The Postulates — Question 4

Question 4

Two opposite null rays constrain the mixing of space and time

Assume a linear one-dimensional map $x\prime=A(x-vt)$ and $t\prime=B(t-\kappa x)$, with positive A and B. Require both $x=ct$ and $x=-ct$ to map to vacuum rays of speeds $+c$ and $-c$.

1. Write the two constraints imposed by the rays.

2. Solve for B/A and $\kappa$, without determining the common scale A.

Question 4 - Solution

Two opposite null rays constrain the mixing of space and time

(a) Method and calculation

$$A(c-v)=cB(1-\kappa c).$$$$A(c+v)=cB(1+\kappa c).$$

(b) Interpretation and further calculation

$$2Ac=2cB\quad\Rightarrow\quad\boxed{B=A}.$$$$2Av=2c^2B\kappa\quad\Rightarrow\quad\boxed{\kappa=v/c^2}.$$

Physical check

Light-speed invariance fixes the time-space mixing here. Reciprocity and the inverse map are still needed to fix A.