[Halc (OP)]

In the original frame, that hour isn't enough to contract the object by 1000th, never mind to 1000th of the initial length. The functionality of the ship is practically halted.

So what?  I smash an egg with a hammer, and the egg is very much broken, even in a frame like you describe where the 'functionality is slowed' a thousand fold.  Observing the destruction in such a frame doesn't change what happened to the egg, and what you're proposing doing to our object will blow it into considerably more smithereens than detonating a series of nuclear bombs along its length.  If you hit even higher acceleration, it will eventually produce enough tidal stress to pull the nucleus of each atom apart.  The faster you go, the more violence you're doing to it.  I don't know why you're pursuing this proposal since it clearly violates the rigidity conditions.

Are you claiming that accelerated reference frames are less real?

Of course they're less real. Take a rotating frame as an example of a fake frame.

Rotating frame are quite real as well.  You live in a rotating accelerating frame.  The only sort of frame that doesn't exist anywhere is the kind described by SR, which is a pure inertial frame.

Imagine that you're in a space station made of a rotating ring designed to produce artificial gravity. You have a series of clocks round the ring which you want to synchronise, so you synchronise the first pair, then the next (meaning one of the first pair plus the next clock round from there), then the next, and so on all the way round to the start. Have you got a frame for the whole ring in which there's a single unified moment?

Even with inertial (SR) frames, the synchronization of clocks not in each other's presence is frame dependent: Not real.
A rotating frame is more real in this sense since it is absolute.  You put out a signal from the center of rotation and all clocks will zero themselves when they get the signal.  You have just objectively synced the clocks in that frame, and you didn't even have to know the angular rate to do it.  Of course if the clocks are not moving in that frame, the rate at which they run is a function of the angular rate and distance from said center, so only clocks at the same radius (like the situation you describe) will stay in sync.

No - you can see it break catastrophically between the first and last clock. It breaks because the speed of light across each clock is different relative to that clock in opposite directions round the ring.

You seem to be attempting to sync clocks in a rotating frame using some sort of (Newtonian?) reference frame rules.  Use rotating rules when using a rotating frame.

Rotating frames are bogus. We know from such rings that the actual speed of light relative to objects varies in different directions, and that's a crucial piece of knowledge which must be applied to everything else.

That's why you use rotating rules.  Speed of light is anything but constant in a rotating frame, so you can't try to use some method that assumes a constant speed of light like you sort of describe above.  Its properties being different doesn't make the frame bogus.

In an accelerated frame where an object is actually accelerating through space, we know that the speed of light relative to different parts is not the same, so the frame provides a distorted representation of reality.

Not distorted.  Just not inertial.

You are free to label accelerating frames to be contrived if you like and only consider things from the perspective of inertial frames.  What appears to various observers and happens to various objects is exactly the same, but the mathematics is far more complicated.  You're not seeing problems that are obvious in accelerating frames. The work for such frames has been done.  Might as well take advantage of it.

With inertial frames too, we know that one of them must be a true representation of reality (because it provides the correct speeds for light relative to an object in every direction)

You are also free to consider things in only that frame, which is going to be difficult since I did not specify what the absolute velocity of the object is before we start to move it that light-hour.  Yet another needless complication.
Are you claiming that these 'more real' frames change the answer to the question posed by this topic?  I have no idea why you're going on about any of these things.

In 4D models, light must actually travel at zero speed because it has no option other than to reduce all the paths it follows to zero length.

Light travels at c in the spacetime model, which is a 4D model.  Paths are not zero length.  Perhaps you're thinking of a different model, but '4D' doesn't ring any other bells.


At this point the post actually seems to get back on topic:

The back end is stopped in the ship's frame, as is all the rest of it.  It is perhaps moving at 452 after the first moment in the initial frame, but that isn't the object's frame.  No, you cannot simultaneously cease acceleration of all of the object's parts in that frame.  It would indeed break.

If the object is stationary in the initial frame, you can't instantly have it with the back end moving at 452km/s without the contraction being wrong when the trip begins.

I had it accelerate in a small amount of time, not instantly.  The contraction is exactly correct the whole way.  That contraction is how I compute the required acceleration of all the other parts, so if there is any stress, it is because I computed the needed acceleration wrong, not because I cannot accelerate the rear of the object as hard I as I like.

The error may be small and trivial at this speed, but when you apply the method to shorter ships, the scale of the error will grow and cause damage.

Kindly illustrate with an example, because the motion of a shorter object (say one meter) is exactly the same as the motion of the trailing meter of the object.  So there you have your shorter object where the scale of the error becomes obvious.  Show me where it breaks.

I was looking for such a solution.  It seems that it doesn't exist.  I invite you to make a description of how that would work, or in particular, how you would get the speed of any part of the object over 452 km/sec without overshooting your destination.

As soon as any part gets to the place where you want it to stop, you stop it there and it will sit there comfortably, so the speed it moves at to get there can be as high as you like.

That would entail decelerating the front of the object while the rear is still accelerating.  It will break under compression.

You have a solution which you consider viable, and I say you can get the tail end to its destination faster by using the caterpillar method.

Only by using a singularity.  The method doesn't work with finite acceleration.  Solutions that approach a singularity are acceptable, but ones that require a singularity are using it to hide things.

I know it could be done with a relatively simple simulation, but I've got thousands of other simple simulations that I'd like to run too, and each one takes a long time to build - even if it's only a few hours work (which is never guaranteed - a simple bug can take a week to find sometimes) , it all adds up to lost years, and life's too short for that. I need better tools to automate all the tedious fiddling involved in these builds, so writing those tools up front is the fast route forward.

My caluculation didn't involve any simulation, and was really trivial.  I accelerated the rest of the universe (at least the parts in that 100 LY) to 452, which contracted the universe just enough to get the front of the stationary object even with the finish line.  Then I decelerated the universe to expand it back to normal again.  Doing it that way takes zero time, but entails a 55.3 year difference in relativity of simultaneity of the events before and after either of those accelerations.  Why write a simulation when the problem is that trivial?

The caterpillar method used a singularity to make contraction computation undefined, thus allowing it to use a higher speed.

Not quite. Each atom is accelerated to a fraction under c and the "2D" part is never quite 2D, so I don't see any singularity there.

The slow caterpillar method used a slow speed of about 3150 km/sec.  Your fast method indeed has no singularity, just a huge bomb of tidal force tearing it to pieces instantly. Playing that destruction in slow motion over an hour doesn't change the fact that it happens.

How are you going to stop the rear if you get it up to such a speed?  That was the part I couldn't solve.

But we resolved that months ago - you stop each atom where it's supposed to end up, so the last atom stops before the one ahead of it stops and it all lengthens back out.

That only works using the singularity to hide the contraction as an undefined value.

That isn't the tough part. The tough part is visualising the limit on how the faster movement of the tail allows faster movement further forward and whether it leads to the front end being able to move faster than it does with your method.

I have less trouble visualizing it.  You just have to find a frame that makes what is going on obvious.

It also takes more than 55 days using this method since it takes longer than that for the tail end of the acceleration wave to get to the front.  Sure, we stop the rear using the singularity that explodes the object, but if the wave moves that slow, what's the point?

Well, I'd recommend parking that for now and returning to it later with the right tools so that it can all be resolved at a fraction of the time cost.

I will perhaps move on to even longer objects then.  You're the only one contributing in a useful way to this topic.

[Halc (OP)]

Halc;

I think you overlooked simultaneity. (If the graphic is correct)

You identified few of the items in the graph, so I'm not sure I know what you intend to convey with it.
F seems to be the points simultaneous in some frame, but that frame isn't identified.
S is perhaps the object, but it depicts inertial motion, and my object is never inertial.
What I presume to be events E1, E2, and E3 are also not identified, nor the Sx line.
The blue lines look like light signals sent from E2 and E3 for some reason, but my topic doesn't involve signals being sent anywhere.
I suspect you have not familiarized yourself with the topic.  Read the OP.

As for your fantasy physics idea:
NASA  has explained the need for an AI program that enables the space probes like those for Mars, to be autonomous. In the landing phase, the machines have to make adjustments in seconds, not minutes. Remote operations are not capable of precise control for material structures, even if they could be built.

I never suggested sending live remote commands to any distant places.  The sequence of accelerations would have to be preprogrammed in advance, at least 50 years in advance given the separation of the various events involved.

Another problem area, if the fragile ‘long stick’ passes near a significant mass, the g-force will not be uniform for its length, resulting in deformation.

I did not posit any masses nearby in this hypothetical scenario.  It just wasn't the point of the topic.  We're assuming flat space just like in SR.

[David Cooper]

In the original frame, that hour isn't enough to contract the object by 1000th, never mind to 1000th of the initial length. The functionality of the ship is practically halted.

So what?  I smash an egg with a hammer, and the egg is very much broken, even in a frame like you describe where the 'functionality is slowed' a thousand fold.

Why are you smashing it with a hammer? All we're doing is accelerating every atom of it up to a fraction under c simultaneously.

If you hit even higher acceleration, it will eventually produce enough tidal stress to pull the nucleus of each atom apart.  The faster you go, the more violence you're doing to it.  I don't know why you're pursuing this proposal since it clearly violates the rigidity conditions.

Why object to that now rather than in the caterpillar method which has an equally severe acceleration being applied to the atoms at the back end?

Rotating frame are quite real as well.  You live in a rotating accelerating frame.  The only sort of frame that doesn't exist anywhere is the kind described by SR, which is a pure inertial frame.

Rotating frames are contrived and broken, as shown when you try to synchronise clocks round the whole ring (or equator) - you can only synchronise them for the equivalent non-rotating frame. Pure inertial frames exist in LET.

Even with inertial (SR) frames, the synchronization of clocks not in each other's presence is frame dependent: Not real.

You can synchronise them all for a specific frame. You can't do that with a rotating frame.

You seem to be attempting to sync clocks in a rotating frame using some sort of (Newtonian?) reference frame rules.  Use rotating rules when using a rotating frame.

If they weren't fake frames, you'd be able to stand between two clocks and synchronise them, then stand between the next pair and synchronise them too, and so on all the way round, ending up with the last and first being found to be in sync when you stand between them, but they aren't.

That's why you use rotating rules.  Speed of light is anything but constant in a rotating frame, so you can't try to use some method that assumes a constant speed of light like you sort of describe above.  Its properties being different doesn't make the frame bogus.

If you're agreeing that the speed of light isn't constant relative to the frame in different parts of the frame, then it's a fake frame. A true frame has the same speed of light relative to any objects in it which are stationary in that frame (after you've corrected for it being slowed by gravity). Accelerated and rotating frames are both contrived abstractions.

Are you claiming that these 'more real' frames change the answer to the question posed by this topic?  I have no idea why you're going on about any of these things.

It came up because you said the whole ship was stationary within its frame at all times with your method, but in reality the back and front ends are moving through space at different speeds and have different contractions acting on them. That was why I had such difficulty visualising how your method worked before, because I hadn't taken in that you were playing such tricks with accelerated frames. Since you made that clear though, I understand what you're doing, and if you're happy to use such a frame and to limit how you move the ship on the basis of applying the rules of that frame, that's fine.

Light travels at c in the spacetime model, which is a 4D model.  Paths are not zero length.  Perhaps you're thinking of a different model, but '4D' doesn't ring any other bells.

If you move a ship at 0.866c between points A and B, you shorten the distance between A and B to half. Move the ship at 0.968c and you shorten the distance to a quarter. These shorter lengths are the physical path lengths through the 4D structure. Higher and higher speeds reduce the distance ever more, tending towards zero. Light is not going to take a longer path from A to B than the fastest moving particle, so it must reduce the distance to zero. In 4D models, the speed of light is zero while it is merely the apparent speed of light that is c. If you want the actual speed of light to be greater than zero, you need to switch to a 3D model (and avoid mixing two incompatible types of model).

At this point the post actually seems to get back on topic:

It was all on topic, every part of it being generated from what came before.

Kindly illustrate with an example, because the motion of a shorter object (say one meter) is exactly the same as the motion of the trailing meter of the object.  So there you have your shorter object where the scale of the error becomes obvious.  Show me where it breaks.

Tell me how fast you have the back end of a metre-long ship move at the start. I assume it will be close to c, while the front end will be stationary. If so, how do you get the back end up to such a high speed and get it to contract to the appropriate length for that speed in an instant?

As soon as any part gets to the place where you want it to stop, you stop it there and it will sit there comfortably, so the speed it moves at to get there can be as high as you like.

That would entail decelerating the front of the object while the rear is still accelerating.  It will break under compression.[/quote]

Why would you need to decelerate the front of the ship? The movement of the back end is independent of it, free to move at a fraction under c to travel a lighthour and then stop in exactly the place you want it to stop. The entire back end of the ship can complete its journey before the front end has even started moving. Indeed, 98% of the ship can have completed its journey before the front end has started moving. But because we want to move the front end as soon as we can, we will start moving it sooner than that.

The method doesn't work with finite acceleration.  Solutions that approach a singularity are acceptable, but ones that require a singularity are using it to hide things.

At no point does it involve a singularity - there is never complete compression of anything to 2D, none of the speeds reach c, and the accelerations can all be spread out over finite lengths of time which are nearly but not quite zero.

My caluculation didn't involve any simulation, and was really trivial.  I accelerated the rest of the universe (at least the parts in that 100 LY) to 452, which contracted the universe just enough to get the front of the stationary object even with the finish line.  Then I decelerated the universe to expand it back to normal again.  Doing it that way takes zero time, but entails a 55.3 year difference in relativity of simultaneity of the events before and after either of those accelerations.  Why write a simulation when the problem is that trivial?

Because the maths gets more complicated when you start trying to move the ship faster by taking advantage of the caterpillar method to move the back faster and potentially allow the front to move faster too. We're back to the business of combining an infinite number of waves, and that makes the maths hard to do (unless you know a way to make it simple, but I don't).

The slow caterpillar method used a slow speed of about 3150 km/sec.  Your fast method indeed has no singularity, just a huge bomb of tidal force tearing it to pieces instantly. Playing that destruction in slow motion over an hour doesn't change the fact that it happens.

There is no such destruction - every atom is accelerated to a fraction under c with this timed for each atom such that when they're moving at full speed they're the right distance apart to sit comfortably. (It's only since this thread was revived that you don't need them to sit the right distance apart as the functionality of the material is practically halted, so no damage will occur if you just move all the atoms simultaneously, but in the original idea, I had them all compress up together into something nearly 2D, and I never used a slow caterpillar method.)

How are you going to stop the rear if you get it up to such a speed?  That was the part I couldn't solve.

But we resolved that months ago - you stop each atom where it's supposed to end up, so the last atom stops before the one ahead of it stops and it all lengthens back out.

That only works using the singularity to hide the contraction as an undefined value.

There's no undefined value. The contracted part might be a meter thick, or a micron, but it's never zero, but I realise now that there would be a major problem trying to decelerate an atom from the rear of the compressed zone without affecting the ones ahead of it too, so that isn't practical, but then we've never been looking for what's practical - we were looking for a fundamental limit ignoring such practical limitations.

That isn't the tough part. The tough part is visualising the limit on how the faster movement of the tail allows faster movement further forward and whether it leads to the front end being able to move faster than it does with your method.

I have less trouble visualizing it.  You just have to find a frame that makes what is going on obvious.

You haven't managed to visualise it and you don't have any frame that makes it obvious. This moves beyond your idea into one that might allow the front end to move faster than in your solution by allowing the back end to move faster than in your solution.

It also takes more than 55 days using this method since it takes longer than that for the tail end of the acceleration wave to get to the front.  Sure, we stop the rear using the singularity that explodes the object, but if the wave moves that slow, what's the point?

We have a series of waves of acceleration propagating at different speeds which allow the front end to move right from the gun, although it will initially move next to zero distance. Any speed that you imagine the ship can have under your method, I can have with mine, but I can improve on the speed at the back, and by doing so, I think the speed of the front can be improved too.

[Halc (OP)]

Why are you smashing it with a hammer? All we're doing is accelerating every atom of it up to a fraction under c simultaneously.

I was being gentle to it.  You are applying vastly more violence to the object, which is more fragile than the egg.  The egg will be some shell and goo, but almost all of the atoms will likely still be intact.  Not so with what you're doing.  We both promise to put all the pieces back together again, but the egg looks like an easier job.

Why object to that now rather than in the caterpillar method which has an equally severe acceleration being applied to the atoms at the back end?

There were no tidal stresses in the standard (55 day) method that has very high acceleration of each end of the object.
The caterpillar method on the other hand has a singularity used to hide the stresses (in the middle more than at the ends).  I rejected that method once that became clear.

Even with inertial (SR) frames, the synchronization of clocks not in each other's presence is frame dependent: Not real.

You can synchronise them all for a specific frame.

As I said, frame dependent.

You can't do that with a rotating frame.

I said how to do it in my prior post.  You were going about it wrong is all.

If they weren't fake frames, you'd be able to stand between two clocks and synchronise them, then stand between the next pair and synchronise them too

That technique relies on properties of inertial frames.

If you're agreeing that the speed of light isn't constant relative to the frame in different parts of the frame, then it's a fake frame.

OK.  Glad to know your terminology.  You can describe events in your frame of choice of course, but I tend to look for one that best simplifies the description of what is going on.

Are you claiming that these 'more real' frames change the answer to the question posed by this topic?  I have no idea why you're going on about any of these things.

It came up because you said the whole ship was stationary within its frame at all times with your method

I did.  You can translate that, if you will to an inertial statement:
At any event at point in the object, in the inertial frame of the object at that event, the other parts of the object are also stationary.  There it is in inertial terms.
This is assuming that there is no infinite acceleration.  It is very high at either end, but if it were infinite, the object would have multiple speeds at that event, and hence no defined speed and thus no defined frame.  So perhaps my object is short of its own event horizon by one meter, which keeps the numbers finite, and adds a millisecond or two to our trip.  I think 3 digits of precision is plenty.
The caterpillar method has a singularity for speed along its entire length at once in an inertial frame, and hence hides the contraction calculation as an undefined value.  Calculus shows that the value does not approach zero as the time of acceleration approaches zero.

but in reality the back and front ends are moving through space at different speeds and have different contractions acting on them.

Only in different inertial frames.  In the object's inertial frame, there is no contraction at all.  This doesn't seem to be a requirement.  It just turned out that way.  Clearly, in other frames, the object moves at different speeds along its length, but no stress has been identified.  The contraction, if you integrate it along its length, is exactly in proportion to the distance between its parts.  The mathematics is considerably more complex to verify this.

Light travels at c in the spacetime model, which is a 4D model.  Paths are not zero length.  Perhaps you're thinking of a different model, but '4D' doesn't ring any other bells.

If you move a ship at 0.866c between points A and B, you shorten the distance between A and B to half. Move the ship at 0.968c and you shorten the distance to a quarter.

First of all, you're thinking of points in space, not event in 4D spacetime.  Move the ship faster and the endpoints of the path are different events.  A and B are not the same two things in these two scenarios.
That said, the two points in space (worldlines in frame X) are unchanged by sending a fast (relative to X) or slow object between them.  From the perspective of the object moving relative to X, different events along those two worldlines are now simultaneous, reducing the spatial separation of the heavily inclined worldlines.

These shorter lengths are the physical path lengths through the 4D structure.

No they're not.  Physical path lengths in a 4D structure are called intervals, and they are frame invariant.  Two events have the same interval between them regardless of the inertial frame chosen.  Any path between them, if straight, will have the same physical path length.  A path between them can be longer if not straight.

At this point the post actually seems to get back on topic:

It was all on topic, every part of it being generated from what came before.

I found suggestions of moving an object faster than it can contract to be off topic.  Using brute force to hold atoms at unnatural separations temporarily is not the sort of solution I'm looking for.

Kindly illustrate with an example, because the motion of a shorter object (say one meter) is exactly the same as the motion of the trailing meter of the object.  So there you have your shorter object where the scale of the error becomes obvious.  Show me where it breaks.

Tell me how fast you have the back end of a metre-long ship move at the start. I assume it will be close to c, while the front end will be stationary.

Well, we accelerated the rear to 452 km/sec in our example, and took say a microsecond to do that.  Less if it helps you illustrate your issue.  I decelerated the object much more slowly (over 55 days) to compensate for the very far end needing to slow down, but you're right, if our object is only a meter long, we don't need to do that.  So if the object is only a meter long, and we're moving it a light-hour, we're going to go a hecka lot faster than 452.
The maximum speed needed is enough to contract the length of something by a factor of 1.079e12, which is the number of meters in a light-hour.  The meter-stick is going to get there in an hour, ± something in the 12th digit or more.  That hardly tells us anything.

One thing to note is that it doesn't take an hour to get the front up to near light speed.  The front end of the object, just like the 100 light year one, is going to approach proper acceleration at c²/length which is 9e16 m/sec² as the tail proper acceleration approaches an arbitrarily high value.  So while proper acceleration is constant over that hour, the actual acceleration is nearly a step function.  The same applies to my super-long object, but the difference between actual and proper acceleration was negligible at 452 km/sec.

If so, how do you get the back end up to such a high speed and get it to contract to the appropriate length for that speed in an instant?

Don't understand this question.  The contraction happens over the course of acceleration, not in an instant.  We're doing it nearly as fast as we can, but it still takes time for light to travel that one meter, so that's a minimum time for the rear to mostly catch up with the front, and after a half-hour, the front an rear match speeds and it starts to expand again back to its original 1-meter length.

The method doesn't work with finite acceleration.  Solutions that approach a singularity are acceptable, but ones that require a singularity are using it to hide things.

At no point does it involve a singularity - there is never complete compression of anything to 2D, none of the speeds reach c

The speed of the object is undefined at two points, and hence the proper length of the object at that moment (in the inertial frame of half the max speed of the object).  That's the singularity.  I didn't say anything was moving at c.

Because the maths gets more complicated when you start trying to move the ship faster by taking advantage of the caterpillar method to move the back faster and potentially allow the front to move faster too.

I simplified that case by using the half-way inertial frame.  Both are pretty trivial cases. You'll notice the code I produced to compute the optimal wave in post 88 did no actual simulation.  For any given speed bump, the time to move the long object is the sum of two simple values: the days needed for the wave to go 100 LY (tDwav), and the days needed to go a light hour at the selected speed (tDmov).

We're back to the business of combining an infinite number of waves, and that makes the maths hard to do (unless you know a way to make it simple, but I don't).

If only the mathematics of those contrived accelerated reference frames worked.  Sorry, you're on your own if you want to do it the hard way.

There is no such destruction - every atom is accelerated to a fraction under c with this timed for each atom such that when they're moving at full speed they're the right distance apart to sit comfortably.

Oh, I thought you were doing it to all of them at once.
Doing it in a wave still has that undefined speed singularity, and not sure why you'd want to do it so fast since it takes nearly 100 years for that wave to make it it the other end of the object.

There's no undefined value. The contracted part might be a meter thick, or a micron, but it's never zero,

In the half-way inertial frame, it is the entire object that has a completely undefined speed, all at once.  That's a singularity.  The thing has no defined proper length.  If you do it as a finite small wave that moves, there is no singularity, and the contracted length of certain parts of the object are always different than actual separation of them, so it breaks.  Again, considering it from that middle frame makes this clear.

but then we've never been looking for what's practical - we were looking for a fundamental limit ignoring such practical limitations.

Indeed.  Not caring about practical.  Einstein never worried about how the wheels on his train going at .6c held together.  That seems to be one of the largest practical limits to making the fastest car: It takes lot of engineering to make a wheel that can turn fast enough to go supersonic speeds.  If it doesn't have wheels, it isn't really a car.

You haven't managed to visualise it and you don't have any frame that makes it obvious.

I've shown you the frames (all inertial) that I used.  I used no calculus so far.

This moves beyond your idea into one that might allow the front end to move faster than in your solution by allowing the back end to move faster than in your solution.

The front has a limit (c²/length) on acceleration.  Faster tail speeds don't help if the front needs too much time to get to those speeds.  Likewise, the tail has that same limit on deceleration, so I begin doing that right away after I get it up to top speed.

We have a series of waves of acceleration propagating at different speeds

If they involve infinite acceleration, then you have that singularity.  If they don't, the wave propagates immediately to the front.  So not sure what you mean by 'different speeds'.

which allow the front end to move right from the gun, although it will initially move next to zero distance.

Yes and yes.  If I jump out of a building, I initially move next to zero distance, but can decelerate abruptly when I hit the street.  The ride at the front is just like that, except less G and drawn out over 55 days.

Any speed that you imagine the ship can have under your method, I can have with mine, but I can improve on the speed at the back, and by doing so, I think the speed of the front can be improved too.

That would be great if it worked.

[guest4091]

Halc;

Thought you understood spacetime graphics.

Maybe I didn't understand all the details as you defined them.

I agree  with Pmb.
Can't imagine the ISS built without engineers!

"All things imaginable are not realizable.'

[Halc (OP)]

Thought you understood spacetime graphics.

I do, or at least the ones that bother to label the items in the graph.
The picture you posted does not in any way depict what I am describing.

[David Cooper]

You can translate that, if you will to an inertial statement:
At any event at point in the object, in the inertial frame of the object at that event, the other parts of the object are also stationary.  There it is in inertial terms.

At any point in the object, the inertial frame in which that point is stationary has other parts of the ship moving through it rather than stationary (except for half way through the journey when the whole thing might be moving at the same speed).

The caterpillar method has a singularity for speed along its entire length at once in an inertial frame, and hence hides the contraction calculation as an undefined value.

There is nothing hidden or undefined about it. The compression is to a specific separation between atoms at which they sit comfortably, all dictated by the speed of movement through space.

but in reality the back and front ends are moving through space at different speeds and have different contractions acting on them.

Only in different inertial frames.[/quote]

In a single inertial frame they are moving at different speeds.

First of all, you're thinking of points in space, not event in 4D spacetime.  Move the ship faster and the endpoints of the path are different events.  A and B are not the same two things in these two scenarios.

I'm talking about 4D Spacetime locations, but to spell things out more precisely than I did last time, we've got three of them. We have two of them at great distance apart, but if we move something at great speed from each of them towards the other, they meet at a third Spacetime location. The faster they travel, the shorter the paths they follow to that meeting point. If they move at a fraction less than c, the distance each one travels is a fraction over zero and the time taken to cover that distance is a fraction over zero too. For light, the distance is zero and the time is zero, and that means there are two paths of zero length connecting our original two Spacetime locations via the third location. They are all therefore zero distance apart. This shows that all[-i] Spacetime locations are zero distance apart.

These shorter lengths are the physical path lengths through the 4D structure.

No they're not.

Yes they are. If X and Z are the starting points for the light and Y is the meeting point, we know that the path from X to Y has zero length and that the path from Z to Y has zero length, so there is a physical path of zero length between X and Z. This is a consequence of having 4D geometry which people may want to hush up, but it is the mathematical reality of it.

I found suggestions of moving an object faster than it can contract to be off topic. Using brute force to hold atoms at unnatural separations temporarily is not the sort of solution I'm looking for.

They are not being held by brute force or by any other kind of force - their functionality is practically halted by their extreme speed of movement through space, so the solution is fully valid.

That hardly tells us anything.

Well, you didn't provide a speed, but the back end of your metre-long ship will have to be doing whatever speed you haven't provided practically instantly (starting from zero speed and with no contraction applied to it), and high contraction will then be needed on it, so either you're going to have that contraction exist on it by magic in an instant or you're going to have to use my method of accelerating it from the back end to contract it down in a fraction longer than zero time in order to get the atoms to comfortable separations for that speed.

If so, how do you get the back end up to such a high speed and get it to contract to the appropriate length for that speed in an instant?

Don't understand this question.

Why not? You have a ship that's stationary in one frame and you have starting points and destinations which are also stationary in that frame. You want to move the ship from the start to the end. You suddenly have the tail end moving at high speed in this frame and decelerating back down to zero speed while the front end slowly accelerates from zero to high speed, and when it reaches the destination point, the front end's moving at high speed and the tail end has stopped. That's what comes out of the numbers that you provided (the speed of the tail going down from 452 to 0 and the front end going up from 0 to 452).

The contraction happens over the course of acceleration, not in an instant.

The contraction on the tail is wrong at the start because the tail is moving at 0, but an instant later it's moving at 452 and it gradually decelerates from there.

The speed of the object is undefined at two points, and hence the proper length of the object at that moment (in the inertial frame of half the max speed of the object).  That's the singularity.  I didn't say anything was moving at c.

There are no points at which the speed of anything is undefined. Each atom has a specific speed and specific accelerations are applied to it at specific times. The ship also has a specific length at all times which can be measured as the distance from the rearmost atom to the leading atom. There is nothing in any of that that breaks the laws of physics, so I don't understand your problem.

I simplified that case by using the half-way inertial frame.

I don't think you have simplified it. You have the back end moving more slowly than it could and you have it decelerating much more gently than necessary, while the result of this is that you're constraining the possibilities for moving the front end. You haven't begun to cover the complex kind of system I'm visualising for moving the ship - it would require a combination of an infinite number of waves of accelerations.

If only the mathematics of those contrived accelerated reference frames worked.  Sorry, you're on your own if you want to do it the hard way.

I'm not asking you to do it. I intend to do it with the help of a simulation to find out what it tends towards as I add more waves. I'm sure there are tricks in maths to get to an answer more directly, but I don't have the right tools for that in my head.

Oh, I thought you were doing it to all of them at once.

In the newest method, I do accelerate them all at once (to a fraction under c, relying on their functionality being practically halted to keep them in place), but in the old method I insist on the atoms being at comfortable separations during the trip at any times when they're not being accelerated, and that means contracting and uncontracting the ship from the rear.

Doing it in a wave still has that undefined speed singularity, and not sure why you'd want to do it so fast since it takes nearly 100 years for that wave to make it it the other end of the object.

We established right at the start that if you use a single wave for the caterpillar and contract the back with it to nearly 2D, the wave propagates forwards at a little under c. If we add in another wave at the opposite extreme, we can have the front end move an instant after the tail starts to move, but the front end can hardly move forwards at all, so this fastest-propagating wave is one which provides almost no acceleration. We can add in a third wave which propagates more slowly, but when it reaches the front end it will allow the front to accelerate significantly. Another wave that propagates more quickly than the third will reach the front more quickly but accelerate it less, while another wave that propagates more slowly than the third will reach the front later but accelerate it more. We want an infinite number of waves like these acting together, and the later each one reaches the front end, the more it accelerates it. The only thing that's undefined about the speed is that I haven't worked out any of the values for the propagation times and the amount of acceleration they provide for the front end. All of these accelerations have to be combined, which sounds easy, but each must be applied to the front end at a different time with the front end at a different location, and that's hard to handle.

In the half-way inertial frame, it is the entire object that has a completely undefined speed, all at once.  That's a singularity.  The thing has no defined proper length.  If you do it as a finite small wave that moves, there is no singularity, and the contracted length of certain parts of the object are always different than actual separation of them, so it breaks.  Again, considering it from that middle frame makes this clear.

I don't understand your objection. Nothing I'm doing breaks the laws of physics - each atom has a viable speed and there are no infinite accelerations. There is a possible problem that I can see building up as more waves are added, but if it is a problem, it will apply as soon as any movement of the front end is made without waiting for the contracted part to reach it after a hundred years, so if that turns out to be an illegal move, so is your method.

[Halc (OP)]

From post 88:

It seems the caterpillar method can get the ship up to 3135 km/sec and do the trip in under 6 days, doing the movement in under 4 days and the wave taking just under 2 days to move up the 100 light-year ship.  That's like 11% of the 55 days I got via the other method.

Did I get all the math right?

I am beginning to wonder if the mathematics is wrong.  The front has a max acceleration else the rear of the object will fall behind, and yet if we ignore the singularity involved, we manage to get the front to move faster than its limit.

So as an experiment, I propose a race between the two objects, one moving as described in post 37 and taking 55 days to get the front to the finish line, and the other the caterpillar method that has the front not moving at all, but it crosses the finish line in 6 days.  Let's hold a race between these two methods, except the first one never turns off the acceleration, but just goes at it indefinitely.  The second one never does the retro-caterpillar move, so it gets up to 3135 km/sec and never goes faster.  Obviously the first object will win in the long run, but the second should get ahead at first, but also cannot, which is a contradiction.  I need to understand that contradiction.

For the most part I am going to use the original frame as my coordinate system, except where explicitly noted.

Object 1 has a tail at just beyond point 0 (0 represents the Rindler horizon)  and the head at point 100 LY. All parts have proper acceleration of c²/D (per post 93) where D is the original distance from point 0. So the tail accelerates at arbitrarily high G force and the head at 0.09513 m/sec² .
Object 2 has a completely different strategy, and only accelerates the rear at first to 3135 km/second, with the parts ahead of it accelerating in a singularity wave that moves fast enough to reach the other end of the 100 LY object in 2 days.  After those 2 days go by, the object coasts permanently at 3135 km/sec.  There is no Rindler horizon for this object since it does not experience continuous acceleration.

Day 2:
Object 1:  Tail is 2 light days from 0.  Head is about 1.4 billion meters from its starting point and currently moving at 16.4 km/sec.  A light hour is about 1.08 trillion meters, so we have a ways to go.
Object 2: Tail has been moving at 3135 km/sec for 172800 seconds, so it has moved about 540 billion meters.  The head of the object has just barely started moving, but the whole thing is now moving in inertial motion.  At that speed, contraction of a 100 LY object would be 2 light days, so things are as they should be.  The object is losing the race since the head of object 1 has been moving from the start, but the whole thing is moving at 270 billion meters a day.

Day 4:
Object 2's tail has traveled the requisite light hour, but we're going to just let it keep going.  The head is 540 billion meters out from its start, halfway there.
Object 1's head is now moving at 32800 m/sec and has moved 5.67 billion meters.  The second object passed it long ago.  It is at maximum acceleration, so how did the other object pass it?

The singularity let us do that of course, but how do we visualize the offense in more concrete terms?  It seems that the infinite acceleration we're giving it puts all of the object that is behind the point of acceleration beyond the event horizon (Rindler horizon) and thus not in the universe.  I plan to kill a bus load of people, so my solution is to toss them in a black hole, thus ending their existence in the universe.  I do the offensive thing that would kill them, then switch the black hole off and find the busload still intact, but different.  Can't hurt them if they don't exist.

Where is the object in the frame of the head of the object, day 4?
With object 1, the object is always stationary in its own frame.  It is moving at 32800 m/sec and the initial tail event (point 0,0) is simultaneous with the head.  All is good.

With object 2, the thing is moving at 3135000 m/sec in the initial frame and it has been in inertial motion for 2 days.  At that speed, the head event is simultaneous with the tail at a time over a year before the tail got moving.  That means that object 2 might well all be moving inertially (all at the same speed, not accelerating) in the original frame, but it certainly isn't in its own inertial frame.  It will not become stationary in its entirety for over a year.  I don't see a problem with that, so I can't point to it as a second way to illustrate the inconsistency.

[Halc (OP)]

You can translate that, if you will to an inertial statement:
At any event at point in the object, in the inertial frame of the object at that event, the other parts of the object are also stationary.  There it is in inertial terms.

At any point in the object, the inertial frame in which that point is stationary has other parts of the ship moving through it rather than stationary (except for half way through the journey when the whole thing might be moving at the same speed).

That makes no sense.  The entire object is stationary in its own inertial frame.  If it is moving, you've chosen the wrong frame.

The caterpillar method has a singularity for speed along its entire length at once in an inertial frame, and hence hides the contraction calculation as an undefined value.

There is nothing hidden or undefined about it. The compression is to a specific separation between atoms at which they sit comfortably, all dictated by the speed of movement through space.

Relativistic contraction is a function of speed, and that speed is undefined at the singularity, hence the contraction (the length that it should be if not under strain) is undefined.

In a single inertial frame they are moving at different speeds.

That's true of all frames except one.  Even in the caterpillar method, there exists one frame where they all move at the same speed, except at the moments of the singularity, where it has no speed at all.

I'm talking about 4D Spacetime locations, but to spell things out more precisely than I did last time, we've got three of them. We have two of them at great distance apart, but if we move something at great speed from each of them towards the other, they meet at a third Spacetime location.

?  Can you draw a picture of this?  How can 4D spacetime locations move?  Points in spacetime don't have a property of speed.  They're fixed points, not worldlines.
I know you're a presentist, but it seem you don't have any understanding at all of the contrasting view.  How can you assert your view is the correct one if you don't understand spacetime?

They are not being held by brute force or by any other kind of force - their functionality is practically halted by their extreme speed of movement through space, so the solution is fully valid.

Speed is not absolute, and their functionality is not changed at all in their own frame.  By increasing your acceleration, you are increasing the violence done to the object at a rate greater than you are decreasing the duration of that violence.  Playing the action in slow motion by describing events in a very different frame does not make that violence any less violent.

Well, you didn't provide a speed

I did.  It is accelerating arbitrarily hard, so it is going as fast as it can to get that meter-rod to move the light-hour.  That is closer to c than can be expressed by 12 digits of precision, and since I'm working with say 3-4 digits of precision, the back effectively accelerates to c.

but the back end of your metre-long ship will have to be doing whatever speed you haven't provided practically instantly (starting from zero speed and with no contraction applied to it), and high contraction will then be needed on it, so either you're going to have that contraction exist on it by magic in an instant or you're going to have to use my method of accelerating it from the back end to contract it down in a fraction longer than zero time in order to get the atoms to comfortable separations for that speed.

No idea what you are talking about.  I'm using the same method as the 55.3 day method for the longer object.  Both accelerate the rear at arbitrarily high levels, but the long object quits accelerating once 452 km/sec occurs, and the short object quits at a higher speed that is very very close to c.  Yes, the rod gets very contracted (by a factor of about 1.079e12) at top speed and travels close to that compression most of the distance, but there is no caterpillar method being used.  I'm accelerating all parts of the object so the strain on the object (and hence the stress) is zero.  The front accelerates at about 9e17 g, which is considerably less than the acceleration of the rear.  I specified the acceleration of every point along the length of the object, which is the same as the acceleration along the length of my longer object.  The only difference is the speed where I reverse the process and begin slowing it down.

If so, how do you get the back end up to such a high speed and get it to contract to the appropriate length for that speed in an instant?

Don't understand this question.

Why not? You have a ship that's stationary in one frame and you have starting points and destinations which are also stationary in that frame. You want to move the ship from the start to the end. You suddenly have the tail end moving at high speed in this frame and decelerating back down to zero speed while the front end slowly accelerates from zero to high speed, and when it reaches the destination point, the front end's moving at high speed and the tail end has stopped. That's what comes out of the numbers that you provided (the speed of the tail going down from 452 to 0 and the front end going up from 0 to 452).

Yes, that's a description of the flight plan, so to speak.  You ask how I get the back end up to such high speed so quickly.  Answer: Same way I slow down the front at the end of the trip: with arbitrarily high acceleration.  The point one meter from the rear is accelerating at nearly 9e17 g, so the very tail must be accelerating far harder than that.  Let's just say 1e1000 g, just to pick a number, since I think 3 digits of precision is enough.  So it doesn't happen in an instant.  It takes time to get up to speed at that acceleration.  It just doesn't take very much time.

The contraction happens over the course of acceleration, not in an instant.

The contraction on the tail is wrong at the start because the tail is moving at 0, but an instant later it's moving at 452 and it gradually decelerates from there.

The tail is a point and has no length to contract.  So I'm unclear what you are trying to convey with this statement.  And as I said, it takes a short but finite time to get up to 452 km/sec at 1e1000 g.  It isn't instantly.

There are no points at which the speed of anything is undefined.

That was referring to the caterpillar method.  Of course the speed is undefined at the point where the wave is, especially in the one frame where there is no wave.  If it is defined, what is it?  The speed (in the original frame) is zero before the singularity, and 3135 km/sec after it, but what about during the singularity?  The question essentially asks for the slope of a triangle wave function at one of the inversion points.  Curves have varying slopes along their lengths.  Angles don't, at least not at the point of the angle.  The slope there is undefined.

Each atom has a specific speed and specific accelerations are applied to it at specific times. The ship also has a specific length at all times which can be measured as the distance from the rearmost atom to the leading atom.

In the halfway frame (of 226 km/sec), the object does not have a proper length, which should be measurable as the sum of proper distances between all the atoms.  Everything in physics has a proper length, and nothing in physics accelerates at an infinite rate except at certain singularities which destroy any object present at it, even despite the slow motion seen by a distant observer.  Even then, the object only seems to approach arbitrarily close to that singularity, and not actually achieve infinite acceleration.

I simplified that case by using the half-way inertial frame.

I don't think you have simplified it. You have the back end moving more slowly than it could and you have it decelerating much more gently than necessary

The half-way frame was for the caterpillar method.  You seem to be going on about the finite-acceleration method, which is simplified by considering the inertial frame of the object at the midpoint of the trip.
I do not have the back moving more slowly than it could be.  Suggest a better speed if you have one, but do so without singularities.  It also does not decelerate slower than it needs to, since it is doing so at the same rate as all points on the object, which is c²/D.

You haven't begun to cover the complex kind of system I'm visualising for moving the ship - it would require a combination of an infinite number of waves of accelerations.

An infinite number of waves would approach a curve as the stairstep singularities become smaller and smaller.  It would approach a valid solution of finite acceleration.  The small velocity steps for each wave would make each wave travel faster and faster, approaching infinite speed as you use more and more of them.  So time for the wave to travel approaches zero.  Thus the front can being to move right away.
I encourage exploration of this method, but I think it will converge on the 55 day answer, not something short.
Edit:  I take that back.  I think it will converge on something even better than 6 days.

I'm not asking you to do it. I intend to do it with the help of a simulation to find out what it tends towards as I add more waves. I'm sure there are tricks in maths to get to an answer more directly, but I don't have the right tools for that in my head.

If you write something, I can review it.

I did a quick one in my head (plus the numbers in post 88), using two waves instead of one.  The original wave was to 3135 km/sec which took 2 days for the wave and 4 days to move one light hour at that speed.  Total time is nearly 6 days.
Lets bump the rear up to 2220 km/sec, at which speed the wave moves to the other end in only 1 day.  Then we immediately bump the rear again by a similar 2nd wave.  The time to move the light hour is now 2.8 days instead of 4, at least for the parts of the object moving at full speed, which none of them do for the whole distance.  The total time to move the object is now 4.8 days (each point is stopped for one day, half speed for 2 days, and full speed for 1.8 days).  Damn... The method seems to be an improvement if I did that right.  Breaking the accelerations into two steps shorted the 6 day trip by 20%.  If we do it in infinite steps, the singularity approaches a defined value for velocity at all times, so the method is valid. 

In my prior post I attempted some sort of argument about how my 55 days solution cannot be improved upon, and I was dissatisfied that I had done that.  This simple 2-step thing shows a trend towards a faster trip than 6 days, not a trend towards the 55 day mark.  I have hope for it.

In the newest method, I do accelerate them all at once (to a fraction under c, relying on their functionality being practically halted to keep them in place)

Answers that change the proper length of the object, however briefly, do not qualify as valid solutions.
The proper length can be integrated, so there is no requirement that the entire object be moving at the same speed at once.  There is no such restriction in nature.  Any accelerating object moves at different speeds along its length, and that alone doesn't cause stress.  It is mathematically necessary in fact.

but in the old method I insist on the atoms being at comfortable separations during the trip at any times when they're not being accelerated

They need to be at comfortable separations at all times, even during acceleration.

We established right at the start that if you use a single wave for the caterpillar and contract the back with it to nearly 2D, the wave propagates forwards at a little under c. If we add in another wave at the opposite extreme, we can have the front end move an instant after the tail starts to move, but the front end can hardly move forwards at all, so this fastest-propagating wave is one which provides almost no acceleration. We can add in a third wave which propagates more slowly, but when it reaches the front end it will allow the front to accelerate significantly. Another wave that propagates more quickly than the third will reach the front more quickly but accelerate it less, while another wave that propagates more slowly than the third will reach the front later but accelerate it more. We want an infinite number of waves like these acting together, and the later each one reaches the front end, the more it accelerates it.

Exactly.  I actually have reasonable expectations of this method, but if it works, the optimal solution will seemingly have a top speed well under light speed.  I just picked an arbitrary value that gave a top speed a little more than 3135, and it resulted in an improvement.  If I picked several steps that added up to a much higher speed, I suspect the total time would get much worse.

The only thing that's undefined about the speed is that I haven't worked out any of the values for the propagation times and the amount of acceleration they provide for the front end.

The propagation time is in my code posted in 88, and it is time relative to the frame before the bump in speed.  The acceleration of any part of the object is undefined (a singularity) for a given bump in speed.  As the velocity bumps approach zero change, acceleration at a given point becomes defined as the size of the bump divided by the frequency at which they occur.  That is a real acceleration.

All of these accelerations have to be combined, which sounds easy, but each must be applied to the front end at a different time with the front end at a different location, and that's hard to handle.

I did it pretty easily with just two bumps, but if you use a lot of little ones (or even just 3), yes, it gets complicated.  Computers are your friend.  Play with a small number of bumps and see where the trends are, and then write something that reflects that correctly.  Now do it in say 1000 steps and vary the top speed up and down until you find an optimal value.  At 1000 steps, the solution will be very close to one with continuous acceleration and no singularities.
Then I have to explain why it works better than every part accelerating at the maximum amount, or I have to explain why the solution isn't valid.  I think it is valid, and my 55 day solution is lousy because it unreasonably insists on the object always being stationary in its own frame.

Nothing I'm doing breaks the laws of physics - each atom has a viable speed and there are no infinite accelerations.

The wave method in post 88 (that yielded a 6 day solution) assumed infinite acceleration.  If you're speaking of some other solution, then I need to know which it is.  The one with a lot of little waves approaches defined speeds and finite acceleration, so that solution is valid, but we have no numbers for it so far.

There is a possible problem that I can see building up as more waves are added, but if it is a problem, it will apply as soon as any movement of the front end is made without waiting for the contracted part to reach it after a hundred years, so if that turns out to be an illegal move, so is your method.

There is no wave in my method, so nothing travels the length.  The length of the object is always consistent with its contracted size in any frame, so there is zero strain, and thus zero stress.  There are also no singularities.  That makes the solution valid, if not optimal.  I really think your method here is going to yield a far better solution, and I don't think that was my opinion until the last couple posts.

[guest4091]

DC 118;

With inertial frames too, we know that one of them must be a true representation of reality (because it provides the correct speeds for light relative to an object in every direction) while all the rest must be false. However, because we can't tell which one is true, we have to treat all inertial frames as potentially true. We shouldn't do that with an accelerated frame though because it's guaranteed not to be true.

[All inertial frames measure light speed as constant, because light speed is independent of any source.
Accelerated motion between two objects interferes with synchronization, which is corrected by periodic updates as in the GPS system.]

In the same way, a ship of any length that's accelerated to a speed that practically stops functionality will not contract significantly

[This has to be as observed from outside. A moving object is affected by length contraction to the same degree as time dilation. That's why an observer moving with the object cannot measure any difference.]

However, all of that is predicated on the idea that light can travel at speeds approaching c in the first place. In 4D models, light must actually travel at zero speed because it has no option other than to reduce all the paths it follows to zero length

[An observers world gets smaller the faster he moves in space. Near zero distance is perceived by the  moving observer only.]

[Relativity defines the propagation speed of light, relative to space, not relative to an object!] 

[jeffreyH]

I haven't read through all replies to this thread but I have to address one point. That clocks at one point in the ship run differently to those in another. This was said to be like those at different altitudes on a planet and was mentioned in connection with uniform gravitational fields. A uniform gravitational field has the value of g the same everywhere. Therefore, clocks will run at the same rate everywhere. The acceleration is linear in this situation. Since you require all parts of your ship to accelerate at the same rate you are producing an equivalent situation and so all your clocks will also run at the same rate.

[David Cooper]

That makes no sense.  The entire object is stationary in its own inertial frame.  If it is moving, you've chosen the wrong frame.

I'm choosing an inertial frame in which part of your object is stationary while other parts are not.

Relativistic contraction is a function of speed, and that speed is undefined at the singularity, hence the contraction (the length that it should be if not under strain) is undefined.

The speed of each atom is defined and so is the local contraction.

Even in the caterpillar method, there exists one frame where they all move at the same speed, except at the moments of the singularity, where it has no speed at all.

And it's a warped, fake frame.

]?  Can you draw a picture of this?  How can 4D spacetime locations move?  Points in spacetime don't have a property of speed.  They're fixed points, not worldlines.

Why are you trying to move the Spacetime locations? They're fixed. The light is moving from two Spacetime locations (X and Z) to a single Spacetime location (Y) to the future of the original two. The light follows zero-length paths (XY and ZY) from the earlier locations to the later one.

Speed is not absolute, and their functionality is not changed at all in their own frame.

We have a frame in which the start and finish line are stationary. The speeds are measured from that frame and have clear finite values at all times. The speed of functionality of material is measured by that frame too - high speed of travel means slowed functionality. If you want to play games where you change frame continually to try to keep the ship stationary within such a chameleon frame, that's fine (the only problems being that it drowns you in complexity and diverts you away from the clearer approach), but I'm looking at this through a single real frame throughout, and if it works in that frame, it necessarily works in all others.

By increasing your acceleration, you are increasing the violence done to the object at a rate greater than you are decreasing the duration of that violence.  Playing the action in slow motion by describing events in a very different frame does not make that violence any less violent.

Can you put a figure on how violent an acceleration is allowed to be before your arbitrary objection kicks in?

Well, you didn't provide a speed

I did.  It is accelerating arbitrarily hard, so it is going as fast as it can to get that meter-rod to move the light-hour.  That is closer to c than can be expressed by 12 digits of precision, and since I'm working with say 3-4 digits of precision, the back effectively accelerates to c.

So you now have the very kind of violent acceleration for your method that you want to deny me the right to use with mine.

I'm accelerating all parts of the object so the strain on the object (and hence the stress) is zero.

If you're taking the rear of the short object up to c in an instant, your separation distances between atoms are going to be way out until the contraction occurs, so you're in the same position as I am with my move-the-whole-shebang-at-nearly-c for-the-whole-trip solution.

You ask how I get the back end up to such high speed so quickly.  Answer: Same way I slow down the front at the end of the trip: with arbitrarily high acceleration.

Isn't that "breaking the egg"?

The tail is a point and has no length to contract.  So I'm unclear what you are trying to convey with this statement.  And as I said, it takes a short but finite time to get up to 452 km/sec at 1e1000 g.  It isn't instantly.

The tail is more than just the atom at the back. It's an arbitrary length starting from the back, but let's make it just the two atoms at the back with one ahead of the other. If you suddenly accelerate them to a fraction under c (as in the 1m-long ship), the separation between them is wildly wrong. Choose a longer ship than that where you don't practically-halt the functionality but still need high contraction, and you're going to have those atoms accelerate together and impact violently.

Of course the speed is undefined at the point where the wave is, especially in the one frame where there is no wave.  If it is defined, what is it?

If you're going to use an overly complex frame, it's your job to convert from a simple frame in which the speeds of all the atoms are defined at all times. Your inability to do that conversion does not break the valid physics in the frame I'm using.

The speed (in the original frame) is zero before the singularity, and 3135 km/sec after it, but what about during the singularity?  The question essentially asks for the slope of a triangle wave function at one of the inversion points.  Curves have varying slopes along their lengths.  Angles don't, at least not at the point of the angle.  The slope there is undefined.

I don't see where the difficulty is. What we're calling a wave is simply a point moving along the length of the ship which triggers the acceleration of the local atoms as it goes. There's nothing in that that breaks the laws of physics.

Everything in physics has a proper length, and nothing in physics accelerates at an infinite rate except at certain singularities which destroy any object present at it, even despite the slow motion seen by a distant observer.  Even then, the object only seems to approach arbitrarily close to that singularity, and not actually achieve infinite acceleration.

What rules for accelerating atoms am I breaking that you aren't breaking too in your 1m-long ship example?

I do not have the back moving more slowly than it could be.  Suggest a better speed if you have one, but do so without singularities.

If your 1m-long ship can get its tail end up to nearly c in an instant without singularities, you should be able to do the same with the back end of your 100ly ship.

Damn... The method seems to be an improvement if I did that right.  Breaking the accelerations into two steps shorted the 6 day trip by 20%.  If we do it in infinite steps, the singularity approaches a defined value for velocity at all times, so the method is valid.

I wouldn't bet on it being valid, but if it breaks, it may also break your 55 day method.

Answers that change the proper length of the object, however briefly, do not qualify as valid solutions.

That depends on the rules you're applying. If you want to conform to the rules of some specific type of chameleon frame, then fine. I'm just looking for methods which don't break the ship (if we ignore the business of how you accelerated an atom hard without destroying it, but you seem to be happy to use extremely hard accelerations too for the 1m-long ship).

but in the old method I insist on the atoms being at comfortable separations during the trip at any times when they're not being accelerated

They need to be at comfortable separations at all times, even during acceleration.

Then you can't accelerate the back end of your 1m-long ship anything like as quickly as you have tried to. Indeed, you can't accelerate them at all without those forces between atoms becoming a little uncomfortable. That's why by my rules (which your method depends on too, even if you don't realise it), you're allowed to have uncomfortable acceleration distances between atoms while accelerating them to new speeds, but when you've finished accelerating them, they must be back to comfortable separations if they aren't moving fast enough to rely on slowed functionality to prevent them from ripping the object to pieces.

If you're speaking of some other solution, then I need to know which it is.  The one with a lot of little waves approaches defined speeds and finite acceleration, so that solution is valid, but we have no numbers for it so far.

The numbers for each "wave" depend on making sure that the front two atoms are the right distance apart for the speed they're moving at. The second atom has to move faster than the first, and the higher the speed of the first, the closer the atom behind it has to get to it. The same applies to the second and third atoms. Maybe the fastest way to work out how to solve this problem is to work out the maximum speed for a two-atom-long ship, then a three-atom-long ship, then a four-atom-long ship, etc. A pattern will emerge from this, and may make it easy to calculate what happens for a ship of any chosen length.

For two atoms, we can accelerate atom 2 to a fraction under c, then do the same to atom 1 as soon as the separation is right for comfortable separation at that high speed. This is the same as compressing the rear.

For three atoms, we do the same thing as before but for atoms 3 and 2 this time. Atom 1 could creep forward a bit as soon as atom 2 has started moving towards it instead of waiting until atom 2 has reached the right separation for atom 1 to move at nearly c too. We can also allow atom 2 to creep forward early though, as soon as atom 3 has begun to move, and that in turn allows atom 1 to creep forward even sooner. The amount of creep of the forward atoms could increase gradually through a range of speeds that maintain correct separation for them back to where the particle behind them is at any point in time based on their own speed. That is the process I'd want to simulate, lengthening the ship by an atom each time and seeing what happens as a result. Note though that the particle behind cannot also be at a comfortable distance from the one ahead of it until they are doing the same speed as each other, so if we were to require it to be, we couldn't accelerate the ship at all. The particle ahead may not be at a comfortable distance from the one behind it either even if the separation is right for its current speed because it takes time for the propagation of forces to adjust to stable levels even after both atoms have stopped accelerating, but again we probably have to ignore this if we're going to accelerate the ship at all.

There is no wave in my method, so nothing travels the length.  The length of the object is always consistent with its contracted size in any frame, so there is zero strain, and thus zero stress.  There are also no singularities.  That makes the solution valid, if not optimal.  I really think your method here is going to yield a far better solution, and I don't think that was my opinion until the last couple posts.

In your method, the front accelerates right from the gun, just as it does with mine. If mine's breaking rules by doing that, so must be yours. My method includes yours as a subset of possible implementations, except for what you're doing at the back end where I think you're accelerating it too quickly by imagining an instant contraction suddenly existing on it right after the gun. If your rear atoms are at the right separation once your instant acceleration has been applied, most of those atoms will have had to teleport backwards to take up the correct positions if your rearmost atom is still in the same place as it was before the gun.

[Halc (OP)]

I'm choosing an inertial frame in which part of your object is stationary while other parts are not.

That doesn't happen with my solution, but I don't think that solution is optimal.

The speed of each atom is defined and so is the local contraction.

Not in the caterpillar method, no.  We're putting infinite acceleration on each atom, giving it an undefined speed.

Even in the caterpillar method, there exists one frame where they all move at the same speed, except at the moments of the singularity, where it has no speed at all.

And it's a warped, fake frame.

It is a nice clean inertial frame.

The light is moving from two Spacetime locations (X and Z) to a single Spacetime location (Y) to the future of the original two. The light follows zero-length paths (XY and ZY) from the earlier locations to the later one.

Those paths are not zero length.  That's what I've been saying.  There is a separation (a frame independent one) between X and Y, and that separation is not zero.

We have a frame in which the start and finish line are stationary. The speeds are measured from that frame and have clear finite values at all times. The speed of functionality of material is measured by that frame too - high speed of travel means slowed functionality.

Agreed.  But slowing down the destruction by a factor of X means you increase the violence done in that time by far more than X.  The object breaks all the harder the faster you change its speed.

Can you put a figure on how violent an acceleration is allowed to be before your arbitrary objection kicks in?

Any acceleration, if done everywhere at once, equally.  This was one of the solutions, to just move the entire object all at once at near c, reducing the time to an hour.
Another disallowed case is infinite acceleration that is not approached by a valid curve, which is the case in the caterpillar method.

The method of moving the rear faster than 452 and not keeping the entire object stationary in its own frame seems to not violate either of these conditions.  This is approximated by a series of small waves which approach waves of zero speed changes, which thus approaches a valid solution.

Well, you didn't provide a speed

I did.  It is accelerating arbitrarily hard, so it is going as fast as it can to get that meter-rod to move the light-hour.  That is closer to c than can be expressed by 12 digits of precision, and since I'm working with say 3-4 digits of precision, the back effectively accelerates to c.

So you now have the very kind of violent acceleration for your method that you want to deny me the right to use with mine.

I didn't do it uniformly. The front acceleration of this meter-long object is many orders of magnitude (like 980 orders of magnitude) lower to allow the rear to catch up so that the contraction of the object always matches its speed.  If the front accelerated as hard as the rear, the object would break.  That seems to be what you have sometimes proposed.

I'm accelerating all parts of the object so the strain on the object (and hence the stress) is zero.

If you're taking the rear of the short object up to c in an instant, your separation distances between atoms are going to be way out until the contraction occurs, so you're in the same position as I am with my move-the-whole-shebang-at-nearly-c for-the-whole-trip solution.

Sorry, I don't follow this.  "Going to be way out"?  What does that mean?  The contraction occurs as the speed increases during the finite acceleration, and since the front accelerates so much less than the rear, the two draw near each other exactly in sync with their contraction.

You ask how I get the back end up to such high speed so quickly.  Answer: Same way I slow down the front at the end of the trip: with arbitrarily high acceleration.

Isn't that "breaking the egg"?

Not if the strain is zero, no.  Speed is always defined (since acceleration is not infinite) and the separation due to movement is always matched by the relativistic contraction.

The tail is a point and has no length to contract.  So I'm unclear what you are trying to convey with this statement.  And as I said, it takes a short but finite time to get up to 452 km/sec at 1e1000 g.  It isn't instantly.

The tail is more than just the atom at the back. It's an arbitrary length starting from the back, but let's make it just the two atoms at the back with one ahead of the other. If you suddenly accelerate them to a fraction under c, as in the 1m-long ship), the separation between them is wildly wrong.

The tail is only the atom in the back.  The atom just in front of it accelerates at a far lower rate.  I do this for both the long and the short object, but with the long one, I cap the acceleration at the much lower speed of 452 km/sec.  The rear one obviously gets to that speed first since it accelerates so much harder than the 2nd atom.
I gave the acceleration of each atom, so you can compute the value yourself and see the vastly lower rate between those two atoms.

It seems that after all these posts, you still don't understand my initial solution.  It is very simple, but apparently not optimal.  But you seem to think it isn't a valid solution.

Of course the speed is undefined at the point where the wave is, especially in the one frame where there is no wave.  If it is defined, what is it?

If you're going to use an overly complex frame

I didn't.  I said the original frame just below where you quote me.

it's your job to convert from a simple frame in which the speeds of all the atoms are defined at all times.

In the frame of your choice, what is the speed of the object at the point where the wave is?  We're talking the pure caterpillar method now.  That speed is undefined in any frame (since if it were defined in one, it would be defined in any frame).  It is delimited (somewhere between A and B), but not defined.

The speed (in the original frame) is zero before the singularity, and 3135 km/sec after it, but what about during the singularity?  The question essentially asks for the slope of a triangle wave function at one of the inversion points.  Curves have varying slopes along their lengths.  Angles don't, at least not at the point of the angle.  The slope there is undefined.

I don't see where the difficulty is. What we're calling a wave is simply a point moving along the length of the ship which triggers the acceleration of the local atoms as it goes. There's nothing in that that breaks the laws of physics.

But the atom has no defined speed at that point.  If all of those points are simultaneous, then the entire object has no defined speed at that moment.  You're right that it doesn't break physics, but all objects break under such physics, as I describe below.

Everything in physics has a proper length, and nothing in physics accelerates at an infinite rate except at certain singularities which destroy any object present at it, even despite the slow motion seen by a distant observer.  Even then, the object only seems to approach arbitrarily close to that singularity, and not actually achieve infinite acceleration.

What rules for accelerating atoms am I breaking that you aren't breaking too in your 1m-long ship example?

I never use infinite acceleration, and never let the proper distance between any parts of the object change for the duration of the motion.

If your 1m-long ship can get its tail end up to nearly c in an instant without singularities, you should be able to do the same with the back end of your 100ly ship.

I did not do it in an instant.  I used an arbitrarily short time.  Yes, we can do it with the super-long object as well if we want, but the acceleration has to be finite so that it always has a proper length.

Damn... The method seems to be an improvement if I did that right.  Breaking the accelerations into two steps shorted the 6 day trip by 20%.  If we do it in infinite steps, the singularity approaches a defined value for velocity at all times, so the method is valid.

I wouldn't bet on it being valid, but if it breaks, it may also break your 55 day method.

Well, 2 steps still isn't valid, but breaking it into an arbitrarily high number of small steps does approach a valid solution.  20 is probably enough to approximate that.  I'd have to play with it to see how many steps are needed before the end time stops varying.
What do you mean 'break my 55 day method'?  Break the object, or break the speed record?  It certainly seem to have the potential to do the latter.

Answers that change the proper length of the object, however briefly, do not qualify as valid solutions.

That depends on the rules you're applying.

I don't see how you can not have this rule and still retain the problem.  Without it, you're moving a line of sand, which can be moved one light hour in an hour, a trivial solution.  If the proper length of a Born-rigid object changes, then the object breaks, by definition.

If you want to conform to the rules of some specific type of chameleon frame, then fine. I'm just looking for methods which don't break the ship (if we ignore the business of how you accelerated an atom hard without destroying it, but you seem to be happy to use extremely hard accelerations too for the 1m-long ship).

That I am because I'm doing it from a mathematical standpoint, not an engineering one.  Indeed, I know of no real way to put the sort of force on any real piece of matter to get it to accelerate arbitrarily hard.  A rail-gun uses EM force to achieve the acceleration desired.  This is how they do it in particle accelerators, and those use far more power than a whole country combined to move a few atoms at far lower acceleration rates than my 'arbitrarily large' values.  My mathematical atoms are 'points at distance X from the end of the object, or from the event horizon that the object nearly reaches, but not quite.

[The 'atoms' need to maintain their proper separation] at all times, even during acceleration.

Then you can't accelerate the back end of your 1m-long ship anything like as quickly as you have tried to.

You keep repeating this, but you're wrong.  If I didn't accelerate the tail that hard, it would lag behind the atom in front of it and the object would break as the proper distance between the two grew to a larger value.

Maybe the fastest way to work out how to solve this problem is to work out the maximum speed for a two-atom-long ship, then a three-atom-long ship, then a four-atom-long ship, etc. A pattern will emerge from this, and may make it easy to calculate what happens for a ship of any chosen length.

I assumed a 2-atom object at first, but those atoms were 100 LY apart.

My method of choice was to assume one wave at first, then two, three, twenty, 100, etc.  At some point, adding more (and smaller) waves will not change the total time, and then we know that we've approached the answer with a continuous wave with no singularities.
One we know that, then we need to know the top speed much like the analysis I did in post 88 where I found the optimal speed for a single wave.

For two atoms, we can accelerate atom 2 to a fraction under c, then do the same to atom 1 as soon as the separation is right for comfortable separation at that high speed. This is the same as compressing the rear.

For small separations of the two atoms, this works great, but not so much for a large separation of the two atoms.

For three atoms, we do the same thing as before but for atoms 3 and 2 this time.

I predict that this method will yield the same total time as the two atom case where the middle of the 3 atoms is missing.  The atom in the middle adds nothing I think, which is why only 2 are needed.  The ones in the middle are interesting, helping you see what is going on, but adding atoms between the initial two doesn't change the end answer.  Adding them beyond the ends (as you describe below) does of course change the answer since that changes the total length, but then you could have done that total length with just 2 atoms again.

Atom 1 could creep forward a bit as soon as atom 2 has started moving towards it instead of waiting until atom 2 has reached the right separation for atom 1 to move at nearly c too. We can also allow atom 2 to creep forward early though, as soon as atom 3 has begun to move, and that in turn allows atom 1 to creep forward even sooner. The amount of creep of the forward atoms could increase gradually through a range of speeds that maintain correct separation for them back to where the particle behind them is at any point in time based on their own speed.

That sounds an awful lot like my initial version.  That converges to 55 days.  I suggest you don't do the creep, but do a lot of tiny waves instead with no creep-ahead.  That would be a different solution, and seemingly a much faster one.  That isn't intuitive, and I was caught by surprise to see the better results.
I think I will try to find some time to attempt to find an optimal speed at which to do that, but not anytime real soon.

Note though that the particle behind cannot also be at a comfortable distance from the one ahead of it until they are doing the same speed as each other

This is not true.  If the object is viewed as a continuous object with defined speed everywhere, that speed and thus the contraction and proper length can be integrated over the length of the object.  So different points (the atoms) along that curved line have a computable proper separation despite a difference in speed due to the non-inertial motion of the object.  As long as that computed proper separation never changes, the distance between any two atoms is comfortable.

so if we were to require it to be, we couldn't accelerate the ship at all.

Indeed, acceleration of any extended object is mathematically impossible without different parts of the object moving at different speeds in some frames.  This is trivial to demonstrate using relativity of simultaneity.  This does not imply that there is ever uncomfortable separation of parts of a rigid object.  There will be a change in proper length if the force that accelerates the object is applied at only one point and not all points.  This can be used to disprove the existence of unobtanium, a finite density material which cannot deform by application of force.

The particle ahead may not be at a comfortable distance from the one behind it either even if the separation is right for its current speed because it takes time for the propagation of forces to adjust to stable levels

Forces never propagate in any Born-rigid object.  All forces that cause any accelerations are applied from outside, never from the material nearby, with which we, if the object is not broken, are in equilibrium.

In your method, the front accelerates right from the gun, just as it does with mine. If mine's breaking rules by doing that, so must be yours.

The discreet wave method did that, but not necessarily this new thing proposed.

My method includes yours as a subset of possible implementations, except for what you're doing at the back end where I think you're accelerating it too quickly by imagining an instant contraction suddenly existing on it right after the gun.

I never said anything about instant contraction.  It takes time to contract, and since the material immediately in front of the rear of the object accelerates so very much less than does the absolute rear, that contraction is exactly in sync with the speed of the object.

[David Cooper]

All inertial frames measure light speed as constant, because light speed is independent of any source.

All inertial frames measure light speed as constant because they assert a particular speed for it and adjust everything else to conform to that requirement.

In the same way, a ship of any length that's accelerated to a speed that practically stops functionality will not contract significantly

This has to be as observed from outside. A moving object is affected by length contraction to the same degree as time dilation. That's why an observer moving with the object cannot measure any difference.

An observer on the ship will see the trip as taking no time due to the practically halted functionality. An observer stationary relative to the start and finish lines will see the ship travelling for an hour without contracting - it will appear to be an unstable length, but its functionality is frozen, as expected for an object moving at such high speed.

An observers world gets smaller the faster he moves in space. Near zero distance is perceived by the moving observer only.

In the 4D non-Euclidean geometry of STR and GTR, those zero-length distances exist. If we sent a ship at nearly c to M32 and back, it would return to us at a different location from the one it left, but the first leg of its trip would be shortened to next to zero length, and so would the second leg of its trip. The time it would pass through for the trip would be next to zero too. Light would make an equivalent trip in zero time of zero distance. That is a mathematical necessity of the model.

Relativity defines the propagation speed of light, relative to space, not relative to an object!

Which Relativity? With LET, yes - it's the propagation speed of light relative to space. With STR and GTR, it's just a constant which represents the apparent propagation speed of light relative to space, but with those models the real speed of light is zero. If you don't want it to be zero, don't use a 4D model.

[Halc (OP)]

In the same way, a ship of any length that's accelerated to a speed that practically stops functionality will not contract significantly

So if I plug .99999999999c into my Lorentz contraction calculation, I will get close to 1 (no significant contraction) because functionality is practically stopped.  Hmm, my calculator doesn't yield that result.  Or did I not use enough 9's?

[David Cooper]

I'm choosing an inertial frame in which part of your object is stationary while other parts are not.

That doesn't happen with my solution, but I don't think that solution is optimal.

Look at your ship before you start it moving. It's all stationary relative to the start line, and relative to the finish line. That's the initial frame, and it's the one I continue to use throughout. As soon as you start moving your ship, you have the back end moving through this frame while the front end begins to accelerate slowly from zero. You have different parts of the ship moving at different speeds relative to this frame. The spacing between the atoms at the back end is immediately wrong - it was right before you started moving the back, but as soon as you started moving it, you broke that. At the moment of the gun, the speed of the rear two atoms has gone from 0 to 452km/s even though they have travelled zero distance at that moment, while the distance between them is the same as it was when they were at rest. Do the same thing with the 1m-long ship and instant acceleration to nearly c and you should recognise the problem. By the time the rear atom has moved 1/1000th of the way towards where the atom ahead of it was itting before the gun, the correct spacing between it and that atom ahead of it can only be realised by moving the atom ahead backwards by almost 999/1000 of the original distance between atoms. You either have to teleport it back there or you have to teleport the rear atom forward the same distance instead (and further forward again to get correct spacing relative to the next atom beyond that, and the same again for the one beyond that, etc.). Your starting move is illegal.

The light is moving from two Spacetime locations (X and Z) to a single Spacetime location (Y) to the future of the original two. The light follows zero-length paths (XY and ZY) from the earlier locations to the later one.

Those paths are not zero length.  That's what I've been saying.  There is a separation (a frame independent one) between X and Y, and that separation is not zero.

For light, that separation is most certainly zero. I refer you to my previous post in this thread. If you're going to use a 4D model, you have a duty to accept its mathematical requirements, and these zero-length paths are unavoidable features of that kind of geometry. The reluctance of most physicists to admit that should not be allowed to mislead you. Recognising that 4D geometry works this way doesn't break the STR or GTR models - it merely forces you to see that any talk of the speed of light being c with such a model is actually the result of mixing incompatible models. Light only moves at c in 3D models. In 4D models light merely has an apparent speed of c while its actual speed is always zero. I don't know why people have so much difficulty accepting this when at the same time they have high-speed particles travelling through the same 4D geometry on near-zero length, near zero-time paths. If you want to ban that too, you're going to have to ditch the time dimension and bring in an absolute frame, at which point you're doing LET instead. You need to make up your mind which model you want to use and stick to its rules. You can't combine them in a single model - they're incompatible.

If the front accelerated as hard as the rear, the object would break.  That seems to be what you have sometimes proposed.

With one set of rules (where things are allowed to sit at uncomfortable separations if their functionality is practically halted), that is what I proposed. With another set of rules, I do not have the front accelerate as hard as the rear.

In the frame of your choice, what is the speed of the object at the point where the wave is?  We're talking the pure caterpillar method now.  That speed is undefined in any frame (since if it were defined in one, it would be defined in any frame).  It is delimited (somewhere between A and B), but not defined.

If I discuss a scenario in which a car drives around an oval track but I don't give you the speed that it's moving at, that speed is undefined. Does that mean it's breaking the laws of physics by going round the track at an undefined speed? No. If it accelerates by an undefined amount and now does each lap faster than the one before, does that mean it's breaking the laws of physics? No. My method has each atom move at legal speeds and accelerate between legal speeds and can have these accelerations apply over lengths of time slightly greater than zero. I don't know what those speeds or accelerations are, but none of them break the laws of physics.

But the atom has no defined speed at that point.

It will do once we've worked out what the ideal speed for it will be.

If all of those points are simultaneous, then the entire object has no defined speed at that moment.

The object is a composite with different parts moving at different speeds. If you want a defined speed for the object at any moment, it will have to be an average speed, and again we won't know what that speed is until we've worked out what it needs to be.

I never use infinite acceleration, and never let the proper distance between any parts of the object change for the duration of the motion.

What are you doing with the back end of the 1m-long ship if that isn't infinite acceleration? Whatever it is, I'm doing the same, except that I don't have to teleport atoms around to shift them to the right separations by magic when the gun goes off.

If your 1m-long ship can get its tail end up to nearly c in an instant without singularities, you should be able to do the same with the back end of your 100ly ship.

I did not do it in an instant.  I used an arbitrarily short time.[/quote]

I do everything in arbitrarily short times too.

What do you mean 'break my 55 day method'?  Break the object, or break the speed record?  It certainly seem to have the potential to do the latter.

If my method is breaking the rules, yours must be too. My method covers a case in which the front end accelerates identically to yours, so if it turns out that I'm not allowed to accelerate it that quickly, you won't be allowed to do so either.

I don't see how you can not have this rule and still retain the problem.  Without it, you're moving a line of sand, which can be moved one light hour in an hour, a trivial solution.  If the proper length of a Born-rigid object changes, then the object breaks, by definition.

I think you're already breaking that rule with the way you accelerate the back end. The same will apply when you halt the front end, because the second atom from the front will have to jump backwards to its destination.

You keep repeating this, but you're wrong.  If I didn't accelerate the tail that hard, it would lag behind the atom in front of it and the object would break as the proper distance between the two grew to a larger value.

I'm not wrong. Your second last atom has to teleport backwards in every single case - it's just a lot more obvious with the 1m-long ship than with the 100ly ship.

For two atoms, we can accelerate atom 2 to a fraction under c, then do the same to atom 1 as soon as the separation is right for comfortable separation at that high speed. This is the same as compressing the rear.

For small separations of the two atoms, this works great, but not so much for a large separation of the two atoms.

It needs to be done with the small separations that would exist between atoms of the real ship.

I predict that this method will yield the same total time as the two atom case where the middle of the 3 atoms is missing.  The atom in the middle adds nothing I think, which is why only 2 are needed.  The ones in the middle are interesting, helping you see what is going on, but adding atoms between the initial two doesn't change the end answer.  Adding them beyond the ends (as you describe below) does of course change the answer since that changes the total length, but then you could have done that total length with just 2 atoms again.

Case 1:-

With two atoms at normal separation, the rear one is almost instantly moving at almost c, so we just treat it as if it's doing c right from the starting gun. By the time it's reduced the distance between it and the lead atom to 0.5 the original length, the front atom should be doing 0.866c. Because the front atom starts moving before that point in time though, working out where both atoms are at the moment when the gap hits 0.5 hard. I'm sure it's dead easy if you know how to apply calculus to it, but I don't. (I could probably work out how to apply it through experimentation, but I don't know how long it would take me to find the right approach.)

Case 2a:-

With two atoms at double-normal separation, by the time the rear one has reduced the distance between it and the lead atom to 0.5 the original length, the front atom should be doing 0.866c, as before, but it must take exactly twice as long to reach that point as in the first case. In this case, the speed of the leading atom will be half that of the equivalent atom in case one at any point in time.

Case 2b:-

We can imagine a version of case 2 where the atom separations are normal rather than double and where the rear atom moves at 0.5c with the rear atom reaching 0.433c at the point when the separation becomes 0.5. This provides the same timings as case 2a. (You'll see how useful this is later.)

Case 3:-

With three atoms at normal separations, the first case applies to the rear two atoms too - it will be identical. The front atom has to accelerate more slowly though. Its relation to the behaviour of the middle atom is clearly different because the middle atom accelerates slowly rather than in a near instant, but its relation to the behaviour of the rear atom certainly looks similar to the original case, but can we prove that it's the same shape of relationship? In cases 1 and 2b, the acceleration of the lead atom is governed by an atom chasing it at a constant speed. In case 3, the acceleration of the lead atom is governed by an atom, the middle one, chasing it at very low speeds initially - much lower than the 0.5c in case 2b. I think that proves that it's a different curve, and if I'm right, then you can't just consider two atoms.

I never said anything about instant contraction.  It takes time to contract, and since the material immediately in front of the rear of the object accelerates so very much less than does the absolute rear, that contraction is exactly in sync with the speed of the object.

The contraction with the 1m-long ship will move some atoms backwards, and they'll then run into other atoms at high energies and break the ship.

[David Cooper]

In the same way, a ship of any length that's accelerated to a speed that practically stops functionality will not contract significantly

So if I plug .99999999999c into my Lorentz contraction calculation, I will get close to 1 (no significant contraction) because functionality is practically stopped.  Hmm, my calculator doesn't yield that result.  Or did I not use enough 9's?

You'll get a contraction figure to next to zero length, but you also get halted functionality which prevents the ship from achieving that contraction - it has insufficient time to contract any significant amount at all. The contraction doesn't suddenly apply in full by magic - it takes time to contract the ship, and we aren't giving it that time.

[Halc (OP)]

You'll get a contraction figure to next to zero length, but you also get halted functionality which prevents the ship from achieving that contraction - it has insufficient time to contract any significant amount at all.

Exactly.  Preventing its contraction is what breaks it, and why I will not consider such a solution.
The thing needs to be held at that unnatural length for an hour in the original frame, so the high speed does not in any way hide from us the incorrect length of the object.

Really, we're working on a viable solution that breaks no rules, and it isn't an obvious one.  Most importantly, it seems to have the potential to be 10x faster than my simple method that restricts itself to always being stationary in its own inertial frame.

[David Cooper]

You'll get a contraction figure to next to zero length, but you also get halted functionality which prevents the ship from achieving that contraction - it has insufficient time to contract any significant amount at all.

Exactly.  Preventing its contraction is what breaks it, and why I will not consider such a solution.

It doesn't break though because its functionality is slowed to a halt - the atoms don't have time to apply any significant contraction.

The thing needs to be held at that unnatural length for an hour in the original frame, so the high speed does not in any way hide from us the incorrect length of the object.

We don't need the incorrectness of the length to be hidden, and maintaining that for an hour is not a problem - we know that something moving that fast will have practically-halted functionality and we'd be shocked if we saw it breaking the laws of physics by contracting.

Really, we're working on a viable solution that breaks no rules, and it isn't an obvious one.

I did say that there are different sets of rules and different winning methods for some or each set. If the only rule is that you get the object there without breaking it and that no severity of acceleration of an atom will break that atom, then this is the fastest solution - the object is moved at a fraction under c and arrives undamaged. Moving to another set of rules does not dethrone this solution as the winner of its category.

Most importantly, it seems to have the potential to be 10x faster than my simple method that restricts itself to always being stationary in its own inertial frame.

The more restrictive sets of rules ban reliance on frozen functionality, requiring atoms to sit at correct separations, but how much of the time are we allowed to break that rule? With the fastest, multi-wave version of the caterpillar method, we try to keep each atom the right distance ahead of the one behind it at all times, but the one behind isn't at the right distance from it because it's going faster - it's impossible to satisfy both atoms at once, and with the leading atom accelerating throughout the entire trip, the one behind it will never feel at a comfortable distance from it. I don't think there are any viable sets of rules for doing what we're trying to do without having some kind of arbitrarily allowed exceptions. If the separations are always to be comfortable, we can't accelerate the ship at all.

With your method where you claim the separations are always comfortable, you're trying to hide the uncomfortable separations by imagining them away through the use of a composite frame that hides their existence. It may be possible to hide their existence if you do that: when you accelerate something, you create contraction forces in it and when you decelerate it you create decontraction forces instead, but viewed from another frame, those forces can be seen as acting the other way round. Pick another frame that accelerates an object from -v to v in an instant and you have no correction of the length being made at all. If you assume that this last way of looking it is the reality in all cases, then you can deny the existence of all such imbalances by changing real frame an infinite number of times in order to make each little change in speed look like a change from -v to v. In LET though, that's illegal - you have to stick to one real frame throughout instead of using a fake, composite frame.

[Halc (OP)]

I did say that there are different sets of rules and different winning methods for some or each set. If the only rule is that you get the object there without breaking it and that no severity of acceleration of an atom will break that atom, then this is the fastest solution

But that wasn't the rule.  The rule was to never change the proper length of the object.  The solution you suggest is a trivial one.  Any object can be moved a light hour in an hour that way.  What's the point in figuring that out?

Most importantly, it seems to have the potential to be 10x faster than my simple method that restricts itself to always being stationary in its own inertial frame.

The more restrictive sets of rules ban reliance on frozen functionality, requiring atoms to sit at correct separations, but how much of the time are we allowed to break that rule?

I don't ban the frozen fuctionality.  I ban the lack of correct separation.  There's no minimum time that can be violated.

With the fastest, multi-wave version of the caterpillar method, we try to keep each atom the right distance ahead of the one behind it at all times, but the one behind isn't at the right distance from it because it's going faster

Not if it is a continuous wave.  One atom is allowed to be faster or slower than the other if the proper separation never changes. You seem not to realize that.

 The discreet waves are illegal of course (since they involve small singularities), but making them smaller is a way of integrating the continuous wave solution that is not illegal.  The solution approaches a valid one.  Just making everything go as fast as you can does not do that.

I don't think there are any viable sets of rules for doing what we're trying to do without having some kind of arbitrarily allowed exceptions.

My 55 day solution had zero exceptions.  The proper length of the thing was fixed the whole time, and didn't even need integration to compute it since it wasn't a curve.  The optimal solution would seem to require integration to compute the proper length of the object en-route.

With your method where you claim the separations are always comfortable

They're not merely comfortable.  They're exactly correct.  There is not a small wiggle margin I'm allowing.

, you're trying to hide the uncomfortable separations by imagining them away through the use of a composite frame that hides their existence.

An inertial frame is not composite, but a curved object will need such a composite frame since in no frame will it be all moving at the same speed.  Anyway, with my 55 day solution, I invite you to point out where the proper separation of two points of the object are different that the original value.
The mathematics is pretty easy with that case.  Every point along its length accelerates at a proper rate of c²/D where D is the distance from a point arbitrarily close behind the tail of the object for acceleration, and a point similarly just beyond the nose of the object for deceleration.  Each point changes from acceleration to deceleration abruptly when 452 km/sec is achieved.  From that description, you can compute the exact location of any point on the object at any time, and notice that the proper separation between any two points is fixed the entire way.

It may be possible to hide their existence if you do that: when you accelerate something, you create contraction forces in it and when you decelerate it you create decontraction forces instead

Since I am accelerating at a modest pace for most of the time, these forces would indeed exist iff there was a change in proper separation, so it is a good thing we don't do that.

but viewed from another frame

Proper length of an object is frame invariant.  Looking at it from another frame won't change the answer.

In LET though, that's illegal - you have to stick to one real frame throughout instead of using a fake, composite frame.

Not sure what the one real frame is going to buy you when computing the proper length of something not stationary.  As I said, that computation is not different from one frame to the next, but for a non-rigid object, it might be different from one moment to the next, like changing the proper length of a rubber band by pulling it.

I suppose you could have a string that is half Born rigid where its proper length is not allowed to change, but it can bend effortlessly in any spaghetti curve you like. Could we move such a (straight) string faster than a similar rod of the same dimensions? I think not. The problem is equivalent to allowing a reduction in the original proper length of the rod, but not allowing an increase of it. More food for thought.