# Minkowski Diagrams Positions in space time We can represent

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```							Minkowski Diagrams Positions in space-time
We can represent the position of a particle moving back and forth along a line with one coordinate, x. To represent a particle’s position in both 1-dimensional space x and time t, we can use a graph with x as the horizontal axis and t as the vertical axis. A particle starts at x = 0 at time t = 0 and does not move. Draw a line representing its “path” through space-time. A path through space-time is called a world-line. If we measure distance along our horizontal axis in light-seconds and distance along our vertical axis in seconds, what does the world-line of a light ray look like? A rocket R moves in the +x direction relative to an observer A on Mars, at a speed 1/2 the speed of light c. Their positions coincide at t = 0. Plot the worldlines of A and R in a space-time diagram. The rocket emits light signals in both the forward and backward directions at t = 2 sec. Draw the corresponding world-lines of the light rays. When does the observer receive the signal? Two radar pulses sent out from the earth at 6:00 AM and 8:00 AM one day bounce oﬀ an alien spaceship and are detected on earth at 3:00 PM and 4:00 PM. You do not know which reﬂected pulse corresponds to which emitted pulse: is the spaceship moving toward earth, or away? If its speed is constant and less than c, when will it (or did it) pass by the earth?

Distance between two points in space-time
Suppose a particle is at x = 0 when t = 0, and assume it cannot move faster than the speed of light. What region of space-time represents the particle’s possible future locations? What region represents places and times the particle could have visited in the past? Illustrate these regions on a space-time diagram. Let d((x1 , t1 ), (x2 , t2 )) be deﬁned by d((x1 , t1 ), (x2 , t2 )) = (t2 − t1 )2 − (x2 − x1 )2 where (x1 , t1 ) and (x2 , t2 ) are points in space-time. What points (x, t) satisfy d((0, 0), (x, t)) > 0? What points satisfy d((0, 0), (x, t)) < 0? What world-line corresponds to the boundary of these regions?

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