ABIAN's TROLLEY (GENERALIZED)
Alexander Abian
Let,
|--a'---M--- b'--|
be a trolley with two very small holes a' and b' at the bottom and an
electric flash bulb M midway between a' and b' as shown in the
picture above.
Let the trolley move on a two dimensional (plane) covered with an
unexposed photographic film P
Let the trolley move ANYWAY you want on P (not necessarily
rectilinearly nor with a constant velocity)
While the trolley is in motion, M gives a flash and two dots
a and b appear on the exposed film.
After a while the trolley is stopped and brought back and a' and a
are made to coincide and b', b, a (and of course with a') are made
to be collinear.
WHICH ONE IS YOUR ANSWER ?
(1) b' coincides with b
(2) b' is to the left of b
(3) b' is to the right of b
P.S. Would it make a difference if while M gives a flash, say,
--
-------------------------------------------------------------------------
ABIAN TIME-MASS EQUIVALENCE FORMULA T = A m^2 in Abian units.
ALTER EARTH'S ORBIT AND TILT TO STOP GLOBAL DISASTERS AND EPIDEMICS.
JOLT THE MOON TO JOLT THE EARTH INTO A SANER ORBIT.ALTER THE SOLAR SYSTEM.
REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT TO CREATE A BORN AGAIN EARTH(1990)
THERE WAS A BIG SUCK AND DILUTION OF PRIMEVAL MASS INTO THE VOID OF SPACE
Alexander Abian
------------------
Sorry for the previously incomplete last sentence which is completed now!
Let,
|--a'---M--- b'--|
be a trolley with two very small holes a' and b' at the bottom and an
electric flash bulb M midway between a' and b' as shown in the
picture above.
Let the trolley move on a two dimensional (plane) covered with an
unexposed photographic film P
Let the trolley move ANYWAY you want on P (not necessarily
rectilinearly nor with a constant velocity)
While the trolley is in motion, M gives a flash and two dots
a and b appear on the exposed film.
After a while the trolley is stopped and brought back and a' and a
are made to coincide and b', b, a (and of course with a') are made
to be collinear.
WHICH ONE IS YOUR ANSWER ?
(1) b' coincides with b
(2) b' is to the left of b
(3) b' is to the right of b
P.S. What would be the answer for the special case where, when M gives
a flash, say, a' is immobile and only b' moves (i.e., describes even an
infinitesimal circular arc , i,e, moves in the direction perpendicular
to a'b'!)
Wayne Throop
: Let,
: |--a'---M--- b'--|
:
: be a trolley with two very small holes a' and b' at the bottom and an
: electric flash bulb M midway between a' and b' as shown in the
: picture above.
:
: Let the trolley move on a two dimensional (plane) covered with an
: unexposed photographic film P
:
: Let the trolley move ANYWAY you want on P (not necessarily
: rectilinearly nor with a constant velocity)
:
: While the trolley is in motion, M gives a flash and two dots
: a and b appear on the exposed film.
:
: After a while the trolley is stopped and brought back and a' and a
: are made to coincide and b', b, a (and of course with a') are made
: to be collinear.
:
: WHICH ONE IS YOUR ANSWER ?
:
: (1) b' coincides with b
: (2) b' is to the left of b
: (3) b' is to the right of b
With the same presumptions as before (y separations are negligable,
light from M is directed y-wards at a' and b'), and with the additional
assumption that any acceleration keeps the trolley Born-rigid, the
answer derived from SR is "any of the above, depending on details
of the motion".
(1) for uniform motion along y.
(2) for uniform motion along the a'b' line.
(3) for sufficiently accelerated motion wrt the film.
The last time this came up, I opined that (3) was ruled out if one is
limited to Born-rigid accelerations. But I'm pretty sure that turns out
not to be the case; I'd appreciate it if folks would let me know what
they think about that case. I was convinced to change my mind by
considering the degenerate case of born rigid acceleration; stopping the
trailing point dead, and giving all the particles further forwards the
appropriate accelerations to maintain born rigidity wrt that point (not
that event: that point). I'm pretty sure this is "possible", and it
means that you can get case 3. I'm not totally sure, but I think you can
even get the dots arbitrarily close together.
Sadly, nobody commented on this the last time it came up in 1996.
Message-ID: <8346...@sheol.org>
References: <abian.8...@pv3449.vincent.iastate.edu>
Wayne Throop thr...@sheol.org http://sheol.org/throopw
Jeremy Boden
<ab...@iastate.edu> writes
>------------------
>
>Let,
>
> |--a'---M--- b'--|
>
>be a trolley with two very small holes a' and b' at the bottom and an
>electric flash bulb M midway between a' and b' as shown in the
>picture above.
>
>Let the trolley move on a two dimensional (plane) covered with an
>unexposed photographic film P
>
>Let the trolley move ANYWAY you want on P (not necessarily
>rectilinearly nor with a constant velocity)
>
>While the trolley is in motion, M gives a flash and two dots
>a and b appear on the exposed film.
>
>After a while the trolley is stopped and brought back and a' and a
>are made to coincide and b', b, a (and of course with a') are made
>to be collinear.
How can you ensure that points a' and a coincide? If you use the
standard "thought experiment" method you would need to emit a flash of
light to ensure that the trolley and film are in the same inertial
frame. This will fog your film.
....
--
Jeremy Boden
Tom Roberts
> [about Abian's generalized trolley]
> With the same presumptions as before (y separations are negligable,
> light from M is directed y-wards at a' and b'), and with the additional
> assumption that any acceleration keeps the trolley Born-rigid, the
> answer derived from SR is "any of the above, depending on details
> of the motion".
>
> (1) for uniform motion along y.
Ummm.... From your description above y is perpendicular to the surface
of the film. I think you mean z (in the plane of the film).
> (2) for uniform motion along the a'b' line.
> (3) for sufficiently accelerated motion wrt the film.
>
> The last time this came up, I opined that (3) was ruled out if one is
> limited to Born-rigid accelerations. But I'm pretty sure that turns out
> not to be the case; I'd appreciate it if folks would let me know what
> they think about that case.
(3) is quite possible:
For simplicity, consider a uniform acceleration of a' and keep the
trolley Born rigid (this implies that b' is not uniformly accelerated).
For definiteness in my description I consider a' to the left of b',
with the trolley initially moving right and the acceleration headed
left, and consider the period near the instant when a' reverses
direction wrt the film (which I assume is at rest in an inertial frame).
Let the light pulse pass through a' at the instant a' is at rest wrt
the film. Born-rigidity implies that at the instant the light passes
through a', the hole b' is at rest in the film frame (simultaneity in
the film frame) -- call this point on the film b". At the time M
emitted this pulse the trolley still had non-zero velocity to the right
wrt the film, so the light pulse has not yet reached b' (=b" in the film
frame, simulteneity in the film frame), and by the time it does reach b',
b' will be to the left of b" (because the trolley is still accelerating
to the left) -- that is b on the film. So (3) applies for any
acceleration as I described above.
> I was convinced to change my mind by
> considering the degenerate case of born rigid acceleration; stopping the
> trailing point dead, and giving all the particles further forwards the
> appropriate accelerations to maintain born rigidity wrt that point (not
> that event: that point). I'm pretty sure this is "possible", and it
> means that you can get case 3. I'm not totally sure, but I think you can
> even get the dots arbitrarily close together.
Stopping the trailing point dead is difficult to analyze; I prefer the
uniform acceleration described above, which should be clear. For larger
accelerations you can get the points a and b closer together. But at some
acceleration the Rindler horizon will come between a' and b' and will
prevent you from maintaining Born rigidity (it would require an infinite
acceleration of b' to keep up with the uniform finite acceleration of a').
This occurs at a uniform proper acceleration c^2/L for a', where L is the
proper distance between a' and b'.
I conjecture that the Rindler horizon will approach b' (from the
right) for increasing acceleration (to the left) as the point
b approaches a. The horizon will prevent b from being to the
left of a. But this is rather loose guess, and I have not
performed the calculations. If this conjecture is true you
can indeed get the dots arbitrarily close together, but not
"backwards".
Tom Roberts tjro...@lucent.com
Wayne Throop
: Tom Roberts <tjro...@lucent.com>
: Ummm.... From your description above y is perpendicular to the
: surface of the film. I think you mean z (in the plane of the film).
Oops. That's right, I already used up "y". So: "along z".
:: (2) for uniform motion along the a'b' line.
:: (3) for sufficiently accelerated motion wrt the film.
::
:: The last time this came up, I opined that (3) was ruled out if one is
:: limited to Born-rigid accelerations. But I'm pretty sure that turns out
:: not to be the case; I'd appreciate it if folks would let me know what
:: they think about that case.
: (3) is quite possible: For simplicity, consider a uniform acceleration
: of a' and keep the trolley Born rigid (this implies that b' is not
: uniformly accelerated). For definiteness in my description I consider
: a' to the left of b', with the trolley initially moving right and the
: acceleration headed left, and consider the period near the instant
: when a' reverses direction wrt the film (which I assume is at rest in
: an inertial frame). Let the light pulse pass through a' at the
: instant a' is at rest wrt the film. Born-rigidity implies that at the
: instant the light passes through a', the hole b' is at rest in the
: film frame (simultaneity in the film frame) -- call this point on the
: film b". At the time M emitted this pulse the trolley still had
: non-zero velocity to the right wrt the film, so the light pulse has
: not yet reached b' (=b" in the film frame, simulteneity in the film
: frame), and by the time it does reach b', b' will be to the left of b"
: (because the trolley is still accelerating to the left) -- that is b
: on the film. So (3) applies for any acceleration as I described above.
Ah. Very good. Thanks.
:: I was convinced to change my mind by considering the degenerate case
:: of born rigid acceleration; stopping the trailing point dead, and
:: giving all the particles further forwards the appropriate
:: accelerations to maintain born rigidity wrt that point
Which, by the way, yields a very similar "verbal analysis" to the
one Tom gives; that is, if a' is stopped dead, light will reach b'
before it it is distanct a'b' away.
But unfortunately:
: Stopping the trailing point dead is difficult to analyze; I prefer the
: uniform acceleration described above, which should be clear. For
: larger accelerations you can get the points a and b closer together.
: But at some acceleration the Rindler horizon will come between a' and
: b' and will prevent you from maintaining Born rigidity
Ah. Yes. That's a flaw in my analysis; stoping the trailing point
dead makes the acceleration of that point high enough to violate
Born rigidity; I was wrong about the degenerate case. Hmmmm.
: (it would require an infinite acceleration of b' to keep up with the
: uniform finite acceleration of a'). This occurs at a uniform proper
: acceleration c^2/L for a', where L is the proper distance between a'
: and b'.
:
: I conjecture that the Rindler horizon will approach b' (from the
: right) for increasing acceleration (to the left) as the point b
: approaches a. The horizon will prevent b from being to the left of a.
: But this is rather loose guess, and I have not performed the
: calculations. If this conjecture is true you can indeed get the dots
: arbitrarily close together, but not "backwards".
I rule out getting them backwards by simply considering
light propogation from M in the film frame. The film-relative
location of M at light emission must be to the right of point a
and to the left of point b.
Hrm. Now I wonder whether that conjecture is true; now that I'm
reminded to think of the Rindler horizon, I wonder if it wouldn't
place a limit on closeness at some fraction of L instead. Hrm.
Most interesting. IMO.
ryk...@my-deja.com
ab...@iastate.edu (Alexander Abian) wrote:
> WHICH ONE IS YOUR ANSWER ?
>
> (1) b' coincides with b
> (2) b' is to the left of b
> (3) b' is to the right of b
>
> P.S. What would be the answer for the special case where, when M
gives
> a flash, say, a' is immobile and only b' moves (i.e., describes even
an
> infinitesimal circular arc , i,e, moves in the direction perpendicular
> to a'b'!)
Rod: If the flash device M is equally spaced between a'
and b' when flashed , the answer is still (1) .
--
Rod Ryker...
It is reasoning and
faith that bind truth .
Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.