Re: BELL'S PARADOX FOR DUMMIES

On Jul 18, 1:04 pm, Uncle Ben <b...@xxxxxxxxxxx> wrote:
On Jul 18, 12:45 pm, Uncle Ben <b...@xxxxxxxxxxx> wrote:





On Jul 18, 11:07 am, Dono <sa...@xxxxxxxxxxx> wrote:

On Jul 17, 6:43 pm, Uncle Ben <b...@xxxxxxxxxxx> wrote:

BELL'S PARADOX FOR DUMMIES

Bell's Paradox is not really a paradox but a prediction of Special
Relativity that confounds many professional physicists.  To help them
isolate what is essential in this investigation, I will present a
worked example revealing the "paradox" in language that is accessible
to the advanced high-school student.  It requires no previous
knowledge of relativity theory except how to relate an event given in
terms of one inertial frame of reference to another inertial frame of
reference by means of the Lorentz Transformation in one dimension.

This presentation is identical in spirit to one that I posted July 15,
10:10 am, in newsgroup sci.physics.relativity, except that I expected
that one to be read by experienced relativists.

Part 1.

Imagine a space platform far from any stars or planets that might
cause gravitational effects.  Let inertial frame S be at rest w.r..t.
the platform, and suppose a firecracker explodes right at the origin
(x,y,z)=(0,0,0) of S.  This is event 1.

Let us agree that the only motion that is going to be considered will
be in the direction of the x axis; We need not bother about space
variables y and z. And let us set a clock to 0 at the moment of event
1.

To make our formulas look pretty, we can without loss of generality
choose units in which c=1, i.e. velocity v=1 means v=186,000 miles/
sec. And let us use as a unit of length the distance light will travel
in 1 second, i.e. 186,000 miles. Let time be in seconds.

So event 1 has x=0 and t=0.

Now imagine a second firecracker event happening at x=1 and t=0.. In
other words, the event is 186,000 miles down the x axis and is
synchronized w.r.t. event 1.

Lastly, imagine that a spaceship is passing the platform at v=1/2 down
the x axis of S. Define a frame S' in which this spaceship is at rest,
having space axes parallel to those of S, velocity 1/2 c in the x
direction w.r.t. S, and time synchronized with event 1 at the origin
of S.

(We will interpret this abstract scheme in terms of rockets and
strings a bit later.)

(Note that we can assign and have assigned coordinates of event 1 as
we wish with respect to each frame S and S'.  And we can assign the
coordinates of event 2 w.r.t. S as we wish, as part of the definition
of the problem.  But this is as far as we can go.  The coordinates of
event 2 w.r.t. S' are already determined and we must calculate them by
the Lorentz transformation of event 2 from frame S to frame S'.

So let's do that: let us calculate the coordinates of event 2 w.r.t..
S'.

The Lorentz transformation equations in one dimension for frames with
a common origin in time and space are as follows:

        Let gamma = (1-v*v)^(-1/2) which you will recognize as one over the
square root of one minus v*v (where c=1).

        Then x' = gamma * (x-vt)
        and t' = gamma * (t-x*v)

W.r.t. S the event 1 has coordinates (t=0, x=0) and event 2, (t=0,
x=1).
W.r.t. S 'the event 1 has coordinates (t'=0, x'=0) and event 2, (t',
x') as yet unknown. We must do the calculation:

        x=1, t=0, v=1/2, and gamma = (3/4)^(-0.5) or 1.155.
        Therefore t' = -1.155/2 = -0.577
        and x' = 0.577

W.r.t. S' the event 2 has coordinates (t'= -0.577, x'= +0.577).

What we have done is to derive (not assume) the SR fact that the event
2, which is simltaneous with event 1 w.r.t. S, is not simultaneous
with 1 w.r.t. S'. Event 2 happens first w.r.t.S'.  This is called the
relativity of simultaneity.

Event 2 is also at a shorter distance from zero position w.r.t. S'
than w.r.t. S; this is the Lorentz contraction.

This movie scene could be described w.r.t. S as two firecrackers on
the space platform going off simultaneously, one on the platform and
the other 1 light second ahead.

W.r.t. S', however, the description is different: the space platform
whizzes backward.  The far firecracker goes off first in the distance
0.577 light-seconds ahead, followed 0.577 seconds later by the near
firecracker, which goes off right at the origin of S'.

Part 2.

Suppose the firecrackers were actually Roman candles that shoot
themselves along the x axis of both S and S'.  Let these be impressive
Roman candles that are guaranteed to fly through space and achieve a
velocity w.r.t. S that reaches 0.6 c in some specified time w.r.t.S..
Expensive!

W.r.t. S' these Roman candles that were whizzing backward have slowed
down, and as they approach velocity 0.5 c w.r.t. S, they actually stop
going backward in S' and start going forward. What is the distance
between these candles w.r.t. S' as the rear candle stops momentarily?

2.1. W.r.t. S, the distance between the candles is still 1 light-
second? Why? Because each of them goes a certain distance to get up to
0.6c, and they start at the same time but with one ahead of the other
by 1 light-second.  What one candle can do, the other can do also.. So
they will still be separated by 1 light-second. (Symmetry under
translation.)

2.2. W.r.t.S', we have to do a backward Lorentz transformation from S
to S' with v reversed.  We start with events simultaneous not in the
rest frame but in the speeding frame and go to the rest frame. You
know enough now to guess that the distance w.r.t. S' will be larger
than 1 light-second, because the Lorentz contraction will be in
reverse from a speeding 1 light-second to something at rest that is
longer.  If a speeding ruler shrinks, a slowing ruler expands. (If you
doubt it, do the calculation following the model I gave you in Part 1
but reverse the sign of v.)

Part 3.

In Bell's Paradox, he imagines not Roman candles but spaceships, and
to make the distance between them something physical, he connects them
with a light string.  In our case the string will be initially at rest
and 186,000 miles long.  Like the Roman candles, the spaceships fire
their rockets at the same time w.r.t. their common rest frame.  When
they get up to some high speed, Bell contends correctly that the
distance between the ships w.r.t. an inertial frame in which at least
one of them is at rest will be longer that the initial rest length of
the string, and if the speed is high enough w.r.t. the launch frame,
it must break.

Comment: This approach to Bell's Paradox shows that you don't need any
advanced concepts in relativity to resolve the question.  You just
have to be clear about what things are relative and what things are
not.  Length, time, mass, simultaneity, and electromagnetic fields are
concepts all of which in SR are defined relative to a frame of
reference.  Events can be described in any frame you like as long as
when you change frames, you use the Lorentz Transformation.

This view is a simple as possible -- but no simpler!

Uncle Ben

So, a lot of talk and no math?
All this just to say that in the frame moving with the string the lead
rocket took off earlier by gamma(v)*v*l_0/c^2 ?
This is not a "proof", it is a story. What if you wanted to know the
exact moment when the string breaks? I.e. the moment when the amount
of stretch in the x direction exceeds the limits given by material
resistance?- Hide quoted text -

- Show quoted text -

This proof is independent of the rocket program and the equation of
motion of the rockets. All we are given about the rocket program is
that the speed evenually gets up to 0.6c.  We are given nothing about
the tensile strength of the string except that if you stretch it by a
lot, it will break.

These are not weaknesses of the proof.  They are strengths, since the
important result does not depend on any of those details. We are given
nothing that does not affect the central question: does the string
break?

Remember in the Discussion of the Wiki article all the details that
came up?  Bell observers, accelerated frames, gravitation, equivalence
principles, Rindler coordinates, and many more.  That just confused
everybody and led into dead ends and circular arguments.

"Rindler coordinates?  Pfft!  We don't need no stinking Rindler
coordinates!"

Uncle Ben

Uncle Ben- Hide quoted text -

- Show quoted text -

How ironic your comment:

All this just to say that in the frame moving with the string the lead
rocket took off earlier by gamma(v)*v*l_0/c^2 ?

When I told you that the lead rocket takes off first with respect to
S' -- QED, you stated flatly "No, that is not what happens in the Bell
statement of the problem. The rockets take off simultaneously."  Or
words to that effect.

If I'm terse, you complain I'm wrong. If I expand, you complain "All
that?"  I don't think anybody could satisfy you.

So on the theory that one of your rules is *Uncle Ben can't win,* I
will quit trying. That's what I was afraid of when I told you to
"fugeddaboutit."

Uncle Ben- Hide quoted text -

- Show quoted text -

I finally figured out why Dono insists that I have stated a problem
different from Bell's
Spaceship Paradox (which is untrue as you will see by reading the Wiki
article, where there is mention of a variation in which the rockets
turn off their motors and coast for a while. But this is cited as
only a minor variation.)

But it is a major point for Dono, because he thinks the string breaks,
not because of the reasons I give as the OP in ths thread, but because
the trailing ship stops its motor first, whereupon the lead rocket
breaks the thread by continuing for some time.

This is incorrect, because the string breaks even if all rocket motors
continue, as in my formulation.

This is a curious argument from one who claims to have written the
Wiki article.

Is there any way to check who wrote the Wiki article? One can infer
indirectly that it is not likely to be Chris Hillman, who withdrew
from the fray in disgust from harassment (his term) by one Rod Ball.
The last discussion participant who said he would post a version of
the article is Edward Schaeffer, posting as "EMS." There is no mention
of "Dono" in the history of the article. Rod Ball has no "user page",
(perhaps his has been deleted), and I couldn't find any means of
discovering his having published anything.

As Artie Johnson used to say as a character on TV's "Laugh-In" before
most of you were born, "Verrrry interesting!"

Dono, is there any evidence on Wikipedia that you wrote the BSP
article? (without revealing your identity, of course).

Uncle Ben
.

Relevant Pages

  • Re: Curvature
    ... depends on the frame of reference ... stretching the string. ... Of course, if you think relativity is bunk, be my guest. ... Bonehead Green claimed Einstein's first postulate: ...
    (sci.physics.relativity)
  • Re: Is "malfunctioning" absolute or relative?
    ... W.r.t. this same frame at the time of launch, ... relativity says the string is trying to contract as determined by ... many instructors introduce relativity as "physics ...
    (sci.physics.relativity)
  • Re: BELLS PARADOX FOR DUMMIES
    ... Relativity that confounds many professional physicists. ... terms of one inertial frame of reference to another inertial frame of ... with a light string. ... their rockets at the same time w.r.t. their common rest frame. ...
    (sci.physics.relativity)
  • Re: Is "malfunctioning" absolute or relative?
    ... W.r.t. this same frame at the time of launch, ... relativity says the string is trying to contract as determined by ... many instructors introduce relativity as "physics ...
    (sci.physics.relativity)
  • Re: Simultaneity
    ... rockets (string endpoints) are moving apart. ... IN THE LAUNCH FRAME the rockets remain the ... concerns within the framework of special relativity, ...
    (sci.physics.relativity)