Light Propagation

Suppose Patricia is driving her car at nearly the speed of light and turns on her headlights. For simplicity in calculations, in the rest frame of an observer on the ground the light takes one second to reach the stop sign 3 × 10⁸ meters away. This ground observer then sees the car reach the stop sign very soon after the initial light reaches the stop sign.

Patricia sees the light moving forward at 3 × 10⁸ m/sec also, but she sees the stop sign approaching her at nearly light speed. Therefore she sees the arrival of the light flash at the stop sign and her arrival there in quick succession.

Call the initial arrival of the light at the stop sign event A and the car’s arrival event B. Will the elapsed time between events A and B be the same for the driver as for the observer on the ground? No, because the ground observer sees both events occur at the same location, at the stationary stop sign, so Δx = 0. As seen by Patricia, these two events occur at two different locations separated by Δx ≠ 0.

Who measures the longer time interval between events A and B? Can you provide a conceptual argument for this non-intuitive result? If the speed of the car is closer to the speed of light, how does the difference in elapsed times measured by driver and ground observer change?

Answer

According to the special theory of relativity (STR), (1) no object can move at the speed of light, (2) the speed of light is the same for all observers, and (3) the space-time interval τ between two events defined by τ² = c² Δt² – Δx² – Δy² – Δz² is the same for all observers, but the Δt and Δx may be different, for example.

For one-dimensional motion τ² = c² Δt² – Δx². The driver has Δx ≠ 0, so her Δt must be greater for the two events than the elapsed time for the observer on the ground. Therefore the driver measures the longer time interval between events A and B.

Suppose the car makes a second run at a great speed. The nearer to the speed of light the car goes, the smaller is Δt for the ground observer, and τ = c Δt is smaller in this case also. But again, as expected, the time interval is longer for the driver. The nearer to the speed of light the car goes in the ground frame, the difference will be the difference in arrival times as observed in the two frames.