The proper time is defined to be the time measured between events in an inertial reference frame in which the two events occur at the same location in that frame: that is, a frame in which the clock does not move, Consider a distant star that is assumed to be at rest with respect to the Earth. An astronaut is assigned the task of travelling to the distant star. The events for this journey are defined as: Astronaut leaves Earth Astronaut arrives at star The astronaut travels to the star from the Earth in an inertial frame 5", while the astronomers on Earth remain in inertial frame S. The astronaut is carrying a clock at his origin. The astronaut is measuring the proper time between events for this journey because the times are measured by a clock that is at the same location (the origin) in reference frame 5. In contrast, astronomers on the Earth do not measure the proper time between the events because the events occur at different positions in frame S. Notice that although the astronomers cannot measure the proper time between the defined events, they can calculate it. [dentifying the proper time cannot be done without reference to events. Based on the derivation in lecture, the equation for proper time is: AT = Anf1- 8 Here, At is the time measured by an observer in any reference frame other than the one in which the proper time is being measured. 1. In the astronomer reference frame on the Earth (5), the astronaut events (leaves Earth, arrives at star) are separated in space by Ax and in time by At. Starting with the above equation for the proper time, show that an equivalent way of expressing the proper time between the two events, from the standpoint of your frame (5) alone, is: At = 4/ (At)?2 = (Ax/c)