Special Relativity - Space-Time Interval

In inertial frame S, two events occur at the same instant in time and 3c·minutes apart in space. In inertial frame S’, the same events occur at 5c·minutes apart. What is the time interval between the events in S’?

A. 0 min
B. 2 min
C. 4 min
D. 8 min
E. 16 min

(GR9677 #50)

Solution:

dS²  =  (dx)² − (cdt)²

dS²  =  dS'²

(3c)²  =  (5c)² − (ct)²

9c²  =  25c² − (ct)² 

t²  =  25 − 9 = 16 

t = 4

Answer: C

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

Special Relativity - Space-Time Interval

In an inertial reference frame S, two events occur on the x-axis separated in time by Δt and in space by Δx. In another inertial reference frame S’, moving in the x-direction relative to S, the two events could occur at the same time under which, if any, of the following conditions?

A. For any values of Δx and Δt.
B. Only if 
C. Only if 

D. Only if 
E. Under no condition

(GR0177 #34)

Solution:

The Invariant Space-Time Interval:  ΔS² = Δx² − (cΔt)² 

For the two events occur at the same time: ΔS²  0
Δx² − (cΔt)²  0
Δx²  (cΔt)²
(Δx/Δt)²  c²
|Δx/Δt|  c

Answer: C