# Spacetime and Timelike Intervals

The difference between a “timelike” spacetime interval and a “spacelike” spacetime interval can be understood in the following way: If the spacetime interval between two events is timelike, there exists a reference frame which measures the proper time between the two events; i.e. it sees the events occur at the same position. It the spacetime interval between two events is spacelike, there exists a reference frame which measures the proper length between two events; i.e. it sees the events occurring simultaneously

Now, suppose that the S reference frame measures the following spacetime coordinates for three separate events:

Event 1: ($x_1 = 300\, \mathrm{m};\, t1 = 3.0\, \mathrm{\mu s}$)

Event 2: ($x_2 = 700\, \mathrm{m};\, t2 = 5.0\, \mathrm{\mu s}$)

Event 3: ($x_3 = 1400\, \mathrm{m};\, t3 = 6.0\, \mathrm{\mu s}$)

Find $(\delta s)^2$ between Event 2 and Event 1. Is this a timelike or a spacelike separation? Find the speed (relative to S) of a reference frame S’ that measures either the proper time or the proper length between the two events. What is this proper time or proper length?

So, finding $(\delta s)^2$:

$$(\delta s)^2 = (c \delta t)^2 - (\delta x)^2$$

$$(\delta s)^2 = [(3 \cdot 10^8\, \mathrm{m/s})(2\, \mathrm{\mu s})]^2 - (400\, \mathrm{m})^2 = 200\, \mathrm{km}$$

So it's a timelike interval and event 1 can affect event 2. How can I begin finding the speed of $S'$?

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