HPS 0410  Einstein for Everyone  Spring 2010 
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For submission
1. Draw a spacetime diagram with the following elements. Be sure to label each one clearly.
2. On the spacetime diagrams below:
(a) An observer A judges the two events
E_{1} and E_{2} to be simultaneous. Draw the worldline of the observer A and a hypersurface of events that A will judge to be simultaneous. How does this hypersurface support A's judgment of the simultaneity of E_{1} and E_{2}. 

(b) An observer B moves relative to A and judges
E_{1} to be later that E_{2}. Draw the worldline of observer B and a hypersurface of events that B will judge to be simultaneous. How does this hypersurface support B's assessment of the time order of E_{1} and E_{2}. 

(c) An observer C moves relative to A and judges
E_{1} to be earlier that E_{2}. Draw the worldline of observer C and a hypersurface of events that C will judge to be simultaneous. How does this hypersurface support C's assessment of the time order of E_{1} and E_{2}. 

(d) If C judges a tachyon to have travelled from E_{1} to E_{2}, what would A and B say about it? 
For discussion in the recitation
A. The relativity of simultaneity is revealed most simply in the following thought experiment in which two observers in relative motion judge the timing of two explosions by means the light signals they produce:
Draw a spacetime diagram of this experiment, indicating:
The planet observer's worldline and associated hypersurfaces of
simultaneity.
The spaceship observer's worldline and associated hypersurfaces of
simultaneity.
The worldlines of the front and rear of the spaceship.
The two explosion events.
The light signals emitted by the explosions.
B. At sunrise of Day 1, a monk commences a long walk up the narrow, winding road from the monastery in the valley to the mountain top. It is a hard, tiring climb, so he stops frequently to rest and even reverses his direction from time to time. He arrives at the mountain top just at the moment of sunset. At sunrise on Day 2, the monk commences the return journey. This time the journey is far easier. Rather than hurry to complete it quickly, the monk decides to pause frequently to admire the wildflowers, inhale the mountain air and absorb the splendor of the view. He arrives in the valley at the moment of sunset.
Is there any moment on the two days at which the monk is in exactly the same position on the road?
At first it seems impossible to determine an answer to this question from the information given. Whether there is such a moment seems to depend on the details of the monk's progress up and down the mountain. Drawing spacetime diagrams rapidly solves the problem, however. To see how, draw plausible world lines for the monk's two journeys on the spacetime diagrams here.
Explain how they make it obvious that the moment specified in the question must always exist no matter what the details of the monk's progress. (Hint: To see this, imagine the two spacetime diagrams superimposed.)