Some Consequences of the Lorentz Equations — Question 1

Question 1

Interval signs distinguish three physically different connections

Relative to an origin event O, three events have $(ct,x)=(5,3)$, $(3,5)$ and $(4,4)$ metres, with no transverse displacement.

1. Calculate and classify the three invariant intervals.

2. Identify which events can be reached from O by a massive inertial traveler or a vacuum light ray.

Question 1 - Solution

Interval signs distinguish three physically different connections

(a) Method and calculation

$$s_A^2=5^2-3^2=\boxed{16\,\mathrm m^2\quad\text{timelike}}.$$$$s_B^2=3^2-5^2=\boxed{-16\,\mathrm m^2\quad\text{spacelike}}.$$

(b) Interpretation and further calculation

$$s_C^2=4^2-4^2=\boxed{0\quad\text{null}}.$$$$\boxed{u_A=3c/5<c},\quad u_B=5c/3>c,\quad\boxed{u_C=c}.$$

Physical check

A future spacelike event cannot be reached by either a massive traveler or a light signal from O.