Does Special Relativity contain contradictions?

[David Cooper]

No, the fact that they're physically identical has nothing, in itself, to do with the speed of light. They are physically identical because each can be transformed into the other by viewing from a different inertial reference frame. And the laws of physics are the same in all inertial reference frames.

Your transformation is changing the speed of light relative to the arena - that's what happens when you change frame, and you are changing frame.

The error that you appear to be making is in failing to remember that the concept of speed, including the concept of the speed of light, is only meaningful when expressed in terms of how it is measured. This is true of all concepts in physics.

The error you're making is in thinking that the speed of light is c in all directions relative to objects A and B even when A and B are not co-moving. That is just the result of brainwashing, and it's irrational - it leads to the contradictions I spelt out in my previous post.

Do you see how this works?

Yes - it generates contradictions. By reversing the roles between clocks 3 and 4, you end up asserting that clock 3 is ticking faster than clock 4 in one experiment, and that clock 4 is ticking faster than clock 3 in the other, but that means you're changing the speed of light relative to them between doing the experiments. When both experiments are done simultaneously, as in the newest version (with Einstein and Lorentz helping out), your lovely trick results in clock 3 ticking faster than clock 4 while clock 4 ticks faster than clock 3, and it results in our clocks and their clocks not ticking at the same rate as each other while they are side by side and co-moving. That's the contradictions that come out of the set 2 model idiocy.

To get your head around this...

...you have to overcome the SR brainwashing and start to apply the rules of reasoning to things in the way that mathematicians do.

Every competent mathematician in the world will agree with this, and they'll also agree that if my clock was ticking faster than yours during the first leg of the first experiment, then your clock was ticking faster than mine during the first leg of the third experiment.

No, what they'll agree is what I explained above

They'll side with me rather than you, because your way generates contradictions.

that when you say such things as "Clock 3 ticks at a constant rate throughout" and "clock 4 ticks at a different constant rate throughout" you have to specify which other clock you're measuring them against and the reference frame against which you're making those measurements.

They don't accelerate, so I don't have to do any such thing. You are all over the place here, trashing the laws both of physics and mathematics.

As I said, and as you have demonstrated perfectly here, clock 3 and 4 swap roles between expts, from the point of view of the reference frames in which the expts are the same.

That illegal move doesn't become any less illegal by making it over and over again. You are cheating by changing the speed of light relative to the clocks, and changing the one which is regarded as moving. However, the new version makes it clearer why you aren't allowed to do this, so it'll be interesting to see if you still want to make the same mistake when the experiments are carried out simultaneously.

It is at rest in the arena where we're doing the experiments throughout - it is ticking at a constant rate because it never accelerates.

A constant rate relative to what?

We have two clocks ticking at constant rates relative to each other as they move at a constant speed relative to each other.

It is my view that you need to better get your head around the process of viewing various moving objects from the point of view of different inertial reference frames.

I'm not going to disable my ability to apply reason - the problem here is that you've done that to shut down your ability. The only question is how tightly you've locked it down and whether the damage can be undone.

In the absence of acceleration, the way to measure the rate at which a clock ticks is to compare it with another clock.

In the absence of acceleration, you know it's constant, but we happen to have another clock to compare it with at all times which is also ticking at a constant rate, so there's no problem, until you make an illegal move by changing frame.

[David Cooper]

David, I've been thinking a bit more how this could be demonstrated to you. An animation would be good.

That's fine - we already have something equivalent on my relativity page. The simulation with the planets and rockets does exactly the same thing as the latest version of the clock experiments where experiment 1 and 3 are done simultaneously. All you have to do is imagine what the clocks are doing. Use mode 2 and don't cheat by changing frame.

[David Cooper]

So simultaneity is not symmetric, and therefore we cannot say that those two clocks both tick faster than the other at the same time. No contradiction.

Parts of simultaneity are symmetric. To see that, you need to look at cases where you're comparing what happened between separation and reunition, with one clock getting from one point to the other without changing speed and the other making the trip in two legs. We then have a guarantee that the former clock ticked faster on average than the latter, and we know too from that that the latter clock must either have ticked more slowly than the former during at least one of the legs. We can then build on that by applying standard reasoning rules to generate conditional truths about what's going on, and by doing this we reveal contradictions.

[David Cooper]

Yes, as you've suggested, we can ask: At the same time as measured by what clock? Because we know that time is a thing that is measured by clocks, so simply saying "at the same time" without mentioning a clock is, strictly speaking, meaningless.

We have two clocks being compared with each other, so I don't understand how you have such difficulty recognising the clocks there.

[Steve3007]

viewtopic.php?p=320773#p320773

That is where you're making your fundamental mistake. All three experiments appear identical, but they aren't. If you're using a set zero model, time doesn't run (or indeed exist at all), so there's no direct problem there, but when we're exploring the contradictions we're concerned with specific models which have running time where we allow some clocks to run faster than others rather than having them all run at the same rate as each other at all times (which would take us into the models with event-meshing failures), we are specifically testing set 2 and set 3 models, and what we find here allows us to eliminate the set 2 ones (including Einstein's original SR).

I'm not really interested in discussing this in terms of model numbers. For clocks C1 and C2 the experiments are identical because in all experiments they move, relative to each other, in the same way. That's a really simple and obvious fact. There are no contradictions in it. I think the underlying reason why you keep on seeing contradictions where none exist is because you simply refuse to accept that physics is an empirical subject, that statements in physics are only meaningful if they can be related to something that can be measured, that the movements of objects are only meaningful relative to another object and that the tick rate of a clock is only meaningful relative to another clock. Your statements fail to take this into account over and over and over again. The most recent posts of yours, to which I am replying here, will demonstrate this apparent blind spot of yours yet again.

There is no such thing as an event meshing failure in physics. As I said earlier, that stems from your misuse of the concept of a 4D spacetime graph and the concept of a worldline, and your weird notion (on your website) that these 4D spacetime graphs have to be "built" like some kind of Lego model.

Yes you are. Every time you change frame, you change the speed of light relative to the arena.

I don't know for sure what you mean by "the speed of light relative to the arena", but I presume you mean the speed of light from any light source as measured against the starting-line reference frame. That speed is measured to be constant. The speed of light, as measured against any reference frame is constant regardless of the movement of the light source relative to that reference frame.

This is the big result of the brainwashing - it is very effective in shutting down people's rational thinking, and you're one of an army of such victims. I've set a simple proof in front of you here, but you appear to be unable to process it.

I disagree that you have provided any kind of proof of anything. I'm not sure you even understand what a proof is.

In experiment 1:-

If my clock is ticking faster than yours during leg 1 of your trip, then clock 3 is ticking faster than clock 4. In experiment 3.
If my clock is ticking faster than yours during leg 1 of your trip, then clock 4 is ticking faster than clock 3.

In all 3 experiments clocks 1 and 2 (yours and mine) are receding from each other at speed v in leg 1. Therefore they each see tick signals from the other as slower than their own tick signals in leg 1.
In all 3 experiments clocks 3 and 4 are receding from each other at speed v. Therefore they each see tick signals from the other as slower than their own tick signals.

Your problem is that you persist in thinking that if clock X measures the tick signals from clock Y as slower than its own, then clock Y must measure the tick signals from clock X as faster than its own. It doesn't. They each measure the other's as slower. This problem seems to me to stem from your apparent inability to get used to the fact that we have to talk about things in terms of how they can be measured, as stated at the beginning of this post.

For reference, see this post:
viewtopic.php?p=320758#p320758

for all the details of the relative speeds of clocks 1, 2, 3 and 4. The relative tick rates that each pair of clocks measures depends on those 2 clocks' relative speeds. Nothing else. You can add in as many other clocks as you like, and have them moving around as you like. Adding new clocks doesn't suddenly change the relative tick rates measured between existing clocks. Think about the simple logic of that for a moment.

If both of these experiments are identical, then clock 3 is ticking faster than clock 4 and clock 4 is ticking faster than clock 3, but that's a mathematical impossibility.

I think your failure to appreciate that tick rates are things that are actually measured by comparing pairs of clocks stems from your attachment to some form of universal time, and failure to realize that time is what is measured by a clock. That's my theory anyway. It is the only reason I can think of for your statements.

The reality is that if my clock is ticking faster than yours during leg 1 of your trip in the first experiment, then your clock is ticking faster than mine during leg 1 of your trip in the third experiment.

When it comes to logical reasoning, you seem to have an entire module missing. You're not alone in that though, because it is very much the norm for people who imagine that SR is valid. Some of them do see the problem though (Minkowski, perhaps) and recognise that set 2 models don't work.

Your assertions do not agree with observed reality.

And that means changing the speed of light relative to the system for each leg, leading to a complete misunderstanding of what's going on.

The speed of light is measured to be the same by all observers. There is no contradiction in that.

Yes it does - you are changing frame and thereby changing the speed of light relative to the system. You're doing bad physics.

I'm not changing the speed of light. See above.

The apparent speed of light relative to them in each case is c. The actual speed of light relative to them may be different for each.

I don't know what you mean by "the actual speed of light". If you're not talking about something that can be measured you're not doing physics.

We're only interested in how fast they're ticking relative to each other. You've playing a game where clock 3 is ticking more quickly than clock 4, then you change frame and claim that clock 4 is ticking more quickly than clock 3, but you're trying to have your cake and eat it by taking both of these things to be true at the same time.

They both measure each other to be slower. This is an empirical fact. See above. I'm not playing any games. I'm sticking to what is measured. i.e. doing physics.

We only need to compare them against each other, but we can include other clocks in that process.

Only if we make measurements against those other clocks.

In experiment 3, I start next to clock 3 and move away from it at the same speed as clock 4, so my clock ticks at the same rate as clock 4.

In experiment 3, the relative speeds of clocks 1 and 3 is v, and the relative speeds of clocks 1 and 4 is zero.

You're playing an irrational game where you change the speed of light relative to the system in order to pretend that experiments 1 and 3 are the same, but they aren't the same.

When you refer to "the system" which observer are you referring to? The speed of light is c as measured by all observers. In all 3 experiments clocks 1 and 2 move at the same speed relative to each other in the first leg and then clock 2 changes reference frames. In all 3 experiments clocks 3 and 4 move at the same speed relative to each other throughout. Throughout it all, all clocks measure the speed of light relative to themselves as c. These are simply the facts.

There's an asymmetry there which you're simply ignoring and pretending doesn't exist. This shows that you have bought into the dogma 100%, because it's controlling you - you appear to be incapable of thinking independently of it any more.

See above.

Let's lock things down more clearly by doing experiments 1 and 3 at the same time with the help of a couple of friends.

OK.

We are at rest in a frame in deep space which I call the arena.

OK. That's what I've called "the starting-line reference frame".

I'm going to stay in the same place in the arena throughout, so my clock will tick along with clock 3 throughout.

OK. You're going to remain stationary relative to the starting-line and clock 3. So let's just call you clock 3.

Einstein is going to move away from me at v, so he will accompany clock 4 throughout.

So let's just call him clock 4. We've added no new observers so far.

You are going to stay with me for a while, then you'll race after Einstein to catch up with him,...

So in leg 1 my speed relative to C4 (Einstein) is -v. In leg 2 my speed relative to C4 is v.

...so your clock will tick at the same rate as mine for the first leg of your trip, and then tick at a different rate during the second leg.

In leg 1 I am stationary relative to C3 (you). In leg 2 the relative speeds of me and C3, as measured by either of us, are +/-xv (where 1 < x < 2). I was previously referring to this as 2v and you were right to point out that this is incorrect. So in leg 2 me and C3 (you) will each measure the others tick signals as slow compared to our own.

Lorentz will initially travel with Einstein but then he'll turn round and come back to me. So Lorentz and I are doing experiment 1 while you and Albert are doing experiment 3.

OK. Lorentz is stationary relative to C4 (v relative to C3) in the first leg and -v relative to C3 in the second leg. So C3 (you) and Lorentz are C1 and C2 in experiment 1. I am C2 in experiment 3. Albert is C4 (moving with C1 in experiment 3).

I don't see the point in adding all these extra people. They're covered by various already existing clocks in the existing 3 experiments. So we've added nothing new. Therefore I see no possibility of any new insights from this. But let's go on and see.

Lorentz's clock runs more slowly than mine over the whole trip, so we can speculate about how that happened.

You are C3. Lorentz is C2 in experiment 1. Therefore Lorentz changes reference frames between leg 1 and leg 2 so that he can be re-united with you. You stay in the same reference frame throughout. So we are simply considering the "twin paradox" that has already been discussed by Halc and Tamminen.

There are five rational possibilities for this when we're testing set 2 and 3 models, as I set out earlier. Those options are:-

(A) My clock ticked more quickly than Lorentz's clock on both legs of his trip.
(B) My clock ticked more quickly than Lorentz's on the first leg and his ticked at the same rate as mine on the second leg.
(C) My clock ticked at the same rate as Lorentz's on the first leg and more quickly than his on the second leg.
(D) My clock ticked more quickly than Lorentz's on the first leg and more slowly than his on the second leg.
(E) My clock ticked more slowly than Lorentz's on the first leg and more quickly than his on the second leg.

None of the above options are valid because none of them refer to what is measured by observers travelling with each clock.

Options (B), (C), (D) and (E) recognise the possibility that the frame in which the arena's at rest is not the absolute frame, but if you deny the existence of an absolute frame, then you can assume that (A) must be correct, so you can ignore the other options if you wish. In the same way, there are five possibilities for what happened with your clock and Einstein's, so we can set them out and give them names in the same way:-

(F) Einstein's clock ticked more quickly than yours on both legs of your trip.
(G) Einstein's clock ticked more quickly than yours on the first leg and at the same rate as yours on the second leg.
(H) Einstein's clock ticked at the same rate as yours on the first leg and more quickly than yours on the second leg.
(I) Einstein's clock ticked more quickly than yours on the first leg and more slowly than yours on the second leg.
(J) Einstein's clock ticked more slowly than yours on the first leg and more quickly than yours on the second leg.

Ditto.

There's some symmetry about this, but it isn't complete. Whenever we do this in the same arena, and all the more obviously when we do this at the same time in the same arena, we should be able to see the nonsense that comes out of changing frame in between the experiments to change the speed of light relative to the arena.

See previous comments about the speed of light as measured by all observers.

Einstein's and Lorentz's clocks were ticking at the same rate as each other during the first leg, and my clock and yours were ticking at the same rate as each other during the first leg too. Logic dictates the following asymmetric things:-

If (A) --> not (F), not (G), not (H), not (I).
Therefore if (A) --> (J).

If (B) --> not (F), not (G), not (H), not (I).
Therefore if (B) --> (J).

If (C) --> not (F), not (G), not (I), not (J).
Therefore if (C) --> (H).

if (D) --> not (F), not (G), not (H), not (I).
Therefore if (D) --> (J).

If (E) --> not (H), not (J).
Therefore if (E) --> (F) or (G) or (I).

All irrelevant because all options are invalid. See above.

These rules come from LET, but they have relevance to set 2 models too. Set 2 models simply don't care and ride the contradictions, but no one who pushes a set 2 model should be denying that it generates contradictions. The contradictions are clear. If my clock is ticking more quickly than Lorentz's during the first leg and Einstein's clock is ticking more quickly than yours during the first leg, then my clock is ticking more quickly than yours while they're co-moving (and side by side), and Einstein's is ticking more quickly than Lorentz's while they're co-moving (and side by side). This renders set 2 models magical, ruling them out from real physics.

As I said earlier, I'm not interested in model numbers.

Those who can't recognise the contradictions here clearly doesn't have an adequate grasp of relativity, so they need to sort themselves out.

I disagree.

[Steve3007]

viewtopic.php?p=320776#p320776

Your transformation is changing the speed of light relative to the arena - that's what happens when you change frame, and you are changing frame.

The speed of light as measured by whom? See previous comments about the invariance of the speed of light as measured by all observers.

The error you're making is in thinking that the speed of light is c in all directions relative to objects A and B even when A and B are not co-moving. That is just the result of brainwashing, and it's irrational - it leads to the contradictions I spelt out in my previous post.

The speed of light is observed to be c relative to all observers, as measure by them. This does not lead to contradictions. It leads to the Lorentz transformations which describe the way that length, time and velocity is measured by various observers.

Yes - it generates contradictions. By reversing the roles between clocks 3 and 4, you end up asserting that clock 3 is ticking faster than clock 4 in one experiment, and that clock 4 is ticking faster than clock 3 in the other...

No. That assertion is not made. See various previous posts. Note the references to measurement in those posts.

...but that means you're changing the speed of light relative to them between doing the experiments.

No. The fact that the speed of light is measured to be the same by all observers relative to them is what leads to the Lorentz Transformations.

When both experiments are done simultaneously, as in the newest version (with Einstein and Lorentz helping out), your lovely trick results in clock 3 ticking faster than clock 4 while clock 4 ticks faster than clock 3, and it results in our clocks and their clocks not ticking at the same rate as each other while they are side by side and co-moving. That's the contradictions that come out of the set 2 model idiocy.

No it doesn't. For details, look up an analysis of the "twin paradox".

They'll side with me rather than you, because your way generates contradictions.

Obviously I disagree.

Since you're now appealing to the judgement of others (competent mathematicians), which of them have sided with you so far? As a starter, I'll cite Bertrand Russell on my team. I've gathered that he was a pretty competent mathematician. I suspect that you will now commit the "No True Scotsman" fallacy. But let's see.

that when you say such things as "Clock 3 ticks at a constant rate throughout" and "clock 4 ticks at a different constant rate throughout" you have to specify which other clock you're measuring them against and the reference frame against which you're making those measurements.

They don't accelerate, so I don't have to do any such thing. You are all over the place here, trashing the laws both of physics and mathematics.

For the above quote, just to be absolutely clear, I've included the passage of mine to which it was a reply. This is because, as shown later in this post, you seem so confused as to whether or not you agree with me that the assessment as to whether a clock ticks at a constant rate is only meaningful by comparison with another clock. Look at the above comment of mine and your reply to it.

As I said, the statement "Clock X ticks at a constant rate throughout" is only meaningful if you specify what other clock you're comparing it to. The only way that you can assess the rate at which a clock is ticking is by comparing it to another clock. If you disagree, describe to me a way to measure the tick rate of a clock without referring to another clock.

In what way am I "trashing the laws both of physics and mathematics" by pointing out this obvious fact to you? Are you sure you actually read the passage which you quoted and replied to here?

That illegal move doesn't become any less illegal by making it over and over again.

Please precisely specify the "illegal move" to which you are referring. Is it illegal to view the experiment with respect to various different reference frames?

You are cheating by changing the speed of light relative to the clocks, and changing the one which is regarded as moving.

As stated numerous times before I am not changing the speed of light. It is the same as measured by all observers. Movement is a relationship between 2 objects. This is true regardless of whether you think that one object is stationary in a reference frame that you regard as absolute. (I've said all this time and time again.)

However, the new version makes it clearer why you aren't allowed to do this, so it'll be interesting to see if you still want to make the same mistake when the experiments are carried out simultaneously.

See previous post.

We have two clocks ticking at constant rates relative to each other as they move at a constant speed relative to each other.

So you admit that the only way to measure the tick rate of a clock is by comparing it to another clock? Are you also "trashing the laws both of physics and mathematics" as you accused me of doing when I said the same thing? Or are you saying here that "we have two clocks ticking at constant rates relative to each other" but we don't need to compare them with each other in order to decide whether we deem them to be ticking at a constant rate?

I'm not going to disable my ability to apply reason - the problem here is that you've done that to shut down your ability. The only question is how tightly you've locked it down and whether the damage can be undone.

Clearly I disagree with this assessment.

In the absence of acceleration, you know it's constant,...

You just know it's constant do you? So now you're apparently back to saying that you can know a clock is ticking at a constant rate without comparing it to another clock! You're all over the place! First, when I point out that the constancy of a clock's ticking is only meaningful by measuring it against another clock, you say I'm "trashing the laws both of physics and mathematics". Then you appear to agree with me. Now you're flip-flopping back again. Which is it? In your view, how do you find out if a clock is ticking at a constant rate?

...but we happen to have another clock to compare it with at all times which is also ticking at a constant rate, so there's no problem, until you make an illegal move by changing frame.

"Which is also ticking at a constant rate"? As measured by what? The other clock! Don't you see this simple, obvious logic?!?

That's fine - we already have something equivalent on my relativity page. The simulation with the planets and rockets does exactly the same thing as the latest version of the clock experiments where experiment 1 and 3 are done simultaneously. All you have to do is imagine what the clocks are doing. Use mode 2 and don't cheat by changing frame.

I disagree that the animations on your web page do this.

viewtopic.php?p=320779#p320779

We have two clocks being compared with each other, so I don't understand how you have such difficulty recognising the clocks there.

OK, so now you're back to agreeing with me when I said that the only way to measure the tick rate of one clock is to compare it with another clock - the thing that you referred to as "trashing the laws both of physics and mathematics"!

[Tamminen]

So simultaneity is not symmetric, and therefore we cannot say that those two clocks both tick faster than the other at the same time. No contradiction.

Parts of simultaneity are symmetric. To see that, you need to look at cases where you're comparing what happened between separation and reunition, with one clock getting from one point to the other without changing speed and the other making the trip in two legs. We then have a guarantee that the former clock ticked faster on average than the latter, and we know too from that that the latter clock must either have ticked more slowly than the former during at least one of the legs. We can then build on that by applying standard reasoning rules to generate conditional truths about what's going on, and by doing this we reveal contradictions.

Simultaneity within a reference frame is symmetric, but simultaneity between frames is asymmetric.

[Halc]

Simultaneity within a reference frame is symmetric, but simultaneity between frames is asymmetric.

It is quite symmetric even between frames

[Steve3007]

This post (one in a conversation between Halc and Tamminen)...

viewtopic.php?p=320629#p320629

...is part of the explanation of the "twins paradox" purely in the context of Special Relativity. But it's also interesting to consider it in the context of General Relativity. In that context, the acceleration that happens in the transition between legs 1 and 2 of the journey is equivelant to a gravitational force, and the concept of gravitational time dilation comes in.

Classically, Newton's second law can be expressed either as:

1.
F = ma

And since acceleration is rate of change of velocity (dv/dt), this is the same as:

F = m dv/dt

Or it can be written as:

2.
F = d(mv)/dt

i.e. Force is rate of change of momentum (mv).

But relativistically the first form is wrong because it assumes constant mass. Using d(mv) instead of dv takes into account the relativistic increase in mass above rest mass as well as the change in velocity. And this shows that as the relative velocity of the object gets bigger, the force required to increase that velocity more gets bigger. So the strength of the gravitational field to which that acceleration is equivalent gets bigger.

So if we imagine the scenario of a spaceship accelerating at a constant 1g (which is often assumed for possible future near-light-speed travel because it is an acceleration that would be comfortable for humans) until it gets close to the speed of light relative to its original rest frame, then the force required to keep that constant acceleration gets greater and greater. So presumably the effect of the gravitational time dilation, relative to that original rest frame, gets greater and greater.

On the other hand, if the force is kept constant then the acceleration will gradually decrease as the spaceship approaches light speed, relative to its original rest frame. So the gravitational time dilation (relative to that frame) will remain constant but will last for much more of the journey.

[Steve3007]

Another thought on this:

In General Relativity, the concept of a force is deprecated. Rate of change of momentum replaces force and the principle of conservation of momentum replaces Newton's second law. So if a rocket was accelerating at 1g in the way that rockets normally do - by throwing mass out of the back and relying on conversation of momentum to thereby accelerate the rocket - then the magnitude of that mass, relative to the original frame, will get greater and greater. So the acceleration will remain at 1g for as long as the mass being thrown out of the back (as measured in the frame of the moving rocket) remains constant.

[Halc]

This post (one in a conversation between Halc and Tamminen)...

viewtopic.php?p=320629#p320629

...is part of the explanation of the "twins paradox" purely in the context of Special Relativity. But it's also interesting to consider it in the context of General Relativity. In that context, the acceleration that happens in the transition between legs 1 and 2 of the journey is equivelant to a gravitational force, and the concept of gravitational time dilation comes in.

That's an interesting way to turn around. Use the little black hole out there to slingshot your way back home. The tides are murder, but think of the fuel savings, and the trampoline I was using was even more murder so I already assumed we had a ship with a good G-couch.

So if we imagine the scenario of a spaceship accelerating at a constant 1g (which is often assumed for possible future near-light-speed travel because it is an acceleration that would be comfortable for humans) until it gets close to the speed of light relative to its original rest frame, then the force required to keep that constant acceleration gets greater and greater.

The force had better remain the same or else it would get quite uncomfortable (fatal) to the humans. The proper acceleration should remain at 1G (resulting in an eventual proper velocity greater than c), but the acceleration in the original frame drops off no matter how much force is applied. Additional force just adds more mass, not more velocity. Maybe there is a difference between force and proper force.

So presumably the effect of the gravitational time dilation, relative to that original rest frame, gets greater and greater.

On the other hand, if the force is kept constant then the acceleration will gradually decrease as the spaceship approaches light speed, relative to its original rest frame. So the gravitational time dilation (relative to that frame) will remain constant but will last for much more of the journey.

My example didn't have continuous acceleration like that, and I don't see where gravity fits in to it all. You described it being used at the transition time (where I had posited accelerations waaaay in excess of 1G) but I don't see where it might be considered elsewhere in the trip. So I don't see how 'gravitational time dilation' fits into this scenario.

[Steve3007]

Gravitational time dilation fits in because a basic principle of General Relativity is that acceleration and gravity are equivalent (the "equivalence principle").

[Halc]

So the acceleration will remain at 1g for as long as the mass being thrown out of the back (as measured in the frame of the moving rocket) remains constant.

This assumes the ship mass remains constant, which it doesn't, because it is obviously discarding it out of its nether regions. This is why rockets can't go really fast, and all the fastest man-made objects broke their records typically from throwing the objects off a cliff.

[Tamminen]

Simultaneity within a reference frame is symmetric, but simultaneity between frames is asymmetric.

It is quite symmetric even between frames

I mean relative to an event. If the event is my arriving at U in our example, the clock at D's starting point reads 10, as also a sychronized clock at U, for an observer in frame D, and the clock in T1 reads 5 for the same observer. But for an observer in frame T1, as his clock reads 5, the clock at D's starting point reads 2.5 as we said before. So the relation is 10 -> 5 in one direction and 5 -> 2.5 in another direction, which means that it is not symmetric in this sense. This is what I mean when I say that simultaneity between frames is asymmetric. Perhaps I should be more precise in my expressions.

[Steve3007]

The force had better remain the same or else it would get quite uncomfortable (fatal) to the humans.

Since force is rate of change of momentum (F = d(mv)/dt) and momentum is mass X velocity (mv) the force, from the point of view of the original reference frame, increases as the mass increases (WRT that frame). But WRT the rocket's current rest frame the force stays the same because the mass, as measured in that frame, remains the same. so the people don't get crushed.