Re: An interesting SR puzzle

From: Jesse Mazer (vze2ztqw_at_mail.verizon.net)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 09:35:38 GMT

RP wrote:

>
>
> glbrad01 wrote:
>
>> "Paul B. Andersen" <paul.b.andersen@deletethishia.no> wrote in
>> message news:cu62db$oiq$1@dolly.uninett.no...
>>
>>> Eli Botkin wrote:
>>>
>>>> "Ben Rudiak-Gould" <br276deleteme@cam.ac.uk> wrote in message
>>>> news:cu0rp6$r7a$1@gemini.csx.cam.ac.uk...
>>>>
>>>>
>>>>> Here's a two-part SR puzzle:
>>>>>
>>>>> 1. I'm standing still (i.e. moving inertially) while whirling a
>>>>> clock around my head at a constant speed. (If you're worried
>>>>> about conservation of momentum, imagine that I'm whirling two
>>>>> clocks: it doesn't make any difference). I'm wearing a
>>>>> wristwatch. Which of these--the clock or the wristwatch--will
>>>>> appear to me to tick faster, and by how much?
>>>>>
>>>>> 2. Now I'm a different person, sitting on the clock as it whirls
>>>>> around, looking at the clock and at the wristwatch of the
>>>>> person doing the whirling. Which (if either) will appear to
>>>>> me to tick faster, and by how much?
>>>>>
>>>>> I already know what the answer to this puzzle is, and why. But I
>>>>> find it very interesting from a pedagogical perspective. Despite
>>>>> doing well in my undergrad SR course, and despite the puzzle's
>>>>> simplicity (no calculation required), I doubt I could have solved
>>>>> part 2 had it been on an exam. I would have become hopelessly
>>>>> mired in trying to figure out the reference frame of the person
>>>>> sitting on the clock, because SR, as it was taught to me, was
>>>>> about relating the reference frames of different observers. I
>>>>> didn't understand at the time that the coordinate system you
>>>>> choose to solve a problem needn't be the rest frame of the
>>>>> measurement device, and that there isn't even a well-defined
>>>>> notion of /the/ rest frame of an object in general. I conjecture
>>>>> that this kind of confusion is quite common.
>>>>>
>>>>> This puzzle seems to highlight this point more clearly than any
>>>>> I've seen before. More generally, it's interesting as an example
>>>>> of a problem involving relative motion for which the Lorentz
>>>>> transformation is really no help at all. And its solution might
>>>>> help to dispel the common misconception that SR can't deal with
>>>>> acceleration.
>>>>>
>>>>> One can also add parts 1b and 2b which ask whether the clock and
>>>>> wristwatch are redshifted or blueshifted, and by what factor. Not
>>>>> until years later did I understand the relationship between
>>>>> Doppler shift and the apparent rate of clocks.
>>>>>
>>>>> In short, I like this puzzle a lot, and I wish it were taught
>>>>> alongside the twin paradox (or instead of it!) in undergraduate
>>>>> courses and textbooks. I'm curious to know if anyone here has
>>>>> taught it, or thought about doing so, or decided against it, or
>>>>> whatever.
>>>>>
>>>>>
>>>>> -- Ben
>>>>
>>>>
>>>>
>>>> Hi Ben:
>>>> I haven't heard this one before. My response is:
>>>> Each person will claim that the other clock is running slower than
>>>> his own.
>>>> Also, if they are viewing each others clocks, the images will
>>>> appear to advance in synchrony with their personal clocks (or
>>>> wristwatchs).
>>>> If you don't agree, I would very much like your take on the correct
>>>> solution.
>>>> Eli
>>>
>>>
>>> The "whirling clock" is accelerated.
>>> That makes all the difference.
>>> The correct answer is they will both agree
>>> that the wrist watch runs faster.
>>>
>>> Paul
>>
>>
>>
>> That is not the correct answer.
>>
>> Brad
>
>
> I agree.
> The correct answer is that in this system GR must be applied.

No, SR can deal with the question of the proper time recorded by an
accelerating clock by considering its path as seen in an inertial frame.
Just integrate [squareroot(1 - v(t)^2/c^2) dt] between two times t1 and
t2 in your frame, with v(t) being the clock's instantaneous velocity at
time t in your frame; this will give the correct answer for the time
elapsed on the accelerating clock during the interval (t1, t2) in your
frame (you also have to take into account the finite speed of light to
figure out how much time you'd actually *see* elapse on the accelerating
clock during that time-interval, but in this problem the answer would be
the same). Then imagine that your own clock is sending signals to the
accelerating clock at the speed of light, and figure out what
time-signal the accelerating clock will be receiving at time t1 and what
time-signal it will be receiving at time t2; this tells you what amount
of time an observer sitting on the accelerating clock would see elapsed
on your clock during the same interval. If the accelerating clock is
maintaining a constant distance from your clock, and a constant velocity
in your frame, it's pretty easy to show that an observer on the
accelerating clock will see your clock sped up by the same amount that
you see the accelarating clock slowed down.

Jesse

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