(b) Alice and Bob are on the platform of a train station. In the station frame of reference, Alice is at rest, while Bob is running alongside the tracks at a speed 4c/5. A train of proper length L is moving in the same direction at speed 3c/5. Bob starts off behind the train, and eventually passes it. Let event E₁ be when Bob just reaches the back of the train, and let E2 be when Bob just reaches the front of the train. (i) Find Ax and At between events E₁ and E2 in the three frames of Alice, Bob, and the train. (ii) Calculate c² (At)² - (Ax)² numerically in the three frames, and thereby show that it has the same value (you can use a calculator for this!). Comment.

(b) Alice and Bob are on the platform of a train station. In the station frame of reference, Alice is at rest, while Bob is running alongside the tracks at a speed 4c/5. A train of proper length L is moving in the same direction at speed 3c/5. Bob starts off behind the train, and eventually passes it. Let event E₁ be when Bob just reaches the back of the train, and let E2 be when Bob just reaches the front of the train. (i) Find Ax and At between events E₁ and E2 in the three frames of Alice, Bob, and the train. (ii) Calculate c² (At)² - (Ax)² numerically in the three frames, and thereby show that it has the same value (you can use a calculator for this!). Comment.