The Postulates — Question 1

Question 1

A Galilean clock convention fails for both light directions

In inertial frame S, two vacuum pulses obey $x=\pm ct$. A proposed frame moving at $v=0.40c$ uses $x\prime=x-vt$, $t\prime=t$.

1. Find the two signed light velocities predicted by this map.

2. Compare with the light postulate and determine whether changing only the spatial origin can repair the discrepancy.

Question 1 - Solution

A Galilean clock convention fails for both light directions

(a) Method and calculation

$$u\prime=\frac{dx\prime}{dt\prime}=u-v.$$$$\boxed{u\prime_+=0.60c},\qquad\boxed{u\prime_-=-1.40c}.$$

(b) Interpretation and further calculation

$$\text{Vacuum postulate:}\qquad u\prime_+=c,\quad u\prime_-=-c.$$$$x\prime\to x\prime+a:\quad\frac{d(x\prime+a)}{dt\prime}=u-v\ne\pm c.$$

Physical check

A constant origin shift changes coordinates but cannot change a measured slope. The time transformation must also change.