Re: Accelerating rocket/light beam question
- From: David Rutherford <drutherford@xxxxxxxxxxx>
- Date: Wed, 01 Apr 2009 11:45:23 EDT
Daryl McCullough wrote:
David Rutherford says...
Place the two rockets (call them A and B) referred to in Fig. 13.7 next
to the rocket (call it C) referred to in Fig. 13.6. Rocket A is placed
next to the top of rocket C, and rocket B next to the bottom of rocket
C. All rockets are, initially, at rest. At t=0, all three rockets
undergo identical uniform accelerations.
If you did that, then (assuming that each rocket's top is
rigidly connected to its bottom) even though
*initially* the top of rocket B coincides with the bottom of
rocket C, and the top of rocket C coincides with the bottom of
rocket A, that will not continue to be the case. The top of
rocket B will recede from the bottom of rocket C, and the top
of rocket C will recede from the bottom of rocket A.
According to your setup, this should be the arrangement of the rockets
at rest and in motion.
B A
Rockets at rest: ---------> --------->
--------->
C
B A
Rockets in motion: -----> ----->
----->
C
The contractions in this post are only meant to show the qualitative
relationship between the rockets at rest and in motion.
If there's a contraction in the rocket lengths, then there should also
be a contraction in the space between the rockets. So I believe the
relationship should look something like this,
B A
Rockets at rest: ---------> --------->
--------->
C
B A
Rockets in motion: -----> ----->
----->
C
As you can see, the space between rockets A and B contracts along with
the rockets (and the space they occupy). So, the top of rocket B won't
recede from the bottom of rocket C, and the top of rocket C won't recede
from the bottom of rocket A. If there was a string connecting the bottom
of rocket A to the top of rocket B, it would also contract by the same
amount as the space between the rockets, so it wouldn't break.
Now I'd like to change the setup, a little bit, to make a point about
the world lines of the rockets. I'll assume, for the moment, that length
contraction applies to accelerated rockets. Let's make the top of rocket
A coincide with the top of rocket C, and the bottom of rocket B coincide
with the bottom of rocket C. Rockets A and B are 1/3 the length of
rocket C, so that the rockets at rest and in motion look like this.
B A
Rockets at rest: ---------> --------->
----------------------------->
C
B A
Rockets in motion: -----> ----->
----------------->
C
Unfortunately, if the world lines of the bottom and top of rocket C are
represented by Fig. 13.6, then the world lines of the bottom of rocket B
and the top of rocket A must also be represented by Fig. 13.6, and not
by Fig. 13.7.
On the other hand, if there is no length contraction, and the world
lines of the bottom of rocket B and the top of rocket A are represented
by Fig. 13.7, then the world lines of the bottom and top of rocket C
must also be represented by Fig. 13.7, and not by Fig. 13.6.
Either way, the world lines of the bottom of rocket B and the bottom of
rocket C would coincide, and the world lines of the top of rocket A and
the top of rocket C would coincide.
You are confusing two different relationships between accelerating
rockets:
1. Rockets A, B, and C are "rigidly connected".
The rockets are not connected to each other, in any way.
This
means that they are accelerating in a straight line in
such a way that the distance between them remains constant (as
measured by a comoving inertial reference frame).
2. Rockets A, B, and C are undergoing identical proper accelerations.
These are *not* the same properties. If (1) is true, then (2) is *false*,
and vice-versa.
(1) is not true.
If, on the other hand, the rockets
are not connected, but are following the same "flight plan"
(blast off at such and such a time, with rocket engines set
to such and such thrust level), then the rockets will *not*
accelerate rigidly. Instead, the distance between the
rockets will increase (as measured in a comoving inertial
frame).
As I showed, above, if the rocket's lengths contract, then I think the
distance between the rockets will decrease.
Let F be the inertial frame in which the rockets are initially
at rest. Let L be the initial distance between the rockets,
as measured in frame F.
Let e_1 be some event (location in space and time) taking place
at the rear rocket after it has begun accelerating.
Let F' be the inertial frame in which the rear rocket is
momentarily at rest at event e_1.
Let e_2 be the event such that (1) e_1 and e_2 are
simultaneous, according to frame F, and (2) the spatial
distance between e_1 and e_2 is L, according to frame F.
Let e_3 be the event such that (1) e_1 and e_3 are
simultaneous, according to frame F', and (2) the spatial
distance between e_1 and e_3 is L, according to frame F'.
You have a choice: (1) If the front rocket travels with
constant acceleration so that it passes through event e_2,
then it will *not* pass through event e_3. So if the rockets
maintain a constant distance apart, as measured in frame F,
then their distance apart as measured in frame F' will be
*larger* than L at the time of event e_1.
In choice (1), assume that the lengths of all three rockets and the
distance between rockets A and B _expand_, in frame F', like this,
B A
Rockets at rest: -------> ------->
----------------------->
C
B A
Rockets in motion: ---------> --------->
----------------------------->
C
A string between rockets A and B would also expand by the same amount as
the interval between rockets A and B. However, there is no stress on a
body, due to length expansion, any more than there is stress on a body,
due to length contraction. So a string between A and B does not break.
If it's possible for rockets and intervals (and strings) to contract,
why isn't it possible for them to expand? Contraction of lengths, in SR,
can be thought of as a result of the fact that, in SR, clocks at forward
positions, in the direction of motion, run behind clocks at rear
positions. Consequentially, a clock at a forward position is,
essentially, where it was at an earlier time, making it look a little
further toward the rear than when at rest. This makes lengths appear
contracted.
The same kind of thing might happen in reverse for accelerated frames.
Clocks in forward positions, for motion in the direction of the
acceleration, run ahead of, and faster than, clocks in rear positions.
Thus, a forward position is where it will be at a later time, making it
look a little further forward of its position when at rest. This makes
lengths appear expanded.
Alternatively, (2) If the front rocket travels with
constant acceleration so that it passes through event e_3,
then it will *not* pass through event e_2. As measured
in frame F, the distance between the rockets decreases
with time.
Same as above, but lengths are contracted, in frame F. But, the string
still doesn't break.
So, basically, the string doesn't break, no matter how you look at it,
period.
--
Dave Rutherford
"New Transformation Equations and the Electric Field Four-vector"
http://www.softcom.net/users/der555
Applications:
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"Proposed Quantum Mechanical Connection"
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