This question relates to the "Twin Trip" example Paul Hewitt uses
in Conceptual Physics (Chapter 15). I have been using this example for the
past two years to demonstrate the idea time dilation.
Here is the core argument. A spaceship and a planet are located in
space. The spaceship emits a flash of light every six minutes. It does
this ten times. If the spaceship stands still, the observer on the planet
will not see the flashes at the same time they are emitted. However, when
they do start arriving to the observer, they should arrive at the same
spacing--frequency--as they were emitted.
Now, if the spaceship moves, the frequency they arrive at the
planet will change. No problem here. Hewitt has the ship move at some
speed so that the frequency they arrive at the planet is one flash every
three minutes. Okay, no problem.
Then, the argument is made that if the ship turns around and goes
the other way at the same speed, the frequency that the flashes will arrive
at the planet will be one flash every 12 minutes. A reciprocal argument is
used here. Since the movement towards the planet gave the observer 1/2 the
frequency, movement away from the planet should give the observer twice the
frequency.
However, when I look at the pictures in the book, and when I draw
this process out on my own, I keep getting the Moving Away Frequency to be
9 minutes.
Let's take the example from t=0 min until t=6min. A flash (Flash A) is
emitted at t=0 min. Now, Flash A has moved 6 min forward from its
original position and also 6 minutes back from its original position--the
light travels in a circle outward, much like the ripple from a rock thrown
in a pond. I would say we have a circle 12 minutes in diameter. If the
ship moved such that the next flash (Flash B) is only 3 minutes from the
forward part of Flash A, isn't the other end of Flash A 9 minutes ahead of
the backwards movement of Flash B? This is where the problem becomes huge!
Using Hewitt's process, if the ship moves away doing ten flashes
and then towards doing ten flashes, then the ship experiences 20 flashes @
6 min = 120 min. The planet experiences (10 flashes @ 3 minutes) + (10
flashes @ 12 mins) = 150 minutes. Hence, time dilation.
If you do the same thing with my numbers, no time dilation. I know
there is time dilation. I believe in relativity. Heck, it's not even a
matter of belief--I know it happens on a daily basis just down the street
from me at FermiLab. However, it was a student that first pointed this out
to me, and I'm running into the same problems he is!
If you have Hewitt, please check out the diagrams on "The Twin
Trip." They are in Chapter 15 (Special Relativity--Space and Time) of
Conceptual Physics. I'm doubting that this has survived the scrutiny of
better physicists than I for over 15 years, so please tell me where I am
going wrong!
Thanks,
Rich Schram
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.