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Gravitational time dilation

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Someone moves to the top of the World Trade Center in New York and moves back to the ground floor after one year according to a clock located at the ground floor. By how much did this person age?

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In this problem, we can use that for weak gravitational fields the metric only slightly deviates from the Lorentzian metric, according to:

ds^2 = (1 + 2 V) dt^2 - (dx^2 + dy^2 + dz^2)

where V is the gravitational potential (denoted as phi in your notes) and we're using c = 1 units. The meaning of this is as follows. The value for the coordinates t, x, y, and z are arbitrary, they are just labels denoting space-time points that you are free to change, but ds^2 evaluated between two neighboring points will always tell you the distance between the points, no matter how you have decided to relabel the space-time coordinates. Here the distance refers to a distance defined in the four dimensional space time: ds^2 = time difference^2 - spatial ...

Solution Summary

This solution provides a detailed, accessible solution fro people without knowledge of the theory of relativity.

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Postulates of the Special Theory of Relativity

1. State and explain the postulates of the special theory of relativity.

2. Suppose a person riding on top of a freight car shines a searchlight beam in the direction in which the train is traveling. Compare the speed of the light beam relative to the ground when:

a.The train is at rest.
b. The train is moving.

How does the behavior of the light beam differ from the behavior of a bullet fired in the same direction from the top of the freight car?

3. Define the following terms:

a. Time dilation.
b. Simltaneity
c. Length contraction
d. Mass-energy equivalence

4. What does the equation E = mc2 mean?

5. A rectangular billboard in space has the dimensions 10 meters by 20 meters. How fast and in what direction with respect to the billboard would a space traveler have to pass for the billboard to appear square?

6. How long would a meter stick appear if it were traveling like a properly thrown spear at 99.5 percent of the speed of light?

7. State and explain the principle of equivalence.

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