Some Consequences of the Lorentz Equations — Question 5

Question 5

A hypothetical superluminal message reverses order in some frames

As a hypothetical test, a signal leaves A at the origin and reaches B with $\Delta x=2c\Delta t$, where $\Delta t>0$. Use $c\Delta t\prime=\gamma(c\Delta t-\beta\Delta x)$, $\beta=v/c$, and $\gamma=(1-\beta^2)^{-1/2}$.

1. Find the boosts for which B precedes A.

2. At $\beta=0.80$, calculate the transformed time difference and state the implication for treating this as an ordinary causal message.

Question 5 - Solution

A hypothetical superluminal message reverses order in some frames

(a) Method and calculation

$$\Delta t\prime=\gamma\Delta t(1-2\beta).$$$$\boxed{1/2<\beta<1\quad\Rightarrow\quad\Delta t\prime<0}.$$

(b) Interpretation and further calculation

$$\beta=0.80:\quad\boxed{\Delta t\prime=(5/3)(-0.60)\Delta t=-\Delta t}.$$$$s^2=c^2\Delta t^2-(2c\Delta t)^2=\boxed{-3c^2\Delta t^2<0}.$$

Physical check

This is a counterfactual spacelike connection, not a permitted local signal in special relativity. Its order is not invariant.