PhysForum Photos
sr_post5_unprimed
Between Events P and Q, the proper time is measured in the frame where the time-like vector from P to Q is stationary, thus Δt' in the frame S' is the proper time. And the proper time is shorter than any other time, like Δt measured in the frame S where the time-like vector from P to Q is seen as moving at speed v = 0.5c.
This is the picture where the red horizontal axis is spatial and the red vertical axis is time for the coordinates of the S frame.
sr_post5_unprimed
Between Events P and Q, the proper time is measured in the frame where the time-like vector from P to Q is stationary, thus Δt' in the frame S' is the proper time. And the proper time is shorter than any other time, like Δt measured in the frame S where the time-like vector from P to Q is seen as moving at speed v = 0.5c.
This is the picture where the red horizontal axis is spatial and the red vertical axis is time for the coordinates of the S frame.