The Postulates — Question 5

Question 5

Preserving a light sphere requires transforming its time coordinate

A spherical pulse in S satisfies $x^2+y^2+z^2=c^2t^2$. Test the candidate map $x\prime=\gamma(x-vt)$, $t\prime=\gamma(t-vx/c^2)$, $y\prime=y$, $z\prime=z$, where $\gamma=(1-v^2/c^2)^{-1/2}$.

1. Expand $c^2t\prime^2-x\prime^2$ and simplify.

2. Show that the transformed pulse is also a sphere at each fixed $t\prime$.

Question 5 - Solution

Preserving a light sphere requires transforming its time coordinate

(a) Method and calculation

$$c^2t\prime^2-x\prime^2=\gamma^2[(ct-\beta x)^2-(x-\beta ct)^2].$$$$=\gamma^2(1-\beta^2)(c^2t^2-x^2)=c^2t^2-x^2.$$

(b) Interpretation and further calculation

$$c^2t\prime^2-x\prime^2-y\prime^2-z\prime^2=0.$$$$\boxed{x\prime^2+y\prime^2+z\prime^2=c^2t\prime^2}.$$

Physical check

Points sampled at fixed t need not be sampled at fixed transformed time. A spatial slice cannot be transformed as if its times were unchanged.