Special Relativity and Simultaneity

General directions

  1. Divide into working groups of three.
  2. Observe the specified time limits
  3. Try to answer all of the questions.
  4. Develop a story with appropriate visual aids (diagrams) that you can tell to others interested in the paradox of relative time.
Now that you have read Dr. Einstein's discussion of special relativity, we will spend some time constructing visual aids and developing a conceptual understanding that will allow us to tell the story of relativity to others.

The principle of relativity, according to Dr. Einstein, is the widely accepted hypothesis that natural phenomena run their course according to exactly the same physical laws no matter what the observational frame of reference. In particular, the principle of relativity tells us that the law of constant speed of light (in vacuo) is independent of the motion of the observer or the motion of the light source. The constancy of the speed of light has been well established empirically.

Adding velocities in classical mechanics (30 minutes)

Consider an observer O standing beside a railroad track watching a long train on which is seated a second observer OT (T for on the train). The train moves along the tracks uniformly, i. e., in a straight line at constant speed. Observer O, facing the train, thinks to herself: I am at rest, and the train and everything on it is moving uniformly to my right. Observer OT looking out the window at O, thinks to herself: If I didn't know better, I could imagine that I am sitting still and everything I see out the window is moving uniformly to my right. Which observer is right? How could they decide? Does it matter?

Suppose the train is traveling at a speed of 60 miles per hour. At the instant OT passes the point where O is standing beside the tracks, OT releases a mouse that runs toward the front of the train at a speed of one mile per hour. How far from OT will the mouse be after one hour has passed? (Remember, this is a thought experiment so you'll need to imagine a long train). How far from O will the mouse be after one hour? Draw a diagram to explain your answers.

Suppose instead that the mouse, when released, runs toward the rear of the train. In this case, how far from OT will the mouse be after one hour? How far from O? Again, draw a diagram to illustrate your answers.

Recalling now that the speed of an object is just the distance traveled by the object divided by the elapsed time, what will be the speed of the mouse in each case as measured by OT? What will be the speed of the mouse in each case as measured by O?

Can you state the general rule for finding the speed of the mouse with respect to observer O, if you know the speed of the mouse with respect to OT, and the speed of OT with respect to O?

Suppose instead of releasing a mouse to run along the floor of the train car, observer OT had released a fly which flew inside of the train at a speed of one mile per hour. Would the results obtained above for the mouse pertain as well to the fly?

Suppose at the instant the two observers pass, observer O beside the train also releases a fly that travels through the air at a speed of one mile per hour. Which fly will reach a point one mile from O first? How long will it take the fly on the train? How long will it take the fly released by O? Explain your answers.

Let's replace the flies by a sound wave. Suppose at the instant the two observers pass, each one yells "Stop!". Will the engineer at the front of the train hear both shouts at the same time? Why?

Now replace the sound wave by a light wave. (Imagine that all the air is removed from inside and outside of the train so that the light is propagated in vacuo.) Suppose at the instant the two observers pass, each one flashes a light toward the front of the train. Will the engineer see the two light flashes at the same time? What does your intuition tell you? What answer does the principle of relativity demand?

The dilation of time in special relativity (40 minutes)

The final conclusion of the section above has been verified repeatedly. The engineer always sees the two flashes of light at the same time, regardless of what our intuition tells us to expect. Accepting the implications of this conclusion is the major hurdle in developing a concept of relativity in space-time.

For example, suppose O and OT both measure the speed of light, one of them along the tracks, and the other on the train. Both have identical meter sticks for measuring distance and identical clocks for measuring time. Suppose each measures the distance traveled and the time required for the light to reach the engineer. (The engineer assists in this by hanging a mirror to reflect the flashes of light back to O and OT.)

Observer O uses her clock to measure the total time interval between the initial light flash and the returning flash of light. Since light travels the same speed in both directions, then the time required for the light flash to reach the engineer is just one half of the total time interval. The returning flash of light also brings back an image of the mile marker the front of the train was passing at the instant the light was reflected, so that O beside the train also has a measurement of the distance traveled by the light in reaching the engineer. From the measured length of the interval and the distance traveled by the light flash, observer O can easily compute the speed of light as the distance traveled divided by the elapsed time.

Observer OT on the train can measure the distance traveled by the light in reaching the engineer by simply using her meter stick to measure the length of the train between her and the engineer. Observer OT can also compute the speed of light as the distance traveled divided by the elapsed time as shown on her clock.

Which of the two observers will measure the greater distance traveled by the light? Draw a diagram to explain your answer. Since the two observes measure different values for the distance traveled by the light, what can we conclude about the time it takes the light to reach the engineer? Remember, each observer must obtain the same value for the speed of light. Which observer will measure the greater amount of time? What can we conclude about the rate at which the two clocks run?

Remember we have assumed that both clocks are identical. How do you rationalize that two identical clocks have run at different rates? For the period of the experiment, which clock runs slower? Which observer O or OT will be older at the end of the experiment? Will either observer experience time running faster or slower?