Answer to Question #152149 in Mechanics | Relativity for Santo Domingo

Question

Consider Bob on Earth. At time t = 0, Alice flies by in her spaceship at constant velocity, heading to the right. (This is the same situation as the first question.) Which diagram represents Bob’s world line in Alice’s frame of reference?

Answer 1

Answer

For Bob and Alice diagram can be shown as

So according theoritical aspect

Now let’s relate this plot to a similar plot but interpreted by 

Bob in his jet. On Alice’s plot (which will be labeled by x 

and t) Bob moves off along the slanted worldline. In Bob’s 

reference frame he sees himself as standing still and Alice 

moving at constant velocity in the –x’ direction (Bob’s 

coordinates will be labeled by x’ and t’).

In Alice’s frame, her own worldline is vertical along the ct 

axis, because she does not move and is always at x = 0 m, at 

Hoover Tower. In Bob’s frame his own worldline is vertical 

as well since he is always at x’ = 0 m – always at the same 

spot within his jet. With this in mind, we can attempt to combine Alice’s and Bob’s 

spacetime plot into one single one. N.B. the two diagrams represent the same 4-dimensional spacetime but plotted in 

different coordinate systems, one which corresponds naturally to Alice’s reference frame and the other to Bob’s. The 

goal is to represent both coordinate systems on the same plot, this will help visualizing the geometry of 4-

dimensional spacetime.

It is easy to find how to properly plot Bob’s time axis within Alice’s plot. Begin by plotting Alice’s 

spacetime plot (with axes x and ct) and add Bob’s worldline but as the time coordinate ct’. This coordinate 

represents the IRF with the same velocity as Bob’s jet. The question is how to add Bob’s spatial direction, x’? This 

will require a little work. To do this we will need to return to the discussion of simultaneous events viewed in both frames. The following diagram represents Bob’s world line in Alice’s frame of reference as