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Assignment 2: Relativity of Simultaneity

For submission Tuesday, January 16, Wednesday January 17.

For Einstein, the big breakthrough in his work on special relativity came when he found a way to reconcile the principle of relativity and the light postulate. He recognized that these principles only seemed irreconcilable because of an unwarranted assumption that we routinely make about space and time. We assume that all observers should agree on which events are simultaneous. Instead, Einstein noticed, we may allow for the possibility that observers in relative motion may disagree about which spatially separated events are simultaneous. This assumption of the relativity of simultaneity allowed him to retain both principle of relativity and light postulate. This assignment will help you to see how.

1. An observer is at the midpoint of a long spaceship. At the same instant he sends light signals to both front and rear of the spaceship. Event A is the arrival of the signal at the rear; event B is the arrival of the signal at the front.

(a) Are the two events A and B simultaneous according to the spaceship observer?

(b) Imagine that there are two good clocks located at the front and the rear of the spaceship and the arrival of the signals is used to reset each clock to the same time. Are the clocks now properly synchronized according to the spaceship observer?

(c) The spaceship is moving rapidly in the direction of its length past a planet. An observer on the planet watches the signaling procedure described above. Does the planet observer judge events A and B to be simultaneous? If not, which happens first?

(d) Does the planet observer judge the two clocks to be set in proper synchrony? If not, which is set ahead of the other?

2. A light signal flashes back and forth between the two ends of the same spaceship. If the light postulate is to hold for the spaceship observer, then the spaceship observer must see light travel at the same speed in all directions. That is, according to the spaceship observer, the signal must take the same time to travel from front to back as from back to front. Assume this transit time is one minute. Then the arrival times of the light signal must be registered as 12:00, 12:02, 12:04, ...etc. at the rear of the ship and 12:01, 12:03, 12:05, ... etc. at the front.

(a) Assume the light postulate also holds for the planet observer. Will the planet observer see the transit time for the forward trip of the light signal to be the same as the transit time for the backward trip? If not, which is longer?

(b) How can the planet observer reconcile the answer to 2.(a) with the readings on the clocks of the moving spaceship that record the transit times for the light signal?

For discussion in the recitation.

A. Imagine you have a gun that can fire a particle at 100,000 miles per second. You are in a spaceship moving at 100,000 miles per second with respect to the earth. You point the gun in the direction of your motion and fire. Would the earthbound observer see the particle travel at 200,000=100,000+100,000 miles per second? Show that he could not since that would violate the principle of relativity, when that principle is combined with the light postulate.^* How rapidly would you (the spaceship observer) see the particle moving?

Recall: The light postulate states that all uniformly moving observers measure the same speed for light whatever may be the velocity of the body that emits the light.