# Gravitational time dilation

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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#### Solution Preview

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.

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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