The Lorentz Transformation — Question 9

Question 9

Spacetime-axis tick marks require invariant calibration

A boost has $\beta=0.60$. Use arbitrary common length units for ct and x. Consider unit events A at $(ct\prime,x\prime)=(1,0)$ and B at $(0,1)$.

1. Find A and B in the unprimed chart.

2. Compare their Euclidean distances from the origin with their invariant intervals.

Question 9 - Solution

Spacetime-axis tick marks require invariant calibration

(a) Method and calculation

$$A:\ (ct,x)=\gamma(1,\beta)=\boxed{(1.25,0.75)}.$$$$B:\ (ct,x)=\gamma(\beta,1)=\boxed{(0.75,1.25)}.$$

(b) Interpretation and further calculation

$$r_{E,A}=r_{E,B}=\sqrt{1.25^2+0.75^2}=\sqrt{17/8}\ne1.$$$$\boxed{s_A^2=+1},\qquad\boxed{s_B^2=-1}.$$

Physical check

A ruler applied to the Euclidean drawing does not calibrate a tilted spacetime axis. The interval calibrates its ticks.