The material that we've been discussing recently (involving people
stationary in inertial frames) is elementary. Much more interesting is
what special relativity says about the conclusions of separated people
who are undergoing identical (as confirmed by accelerometers)
simultaneous finite accelerations along the same straight line. The
answer is that the person on the trailing rocket will conclude that the
leading rocket maintains a constant separation ahead of the trailing rocket.

Many people still believe that the separation INCREASES in that
scenario, according to the person on the trailing rocket. That belief
comes from mutually-contradictory statements in Bell's Spaceship
paradox, as given in the webpage:

https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox .

That webpage makes two mutually-contradictory claims:

First, that the two rockets maintain the same separation, according to
the initial inertial observers, and secondly,
that the two rockets are identically constructed, and thus produce the
same thrust and acceleration when ignited, according to the people on
the trailing rocket. Those two statements can't both be simultaneously
true. If the first statement is true, then the second statement is
false: the two accelerometers can't have the same reading. The leading
rocket will be accelerating faster than trailing rocket, according to
the person on the trailing rocket.

I've written eleven papers (on viXra) on this subject, the first two
fairly long, and the latter ones fairly short. You can find them on viXra:

https://vixra.org

by searching on my full name: "Michael Leon Fontenot". They can be
downloaded (in PDF form) at no charge.

You can also get the two long papers separately on Amazon, and a third
paper on Amazon that contains all of the short papers. They aren't
free, but only cost about $7 (not counting shipping and taxes, etc.) ...
that's just a dollar or so more than printing costs. To find them, you
can just search on Amazon for my full name. (The fourth monograph
returned in that search, "A New Simultaneity Method for Accelerated
Observers in Special Relativity", is now known to be incorrect ... it's
only value is in providing some comfort to those people who can't
tolerate the instantaneous ageing of the home twin, according to the
traveling twin when he instantaneously reverses course in the twin paradox).

The titles of the first two long monographs are

"An Inconsistency Between the Gravitational Time Dilation Equation and
the Twin Paradox"

and

"A New Gravitational Time Dilation Equation".

The third Amazon monograph is titled

"An Accelerated Array of Clocks in Special Relativity: A Meaningful
"NOW-at-a Distance” ",

and contains all the short papers.

If you have any questions, I can be reached at:

PhysicsFiddler@gmail.com

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On 12/17/23 6:46 PM, Mike Fontenot wrote:

the two rockets are identically constructed, and thus produce the
same thrust and acceleration when ignited, according to the people on
the trailing rocket.

NO. Their thrust and acceleration are the same TO OBSERVERS IN EACH
ROCKET. That is, their proper accelerations are equal. Identical
construction MUST yield equal proper accelerations (in an idealized
gedanken like this).

Just think about the symmetry. Singling out the trailing rocket is
unwarranted and WRONG.

You keep screwing this up and writing nonsense.

Tom Roberts

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On 2023-12-18 00:46:33 +0000, Mike Fontenot said:

Many people still believe that the separation INCREASES in that
scenario, according to the person on the trailing rocket.

You belive otherwise but cannot support your belief with correct
mathematics. That your reasoning is faulty is shown many times
both in sci.physics.relativity and on comments on vixra pages.

Mikko

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You are forgetting the length contraction equation (LCE) of special
relativity. If a yardstick is moving away from an inertial observer,
the inertial observer will conclude that the yardstick is getting
shorter by the factor gamma. And if you run the experiment again, but
this time you remove the middle 34 inches of the yardstick, leaving only
the outer two inches, the LCE tells you that the two one-inch pieces of
the original yardstick will still get closer, by the same factor gamma.
And the two separated rockets (with accelerometers showing the same
constant readings) are like the two outer inches of what was once the yardstick: the two rockets will likewise will get closer together. So,
in the Bell scenario, where the initial inertial observers, by
definition, say the separation of the rockets is CONSTANT, that means
that the people on the trailing rocket will say that the leading rocket
is getting farther away, and the accelerometers do NOT show the same
reading.

So the Bell scenario IS different from my scenario, in which the
accelerometers show the same reading, and the separation between the
rockets is constant.

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On 12/19/23 6:46 PM, RichD wrote:

On December 17, Mike Fontenot wrote:

The answer is that the person on the trailing rocket will conclude
that the leading rocket maintains a constant separation ahead of
the trailing rocket.

The wrong answer -

Yes, Fontenot keeps insisting on his WRONG answer.

Assuming the person on the trailing rocket measures the separation simultaneously in their successive instantaneously co-moving inertial
frames, that separation MUST successively increase.

Many people still believe that the separation INCREASES in that
scenario, according to the person on the trailing rocket. That
belief comes from mutually-contradictory statements in Bell's
Spaceship paradox, as given in the webpage:
https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox . First,
that the two rockets maintain the same separation, according to
the initial inertial observers,

This has nothing to do with Fontenot's misreading of that page. It has
to do with simple calculus.

Correct, BY STIPULATION, not 'conclusion'.

No. The stipulations are:
a) the rockets have identical proper accelerations as a
function of their proper times since starting
and
b) they start accelerating simultaneously in their initial
inertial frame.

One CONCLUSION is that they maintain a constant separation when measured simultaneously in their initial inertial frame. This is CONCLUDED from
simple calculus: the first integral of their accelerations shows that
their speeds relative to that inertial frame are always equal when
measured simultaneously in that frame. Integrate their speeds and one
concludes that their separation is constant when measured simultaneously
in that frame. Both integrals are with respect to the time coordinates
of that inertial frame.

I do not know why Fontenot keeps getting this simple calculus wrong. Or
why he keeps ignoring me when I tell him this. He has been obsessing
over this scenario for many years, and refuses to THINK about it.

THINK 1: If the rockets have identical proper accelerations and
start simultaneously in their initial inertial frame, then their
trajectories MUST be congruent in the frame, differing only by their
initial offset. That is, their separation measured in that frame must be constant.

THINK 2: Since their separation is constant in the initial inertial
frame, then in their successive instantaneously co-moving inertial
frames their separation MUST successively increase, as the "length
contraction" factor successively increases with increasing speed
relative to the initial frame.

and secondly, that the two rockets are identically constructed, and
thus produce the

same thrust and acceleration when ignited.

Yes -- that is identical proper accelerations as a function of their
proper times.

Those two statements can't both be simultaneously true.

Sure they can.

I've written eleven papers (on viXra) on this subject,

Fontenot's claims are wrong. Repeat them eleven times and they are still
wrong.

Tom Roberts

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On 12/20/23 11:44 AM, Tom Roberts wrote:

Assuming the person on the trailing rocket measures the separation simultaneously in their successive instantaneously co-moving inertial
frames, that separation MUST successively increase.

That's NOT the way person on the trailing rocket measures the
separation! He measures the separation with a tape measure, and THAT
shows the separation is constant.

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On 12/20/23 1:06 PM, Mike Fontenot wrote:

On 12/20/23 11:44 AM, Tom Roberts wrote:

Assuming the person on the trailing rocket measures the separation
simultaneously in their successive instantaneously co-moving
inertial frames, that separation MUST successively increase.

That's NOT the way person on the trailing rocket measures the
separation! He measures the separation with a tape measure, and
THAT shows the separation is constant.

WHAT tape measure? -- none is specified in the problem.

Add one and you're still wrong:
To qualify as a tape measure, it must execute Born rigid motion, nailed
down only at the trailing rocket -- this makes it always be at rest in
the instantaneously co-moving inertial frame of the trailing rocket, so
this is no different from what I said above. Born rigid motion requires
the leading end of the tape measure to have a smaller proper
acceleration than the trailing rocket, with the difference evenly spread
out along its length. Note the two rockets do not execute Born rigid
motion, because they have equal proper accelerations; the leading rocket
has a larger proper acceleration than the leading end of the tape
measure, and thus pulls away from it. So the tape measure shows the
separation is increasing. This is no different from what I said above.

I do not know why Fontenot keeps getting this wrong. Or why he keeps
ignoring me when I tell him this. Or why he ignores the voluminous
literature on this. He has been obsessing over this scenario for many
years, gets it wrong, and refuses to THINK about it -- see THINK 1 and
THINK 2 of my previous message for a very simple demonstration of my
claims.

Tom Roberts

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On 12/20/23 10:29 PM, Tom Roberts wrote:

On 12/20/23 11:44 AM, Tom Roberts wrote:

Assuming the person on the trailing rocket measures the separation
 simultaneously in their successive instantaneously co-moving
inertial frames, that separation MUST successively increase.

I (Mike Fontenot) respond:

The person on the trailing rocket would want NOTHING to do with the
momentarily co-moving inertial person (the MCMIP) or what the MCMIP says
about the separation of the rockets: that person says that the rockets
started their engines at different times ... the accelerating trailing
person says the rockets were fired at the same instant.

And then I (MLF) said:

That's NOT the way person on the trailing rocket measures the
separation!  He measures the separation with a tape measure, and
THAT shows the separation is constant.

And Tom Roberts (TR) responded:

WHAT tape measure? -- none is specified in the problem.

If the two people on the two rockets want to run a tape measure between
their rockets, they can do it. Since I'm interested in the trailing
person's conclusions about their separation, I'll assume that the end of
the tape is attached to the tail-end of leading rocket, and that the
CASE of the measuring tape is attached to the front-end of the trailing
rocket, so that the tape is free to slide in or out of the case.

Tom wrote:

the leading rocket
has a larger proper acceleration than the leading end of the tape
measure, and thus pulls away from it.

And I (Mike) responded:

That's ridiculous. The spring tension inside the case of the tape
measure is small, and certainly doesn't pull the end of the tape away
from the leading rocket, where it is firmly attached.

Then Tom said:

So the tape measure shows the
separation is increasing.

And I (Mike) respond:

No, it doesn't. The accelerometers on the two rockets show the same
(constant) readings, and that results in the separation between the
rockets being constant, and the tape measure showing a constant reading.

Then Tom said:

Or why he (Mike) ignores the voluminous
literature on this.

The primary relevant literature for this discussion is the Wiki page on
Bell's Spaceship Paradox, and that webpage is a great example of "too
many cooks spoiling the broth" ... it's a collection of multiple
statements that are mutually inconsistent. For example, they say that
the initial inertial observers will observe the separation of the
accelerating rockets to be constant. The length contraction equation
(LCE) says that if a yardstick is moving away from an inertial observer,
the inertial observer will conclude that the length of the yardstick has
gotten smaller at any instant by the factor gamma (where gamma is a
function of speed, and increases with speed). So that means that if the initial inertial observers say the rocket separation is constant, then
the two rockets must actually be getting farther apart (as measured in
the frame of the person on the trailing rocket). And THAT means that
the two accelerometers CAN'T be showing the same readings: the leading
rocket is accelerating faster than the trailing rocket. So when the Wiki article claims that the initial inertial observers say the separation is constant, AND that the accelerometers show the same readings (i.e., that
the rockets "are identical"), they are being inconsistent. If the
initial inertial observers say the separation of the rockets is
constant, then the separation of the rockets, according to the people on
the trailing rocket, is increasing, and the leading accelerometer will
read higher that the trailing accelerometer.

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On 2023-12-21 05:29:31 +0000, Tom Roberts said:

I do not know why Fontenot keeps getting this wrong.

Maybe he just wants to disagree. That is easier if he gets
sometihnig wrong and keeps getting it (rather than some
other thing) wrong.

Mikko

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On 12/20/2023 11:29 PM, Tom Roberts wrote:

On 12/20/23 1:06 PM, Mike Fontenot wrote:

On 12/20/23 11:44 AM, Tom Roberts wrote:

Assuming the person on the trailing rocket measures the separation
 simultaneously in their successive instantaneously co-moving
inertial frames, that separation MUST successively increase.

That's NOT the way person on the trailing rocket measures the
separation!  He measures the separation with a tape measure, and
THAT shows the separation is constant.

WHAT tape measure? -- none is specified in the problem.

Add one and you're still wrong:
To qualify as a tape measure, it must execute Born rigid motion, nailed
down only at the trailing rocket -- this makes it always be at rest in
the instantaneously co-moving inertial frame of the trailing rocket, so
this is no different from what I said above. Born rigid motion requires
the leading end of the tape measure to have a smaller proper
acceleration than the trailing rocket, with the difference evenly spread
out along its length. Note the two rockets do not execute Born rigid
motion, because they have equal proper accelerations; the leading rocket
has a larger proper acceleration than the leading end of the tape
measure, and thus pulls away from it. So the tape measure shows the separation is increasing. This is no different from what I said above.

I do not know why Fontenot keeps getting this wrong. Or why he keeps
ignoring me when I tell him this. Or why he ignores the voluminous
literature on this. He has been obsessing over this scenario for many
years, gets it wrong, and refuses to THINK about it -- see THINK 1 and
THINK 2 of my previous message for a very simple demonstration of my
claims.

Tom Roberts

Wasting your time. But then it is yours to waste.

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You need to completely re-think this, because your claims are wrong. You
MUST learn what Born rigid motion [#] is. And isn't. And you MUST learn
to always specify which frame you are talking about. EVERY TIME.

On 12/21/23 1:48 PM, Mike Fontenot wrote:

The person on the trailing rocket would want NOTHING to do with the momentarily co-moving inertial person (the MCMIP) or what the MCMIP
says about the separation of the rockets:

That's very strange coming from you, as that is how you insist an
accelerated observer determines the "current age of a distant person".

The accelerometers on the two rockets show the same (constant)
readings, and that results in the separation between the rockets
being constant,

You keep saying stuff like this, WITHOUT SPECIFYING WHICH INERTIAL
FRAMES ARE INVOLVED. The separation is constant IN THE INITIAL INERTIAL
FRAME. But not in any instantaneously co-moving inertial frame of either rocket.

and the tape measure showing a constant reading.

No, it doesn't. See below.

First, let me show that the separation is constant in the initial
inertial frame:

*** Since they have equal proper accelerations [@], their
*** trajectories relative to the initial inertial frame
*** are identical, except for their initial offset. Their
*** separation therefore remains constant when measured
*** simultaneously in that frame.

[@] aka identical rockets or equal values displayed
by onboard accelerometers.

And you already know that being constant in the initial frame means
increasing separation in the successive instantaneously co-moving
inertial frames of either rocket:

[...] if the initial inertial observers say the rocket separation is constant, then the two rockets must actually be getting farther apart
(as measured in the frame of the person on the trailing rocket).

BEWARE: your "actually" is completely misplaced. You CANNOT talk like
that in relativity. There is no "actual", there are just values measured
in different inertial frames. This may be the core of your confusions.

Let's see how a tape measure is applied between rockets. Note the tape
measure must extend between the rockets on its own, and can be attached
to just one rocket, so the position of the other rocket can be read on
the tape. Any spring in the tape case is irrelevant.

To qualify as a tape measure it must execute Born rigid motion [#]. If
you affix the tape to the trailing rocket, to show what the trailing
observer measures (with an assistant at the leading rocket to read the
tape), then the leading rocket has a larger proper acceleration than the leading end of the tape, and the tape measure shows the rockets have
increasing separation. If you affix the tape to the leading rocket, to
show what the leading observer measures (with an assistant at the
trailing rocket to read the tape), then the trailing rocket has a
smaller proper acceleration than the trailing end of the tape, and the
tape measure shows the rockets have increasing separation.

[#] Born rigid motion is specified as an object
maintaining constant proper length. For an
acceleration along an axis, the object's leading
end must have a smaller proper acceleration than
its trailing end.

Bottom line: the two rockets do NOT execute Born rigid motion, because
their proper accelerations are equal and they are separated along their direction of acceleration. The tape measure, however, MUST execute Born
rigid motion, or it is not a tape measure.

[... more confusion due to not specifying which frames are involved]

You need to completely re-think this, because your claims are wrong. You
MUST learn what Born rigid motion is [#]. And isn't. And you MUST learn
to always specify which frame you are talking about. EVERY TIME.

Tom Roberts

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Tom says:

That's very strange coming from you, as that is how you insist an
accelerated observer determines the "current age of a distant person".

No, that is not at all how I say an accelerating observer determines the current age of a distant
person ... it involves NO inertial observers at all.

An accelerating observer (him, the AO) determines the current age of a
distant person (her, the DP) by asking the helper person, the HP, who is
moving along the same line as the AO, and accelerating with the same acceleration as the AO, and who happens to be momentarily co-located
with her at that instant.

The details are explained in my Amazon monograph:

An Accelerated Array of Clocks in Special Relativity: A Meaningful "NOW-at-a-Distance".

On Amazon, you can search on my full name (Michael Leon Fontenot), and
it will pop up all three of my monographs, plus an older one which is no
longer of any value. You can also get the same thing on viXra for free,
by searching on my full name there, and it will list all of my viXra
papers. That's more complicated, though, because there are a LOT of
them ... the Amazon one with the "Accelerated Array ... " title is an
easier way to get the collection. The two other monographs on Amazon
are the first one, which shows an error that Einstein made in his1907 gravitational time dilation equation [and its equivalent acceleration
version], and the second one where I give the corrected acceleration
version].

_______________________________________


Here are some additional responses to your last post:

In the current situation, of interest to me, (with finite, and constant, accelerations), the person on the trailing rocket knows that his
accelerometer shows the same constant value that the accelerometer shows
on the leading rocket, and so he knows that the separation between the
two rockets is constant. And he doesn't care what any inertial observers
say about anything! In that scenario, the initial inertial observers
will say that the separation between the rockets is decreasing,
according to the length contraction equation (applied at each instant).

The scenario in the Bell Paradox is DIFFERENT! There, the initial
inertial observers say that the separation between the two rockets is constant. In that case, (in spite of what some of the authors of the
Wiki page on the Bell Paradox say), the accelerometers on the two
rockets will show different accelerations (the leading rocket's
accelerometer will show a greater acceleration the reading on the
trailing rocket's accelerometer), and the separation between the two
rockets, according to the people on the rockets, will be increasing.

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On 12/23/23 6:13 PM, Mike Fontenot wrote:

In [the Bell paradox scenario], [...] the accelerometers on the two
rockets will show different accelerations

This is JUST PLAIN WRONG. Equal proper accelerations is stipulated in
the setup. (IOW: the rockets are identical.)

Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

You are apparently too invested in your mistakes to re-think this and
resolve your errors. Your problem, not mine.

Tom Roberts

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Tom Roberts wrote:

This is JUST PLAIN WRONG. Equal proper accelerations is stipulated in
the setup. (IOW: the rockets are identical.)

It WAS stipulated in the setup that the separation of the rockets,
according to the initial inertial observers, is constant. AND, it was stipulated in the setup that that the accelerometers on the two rockets
show the same (constant) readings, and thus that the separation of the
rockets is constant, according to the people on the trailing rocket.
But those two stipulations are inconsistent ... they cannot both be simultaneously true ... the length contraction equation (LCE) guarantees
that.

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On 12/24/23 1:14 PM, Mike Fontenot wrote:

Tom Roberts wrote:

[the Bell spaceship paradox] Equal proper accelerations is
stipulated in the setup. (IOW: the rockets are identical.)

It WAS stipulated in the setup that the separation of the rockets,
according to the initial inertial observers, is constant. AND, it
was stipulated in the setup that that the accelerometers on the two
rockets show the same (constant) readings, and thus that the
separation of the rockets is constant, according to the people on
the trailing rocket.

Nope. "according to the people on the trailing rocket" is YOUR mistake.
For instance that phrase does not appear in the Wikipedia article
referenced below. In adding it you have just confused yourself.

It should be replaced with "according to (people in) the initial
inertial frame".

But those two stipulations are inconsistent ... they cannot both be simultaneously true ... the length contraction equation (LCE)
guarantees that.

Yes. YOUR addition "according to the people on the trailing rocket" is
the source of the inconsistency. I have no idea why you think it applies.

Just THINK about it -- it makes no sense unless the rockets are
identical. That implies equal proper accelerations, and onboard
accelerometers read the same value.

Looking at https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox , I
do not see how you can misread it so badly. That page unequivocally says
the string will break, several times; equivalently your tape measure
will show the separation is increasing. Moreover, it repeatedly mentions
"equal accelerations" of the rockets, meaning either proper
accelerations or with respect to the initial inertial frame.

I repeat:
Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

You need to completely re-think this, because your claims are wrong. You
MUST learn what Born rigid motion [#] is. And isn't. And you MUST learn
to always specify which frame you are talking about. EVERY TIME.

[#] Born rigid motion is specified as an object
maintaining constant proper length. For an
acceleration along an axis, the object's leading
end must have a smaller proper acceleration than
its trailing end.

Tom Roberts

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Here is what you're missing, Tom:

The length contraction equation (LCE) says that the initial inertial
observers (IIO's) will say that the separation of the two rockets
(whenever they are moving wrt the IIO's) is LESS than what the people on
the rockets say the separation is.

So, IF the initial inertial observers (the IIO's) say that the
separation of the rockets is CONSTANT, then the separation must be
increasing, according to the people on the rockets.

But in the Wiki article, it says that the initial inertial observers
(the IIO's) say the separation is constant, AND the Wiki article says
that the rockets are identical (i.e., that the two rockets are
undergoing equal acceleration according to their accelerometers, so
their separation is constant, according to the people on the rockets).
That VIOLATES the LCE, so the Wiki article is WRONG. If the IIO's say
the separation is constant, then the accelerometer on the leading rocket
must read HIGHER than the accelerometer on the trailing rocket, and the separation must therefore be INCREASING, according to the people on the rockets.

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On 12/25/23 12:04 PM, Mike Fontenot wrote:

in the Wiki article, it says that the initial inertial observers (the
IIO's) say the separation is constant, AND the Wiki article says that
the rockets are identical (i.e., that the two rockets are undergoing
equal acceleration according to their accelerometers,

Yes.

so their separation is constant, according to the people on the
rockets).

NO! Justify this claim. Show your work.

[You'll find that you cannot justify this claim, for the
simple reason that it is WRONG.]

I have no idea where you got this notion.

I repeat:
Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

Tom Roberts

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You agreed with this:

"the Wiki article says that the rockets are identical (i.e., that the
two rockets are undergoing
equal acceleration according to their accelerometers,"


And yet you disagreed with this:

"so their separation is constant, according to the people on the rockets)."


So you seem to be saying that when the two rockets are undergoing equal accelerations, their separation isn't constant.


If that IS what you're saying, I certainly don't agree.

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On 12/25/2023 1:04 PM, Mike Fontenot wrote:

But in the Wiki article, it says that the initial inertial observers
(the IIO's) say the separation is constant, AND the Wiki article says
that the rockets are identical (i.e., that the two rockets are
undergoing equal acceleration according to their accelerometers, so
their separation is constant, according to the people on the rockets).
That VIOLATES the LCE, so the Wiki article is WRONG.

The initial observers see the rockets go faster and faster and are
subject to length contraction, as is the string. The string contracts in
length but the separation remains the same, therefore (according to
them) the string breaks.

That's one thing I think is cool about SR. Different observers get the
same result but for different reasons. One observer observes the string
length contracting and breaking, another observer observes the rockets
getting farther apart and the string breaking.

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On 12/25/23 2:25 PM, Mike Fontenot wrote:

[... more confusion and errors]

I repeat:
Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

Tom Roberts

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The length contraction equation (LCE) says that any inertial observer
(she) will measure the length of a yardstick that is moving away from
her to be getting shorter, by the factor gamma. Two separated rockets
with the same accelerometer readings are no different than the yardstick
as far as the LCE is concerned.

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You still owe me a response to this:
__________________________________________

You agreed with this:

"the Wiki article says that the rockets are identical (i.e., that the
two rockets are undergoing
equal acceleration according to their accelerometers,"


And yet you disagreed with this:

"so their separation is constant, according to the people on the rockets)."


So you seem to be saying that when the two rockets are undergoing equal accelerations, their separation isn't constant.


If that IS what you're saying, I certainly don't agree.

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On 12/26/23 10:52 AM, Mike Fontenot wrote:

The length contraction equation (LCE) says that any inertial observer
(she) will measure the length of a yardstick that is moving away from
her to be getting shorter, by the factor gamma.

Yes, when she measures it using standard rulers at rest in her frame,
marking the endpoints simultaneously in her frame. Because the yardstick
is executing Born rigid motion [#].

[#] If the yardstick does not execute Born rigid motion,
it is no longer a yardstick.

Two separated rockets with the same accelerometer readings are no
different than the yardstick as far as the LCE is concerned.

Not true. You are applying a statement valid for inertial motion to
the accelerating rockets. THAT'S INVALID.

In particular, you are assuming the rockets execute Born rigid motion,
like a yardstick would do [#], when they DON'T:

The yardstick's leading end has a smaller proper acceleration
than its trailing end, as required for Born rigid motion with
acceleration along an axis; the rockets, on the other hand,
have equal proper accelerations.

You MUST do this exercise:
Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

SR asserts the only different between their trajectories,
measured in the initial inertial frame as a function of
its time coordinate, is their constant separation.
IOW: their trajectories have the same shape in that frame.

So you seem to be saying that when the two rockets are undergoing
equal accelerations, their separation isn't constant.

YOU KEEP DOING THIS, AND IT TURNS WHATEVER YOU SAY INTO NONSENSE. You
MUST specify which frame you are discussing.

I am saying that their separation IN THE INITIAL INERTIAL FRAME is
constant, and therefore their separation in successive instantaneously co-moving frames of either rocket cannot be constant.

You just keep repeating the same mistakes, so from now on I am just
going to repeat the above exercise, until you attempt to do it.

Tom Roberts

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The people on the two rockets don't CARE what any inertial observers
think. The people on the rockets care that the accelerometers on their
rockets show the same readings, and that the separation between the
rockets doesn't change.

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On 12/27/23 7:51 AM, Mike Fontenot wrote:

The people on the two rockets don't CARE what any inertial observers
think.  The people on the rockets care that the accelerometers on their rockets show the same readings, and that the separation between the
rockets doesn't change.

Except to them their separation does change.

Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

SR asserts the only different between their trajectories,
measured in the initial inertial frame as a function of
its time coordinate, is their constant separation.
IOW: their trajectories have the same shape in that frame.

Tom Roberts

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I (Mike Fontenot) wrote:

The people on the two rockets don't CARE what any inertial observers
think.  The people on the rockets care that the accelerometers on
their rockets show the same readings, and that the separation between
the rockets doesn't change.

And Tom Roberts responded:

Except to them their separation does change.

Einstein didn't agree with you.

See:

https://einsteinpapers.press.princeton.edu/vol2-trans/319

In that paper, the separation of the two clocks undergoing equal
accelerations (and with no gravitational fields) is constant.

Also, Einstein showed the equivalence between a scenario with two
separated rockets undergoing equal accelerations versus a scenario with
two stationary clocks in a gravitational field that is constant in both
time and space.

See the first equation given in:

https://en.wikipedia.org/wiki/Gravitational_time_dilation

It says, in particular, that for two clocks in a constant and uniform gravitational field of force per unit mass “g”, separated by the
constant distance “d” in the direction of the field, the clock that is closer to the source of the field will run slower than the other clock,
by the factor exp(g d).

The equivalence principle then says that for two clocks that are
accelerating with the same acceleration “A”, separated by the constant distance “d” in the direction of the acceleration, the trailing clock
will run slower than the other clock, by the factor exp(A d). The two
values “g” and “A” are numerically the same.

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The reason that result is important is that it allows an accelerating
observer (undergoing a constant acceleration) to set up an arbitrarily
long array of clocks having that constant separation, along any given
straight line passing through him, which he can then use to tell him the current age of a distant person who is important to him (like his twin
that he left long ago). I.e., it gives him a meaningful
"NOW-at-a-distance".

I give all the details in my Amazon book (the one with the blue cover), entitled:

An Accelerated Array of Clocks in Special Relativity: A Meaningful
"NOW-at-a Distance”.

You can find it easily on Amazon by searching on my full name: "Michael
Leon Fontenot", or on "An Accelerated Array of Clocks". It's priced at
$7.07, which is about a dollar over the printing cost. (Amazon doesn't
print any books until they are ordered).

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On 12/27/23 2:29 PM, Mike Fontenot wrote:

The people on the two rockets don't CARE what any inertial
observers think. The people on the rockets care that the
accelerometers on their rockets show the same readings, and that
the separation between the rockets doesn't change.

Tom Roberts responded:

Except to them their separation does change.

Einstein didn't agree with you. https://einsteinpapers.press.princeton.edu/vol2-trans/319 In that
paper, the separation of the two clocks undergoing equal
accelerations (and with no gravitational fields) is constant.

Don't you read your references????

He says "... one restricts oneself to the case in which 7 is so small
that terms of the second or higher power in 7 may be neglected. Since we
are going to restrict ourselves to that case, we do not have to assume
that the acceleration has any influence on the shape of the body."

Note also that this was written 1900-1909, well before the understanding
of GR. He is talking in terms of "clocks running more slowly", which is inconsistent with GR. Indeed it was written before Born rigid motion was described.

Note that the two rockets have identical proper accelerations, and
therefore do NOT execute Born rigid motion -- their separation
is not constant in their subsequent inertial rest frames (being constant
would be Born rigid motion).

https://en.wikipedia.org/wiki/Gravitational_time_dilation It says,
in particular, that for two clocks in a constant and uniform
gravitational field of force per unit mass “g”, separated by the
constant distance “d” in the direction of the field, the clock that
is closer to the source of the field will run slower than the other
clock, by the factor exp(g d).

That, too, is WRONG. Identical clocks ALWAYS RUN AT THE SAME RATE,
regardless of their motion or location in a gravitational field. It is COMPARISONS of clock that show this, not the clocks themselves.

The equivalence principle then says [...]

The equivalence principle is a LOCAL principle only. You cannot extend
it to the case of two rockets separated by astronomical distances.

You STILL have not responded to my challenge, and until you do you won't understand this:

Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations. You are claiming they do have differently-shaped trajectories, which is ABSURD. (See the "***" paragraph of my previous
post, and its [@] footnote.)

SR asserts the only different between their trajectories,
measured in the initial inertial frame as a function of
its time coordinate, is their constant separation.
IOW: their trajectories have the same shape in that frame.

Tom Roberts

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On 1/1/24 10:42 AM, Tom Roberts wrote:

Please explain how, in the initial inertial frame, two identical rockets
can have differently-shaped trajectories simply because they are started
at different locations.

Because the length contraction equation (LCE) DEMANDS it!

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On 1/1/24 11:58 AM, Mike Fontenot wrote:

On 1/1/24 10:42 AM, Tom Roberts wrote:

Please explain how, in the initial inertial frame, two identical
rockets can have differently-shaped trajectories simply because
they are started at different locations.

Because the length contraction equation (LCE) DEMANDS it!

That's nonsense. SHOW YOUR WORK. You'll find that you are assuming
constant separation in their subsequent instantaneously co-moving
inertial frames, without justification. That assumption is wrong.

Just THINK about it: location in an inertial frame cannot possibly
affect the shape of identical rocket trajectories measured in that
frame. Translation invariance applies.

Apparently you are too psychologically invested in your mistake to even consider learning about Born rigid motion, when it applies, and when it doesn't. How sad.

Tom Roberts

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On 12/30/23 2:40 PM, Mike Fontenot wrote:

The reason that result is important is that it allows an
accelerating observer (undergoing a constant acceleration) to set up
an arbitrarily long array of clocks having that constant separation,
along any given straight line passing through him, which he can
then use to tell him the current age of a distant person who is
important to him (like his twin that he left long ago). I.e., it
gives him a meaningful "NOW-at-a-distance".

I keep telling your "that result" is wrong.

But even if it were correct this would not work, because none of the
clocks along the rocket's direction of acceleration are synchronized --
the rocket would observe clocks ahead of the rocket to accumulate proper
time faster than the rocket's clock, and clocks behind the rocket to
accumulate proper time more slowly than the rocket clock. (It does not
matter how the rocket observes the distant clocks, as long as they use a consistent method.)

Not to mention the impossibility of constructing such an
array of clocks....

The basic problem is that if these clocks all have equal proper
accelerations, then they don't execute Born rigid motion, and their
separations will vary. If they are given different proper accelerations
such that they do execute Born rigid motion, the previous paragraph
still applies.

Tom Roberts

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I (Mike Fontenot) wrote:

The reason that result is important is that it allows an
accelerating observer (undergoing a constant acceleration) to set up
an arbitrarily long array of clocks having that constant separation,
 along any given straight line passing through him, which he can
then use to tell him the current age of a distant person who is
important to him (like his twin that he left long ago).  I.e., it
gives him a meaningful "NOW-at-a-distance".

And Tom Roberts responded:

I keep telling your "that result" is wrong.

But even if it were correct this would not work, because none of the
clocks along the rocket's direction of acceleration are synchronized ...

And I (Mike) responded:

That's true, the clocks on the rockets farther in the direction of the acceleration run faster, but they are faster by a known factor, so the
given observer (the "GO", whose viewpoint we want to determine) can
compute (for each instant in HIS life) all the clock readings of each of
the other rockets. So that still establishes a "NOW-at-a-distance" for
him: he can (eventually) determine the current age of any particular
distant object that he cares about (like his home twin whom he left long
ago). All he has to do is arrange for his "helper friend", on the
rocket that happens to be momentarily stationary with respect to her, to observe her age at the given common instant in the frame of the array of clocks. The helper friend then relays that observed age of the home
twin to the given observer (the "GO").

The basic problem is that if these clocks all have equal proper accelerations, then they don't execute Born rigid motion, [...]

Born rigid motion concerns the views of inertial observers, which is
irrelevant in my analysis. My analysis concerns only the views of
observers who all have the same acceleration, as confirmed by their accelerometers). In that case, as confirmed by Einstein's 1907 paper,
and by his equivalence principle example, the separation of the rockets
is constant.

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