Special Relativity - Space-Time Interval

Two observers O and O’ observe two events A and B. The observers have a constant relative speed of 0.8c. In units such that the speed of light is 1, observer O obtained the following coordinates:

Event A: x = 3, y = 3, z = 3, t = 3 

Event B: x = 5, y = 3, z = 1, t = 5

What is the length of the space-time interval between these two events, as measured by O’.

A. 1
B. √2
C. 2
D. 3
E. 2√3

(GR8677 #21)

Solution:

dS²  = dr² + (cdt)²  =  dx² + dy² + dz² − (cdt)² 
dS² = dS'² 

dx² = (5 − 3)² = 4
dy² = (3 − 3)² = 0
dz² = (1 − 3)² = 4
dt²  = (5 − 3)² = 4
c = 1

dS'² = 4 + 0 + 4 − 4 = 4
dS' = 2

Answer: C