityHL ONLY

1. The Space-Time Interval

In Galilean relativity, the spatial distance between two events ($\Delta x$) and the time between them ($\Delta t$) are absolute—everyone agrees on them. In Special Relativity, different observers measure different distances and different times. However, they all agree on a combined mathematical quantity called the Space-Time Interval ($\Delta s$).

Invariant Space-Time Interval:

$$\Delta s^2 = (c\Delta t)^2 - \Delta x^2$$

No matter what reference frame you are in, $\Delta s^2$ yields the exact same number. It is an invariant property of the universe.

2. Simultaneity

A direct consequence of the Lorentz transformations is that simultaneity is relative. Two events that happen at the exact same time but at different locations in one reference frame will happen at different times in a frame moving relative to the first.

⚠ Conceptual Rule

If two events are spatially separated ($\Delta x \neq 0$), observers moving at different relative speeds will disagree on the order of events. "Now" is only a local concept; there is no universal "now" across the universe.

3. Spacetime (Minkowski) Diagrams

Spacetime diagrams are visual tools used to map events in 1D space and time.

Anatomy of a Spacetime Diagram:

  • Vertical Axis ($ct$): Represents time. It is multiplied by $c$ so that the units (meters) match the x-axis.
  • Horizontal Axis ($x$): Represents 1D spatial position.
  • Worldline: The path an object traces through spacetime. A stationary object has a perfectly vertical worldline. A moving object has a slanted worldline.
  • Light Cone: Light ($v = c$) always travels at a $45^\circ$ angle on these diagrams (since $x = ct$). The worldline of any object with mass must be steeper than $45^\circ$ because it cannot exceed $c$.