gen-define-the-spacetime-interval-and-give-one-short-worked-exam-702a

Define the spacetime interval and give one short worked example.

# Spacetime Interval The **spacetime interval** between two events is an invariant quantity in special relativity, meaning all observers measure the same value regardless of their relative motion. ## Definition For two events separated by spatial distance Δ**r** and time Δt, the spacetime interval is: $$\Delta s^2 = -c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ Or equivalently: $\Delta s^2 = -c^2\Delta t^2 + |\Delta \mathbf{r}|^2$ **Note:** Some texts use the opposite sign convention $(+---)$ instead of $(-+++)$. The interval can be: - **Timelike** if $\Delta s^2 < 0$ (events can be causally connected) - **Spacelike** if $\Delta s^2 > 0$ (events cannot be causally connected) - **Lightlike/null** if $\Delta s^2 = 0$ (events connected by light signal) ## Worked Example A lamp flashes at position x = 0. Observer A (at rest) measures the flash at t = 0 and sees the reflection from a mirror at x = 3 light-seconds at t = 6 seconds. **Calculate the spacetime interval between these events:** - $\Delta t = 6$ s, $\Delta x = 3$ light-seconds, $c = 1$ light-second/s $$\Delta s^2 = -(1)^2(6)^2 + (3)^2 = -36 + 9 = -27 \text{ s}^2$$ This is **timelike** (negative), confirming the events can be causally connected (as expected for a light signal).