Assignment 5: Spacetime

(new version February 1)

For submission Tuesday February 6, Wednesday February 7

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 E_past which is in the past of O and can causally affect O.
An event E_future which is in the future of O and can be causally affected by O.
An event E_elsewhere 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.

For the diagram consider a two dimensional space with time the third dimension in your diagram.

2. (a) Draw a spacetime diagram showing the worldline of an observer A, a hypersurface of events that A will judge simultaneous and two events E₁ and E₂ that lie in the hypersurface and therefore are simultaneous for A.

(b) On the same diagram, draw the world line of an observer B, moving relative to A, such that B would judge E₁ to be later that E₂.

(c) On the same diagram, draw the world line of an observer C, moving relative to A, such that C would judge E₁ to be earlier that E₂.

(d) If C judges a tachyon to have travelled 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:

Draw a spacetime diagram of this experiment, indicating:

The planet observer's worldline and his hypersurfaces of simultaneity.
The spaceship observer's worldline and his 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. We saw in Assignment 3 how the relativity of simultaneity made it possible for observers in relative motion each to judge the other's clock to have slowed. Draw the spacetime diagram that accompanies the analysis of that assignment.

C. 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.)