[Halc]

OK, I’ll try to comment on the whole thing this time.

In an LET universe, when an object accelerates, it contracts in length regardless of whether it's being pulled or pushed up to speeds.

A pulled thing will stretch due to strain.  A long railroad train is usually one or two railroad-cars longer in length than when the same train is being pushed.
Speed, not acceleration, causes relativistic contraction, and that is true in both LET and elsewhere.

If you take an elastic band and accelerate every atom of it up to 0.86c simultaneously in an instant, it will find itself to be stretched to twice its unstressed length, so it will shorten.

Yes, but that was neither pushed nor pulled.  The acceleration was applied everywhere, so all the strain is due to the instantaneous proper length expansion.

The question is whether that stress could be detected in any way during an acceleration, and if it can, it would be different for a deceleration from 0.86c to zero, because that would lengthen it instead and lead to it going loose (or to a solid rod being compressed after the deceleration and needing to extend).

It can be detected, but you are pushing and pulling your rods, and the strain from doing that is going to completely overwhelm the miniscule strain from the gradual relativistic length changes. 

The acceleration of the ship as a whole is constant, but the ends will either contract in or extend out, and I'm looking to see if that can be detected and if the two different things can be told apart.

From different frames, yes, they’ll be different.  You can get any data you want by selecting the frame that gives that data.  That seems to be what you’re doing here.  You never say in which frame these measurements are to be taken.  Ship frame?  The rods are static in that frame.

The measurements are made in the ship's frame of reference,

Oh wow, I never got that.  Yes, they’d have to be if you’re trying to detect the aether.  Any other frame probably begs that you already know it.  But nothing is going to be different in ship frame between accelerating and decelerating.  Principle of relativity would be violated, and you’d win a Nobel prize.  Surely you’re aware of this.

but LET says that it is either contracting or extending (= uncontracting). Throw off your SR glasses for this and try to see it through LET. You may be the only other person here capable of thinking through this stuff properly, so I'd certainly value your help.

I can do that.

First of all, your ship has to face in the way it is thrusting.  I don’t totally get that from your descriptions where I wasn’t sure in the deceleration case which way the ship is facing.  If it is facing forward during deceleration, then you presume to know which way is the aether, and you’re oriented your ship that way.  So it always faces in the direction of thrust just like the space shuttle does when it drops out of orbit.  I’d say it always accelerates forward, but you have a different definition of acceleration which really inhibits communication.  FL and FH are always compressed by strain, and RL and RH are always stretched by strain.  Both are contracted by speed, but that isn’t strain.

An automobile violates this.  It faces away from its thrust when decelerating.  The occupants holding plumb lines can very much tell the difference.  The automobile presumes to know the direction of the road-wind, and is thus no proof at all that the road is stationary.
The physics of your ship is much more like the asteroids video game and far less like the physics in star wars.  Yes, there is aether in asteroids, but detecting it is subtle (assuming the ship cannot see the screen edge) if you play a version with no friction.  I’ve seen some versions with friction, which of course totally different physics.

Rod RH (already stretched by the acceleration) has a bit of extra effective stretch added to it by length contraction,

How does contraction add to stretch?  Wouldn’t they potentially cancel if they happen to be equal?  Sorry to interrupt mid-sentence...

If you have a piece of rubber a lightyear long which is capable of being stretched to twice its normal length without breaking, accelerating every part of it to 0.866c in a second would leave it in a stretched state due to length contraction. It will then take a good few years to contract, although it might break up in the attempt.

This analogy is inappropriate.  Your ship is in steady state, not accelerating from a stop to some speed in an instant.  The rubber rods are respectively pulled and pushed by that acceleration, and are not getting thrust applied to their parts all at once.  Your rods would not in any way behave like this piece of rubber you’re describing here.  They can be modelled as rubber, which is just a metal rod with more obvious strain resulting from a given stress.  I can even ignore the fact that a long rod will buckle if you push too hard on it.  The trailing rods have a finite length limit that is a function of acceleration.  A trailing rod cannot be longer than the distance to the Rindler horizon, even given infinite strength.   The rod in front can be as long as you want, so long as you don’t expect the ship to turn easily.

All that said, you have answered my question.  The trailing rods have additional strain from the fact that the ends of them need to accelerate harder than the rest of the ship.  That strain adds to (not cancels out) the strain put on them by the thrust of the ship.  Still, that strain is fixed and a function of acceleration, not a function of speed.  It isn’t going to change as your speed goes up and down.

This was the thing that confused me through my whole reply earlier, leading me to wonder if you’re facing the ship in the direction of travel rather than in the direction of thrust.  You are always facing thrustward, as a spacecraft should.  I am re-interpreting things in that light.

The length contraction should accelerate the end of the rear rods more than the ship as a whole, and that's the effect we want to detect. If RL contracts more quickly than RH because of the large mass on the end of RH, they should contract at different rates and show up the length contraction that can't normally be detected, making it visible to observers in all frames.

How do RL and RH contract at different rates?  Under constant proper speed change, the two rods will be stretched to some fixed strain, and the two marks will always line up.  The weight on the end of one rod is why the rods are different lengths, yes, but the weight does not affect length contraction except that it accelerates a little harder, being a tad further behind.  The point where the two marks line up are always accelerating at the exact same rate as each other, but still more than the ship as a whole, as you point out.
This is going to be true whether the ship is accelerating or decelerating.

Case 1 and case 2 seem identical to the people in the ship, unless they see different behaviour in the rods regarding whether the ends of the shorter ones are level with the marks on the longer ones.

Yes, that would be a difference.  But they’ll not see it.  Each shorter rod should always meet the mark on the longer rod in both cases. I am attempting to see how you might think that is not so.

If we slow something from 0.86c to zero in an instant, it will be compressed, and it will immediately extend as a result.

This is a different case than the steady acceleration.
If you are coasting and go from 0.86c to 0 instantly, the rear rods will break off right at the ship and keep going at .86c.  The front rods will also continue to move for a while, compressing their length to almost nothing, before they spring back to their proper length.  A maximum strength front rod must still compress to about 0.28 its length before the end beings to decelerate.

Anyway, I realize you’re not proposing this, just trying to clarify the parts I could not parse.
I’ll have a second go at it.

FL reacts more quickly than FH because it doesn't have a great mass at the end to push forward.

In the abrupt stop, the two react (begin to decelerate) at almost the same time.  FH actually begins first because it is closer to the ship, and it reacts harder (greater peak acceleration) because the rod is going to compress further than FL due to that weight that is going to have a shorter distance to achieve its deceleration.  Again, none of this is relevant to the steady thrust scenario.

I’m going to comment again on the original paragraph that I edited off.  You’ve clarified things, thanks.

This is the deceleration case, where you propose a difference in local observation.  What we locally observe is that the two marks always line up, both front and back.  Your job I suppose is to explain why that would not be the case during deceleration.

Rod RH (already stretched by the acceleration) should have a bit of effective compression added to it by length extension (or decontraction), as does RL, removing some of the stretch and allowing the rod to lengthen, but I'm not sure how it would react. Is it a hindrance as before, or is it now going to help extend the rod more quickly?

The ends of the R rods are moving faster than a decelerating ship, so they already have the momentum needed to extend the rods without the extension effect adding or subtracting to the stress and strain on the rods.  So it doesn’t have any additional compression effect on the rods.  It is not pushing the rods out, because the rod ends are already moving out and require no force adjustment to do their extending.  So it seems to be neither help nor hindrance.

If the latter, then it could hide the effect we're trying to see, but remember that it should still show up when the ship is momentarily stationary (moving from deceleration to acceleration), because at that point the ends of the shorter rods would line up with the marks.

There should never be a time in any frame where the ends of the short rods don’t line up with the marks on the longer.  Anybody in any frame can look at that because they’re in each other’s presence.  See my remark below about the validity of this statement.

Either way then, we should have a method by which the absolute frame could be identified, unless there's a fault somewhere in the argument (which I fully expect to be the case, but if it turns out that there isn't, it would be a shame to miss the experiment that finds the aether by assuming that no such experiment can exist).

My statement just above makes that very assumption, so take it as exactly that, just an assertion that comes from Galileo's time.  If we’re questioning that assumption, it would be begging to assume the principle.  Still, the comment about no compressive force resulting from deceleration is the one that applies here.  That’s why the rods don’t change alignment between acceleration and deceleration.

[David Cooper]

Thanks for giving it a go, Halc. I think I've found the answer (having had the unfair advantage of 24 extra hours to think about it), but before I can explain it, I need to make sure that you understand the experiment. From the point of view of the people in the ship, it's always moving in the same direction, but it may be moving backwards and decelerating relative to the absolute frame, or it may be moving forwards and accelerating relative to that frame (or it could be doing the first of those thing until it stops relative to the absolute frame, at which point it will start to do the second thing). While it's doing the first thing (decelerating), there should be length extension, and while it's doing the second, there will be contraction instead. For the rearward-pointing rods, extension should reduce the amount of acceleration acting on them, while contraction should increase it. If you increase the acceleration force, the rod with the large mass on the end will lengthen more than its partner, whereas if you decrease the force, it will shorten more than its partner, which means the end of the rod without the mass on its end will move relative to the mark.

The solution to this though is to recognise that while the force changes due to length contraction/extension and doesn't match up to the constancy of acceleration of the ship as a whole, the change in force is different for the different rods due to the difference in tension in each - a higher amount of tension in the rod with the large mass on its end leads to a greater adjustment force induced in it by the length contraction, so the two rods lengthen or contract in equal measure, completely hiding the effect from view in all cases.

I got there by replacing the rods with elastic to multiply the differences and make things easier to see clearly. If you imagine one piece of elastic with a weight on the end of it and the other without, the one with the weight may stretch out to twice the length under acceleration while the other one hardly stretches at all. Add in some length contraction and it's acting on atoms in the stretched elastic that are much more strongly trying to pull in, so the length contraction is acting on a lot more force and its effect is proportional to the amount of force it's acting on. In the unweighted elastic, there's hardly any force there for it to affect, so it has less input. The key thing is that it isn't an input of force, but something that adjusts the force that's already there, so it affects it in direct proportion to the amount of force that's active there. It is not an equal additional acceleration force in both rods, and the mistake I made at the start was in assuming that it would be an equal input of force to each.

[David Cooper]

This is just another example of how the absolute frame remains hidden from us - every experiment we try to design to reveal these physical differences for different speeds of travel through the fabric of space is masked in one way or another. It looks impossible to design any experiment that should be able to detect the absolute frame (or the aether wind).

Mad Aetherist frequently points to MMX experiments with gas in them providing different results from ones using a vacuum, but all the interactions between the light and gas should obey the same rules and mask the effect that Cahill claims has been detected. I doubt that anything unusual is going on there that could make a difference: the gas molecules and their atoms will length contract like anything else, and their rate of interaction with any photons they encounter will be slowed by their speed of movement through space. What mechanism could there be that would enable a gas-filled MMX to detect the aether that wouldn't also interfere with the speed of communications through air?

[Halc]

From the point of view of the people in the ship, it's always moving in the same direction,

It is always thrusting in the same direction.  They can’t detect motion at all, and for all they know (without  a window), they’re sitting in a building on a planet.  They detect a force, and that’s it.  The rods and stuff would all behave the same from their POV if it were uniform gravity and no speed whatsoever.

For the rearward-pointing rods, extension should reduce the amount of acceleration acting on them, while contraction should increase it.

This is incorrect. Both increase the acceleration.  In both cases, the R rods experience more g force than the main ship (and the F rods less g force).  In the decelerating case, it is because the far ends of the R rods are already moving faster than the ship, but are slowing more than is the ship, to eventually match the ship’s speed when it reaches the absolute frame.

If you increase the acceleration force, the rod with the large mass on the end will lengthen more than its partner, whereas if you decrease the force, it will shorten more than its partner, which means the end of the rod without the mass on its end will move relative to the mark.

That it will, but our example has constant acceleration of the ship, so the marks remain aligned forever.

The solution to this though is to recognise that while the force changes due to length contraction/extension

No it doesn’t.  The force never alters.  The force is a function of acceleration, not of speed, so it never wavers for any given point on the ship or a rod.  It is different at one point than another, but always constant at a given point.

I got there by replacing the rods with elastic to multiply the differences and make things easier to see clearly. If you imagine one piece of elastic with a weight on the end of it and the other without, the one with the weight may stretch out to twice the length under acceleration while the other one hardly stretches at all.

Yes, the effect is more noticeable with less stiff materials.  A real science experiment would probably still use some form of metal which has very predictable strain properties as opposed to elastic which tends to permanently deform after not much time under stress.  The weight scales for vegetables in the stores use metal springs for this reason, not elastic ones.  For the thought experiment, elastic works fine.

[David Cooper]

From the point of view of the people in the ship, it's always moving in the same direction,

It is always thrusting in the same direction.  They can’t detect motion at all, and for all they know (without  a window), they’re sitting in a building on a planet.  They detect a force, and that’s it.  The rods and stuff would all behave the same from their POV if it were uniform gravity and no speed whatsoever.

Coincidence that the two cases look the same. The physics is very different as no length contraction is involved in the gravity case. The ship has windows though in any case, and this is happening in deep space with the ship not tied to or resting on anything.

For the rearward-pointing rods, extension should reduce the amount of acceleration acting on them, while contraction should increase it.

This is incorrect. Both increase the acceleration.

If contraction acts on the rearward-pointing rods, the ends are pulled in towards the ship. If extension acts on them, the ends are pushed out away from the ship. In the former case, that increases the acceleration of the ends, while in the latter case it decreases their acceleration (compared with the acceleration acting on the ship as a whole).

In both cases, the R rods experience more g force than the main ship (and the F rods less g force).  In the decelerating case, it is because the far ends of the R rods are already moving faster than the ship, but are slowing more than is the ship, to eventually match the ship’s speed when it reaches the absolute frame.

If you're analysing events from the absolute frame and you start with the ship at rest, when the acceleration begins, we get a contraction which means that the ends of the rearward-pointing rods must be accelerating more. In the opposite case where the ship is moving backwards at a constant speed, then decelerates (while thinking it's accelerating forwards), the extension of the rearward-pointing rods must reduce the acceleration acting on their outermost ends.

If you increase the acceleration force, the rod with the large mass on the end will lengthen more than its partner, whereas if you decrease the force, it will shorten more than its partner, which means the end of the rod without the mass on its end will move relative to the mark.

That it will, but our example has constant acceleration of the ship, so the marks remain aligned forever.

Only because the extra or reduced acceleration from the length extension/contraction is masked by it varying in strength according to the different amount of force running through the rods.

The solution to this though is to recognise that while the force changes due to length contraction/extension

No it doesn’t.  The force never alters.  The force is a function of acceleration, not of speed, so it never wavers for any given point on the ship or a rod.  It is different at one point than another, but always constant at a given point.

There are two factors involved in producing the acceleration force - one is the constant acceleration of the ship and the other is length contraction/extension. The latter one will be visible to observers at rest in the absolute frame and they will see the truth of what is going on, but they will not know it to be the truth as they won't know which frame is the absolute frame.

[Halc]

They can’t detect motion at all, and for all they know (without  a window), they’re sitting in a building on a planet.  They detect a force, and that’s it.

Coincidence that the two cases look the same. The physics is very different as no length contraction is involved in the gravity case.

Of course there is, else you’d have a local test for dilation when light speed is measured at greater than c in a gravity well because clocks run slow down there but the lengths are unaltered.
The two cases look the same not by coincidence, but because the one was derived from the other.

For the rearward-pointing rods, extension should reduce the amount of acceleration acting on them, while contraction should increase it.

You’re slipping up David.  I just realized you used the word acceleration correctly when discussing an instance where extending is going on.  That’s like cheating on your lover.

This is incorrect. Both increase the acceleration (edit: deceleration).

If contraction acts on the rearward-pointing rods, the ends are pulled in towards the ship. If extension acts on them, the ends are pushed out away from the ship.

Yes, but all that reduced deceleration was done when the ship started decelerating and the ends of the R rods were allowed to continue at a greater speed than the already decelerating ship.  The force (the push for extending that you’re talking about) is a function of the second derivative of speed, not the first derivative.  That means it is felt when the ship starts accelerating/decelerating, but during steady deceleration, that component is absent and all you have remaining is the R ends going faster trying to slow down (tension) to match the ship’s speed, which they will when it gets to the absolute frame.
The F rods initially decelerated to a speed less than the ship (momentarily reducing the steady state compression from a nonzero second derivative), but after that (steady state) spend their time decelerating less until the ship speed falls enough to match it.

If you're analysing events from the absolute frame and you start with the ship at rest, when the acceleration begins, we get a contraction which means that the ends of the rearward-pointing rods must be accelerating more. In the opposite case where the ship is moving backwards at a constant speed, then decelerates (while thinking it's accelerating forwards), the extension of the rearward-pointing rods must reduce the acceleration acting on their outermost ends.

They’re going faster and have to stop at the same time as the ship, so they have to be decelerating more than the ship, not less.  It is no different than running the acceleration case in reverse.  The situation is totally time symmetric, so the R rods are going to have the same g force in both cases.

You asked for my opinion.  That’s it, and it is with my etherist hat on as best I can.

There are two factors involved in producing the acceleration force - one is the constant acceleration of the ship and the other is length contraction/extension. The latter one will be visible to observers at rest in the absolute frame and they will see the truth of what is going on, but they will not know it to be the truth as they won't know which frame is the absolute frame.

Agree, but the truth is that the R ends are going faster than the ship, and are thus decelerating harder.  The observers in the rest frame will see this even if they don’t know they’re in that frame.

[Halc]

The physics is very different as no length contraction is involved in the gravity case.

Of course there is, else you’d have a local test for dilation when light speed is measured at greater than c in a gravity well because clocks run slow down there but the lengths are unaltered.

This reply I gave here is hasty.  I looked up this sort of thing and the answer is quite complicated.  Similar to the accelerating ship, the length will be contracted only in the direction of acceleration, meaning a ruler held vertical will be shorter than the same ruler outside a gravitational field.  It is actually longer if held horizontal.

The calculations involved are quite different for a free-falling observer in the gravity field vs a stationary one (the latter being the case for a planet or for an accelerating ship).

[opportunity]

These are one of the long questions based on the fundamentals used in the question. The question asks for a completeness of relativity that should already be implied in the a-priori definitions of time and space, right?

[alancalverd]

Relativity is very simple as long as you remember that it is relativity. If observers are not moving with respect to one another, there  can be no relativistic effect between them. If they are, then Einstein's equations seem to have been tested to an extraordinary degree of precision and found correct.

[opportunity]

Time and space are already relative without asking the allegiance of a theory that can't explain everything such as Einstein's.

sh1t, why do I have to say that?

[David Cooper]

...The physics is very different as no length contraction is involved in the gravity case.

Of course there is, else you’d have a local test for dilation when light speed is measured at greater than c in a gravity well because clocks run slow down there but the lengths are unaltered.

I'm not talking about a detectable difference, but an underlying one. In the acceleration case, you get more and more length contraction the longer you go on accelerating. If you're just sitting on the surface of a planet, you get no change in length no matter how long you sit there.

For the rearward-pointing rods, extension should reduce the amount of acceleration acting on them, while contraction should increase it.

You’re slipping up David.  I just realized you used the word acceleration correctly when discussing an instance where extending is going on.  That’s like cheating on your lover.[/quote]

It was right for one of them, and I assumed you could convert for the other if you wanted to.

This is incorrect. Both increase the acceleration (edit: deceleration).

If contraction acts on the rearward-pointing rods, the ends are pulled in towards the ship. If extension acts on them, the ends are pushed out away from the ship.

Yes, but all that reduced deceleration was done when the ship started decelerating and the ends of the R rods were allowed to continue at a greater speed than the already decelerating ship.

The reduced deceleration acts for as long as the ship is decelerating, and the rods continue to extend in length until the point where no contraction is acting on them, at which point they are at maximum distance from the centre of the ship. Once the deceleration turns becomes acceleration, They contract and move closer to the centre of the ship. This is not something that all happens at the instant that the deceleration/acceleration begins.

The force (the push for extending that you’re talking about) is a function of the second derivative of speed, not the first derivative.  That means it is felt when the ship starts accelerating/decelerating, but during steady deceleration, that component is absent and all you have remaining is the R ends going faster trying to slow down (tension) to match the ship’s speed, which they will when it gets to the absolute frame.

That can't be right because when the deceleration of the ship become acceleration (when it is momentarily at rest in the absolute frame, the extension turns into contraction. There has to be an on-going reduction in deceleration up to that point and a new, then on-going increase in acceleration after that point - if that wasn't the case, it would just adjust to a fixed length for the rest of the acceleration.

The F rods initially decelerated to a speed less than the ship (momentarily reducing the steady state compression from a nonzero second derivative), but after that (steady state) spend their time decelerating less until the ship speed falls enough to match it.

When the deceleration begins, they are compressed, but the length extension reduces their shortening all the way to the point when the ship stops in the absolute frame, and then they are more shortened after that as the contraction kicks in, so again this is not something that all happens at the start of the deceleration.

They’re going faster and have to stop at the same time as the ship, so they have to be decelerating more than the ship, not less.

I wasn't discussing the ship stopping, but clearly if you switch the rockets off during acceleration, the ends of the trailing rods should (in the absence of the front rods) continue to accelerate the ship for a moment due to their faster movement. During a deceleration though, when the rockets are switched off, the end of the rods will be slower than the ship, so the ship will appear to be decelerated for a moment from the point of view of the people inside it. That would show up the absolute frame, so either relativity breaks or I've made another mistake somewhere.

You asked for my opinion.  That’s it, and it is with my etherist hat on as best I can.

Thanks for going through it - you've just brought something else out that needs clearing up, and it's a much simpler thought experiment:-

We now have a ship with a single rod sticking out behind it and a large mass on the end. Under deceleration (which still feels like acceleration to the people in the ship), length extension (contraction being lost) leads to the mass moving away from the ship, whereas under acceleration the rod will be contracting instead, leading to the mass moving towards the ship, so when the rockets are turned off, what happens? Do the people in the ship measure a momentary deceleration of the ship in the former case? Of course, there's tension in the rod when the rockets are firing, so the mass will always pull the ship towards itself a bit when the rockets are switched off because that tension force has to be removed, but in one case there will be a tiny bit of extra force added to that, while in the other case it will subtract instead.

How can this effect be masked?

Edit: It may be a synchronisation issue, because those are also reversed for the two cases.

[Halc]

Of course there is, else you’d have a local test for dilation when light speed is measured at greater than c in a gravity well because clocks run slow down there but the lengths are unaltered.

I'm not talking about a detectable difference, but an underlying one. In the acceleration case, you get more and more length contraction the longer you go on accelerating. If you're just sitting on the surface of a planet, you get no change in length no matter how long you sit there.

And there’s me making the derrivative mistake.  Yes of course, it is not equivalent to length contraction over time, which is not a local effect.  There is dilation of sorts in a static field, just as there is static dilation on an accelerating ship.  I mentioned that in the prior post where I found myself unable to justify the statement you quoted here.  Anyway, standing on a planet (not in freefall), length is contracted a fixed amount in the vertical direction, and is actually a bit longer in the horizontal direction.  I imagine both these would be true locally on a ship, but the don’t necessarily translate to any kind of absolute change.
The effect might be part of why you cannot orbit near a black hole, where these effects are very significant.  It takes constant acceleration (power) to keep out of a black hole beyond a certain limit that is at a radius half again where the event horizon is.  I digress...

The reduced deceleration acts for as long as the ship is decelerating, and the rods continue to extend in length until the point where no contraction is acting on them, at which point they are at maximum distance from the centre of the ship.

We can repeat all we want.  So I guess we agree to disagree on this point.
Tell me what the observer will see then.  I imagine you have a ship decelerating until it changes to acceleration, and there is some change (the marks suddenly line up differently?) and you have your empirical test for being stationary, at least in the dimension along which the ship is running.  You’d have to do it 3 times to get all 3 dimenstions.
The lines don’t budge if my argument is correct.  There is no observed change when the mode changes to acceleration.

That can't be right because when the deceleration of the ship become acceleration (when it is momentarily at rest in the absolute frame, the extension turns into contraction.

Yes, it does, but that is no change to the derivative of velocity, only the velocity itself, moving in a straight unbroken line from negative into positive territory, with no discontinuity in the line.

I wasn't discussing the ship stopping, but clearly if you switch the rockets off during acceleration,

Stopping = reaching speed zero, not shutting anything off.  The plan was to keep the engines on and go right into acceleration as you described it.  You can’t shut off at the absolute frame if you haven’t yet measured it, which was the point of this experiment.

Under deceleration (which still feels like acceleration to the people in the ship),

Funny that.  Almost like Einstein was correct, eh?  Sorry.  My ether-hat fell off there for a second...

so when the rockets are turned off, what happens? Do the people in the ship measure a momentary deceleration of the ship in the former case? Of course, there's tension in the rod when the rockets are firing, so the mass will always pull the ship towards itself a bit when the rockets are switched off because that tension force has to be removed, but in one case there will be a tiny bit of extra force added to that, while in the other case it will subtract instead.

The force due to the tension and strain will cause a brief deceleration of the main ship.  Newton says that.  It will cause the rod to vibrate actually, but it dies down.  So we account for that, and are left with the fact that the weight out there is going faster than the ship when we shut off the thrust.  It is moving in or moving away depending on if it was accelerating or decelerating, and the final speed will be a result of that momentum being transferred to the ship.  Yes, a difference between the two cases, but only if you make the computation in the known rest frame.  The people on the ship who are not at rest will feel no difference because in their frame, the weight at the end of the strained rod is stationary the whole time.  It isn’t a test for determining the rest frame if you need to make your measurements from it to detect it.

[David Cooper]

It was indeed a synchronisation issue. It's all to do with the time that it takes for the removal of the acceleration force to propagate to the weight at the end of the rod. In the case where the ship is decelerating and then switches off the rockets, the weight at that moment is moving more slowly than the ship, but it takes a long time for the removal of force to reach the end of the rod by the weight, so it's still accelerating the weight right up to that time. In this case, the effect of the length extension is cancelled out by the extra time the weight spends accelerating.

If the ship is accelerating, length contraction ensures that the weight is moving faster than the ship when the rockets are switched off, but this time the propagation of the loss of the acceleration force travels the length of the rod much faster, giving the weight less time to continue accelerating.