Laboratory for 1/26/98

The Problem

Consider Beldar and Premat Conehead living on Earth. Beldar takes a trip to their home planet Remulac, which is at rest with respect to the Earth. Clocks on Earth and Remulac are synchronized in the Earth-Remulac inertial reference frame. Beldar quickly accelerates to a speed of .8660c ( Ì = 2.000 ) and travels for 4 years Earth-Remulac time. He quickly decelerates and spends only a short time on Remulac before returning at the same speed. The total trip takes eight Earth-Remulac years.

The Space-Time Diagram

The space-time diagram we will use plots time (in years) on the vertical axis and displacement (in lightyears) along the x direction on the horizontal axis, with equal spacing given for both years and lightyears. A world line on such a diagram represents the position vs time history of any object. The world line of a light signal on this diagram is represented by a 45o line and is called a light line.

Plotting

Carefully plot both Premat's and Beldar's world lines. Premat will be at rest in your diagram, so her world-line will be a vertical line along the vertical axis at x = 0. Beldar's world line will be a straight line whose slope can be found by simply noting how far along the x axis he travels in a given time. Note that Beldar's world line will reverse slope at the end of four of Premat's years.

Premat and Beldar have agreed to keep in touch by sending each other light signals at the end of each year, according to their own clocks. Place dots on the world lines representing one year intervals, as measured in the rest frame of each. For Premat, the dots simply match the divisions on the vertical time axis. The location of Beldar's time dots can be determined by considering the time elapsed in the Earth-Remulac frame for each of Beldar's years.

Draw light lines from each of these dots until they intersect the other's world line. It will enhance clarity if a color scheme is adopted to distinguish the world and light lines of Beldar and Premat.

Analysis

The relativistic equations can now be used to confirm and elucidate our graphical results. Show all calculations explicitly. Where appropriate, explain your choices for times and locations; that is, define the events in this exercise clearly. For simplicity, assume that both Beldar's and Premat's clocks reads zero when he leaves Earth.

1. Using the relativistic Doppler shift equation, determine how often each expects to receive signals from the other while Beldar is traveling away from Earth. Is this consistent with your space-time diagram? Note that Beldar will not receive a signal from Premat during the first part of the trip. It will be necessary to extrapolate the light line representing Premat's first signal to verify that Beldar would have received the signal at the time calculated.

   Repeat the calculation for the return portion of Beldar's trip? Is this consistent with your space-time diagram?

2. From your graph, how many signals does each receive from the other for the entire trip? The discrepancy represents a real difference in the passage of time on Earth and on Beldar's spaceship. How does this compare with the time dilation factor?

   Note that infinite acceleration was assume at the turn around point. In a more realistic calculation with finite accelerations (which is beyond the scope of this course), the difference in total elapsed time between the two frames would not be as simple as the results above. For such a case, Beldar's world line would not have the point inflection at the turn around point. Instead, his Earth to Remulac world line would curve smoothly upward and over matching the return trip world line.

   On a separate space-time diagram sketch Beldar's world line for a finite acceleration, making the curved section symmetric and comparable in length to the straight sections for clarity. Place about 5 dots along each straight section representing arbitrary but equal time intervals. How would the spacing of the dots representing the same time intervals placed on the curve section compare to those on the straight sections? (Consider what observers in the Earth-Remulac frame would claim is happening to Beldar's clock rate during the deceleration and acceleration portions.) Place dots along the curved section representing this fact. What would you deduce concerning the total number of dots along this realistic curved world line compared to the number on the original idealized world line? Would the descrepancy in aging be less or more in the realistic version? (Note that Beldar won't actually reach Remulac)

3. A description (in the Earth-Remulac frame) of the behavior of Beldar's clock must be made by either a series of observers or by one observer (e.g. Premat) who carefully takes into acccount the doppler effect on their observations. As calculated by Premat, describe the behavior of Beldar's clock during the two non-accelerating portions and the accelerating portion of his trip. This is most easily done by considering your space-time diagram and imagining a series of observers in the Earth-Remulac frame which are positioned along Beldar's path.

4. We now examine the behavior of Premat's clock according to Beldar. First, describe what Beldar calculates for the passage of time on Premat's clock during the non-accelerating portions of the trip. Does this differ from Premat's observations of Beldar's clock? Is this what you would expect on the basis of the first principle of relativity?

   Understanding Beldar's view of Premat's clock during the acceleration portion of his trip requires a little work. During the first non- accelerating portion, Beldar claims that Premat's clock is not synchronized with the clocks on Remulac. Use the relativistic transformation equations to determine the difference in the clocks according to Beldar. Be certain to carefully explain your calculations. State clearly what your calculated values represent.

   Which clock "leads" the other? What happens to this difference as Beldar deccelerates during his approach to Remulac? What about during his short stay on Remulac? Finally, what is the situation as he accelerates and reaches cruising speed on his return trip to Earth? During what portion(s) of the trip is there assymetry between what Premat observes on Beldar's clock and what Beldar observes on Premat's clock? During what portion(s) of the trip would you say that the discrepency in aging between Beldar and Premat occurred?