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Title: GR8677.021
Post by: conantee on December 31, 2009, 11:06:44 AM
21. Two observers O and O^\prime observes two events, A and B. The observers have a constant relative speed of 0.8 c. 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^\prime?
(A) 1
(B) \sqrt{2}
(C) 2
(D) 3
(E) 2 \sqrt{3}


Title: Re: GR8677.021
Post by: conantee on December 31, 2009, 11:53:43 AM
First, the space-time interval is independent of observer. So, don't worry about the speed of observer O^\prime.
Next, let's calculate the space-time interval between two events. You should know that space and time are treated differently. Instead of usual cartezian coordinates, time is treated in imaginary axis. The "distance" in space-time is then can be calculated from:
(\Delta D) ^2 = (\Delta x)^2 + (\Delta y) ^2 + (\Delta z) ^2 - (\Delta t)^2 where speed of light is 1

In our problem \Delta x = 5-3 = 2, \Delta y = 3-3 = 0, \Delta z = 3-1 =2, \Delta t = 3-5 = -2
Hence, (\Delta D)^2 = 2^2 +0^2+ 2^2 - (-2)^2 = 4. So, \Delta D = 2.

The answer is (C). 24 of 100 people got the right answer.