[David Cooper]

I don't understand your point at all.  Of course its possible, at least mathematically, and this is a mathematical exercise after all, not an engineering one.  I'm not really concerned about accelerating individual atoms.  I consider the object to be a homogeneous rod, perhaps with length marks along it.

If you have a "ship" made of only two atoms and you accelerate them (in any direction), the amount of force each receives from the other will vary momentarily and will only settle down when you stop accelerating them. That is stress on a two-atom "ship" - it's unavoidable. If you want to avoid all stress, you can't accelerate it. The two atoms are both applying forces to each other, and if you move them a little, those forces are momentarily being applied in the wrong direction.

This creates a problem for us though, because if we are allowed to have some stress, how can we limit it? We could just have the whole ship accelerate to a fraction under c and maintain that speed for a fraction over one hour, then stop the whole ship in an instant, and all we need to do is hold each atom in place so that the ship can't contract in length. It's only if we let go of them at any point during that year that the ship will be able to contract and will rip itself into fragments, although by moving it at nearly c, that hour gets converted into such a short time that no contraction may occur, meaning that by the time we've stopped it again, the whole ship is completely undamaged. This lets us move every part of it a whole lighthour in a time just a fraction over one hour, and the accelerations all take place in a greater-than-zero length of time.

'When we stop accelerating' is a frame dependent thing.

I was talking about stopping an individual atom, and there's no need to worry about trying to stop accelerating any two atoms simultaneously - you would just stop accelerating them once they are in places where they will sit comfortably without further inputs of force.

The fault in the description was that it posited zero time for the acceleration, and that is unreasonable.  It can be done over a finite time to give finite acceleration, but the interval of time to do it shortens as the wave moves forward, and it shortens to zero before it gets to the other end of the object.

How can it shorten to zero? Why can't you just start accelerating each particle sooner than the one behind it and have the propagation of the wave accelerate to accommodate this?

I presumed the ship had to be always stationary in its own frame along its entire length during the whole trip.

I don't see how it would be possible for it to be stationary in its own frame when it has parts moving at different speeds, other than by being stationary on average in one frame (which will always be the case no matter what you do).

The 55 day thing required 2 steps.

Step 1: Compute the exact speed of the ship.  There are two ways to do this.

1A: Use a lorentz conversion to compute the speed the universe would need to go to get it the 100 LY distance to contract by a light hour.  That gives me the speed the object needs to go (~452 km/sec).

I don't understand why this should be the speed of the ship. If you have a constant acceleration for the front end of the ship, what's the back end doing? What are the speeds of the front and back ends of the ship at 0, 5, 10, ... 45, 50, 55 years into the trip (or use some other time gap if you've already got a similar set of numbers). This would make it possible for other people to visualise how your ship is moving.

We can then imagine a ship of 100 atoms in length all ending up one atom further along from where they started.

Yea, but who can relate to the bazillionth of a second it takes to do that?  The relativity isn't going to be apparent.

You can contain all the action on the top of a desk. Light only moves about 30cm in the tick of a 1 gigahertz processor, and an object 30cm long moving at 0.866c will be contracted to 15cm in length. there's no need to go big to illustrate relativity. But if you want to, you can spread the 100 atoms out over a hundred lightyears and have them sit comfortably a lightyear apart. What matters is that you find ways to provide an illustration of what different parts of the ship are doing - what speeds they're moving at and when. In the absence of diagrams, that needs a table.

I can have length-marks on my object, which seems to serve the same purpose as your discreet atoms, but I found no need to refer to them to compute the times required by the various methods, or to demonstrate that the caterpillar method doesn't work at all.

You may think there's no need to refer to them, but I can't follow what different parts of the ship are doing in your 55 day version.

I twas trying to do that, finding a faster way by using multiple waves.  It doesn't work.  The waves catch up to each other, which wasn't at all obvious at first.

With each wave propagating at a lower speed than the one that set out before it, that can't be possible.

So I find it unreasonable to say that since it happens in zero time that one can get from one speed to another speed without hitting the speeds in between.
Avoiding the length contraction by doing the acceleration 'while God blinks' so to speak seems a cheat.

It isn't cheating - we can make the acceleration take a finite time longer than zero and still not have to worry about the length contraction because it's so quick that the particles have no chance to respond to the momentary contraction forces. Objects are not required to be the length that the length contraction formula says they would settle to if they're left long enough to settle.

These forces do not exist when doing it over a millisecond, atom by atom.  At no point are two atoms in a stressful arrangement.  The argument would indeed have merit if this temporary force existed, but it doesn't.  I showed that it doesn't.  Two atoms cannot exert a force during a pair of event separated in a space-like manner.

As soon as you move one atom towards another, you run it into a strengthening force from the other atom, but you also sent ahead a strengthening force toward the other atom which will propagate towards it at c, and that may start to accelerate the next atom before we start trying to accelerate it directly. Even if it doesn't though, it will still add force and lead us to need to put less direct acceleration force into it (although we aren't saving energy as we had to put extra energy into the first atom due to the increasing force it was running into from the second atom).

Well, no.  It sort of works exactly because this doesn't happen.  Acceleration of an atom doesn't result in any force against its unaccelerated neighbor.  What does this is displacement, and displacement takes time.  For it to put a force on the neighbor would be to have a causal effect at greater than light speed, which cannot happen.

The change in force form the moving atom will reach the other atom at c. The change in force from the stationary atom upon the atom moving towards it will apply instantly because the moving atom is running into that strengthening force.

Your approach is also on the discreet atomic level rather than the homogeneous mathemeatical level, but if we model the 'ship' as a series of discreet points that are accelerated individually, then there really is no length of the object, just spacings between the atoms which are to be ignored for the duration that a force is applied to them.  If we allow that, even for a shorter duration than the speed of light between adjacent atoms, then the length of the object seems meaningless.  The rules are to be ignored while we briefly take tongs to each atom in turn and change its velocity.

If you accelerate a ship and the length changes, you necessarily have different parts of it moving at different speeds and different length contractions applying to it in different places, so how are you going to stop that reaching the level of individual pairs of atoms? You can't do it on a whole-ship basis, and any other basis in between uses arbitrary divides.

It apparently doesn't work in any frame, but it isn't so intuitive in say the initial rest frame.  The one (middle) frame just made it real obvious why it didn't work, and yes, per relativity, the other frame thus must also not work.

I see it working in all frames.

Consider just two waves. We have one where we accelerate atoms nearly to c, but we have a second wave where we accelerate them to 0.5c. The latter acceleration will propagate from atom to atom at a higher speed than the former, with both propagating at speeds higher than c, but these things are fully possible in the frame of reference in which the starting speed is zero.

No, they're not possible in that (or any) frame.  It just wasn't initially obvious to either of us.  It became more apparent when I started to attempt optimizations and was running into so much trouble.

The front end is moving slowest, so any problem caused by a wave not propagating fast enough must be possible to solve by increasing its propagation speed as it goes along. I can't see any way for this to fail to work. We only need it to work in one frame to know that it must work in all frames, and so long as we aren't moving anything faster than the speed of light (or even just reaching it), it should be fine. Instant accelerations (from one speed to another) are possible for individual particles, so they should also be possible for atoms, but even if you want to make them take a longer-than-zero time for each acceleration, that can still be achieved by starting the accelerations sooner. The back end of the ship moves fastest and the front end moves slowest, so if the front end isn't starting any acceleration soon enough, we start that acceleration sooner.

[yor_on]

You can accelerate something without 'stress', ideally at least. You just need a geodesic ending into a 'infinity' avoiding 'tidal forces'. That is also called a 'gravitational acceleration'. Locally measured there shouldn't be any 'stress' applied anywhere in such a 'ship'. What you can't avoid though should be a 'length contraction', but just as with the 'gravitational acceleration' this 'length contraction' you find another object to have is a result of frames of reference interacting. Locally defined you still live in proper time finding a proper length.
=

This is if you go by local experiments.
If you have another opinion you also need to show how you will measure that 'stress' in a 'non local' manner.

a non gravitational acceleration is quite another thing. It will be local,  locally measurable bringing with it a 'stress' on the 'ship'. That should hold for a 'atom' too I think. When it comes to 'point particles' I'm not as sure, you need length, width and height, as well as 'time' to define a SpaceTime. But mathematically with 'point particles'?

Is light length contracted? If you think of it in a 'real acceleration' then light will blue respectively red shift depending on where in the ship you measure it from. Ideally those two should even themselves out, leaving the intrinsic properties of light unchanged. That's the only properties I can think of when it comes to light. A 'length contraction' doesn't make sense for it. Ok :) forget the momentum for this. Actually, thinking of light this way makes a 'time dilation' unnecessary too, possibly? Now, that is weird, isn't it? Or maybe not, depending on views :)

[Halc (OP)]

I don't understand your point at all.  Of course its possible, at least mathematically, and this is a mathematical exercise after all, not an engineering one.  I'm not really concerned about accelerating individual atoms.  I consider the object to be a homogeneous rod, perhaps with length marks along it.

If you have a "ship" made of only two atoms and you accelerate them (in any direction), the amount of force each receives from the other will vary momentarily and will only settle down when you stop accelerating them.

The other atom will not notice the acceleration of the first before it too is accelerated.  I suppose the first one can feel the force from the 2nd as it moves through the essentially static EM field generated by that 2nd atom.  But all this action takes place outside each other's light cone.  I posted that the discrepancy between the field of the distant atom and what the atom over there is actually doing outside our light code is counterbalanced by photon or something equivalent to a graviton that corrects for the discrepancy.  Thinking about such things is how GR needed to posit such particles.

So there are these photons or whatever that result from acceleration, and those exert force.  I can accept that, but at not point can two particles be allowed to exist at an unnatural distance from each other for the duration that light takes to travel between them.
That alone admittedly doesn't preclude infinite acceleration.  It doesn't even have to be infinite since it just has to take less time than light takes to allow the particles to notice the new velocity of its neighbors.
My protest against the method is that the solution cannot be approached by increasing acceleration to an arbitrary value.  Using a singularity is a cheat because it is taking advantage of the fact that the length of the object is undefined at that singularity, and in particular, that length is not something that is approached by increasing acceleration.

That is stress on a two-atom "ship" - it's unavoidable. If you want to avoid all stress, you can't accelerate it. The two atoms are both applying forces to each other, and if you move them a little, those forces are momentarily being applied in the wrong direction.

It seems that slow and steady acceleration seems quite stable.  The acceleration varies from one end to the other, but the object is mathematically Born-rigid the entire way. The entire length of the accelerating region is stationary and unvarying in length in its own frame. Yes, limited light speed puts some stress on the atoms as you describe, but only because the various atoms have no way of detecting what the other ones are doing, only what they have done some time ago.

This creates a problem for us though, because if we are allowed to have some stress, how can we limit it? We could just have the whole ship accelerate to a fraction under c and maintain that speed for a fraction over one hour, then stop the whole ship in an instant, and all we need to do is hold each atom in place so that the ship can't contract in length.

That violates Born rigidity.  The wave thing does not, but a finite wave much change form as it moves, which wasn't apparent to either of us at first.  Holding each atom at an unnatural separation from its neighbor for an hour (using the unlimited force with which we've endowed our propulsion) would work, but it would be stress.  The whole thing would be under massive tension stress, balanced mostly except near the ends.  No force applied in any direction will relieve that tension stress.  Force is OK but stress isn't a force since it doesn't cause acceleration (except again at the ends, which are for the most part not in the light cone of the vast majority of the object).

It's only if we let go of them at any point during that year that the ship will be able to contract and will rip itself into fragments

It was already ripped to fragments when the acceleration was done.  I suppose it can always be reassembled by using force to put each atom back in its rest frame and then letting go.

although by moving it at nearly c, that hour gets converted into such a short time that no contraction may occur, meaning that by the time we've stopped it again, the whole ship is completely undamaged.

It takes time in frames other that the one of the moving object.  I agree that it comes out undamaged since we've put every fragment back in place.  A broken egg doesn't stay broken if every atom is put back exactly where it used to be.  The problem is trivial if we allow such things.

The fault in the description was that it posited zero time for the acceleration, and that is unreasonable.  It can be done over a finite time to give finite acceleration, but the interval of time to do it shortens as the wave moves forward, and it shortens to zero before it gets to the other end of the object.

How can it shorten to zero?

I think I was mistaken when I said that.  A wave initiated over a finite time is a bunch of small discreet accelerations, and since they are small, their waves propagate at more speed than the aggregate.  So while acceleration to almost c creates a wave that supposedly moves at just over c, in fact the little waves move much faster, approaching infinitely fast.  It gets to the front of the object right away, but also at a much smaller accleration rate.  Anyway, the wave doesn't shorten to zero size.  Rather the opposite.  Any non-infinite acceleration propagates at a rate that approaches infinite speed since it is made up of acceleration quanta that approach infinitely small speed changes, and yet the aggregate speed of the wave must be much less, a contradiction.

Why can't you just start accelerating each particle sooner than the one behind it and have the propagation of the wave accelerate to accommodate this?

That's exactly what happens naturally.  There is no wave then since the whole thing starts to move at once, and the entire object is stationary in its own frame at all times.  Funny thing is that it is not all moving at the same speed in any other frame.  Only its own frame.

So this begs a different problem:  How long does it take to accelerate a 100 light year object to say 1%c?  Answer: depends where the clock is.

I presumed the ship had to be always stationary in its own frame along its entire length during the whole trip.

I don't see how it would be possible for it to be stationary in its own frame when it has parts moving at different speeds, other than by being stationary on average in one frame (which will always be the case no matter what you do).

But it is.  In its own frame, no part is moving at a different speed than any other part.  They're all stopped in fact.  Not true of the infinite-acceleration wave, but that involves discontinuities.

The 55 day thing required 2 steps.

Step 1: Compute the exact speed of the ship.  There are two ways to do this.

1A: Use a lorentz conversion to compute the speed the universe would need to go to get it the 100 LY distance to contract by a light hour.  That gives me the speed the object needs to go (~452 km/sec).

I don't understand why this should be the speed of the ship. If you have a constant acceleration for the front end of the ship, what's the back end doing?

Constant acceleration to the same speed as the front.  It takes less time for the back to do this since it accelerates harder.

What are the speeds of the front and back ends of the ship at 0, 5, 10, ... 45, 50, 55 years into the trip (or use some other time gap if you've already got a similar set of numbers). This would make it possible for other people to visualise how your ship is moving.

Speed of the rear (km/sec) is 452, 411, 370 ... 82, 41, 0.  Speed of the front is those same numbers, but in reverse.  The 452 figure is just after the high acceleration finishes after say one minute, and one minute before the front decelerates hard to 0.
For points other than the front or the rear, the peak speed is reached at a proportional time relative to the distance from the rear.  So a point an 11th of the way forward will reach peak speed in 5 days and ramp evenly down from there.  The midpoint will accelerate constantly for half the time and decelerate constantly for the other half.

You can contain all the action on the top of a desk. Light only moves about 30cm in the tick of a 1 gigahertz processor, and an object 30cm long moving at 0.866c will be contracted to 15cm in length. there's no need to go big to illustrate relativity. But if you want to, you can spread the 100 atoms out over a hundred lightyears and have them sit comfortably a lightyear apart.

Or just put marks each light-hour on the object.  I'm moving my big object a lot less than 1% of its length, and the relativistic effects are a lot more than just contraction.  The other effects are not readily apparent to me when working with a ruler and femptoseconds.  Maybe its just me.

What matters is that you find ways to provide an illustration of what different parts of the ship are doing - what speeds they're moving at and when. In the absence of diagrams, that needs a table.

OK, I gave an initial 'table' with crude figures (for speed) above.  A computer printout would be more accurate, especially when illustrating an attempt at a wave, showing where it fails.

So I find it unreasonable to say that since it happens in zero time that one can get from one speed to another speed without hitting the speeds in between.
Avoiding the length contraction by doing the acceleration 'while God blinks' so to speak seems a cheat.

It isn't cheating - we can make the acceleration take a finite time longer than zero and still not have to worry about the length contraction because it's so quick that the particles have no chance to respond to the momentary contraction forces.

I would accept that if the solution could be approached by arbitrarily high acceleration, but it isn't, and that makes it a trivial cheating answer to a real question posed in the OP.  I accept the instant speed thing on the ground where they work:  Sure, the stress is unnoticed in the sufficiently short time since light cannot travel to the next atom in that time. 

As soon as you move one atom towards another, you run it into a strengthening force from the other atom, but you also sent ahead a strengthening force toward the other atom which will propagate towards it at c, and that may start to accelerate the next atom before we start trying to accelerate it directly.

Of course not.  Our waves move faster than light.  We'll be accelerating it directly before the motion of the first atom is noticed by the 2nd.  We'd violate rigidity if that were not so. Still, argument from the atomic level is probably still a physical violation since there is no way to apply that sort of force to a single atom without affecting any of its neighbors.  That's why I've been going for idea mathematical solutions, not practical 'quantum' ones.

If you accelerate a ship and the length changes, you necessarily have different parts of it moving at different speeds and different length contractions applying to it in different places

Agree, this is true whether we're doing a wave or not.  This does not itself cause stress since the parts of the object moving at a different speed are outside of the causal cone.

so how are you going to stop that reaching the level of individual pairs of atoms? You can't do it on a whole-ship basis, and any other basis in between uses arbitrary divides.

If it is a continuous curve (such as it is in the 'rest frame' moving the ship as a whole), then you need to integrate the contraction over the length of the object moving at varying speeds. I didn't bother to do that since it was much simpler to use the frame of the object.
No integration is needed for the wave method since there are only two discreet chunks of objects, one at each speed.

[The finite-acceleration wave method] apparently doesn't work in any frame, but it isn't so intuitive in say the initial rest frame.  The one (middle) frame just made it real obvious why it didn't work, and yes, per relativity, the other frame thus must also not work.

I see it working in all frames.

Then you're not thinking it through.  You don't show how you arrive at this conclusion, so I have a hard time pointing out where it fails.

[PmbPhy]

I don't understand your point at all.  Of course its possible, at least mathematically, and this is a mathematical exercise after all, not an engineering one.  I'm not really concerned about accelerating individual atoms.  I consider the object to be a homogeneous rod, perhaps with length marks along it.

It makes a difference whether you take into account an objects stress when its accelerating. The answer depends on it.
Saying this is just a math question is wrong. Math is the description of nature, not the other way around. I could say that an object is moving at 3 times the speed of light "mathematically" but physically its wrong.

[David Cooper]

The other atom will not notice the acceleration of the first before it too is accelerated.

You're right - I was forgetting that the acceleration wave is propagating faster than light.

We could just have the whole ship accelerate to a fraction under c and maintain that speed for a fraction over one hour, then stop the whole ship in an instant, and all we need to do is hold each atom in place so that the ship can't contract in length.

That violates Born rigidity.  The wave thing does not, but a finite wave much change form as it moves, which wasn't apparent to either of us at first.  Holding each atom at an unnatural separation from its neighbor for an hour (using the unlimited force with which we've endowed our propulsion) would work, but it would be stress.  The whole thing would be under massive tension stress, balanced mostly except near the ends.

With the ship travelling at just a fraction under c, the time of one hour for us would to that ship appear to be an infinitesimal moment, not giving it enough time to shorten - we don't need to support each atom at all during that hour but can just let them all drift, and while they will pull together a beyond-microscopic amount during that hour, we will decelerate the whole thing to a halt before it does any damage, at which point it will push back out the same amount tiny amount. This allows any ship to be moved anywhere at nearly c with all parts moving at practically the same speed. There is no significant contraction of the object because its functionality is as good as halted for the entire trip.

Anyway, the wave doesn't shorten to zero size.  Rather the opposite.  Any non-infinite acceleration propagates at a rate that approaches infinite speed since it is made up of acceleration quanta that approach infinitely small speed changes, and yet the aggregate speed of the wave must be much less, a contradiction.

I'm not managing to convert that into anything that I can visualise, so I can't see the contradiction (which I'm not saying isn't there). I probably won't get it without a diagram (and one that shows things moving).

Why can't you just start accelerating each particle sooner than the one behind it and have the propagation of the wave accelerate to accommodate this?

That's exactly what happens naturally.  There is no wave then since the whole thing starts to move at once, and the entire object is stationary in its own frame at all times.  Funny thing is that it is not all moving at the same speed in any other frame.  Only its own frame.[/quote]

It may be that trying to describe it as waves is limiting the ability to represent what can actually be done.

I presumed the ship had to be always stationary in its own frame along its entire length during the whole trip.

I don't see how it would be possible for it to be stationary in its own frame when it has parts moving at different speeds, other than by being stationary on average in one frame (which will always be the case no matter what you do).

But it is.  In its own frame, no part is moving at a different speed than any other part.  They're all stopped in fact.  Not true of the infinite-acceleration wave, but that involves discontinuities.

That doesn't work - you have the back end moving at a different speed from the front end, so the material at different places along your ship are contracted to different extents and you don't have a single frame for the whole ship to be stationary in, unless you're using some weird kind of frame which pretends that they're all moving at the same speed by being a mixture of a long series of real frames. If you're doing that, then you're going to get into horrific mathematical complications which will make it very hard to work out what's going on, not least because the speed of light relative to each part (in the direction of travel) is different in real frames.

I don't understand why this should be the speed of the ship. If you have a constant acceleration for the front end of the ship, what's the back end doing?

Constant acceleration to the same speed as the front.  It takes less time for the back to do this since it accelerates harder.

If you have the back end accelerating harder, it must be moving faster than the front end. It's only when you stop the acceleration that the two ends of the ship can settle to moving at the same speed (after a bit of compression while the extra momentum form the tail end is shared out with the front).

What are the speeds of the front and back ends of the ship at 0, 5, 10, ... 45, 50, 55 years into the trip (or use some other time gap if you've already got a similar set of numbers). This would make it possible for other people to visualise how your ship is moving.

Speed of the rear (km/sec) is 452, 411, 370 ... 82, 41, 0.  Speed of the front is those same numbers, but in reverse.  The 452 figure is just after the high acceleration finishes after say one minute, and one minute before the front decelerates hard to 0.

So, you start the process by instantly having the rear part move at 452, but you immediately begin to decelerate it while you accelerate the front part gradually. By the half way point, the whole ship will be moving at the same speed, but at all other times, different parts are moving at different speeds and will be differently contracted. Have I understood that correctly?

For points other than the front or the rear, the peak speed is reached at a proportional time relative to the distance from the rear.  So a point an 11th of the way forward will reach peak speed in 5 days and ramp evenly down from there.  The midpoint will accelerate constantly for half the time and decelerate constantly for the other half.

What happens to the length of your ship through the course of this process? The initial acceleration of the rear part will lead it to want to be contracted, while the contraction required for the rest goes down for each section all the way to the front. By the end of the process, the opposite occurs, so the length is the same at the end as it was at the start, except that when it started, the sudden acceleration of the tail from 0 to 452 made it the wrong length for a moment (so it was too long for the atom-to-atom separation distances to be comfortable). Half way through the process, I'm imagining the whole ship moving at the same speed (226). The ship should be at its shortest length at this point because the speed of the front has caught up with the speed of the rear, and from now on it will lengthen out again. I don't know how to apply maths to this to test whether the length of the ship is always right for the sum of all the differently-contracted parts at every moment in the process, so I don't know if the numbers match up, but the whole methodology looks a bit suspect.

I see it working in all frames.

Then you're not thinking it through.  You don't show how you arrive at this conclusion, so I have a hard time pointing out where it fails.

I arrive at that conclusion because I can see the back end being able to accelerate up to a fraction under c where it can all concertina up into almost a 2D object, and I can see the potential for everything ahead of the 2D compression zone to move forward a little before that 2D compression zone catches up with it, and this delays the formation of the 2D compression zone a bit because the second last atom will move forwards a bit while the rearmost atom closes in on it. I realise now that the limit of this though ends up being the case where the rearmost atom doesn't actually close in on the one ahead of it at all because the one in front is moving at the same speed, as is the one ahead of that, and all the way to the front, so we end up with the whole thing moving at next to c, and it's stable in transit because its functionality is slowed to a halt. This works at the highest speed, and it works at the lowest speed, but there may be some speeds in between where it breaks.

[Halc (OP)]

With the ship travelling at just a fraction under c, the time of one hour for us would to that ship appear to be an infinitesimal moment

Not sure why you're considering this case since at that speed, it takes the wave almost 100 years to get to the other end.  Hardly an optimal solution.

It may be that trying to describe it as waves is limiting the ability to represent what can actually be done.

Indeed, it isn't a wave anymore.  The only way to do a wave is with infinite acceleration (abrupt change in velocity to something else), and it only works because the singularity makes certain values (notably length of the object) undefined.

In its own frame, no part is moving at a different speed than any other part.  They're all stopped in fact.  Not true of the infinite-acceleration wave, but that involves discontinuities.

That doesn't work - you have the back end moving at a different speed from the front end, so the material at different places along your ship are contracted to different extents and you don't have a single frame for the whole ship to be stationary in.[/quote]How is stopped a different speed than the stopped at the other end?  That single frame is for the whole object.  I don't call it a ship since it is easier to visualize the forces needed being applied by the rail gun outside the object.

Yes, different parts of the object are moving at different speeds in other frames where they are not stationary, and hence the contraction factor isn't constant over the length of the object in those other frames.  There is no contraction in the object frame since it is everywhere stationary in it.

unless you're using some weird kind of frame which pretends that they're all moving at the same speed by being a mixture of a long series of real frames.

It is a standard accelerating reference frame.  Different stationary points in space accelerate at different rates in such a frame, and the reference frame is bounded by an event horizon to the rear, beyond which events are not part of the frame at all.  The object cannot extend beyond that event horizon, at least not while remaining Born rigid.
I learned a bit about general relativity when researching this topic.

If you're doing that, then you're going to get into horrific mathematical complications which will make it very hard to work out what's going on, not least because the speed of light relative to each part (in the direction of travel) is different in real frames.

Speed of light isn't really a meaningful thing in an accelerating reference frame.  I suppose I can shine a light to a mirror further forward and time the return of that signal, and the observer up there can similarly do such a measurement via a mirror by me.  We won't measure the same duration.  I will measure a longer elapsed time.

If you have the back end accelerating harder, it must be moving faster than the front end.

In an inertial frame, yes. The frame of the object  is not an inertial frame.

It's only when you stop the acceleration that the two ends of the ship can settle to moving at the same speed (after a bit of compression while the extra momentum form the tail end is shared out with the front).

If we stop the acceleration all at once in the object's frame, then there is nothing to settle and no extra compression or momentum to deal with.  The object is already stopped in its own frame (and always has been), so nothing needs to be fixed.

So, you start the process by instantly having the rear part move at 452

Quickly at least.  Instantly isn't necessary.  We do it in an arbitrarily short time.

but you immediately begin to decelerate it while you accelerate the front part gradually. By the half way point, the whole ship will be moving at the same speed.

In the initial inertial frame, the object is moving at top speed at the half way point, and the ends are moving at half speed.

but at all other times, different parts are moving at different speeds and will be differently contracted. Have I understood that correctly?

At all times the different parts are moving at different speeds.  This is true of any accelerating object .  I just made it more obvious by making the object stupidly large.  When it stops accelerating, it needs to do it along its length simultaneously in its own frame, not simultaneously in the original frame, which wouldn't work since it is moving at different speeds in that frame.
In our 55 day trip, we immediately start decelerating, and don't stop at all until we're stationary back in the original frame.

For points other than the front or the rear, the peak speed is reached at a proportional time relative to the distance from the rear.  So a point an 11th of the way forward will reach peak speed in 5 days and ramp evenly down from there.  The midpoint will accelerate constantly for half the time and decelerate constantly for the other half.

What happens to the length of your ship through the course of this process?

Frame dependent question. In the original inertial frame, the object contracts at first, and expands again towards the end, as you would expect of an object that moves at relativistic speed.  We're only going a lousy 452 km/sec max in the middle when the object as a whole achieves maximum average speed, but it is never going all the same speed in the original frame except at the endpoints.

The initial acceleration of the rear part will lead it to want to be contracted, while the contraction required for the rest goes down for each section all the way to the front. By the end of the process, the opposite occurs, so the length is the same at the end as it was at the start, except that when it started, the sudden acceleration of the tail from 0 to 452 made it the wrong length for a moment (so it was too long for the atom-to-atom separation distances to be comfortable).

Did not.  The material nearby is also going nearly that speed, so it all contracts exactly the amount it needs to.  It is always the correct length.  No strain.

Half way through the process, I'm imagining the whole ship moving at the same speed (226).

No, the middle reaches peak speed at that point.  All points need to average 226 the whole way, so the middle is no exception, accelerating cleanly from 0 to 452 and back down again.  So at 27.6 years, the ends are moving at 226 but the middle is moving at 452.

The ship should be at its shortest length at this point because the speed of the front has caught up with the speed of the rear, and from now on it will lengthen out again.

Correct.

I arrive at that conclusion because I can see the back end being able to accelerate up to a fraction under c where it can all concertina up into almost a 2D object,

Well, the rear of the ship does that quickly as you say, but it takes well over 100 years for the whole thing to compress to a 2D object like that.  The front is accelerating at about 0.3 m/sec per hour, so it takes a wicked long time to get the front up to enough speed to consider the object compressed to negligible length.  The front cannot accelerate faster.  It is a function of the proper length of the object and has nothing to do with the power we're applying to the thrust.

and I can see the potential for everything ahead of the 2D compression zone to move forward a little before that 2D compression zone catches up with it, and this delays the formation of the 2D compression zone a bit because the second last atom will move forwards a bit while the rearmost atom closes in on it.

Like that, yes.

I realise now that the limit of this though ends up being the case where the rearmost atom doesn't actually close in on the one ahead of it at all because the one in front is moving at the same speed, as is the one ahead of that, and all the way to the front,

During accleration, the atoms behind are always faster than the ones ahead of them since they are accelerating harder.  From some inertial frame where things are speeding up, the object always keeps compressing.

so we end up with the whole thing moving at next to c, and it's stable in transit because its functionality is slowed to a halt. This works at the highest speed, and it works at the lowest speed, but there may be some speeds in between where it breaks.

It is stable and retains its proper length the whole way, so it doesn't break.  That was the initial condition, not a conclusion.  I accelerate each piece enough so the accelerating rear-most piece stays the same proper distance from it at all times.

[yor_on]

If you're only considering the 'stress' created by a length contraction, ignoring compression waves/stresses, then it doesn't matter. Everything should break just before 'c', doesn't matter if it's a spinning disk or a 'rod'. It's mass will reach infinity at whatever 'edge' you define, be it a rod or a plate. Presuming that you can accelerate a object equally over its whole mass/density, which to me seems pretty impossible in itself ( I would really like to see how that is thought to work btw ) the idea seems to be that everything becomes this 'edge'? In that case we might think of it as every 'atom' having a equivalent acceleration. Well, seems to me you're setting up a 'black hole scenario' if so :)  For this it doesn't matter if you constantly accelerate at one gravity to then, some years later, close in to 'c'. The rods mass must reach infinity as it nears the speed of light in a vacuum.

[yor_on]

Hmm, would you mind explaining how you think writing that the rod would break if every atom in it had a equal acceleration? If you on the other hand presume slightly different accelerations inside the material you must include stress. A equal acceleration of every atom should give a equal length contraction at every point of this rod. It's a weird idea :) though that seems to go against the lack of simultaneity relativity discuss. You could treat the atoms as related to each other relative both time dilation's and LorentzFitzGerald contractions.
=

Actually, presuming identical atoms in a identical space having a identical acceleration, seems to me to state that they also must share a exact same frame of reference. That would be turning accelerations upside down sort of :) Doesn't mean it has to be wrong, but it's funny :)

You could argue that this is what a perfect non spinning sphere, of a perfectly distributed density of one gravity, actually have. And as that could be seen as the equivalence to a uniform acceleration of one gravity? Not sure though. Accelerations always seem to come in 'steps', even with a planet. Mass invariant, with gravity's potential changing inside the planet.

[David Cooper]

With the ship travelling at just a fraction under c, the time of one hour for us would to that ship appear to be an infinitesimal moment

Not sure why you're considering this case since at that speed, it takes the wave almost 100 years to get to the other end.  Hardly an optimal solution.

I wasn't doing it with a wave, but with the whole ship being accelerated to a fraction under c at the same time. The whole thing can then travel for an hour and it won't have time to contract significantly because it's functionality is practically halted by its high speed of travel, so when you halt it an hour later, it is still almost the same length as when it started and will make an infinitesimal correction in an infinitesimal amount of time to get back to full length. No damage done. This is the fastest way in principle to move objects, including long ones, and it's a lot less interesting than what I was hoping to find.

unless you're using some weird kind of frame which pretends that they're all moving at the same speed by being a mixture of a long series of real frames.

It is a standard accelerating reference frame.

That is why I had trouble working out what you were doing - I assumed you would be using real frames rather than contrived ones in which the speed is claimed to be the same relative to each part of the ship while in the real universe it varies. If you're allowed to use such contrived frames, you can design some really warped ones to cover all the action in any caterpillar solution too and assert that the entire ship is stationary in the ship's frame at all times, though clearly you want to stick to the particular contrived frames used that are accepted in GR, so that's fair enough as an exercise.

Different stationary points in space accelerate at different rates in such a frame, and the reference frame is bounded by an event horizon to the rear, beyond which events are not part of the frame at all.  The object cannot extend beyond that event horizon, at least not while remaining Born rigid.
I learned a bit about general relativity when researching this topic.

Well, now I can see why you're aiming for that specific kind of solution, and if your requirement is to have the whole ship stationary in an officially recognised GR frame, then you likely do have the best solution of that kind.

If we stop the acceleration all at once in the object's frame, then there is nothing to settle and no extra compression or momentum to deal with.  The object is already stopped in its own frame (and always has been), so nothing needs to be fixed.

Except that you have the back end instantly moving at 452 without any time for it to contract to a comfortable length (as observed from the inertial frame in which the journey begins with the whole ship at rest).

I arrive at that conclusion because I can see the back end being able to accelerate up to a fraction under c where it can all concertina up into almost a 2D object,

Well, the rear of the ship does that quickly as you say, but it takes well over 100 years for the whole thing to compress to a 2D object like that.

We aren't worried about compressing the whole ship. The point of the caterpillar method is that we should be able to move the entire ship using your method as a starting point, but add in the caterpillar compression to the rear to reduce journey time for the rear while still delivering the front end to its destination in 55 days. This must produce a viable method of moving the ship without leaving the atoms at uncomfortable separations for any extended length of time, but merely fails to comply with any GR-approved frame for the whole ship to be stationary in at all times.

The front is accelerating at about 0.3 m/sec per hour, so it takes a wicked long time to get the front up to enough speed to consider the object compressed to negligible length.  The front cannot accelerate faster.  It is a function of the proper length of the object and has nothing to do with the power we're applying to the thrust.

I can see that there will be a limit to the speed you can get the front end up to under this rule (of not leaving the atoms at uncomfortable separations for any extended length of time), and your method may well have identified that limit for the caterpillar method too, but I wouldn't want to bet on that. Because I can get the back end moving faster, the length is shortening faster and the limit may be shifted by that.

I realise now that the limit of this though ends up being the case where the rearmost atom doesn't actually close in on the one ahead of it at all because the one in front is moving at the same speed, as is the one ahead of that, and all the way to the front,

During accleration, the atoms behind are always faster than the ones ahead of them since they are accelerating harder.

Not in the case I was thinking about there where I was referring to the limit under rules that allow you to have atoms sit at uncomfortable separations for extended periods so long as their functionality is practically halted (such that they won't contract or extend). There are three different sets of rules here, so we're tripping up over category boundaries.

I accelerate each piece enough so the accelerating rear-most piece stays the same proper distance from it at all times.

Fair enough, but you can certainly move the rear faster than that while still following that rule, so the only thing stopping you doing that is your desire to stick to a GR-approved frame in which the whole ship is stationary. We have three categories with different rules applying, and you have identified the fastest method for the category with the greatest constraints on what's allowed (although you may have to adjust the way you start the back end moving, because for shorter ships you're going to have trouble with an instant acceleration to high speed where the lack of contraction on it is wrong), while I've identified the fastest method for the category with the least constraints. The remaining category is the one in between in which the tail compresses to nearly 2D while the front end may move at the same speed as in your accelerated frame method, so the main remaining interest for me is whether that limit should apply to the front end or if it can accelerate faster due to the compression of the back end. The back end needn't immediately compress to nearly 2D though, because the particle ahead can accelerate early to enable the one ahead of it to accelerate early, and this chain of early accelerations will propagate all the way along the ship, potentially allowing the front end to move a bit faster than in your case.

[Halc (OP)]

I wasn't doing it with a wave, but with the whole ship being accelerated to a fraction under c at the same time. The whole thing can then travel for an hour and it won't have time to contract significantly because it's functionality is practically halted by its high speed of travel, so when you halt it an hour later, it is still almost the same length as when it started and will make an infinitesimal correction in an infinitesimal amount of time to get back to full length. No damage done.

In the original frame, it is held at its full length for an hour while it should have contracted to say a 1000th that length.  Damage is very much done.
In its own frame, it's like I spread your body all over Earth in tiny pieces, which is OK since I promise to put the pieces back in 3 seconds.  It breaks.  Such solutions are not even close to following the premise.

This is the fastest way in principle to move objects, including long ones, and it's a lot less interesting than what I was hoping to find.

Fastest way to move a lengthy load of sand perhaps, but not my rigid object.

It is a standard accelerating reference frame.

That is why I had trouble working out what you were doing - I assumed you would be using real frames rather than contrived ones in which the speed is claimed to be the same relative to each part of the ship while in the real universe it varies.

Are you claiming that accelerated reference frames are less real? Sure, they have different properties than inertial or rotating frames, but they're all equally natural.  Per the equivalence principle, you live in such a frame, and there is no avoiding it.  Everything the object is doing in my descriptions also happens to you and more rigid things like buildings and such.

If you're allowed to use such contrived frames, you can design some really warped ones to cover all the action in any caterpillar solution too and assert that the entire ship is stationary in the ship's frame at all times, though clearly you want to stick to the particular contrived frames used that are accepted in GR, so that's fair enough as an exercise.

Suit yourself.  You are welcome to compute all the locations, speeds, lengths and stresses in say the one (unspecified) absolute frame, but the stresses will work out to exactly zero or you've done it wrong.  The mathematics of that is beyond me.
As for contrived frames, I think we used a composite frame to work through the wave method, but I eventually found that it was much simpler to use the inertial 226 km/sec frame.  That wasn't obvious at first.

Well, now I can see why you're aiming for that specific kind of solution, and if your requirement is to have the whole ship stationary in an officially recognised GR frame

I never had that requirement.  I just haven't found any better solutions that don't involve singularities.

If we stop the acceleration all at once in the object's frame, then there is nothing to settle and no extra compression or momentum to deal with.  The object is already stopped in its own frame (and always has been), so nothing needs to be fixed.

Except that you have the back end instantly moving at 452 without any time for it to contract to a comfortable length (as observed from the inertial frame in which the journey begins with the whole ship at rest).

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.

We aren't worried about compressing the whole ship. The point of the caterpillar method is that we should be able to move the entire ship using your method as a starting point, but add in the caterpillar compression to the rear to reduce journey time for the rear while still delivering the front end to its destination in 55 days.

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.  I found the 452 figure right away (working backwards from a target Lorentz contraction) and only later computed how long it would take to move a light hour averaging exactly half that speed.

Later on I computed the same number using A= c²/length :  c2 is 9e16 (units of meters and seconds) / 9.46e17 meters (100 LY) giving us a max acceleration of 0.09513 m/sec2.  That is the max acceleration of the front and the max deceleration at the rear, and it takes 55.3 days at that acceleration to go a light hour, topping out at 452 km/sec after those 55.3 days.  Go any faster and it takes longer that 55 days to get that fast and you can't stop it in time.

I can see that there will be a limit to the speed you can get the front end up to under this rule (of not leaving the atoms at uncomfortable separations for any extended length of time), and your method may well have identified that limit for the caterpillar method too

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

Fair enough, but you can certainly move the rear faster than that while still following that rule, so the only thing stopping you doing that is your desire to stick to a GR-approved frame in which the whole ship is stationary.

Nope.  I invite other solutions, but the singularity was too much. Acceleration needs to be finite.
Clearly it is allowed for different parts of the object to be moving at different speeds in other frames.  No reason why other parts of the object need to be stationary in the frames any given parts.  So maybe there is a better solution, and we just haven't identified it yet.  I suppose I need a proof that my 55.3 day thing is optimal, but I don't have that proof yet.

We have three categories with different rules applying, and you have identified the fastest method for the category with the greatest constraints on what's allowed (although you may have to adjust the way you start the back end moving, because for shorter ships you're going to have trouble with an instant acceleration to high speed where the lack of contraction on it is wrong), while I've identified the fastest method for the category with the least constraints. The remaining category is the one in between in which the tail compresses to nearly 2D while the front end may move at the same speed as in your accelerated frame 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.

so the main remaining interest for me is whether that limit should apply to the front end or if it can accelerate faster due to the compression of the back end. The back end needn't immediately compress to nearly 2D though, because the particle ahead can accelerate early to enable the one ahead of it to accelerate early, and this chain of early accelerations will propagate all the way along the ship, potentially allowing the front end to move a bit faster than in your case.

I encourage investigation of such a solution.  The wave thing worked best at around 3150 km/sec, hardly a speed worthy of massive contraction, but it sure got the job done a lot faster than 55 days.  But it only worked with that singularity, not if you approached it.

[David Cooper]

In the original frame, it is held at its full length for an hour while it should have contracted to say a 1000th that length.  Damage is very much done.

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. Think about a planet orbiting a star. If the star is stationary, you could watch the planet take a year to go round it once. Now have the star and planet move past you at 0.866c while you are stationary and you will see the planet take two years to go round the star instead of one. Increase the speed to 0.969c and you will see it take four years to orbit once. The faster we move the star and planet past you, the more we slow those orbits, and as we get near to c, the orbiting (and all other functionality) practically stops. In the same way, a ship of any length that's accelerated to a speed that practically stops functionality will not contract significantly - we can pick a theoretically achievable speed that will keep the amount of actual contraction so low that no damage will be done by the time we've stopped the ship again, even if the journey lasts a billion years. This solution works.

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. 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? 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. 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. 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. 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.

Sure, they have different properties than inertial or rotating frames, but they're all equally natural.  Per the equivalence principle, you live in such a frame, and there is no avoiding it.  Everything the object is doing in my descriptions also happens to you and more rigid things like buildings and such.

When an accelerated frame is actually stationary and the difference in the speed of light across objects relative to them in different directions is caused by gravity, again that frame is a misrepresentation of reality. A clock higher up is ticking more quickly because the speed of light across it is either higher or more even, but the true frame has its time running faster still, and it runs at that higher speed at all depths in the gravity well.

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. These 4D models provide the only semi-reasonable excuse to declare frames that I label as fake to be valid, but they don't stand up to scrutiny when you push them into a corner to see if they actually work as claimed (due to event-meshing failures in dynamic versions and fake causality in static versions where nothing ever had the opportunity to cause anything). That's a discussion for elsewhere though as some proofs are not welcome outside of the backwaters.

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. 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.

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. 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. I haven't said that you can get the front end to its destination faster than your method (other than through the recently found method where the whole thing moves at a fraction under c and holds together for an hour unsupported due to it's practically-halted functionality), but it is not yet clear to me that the front end will be unable to go a bit faster than with your method if the back end is moving at much higher speed (given that that frees up the bit ahead of the back end to go a bit faster too, and so on, potentially all the way up to the front). I'm not going to spend time trying to do the maths for it though when it's a better use of time to focus on building tools that will make that maths easier to apply, so I'm going to put this on the shelf for later. 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.

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.

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 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. A simulation could resolve that, but it could also provide misinformation if there's an unrecognised bug in it. A top mathematician might notice a way to work through this on the back of an envelope at breakfast, but I haven't spent my life collecting the right algorithms to do that.

I encourage investigation of such a solution.  The wave thing worked best at around 3150 km/sec, hardly a speed worthy of massive contraction, but it sure got the job done a lot faster than 55 days.  But it only worked with that singularity, not if you approached it.

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.

[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.

[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 48 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.

[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 48:

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 20 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 c?/D (per post 59) 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/sec? .
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.

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 48), 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 48 (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.

[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 48 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.