HPS 0410 Einstein for Everyone Spring 2017

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# Assignment 5: Spacetime

For submission

1. Draw a spacetime diagram with the following elements. Be sure to label each one clearly.

An event O.
A worldline of an observer A that passes through O.
The light cone at O.
The hypersurface of all events simultaneous with O (for observer A).
An event Epast which is in the past of O and can causally affect O.
An event Efuture which is in the future of O and can be causally affected by O.
An event Eelsewhere which is outside the light cone of O and cannot be causally affected by O.
A timelike curve through O.
A spacelike curve through O.
A lightlike curve through O.

2. On the spacetime diagrams below:

 (a) An observer A judges the two events E1 and E2 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 E1 and E2. (b) An observer B moves relative to A and judges E1 to be later that E2. 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 E1 and E2. (c) An observer C moves relative to A and judges E1 to be earlier that E2. 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 E1 and E2. (d) If C judges a tachyon to have traveled from E1 to E2, 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.

The light postulate is an essential part of the above analysis that leads to the relativity of simultaneity. Just how does it figure in the analysis?

B. In a Newtonian space and time, the light postulate does not hold. Instead light conforms with an emission theory. That is, if light is emitted by a moving source, then the speed of the source is added to the speed of light (if the source moves in the direction of the light propagation); or subtracted from it (if the source moves in a direction opposite to that of the light propagation). Show that the above analysis does not yield the relativity of simultaneity if light conforms to an emission theory.

C. In the spacetime diagram below, two rods A and B approach one another from opposite directions.

(a) According to the judgments of simultaneity of the observer shown, the two rods have the same length: they coincide exactly when they pass by each other. Draw the observer's hypersurface of simultaneity for this moment and show that the observer will judge them to have the same length.

(b) Draw hypersurfaces of simultaneity for an observer who moves with rod A. Show that this observer will judge rod B to be shorter than rod A.

(c) Draw hypersurfaces of simultaneity for an observer who moves with rod B. Show that this observer will judge rod A to be shorter than rod B.

D. How does the analysis of C relate to the problem of the car in the garage?

E. 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.)