Can't Tell If you are shrinking?

[TheScienceOrca]

I don't understand how this makes any difference. Suppose we add a measuring tape to my description in my previous post. In phase 1 (Bob is on the planet), the cubes all move in a certain way relative to the measuring tape, and Bob records all those motions very precisely. Then in phase 2 (Bob is flying at a velocity close to $c$ away from the planet), if Bob starts the cubes out the same way he did in phase 1 (which seems to be what you intend--after all, if he starts the cubes moving differently, of course they're going to move differently), then all their motions relative to the measuring tape in phase 2 will be *exactly* the same as in phase 1. So Bob won't be able to tell the difference.

Read my post, yes you are right, but the time it took the shrink changes based on v1.

time from v1 to v2 is shorter if v1 is closer to v2 as nothing can accelerate instantly (teleport)

 

[Orodruin]

As you have been told repeatedly, your experiment will not work the way you believe. Yet you continue to argue rather than trying to understand which part of the theory you do not understand. This is not scientific discussion, this is us explaining things to someone who does not want to listen and that is tedious and frustrating. I am done with this thread.

 

[Oldtheorist]

I'm going to reread this when it's not 1 am local but I believe you are not considering time dilation along with length contraction. Time dilation will allow you who are traveling at .9c to launch a projectile that's also traveling at .9c RELATIVE TO YOURSELF and in the same direction of travel, but the projectile itself will never exceed c - not when measured by you or by an outside observer.

 

[Chronos]

I'm not sure what you're saying here. The reduction in distance between the two rockets is happening as though one of them is traveling at 1.8c relative to the other, yes? Why would an external observer not conclude that?

The clocks will disagree.

 

[Orodruin]

The clocks will disagree.

Careful here, he is arguing that an external observer will see the distance between the objects shrink at speed 1.8c, which is true in the frame of the external observer. In this frame there is only one time, the time coordinate of the frame. Then there is a terminology issue in calling this "relative velocity" since this is typically reserved for how fast something moves relative to a given observer.

 

[TheScienceOrca]

I'm going to reread this when it's not 1 am local but I believe you are not considering time dilation along with length contraction. Time dilation will allow you who are traveling at .9c to launch a projectile that's also traveling at .9c RELATIVE TO YOURSELF and in the same direction of travel, but the projectile itself will never exceed c - not when measured by you or by an outside observer.

Right because the time of the universe relative to the projectile is now traveling VERY slowly.

Will this result in even more shrinkage of the projectile or since it is never over c, it won't shrink more.

 

[pbuk]

I had chosen not to join in on this thread TheScienceOrca: you don't explain what you are thinking very well, you don't seem to listen effectively to what other people are saying and you sometimes come over as arrogant.

I am very sorry I have aspergers

I now understand why your posts come over this way: don't worry, lots of other people on this forum also have Asperger's syndrome and nearly everyone in Physics has experience (admittedly not always successful) of working with people with AS.

So I'll see if I can help...

1) No object can travel faster than the speed of light regardless of frame of reference

Let's get away from the words "frame of reference". I don't think you have the maths skills to work with this concept and it is not necessary to an understanding of Special Relativity.

So I'll restate this as: Let c be the speed of light in a vacuum. No object or information can be measured by any observer to be traveling faster than c, and no object with mass can be measured by any observer to be traveling at c.

objects simply can not go faster than the speed of light.

This statement does not make sense: the whole point of SR is that measurements of velocity are relative to the observer, an object does not "simply go" at any speed.

A mass shrinks in the direction of travel relative to the velocity, as the mass approaches c it will continue to shrink in the direction of motion.

This is not correct. Objects moving away from an observer at relativistic speed are seen by that observer to contract in the direction of motion. No change is observed by the object itself.

So two identical objects traveling in the same direction one at speed of .5c and one at a speed of .2c, the object traveling at .5c will be shorter in the direction of motion relative to the object traveling at .2c. Even that object at .2c would be shorter in the direction of motion than an identical object at .1c.

No, the lengths of all the objects remain the same. Each object will measure its own length to be the same. Each object will measure the other objects' lengths as different, the amount of the difference and whether it is an apparent enlargement or contraction depends on how they take the measurement and what they measure their relative velocities to be.

In the center of his experiment room he has a device which can shoot 1cm blocks at close to c in 4 directions 3 all perpendicular to each other and the fourth a vector sum of the 3 other. These blocks are shot across the ships floor.

That doesn't work: if the four blocks are traveling in the directions you describe then a maximum of two of them can be shot across the floor.

I suggest that instead Bob wants to fire 4 blocks at the same speed (measured by Bob) across the (frictionless) floor at 0, 90, 180 and 270 degrees to the direction of travel away from Alice.

Now this is important, the relative shrinkage is also relative to the relative velocity between the block floor and the block projectile.

No, the relative shrinkage (measured by Bob) is ONLY relative to the relative velocity between the floor (and therefore Bob) and the blocks. As Bob fired all the blocks away from him at the same speed from his point of view he measures their shrinkage to be the same.

However because Alice measures the blocks to be traveling at different speeds, she measures their shrinkage as correspondingly different.

This way he can find the relative shrinking at this exact velocity, he will not know what that velocity is, but it will come into use later. The only way he can know which speed he is at is if when the block shot does not change in size relative to the blocks on the floor he will know he is traveling at c.

If Bob's ship is now fires his rockets and moves from the planet he will be able to prove that he is moving by doing the following.


Running the same experiment again.

If the 1cm block shrinks to the shortest shrinkage (speed of light) faster than earlier, bob will know he is moving faster than before, this is why;

Lets say the time to the shortest shrinkage the projectile could become on the planet was 1 second. This is because it took 1 second for the 1cm block to get to the speed of light.

Two problems with this:
1. The block can never get to the speed of light.
2. Bob fires the blocks at a certain speed v across a frictionless surface: their velocity does not change over time.

 

[pbuk]

You should also be aware that there are two different apparent length contraction effects. I won't go into details, I'll just mention that

1. If you measure length by recording the time the two ends of a relativistic object are observed to pass by stationary synchronized clocks you will always measure a length contraction (the Lorentz contraction).
2. If you measure length by taking a photograph of a relativistic object passing a stationary ruler you will measure a contraction if the object is traveling away from you, an expansion if the object is traveling towards you, and a weird distortion if the object is traveling past you. This is the Penrose-Terrell effect (it also goes by other names).

Because of this we need to agree on what we mean when we say "observe", "measure" or "see" the length of an object moving at relativistic velocity.

 

[TheScienceOrca]

Thank you for the in depth reply, I feel like I have a better grasp on relativity. Due to the time dilation objects can move at faster than the speed of light relative to other objects as the time simply changes for the universe relative to the object?

Let me know if this is right, and thanks for the explanation, that is exactly what I wanted to hear and now I know why I am wrong.

 

[TheScienceOrca]

You should also be aware that there are two different apparent length contraction effects. I won't go into details, I'll just mention that

1. If you measure length by recording the time the two ends of a relativistic object are observed to pass by stationary synchronized clocks you will always measure a length contraction (the Lorentz contraction).
2. If you measure length by taking a photograph of a relativistic object passing a stationary ruler you will measure a contraction if the object is traveling away from you, an expansion if the object is traveling towards you, and a weird distortion if the object is traveling past you. This is the Penrose-Terrell effect (it also goes by other names).

Because of this we need to agree on what we mean when we say "observe", "measure" or "see" the length of an object moving at relativistic velocity.

Ah I see, well if it is true to the question I asked above there would be no need to rewrite the concept as it would be wrong.

The whole concept was based off objects not being able to go as fast relative to bob as they would when he is not moving as they would be limited by the speed of light.

 

[pbuk]

the time simply changes for the universe relative to the object?

I think I would rather say that "time is different for other observers in the universe which are moving relative to the object", but you seem to be on the right track now.

There must be a good non-mathematical primer for SR on the web (with animations of passing trains etc), can anyone recommend one to help TheScienceOrca reinforce correct understanding?

 

[phinds]

Careful here, he is arguing that an external observer will see the distance between the objects shrink at speed 1.8c, which is true in the frame of the external observer. In this frame there is only one time, the time coordinate of the frame. Then there is a terminology issue in calling this "relative velocity" since this is typically reserved for how fast something moves relative to a given observer.

Exactly. I'm speaking of "closing speed" which does not need any transformations to bring it down below c. It IS greater than c in the frame of an external observer. This has been discussed many times on this forum and I was surprised to see Chronos apparently contradicting it.

 

[PeterDonis]

Ok the conditions are as follows

MrAnchovy's comments on these look useful to me. I don't really have anything to add here; I think it's more fruitful just to directly comment on your experiment.

He fires the machine and with his extremely accurate machines measures the size of the projected block relative to the blocks that are on the floor.

Now this is important, the relative shrinkage is also relative to the relative velocity between the block floor and the block projectile.

The faster the projectile is relative to the floor the greater the relative shrinkage.

Ok so far.

Running the same experiment again.

If the 1cm block shrinks to the shortest shrinkage (speed of light) faster than earlier

Which won't happen; all 4 blocks will behave exactly the same, according to all Bob's measurements, as they did in the baseline run of the experiment. This is what you don't seem to get; your proposed experiment will not give the results you claim it will.

 

[PeterDonis]

Read my post, yes you are right, but the time it took the shrink changes based on v1.

No, it doesn't.

time from v1 to v2 is shorter if v1 is closer to v2 as nothing can accelerate instantly (teleport)

This is correct as a matter of logic, but it is based on an incorrect premise: you are assuming that "v1 is closer to v2" during the second run of the experiment. It isn't; the "distance" between v1 and v2 is the same during both runs, because of the way velocities transform between frames in relativity. This is what people have been trying to explain to you.

 

[PeterDonis]

The whole concept was based off objects not being able to go as fast relative to bob as they would when he is not moving as they would be limited by the speed of light.

And this is not correct; the speed of light does not "limit" velocities in this sense. You are assuming that velocities add linearly; in SR, they don't. If you want something related to velocity that does add linearly, you need to use the "velocity parameter" (also called the "rapidity") $\alpha$, which is defined by $v = \tanh \alpha$, where $v$ is the ordinary velocity (note that I'm using units in which $c = 1$) and $\tanh$ is the hyperbolic tangent function; $\alpha$ is what adds linearly the way ordinary velocities do in non-relativistic mechanics.

If you look up the $\tanh$ function, you will see that as $v \rightarrow 1$, $\alpha \rightarrow \infty$. So the speed of light corresponds to an infinite velocity parameter $\alpha$, and any finite number is the same "distance" from infinity, so the speed of light does not limit velocities in relativity the way you are thinking it does.

 

[TheScienceOrca]

I think I still had a Newtonian view of everything so it really messed me up in relativity, but now I think I can get it!

 

[TheScienceOrca]

I didn't know that the shrinkage was based on relativity and that objects can move in reference to you at speeds over c even though they are not traveling over c.

It makes complete sense now.

 

[PeterDonis]

objects can move in reference to you at speeds over c even though they are not traveling over c.

This may be a misunderstanding on your part, or it may just be bad phrasing. Normally, an object moving "in reference to you" means the velocity that you measure the object to have, relative to you. That can never be greater than $c$ (and it can only be $c$ for light or other massless objects, not for any object with nonzero invariant mass).

What can be greater than $c$ is the "closure speed" between two objects, both moving relative to you, as measured by you. But if, for example, object A is moving at $0.8c$ relative to you, and object B is moving at $0.8c$ relative to you in the opposite direction, then, while the closure speed between A and B, relative to you, is $1.6c$, the speed of B relative to A (meaning, the speed A measures B to have in reference to A), or of A relative to B (meaning, the speed B measures A to have relative to B), is *not*; it is only $0.976c$ (approximately).

Once again, this is because of the way velocities add in relativity. One way of seeing it is to use the fact, which I mentioned in a previous post, that $v = \tanh \alpha$, and that $\alpha$ adds linearly. So if we have $\alpha_A$ and $\alpha_B$ as the velocity parameters of A and B, relative to you, then the velocity parameter of A relative to B (or B relative to A) is $\alpha_{AB} = \alpha_A + \alpha_B$, and using the rule for adding hyperbolic tangents gives us (again, I'm using units where $c = 1$):

$$
v_{AB} = \frac{v_A + v_B}{1 + v_A v_B}
$$

If you plug in $v_A = v_B = 0.8$, you will find $v_{AB} \approx 0.976$, as I said.

 

[TheScienceOrca]

What would Alice see standing on a planet if a ship going .9c flies by parallel to her and shoots a bullet in the same direction of travel .9c.

You earlier stated that the cube would fly relative to the fast moving ship and not be limited by the speed of light.

So would Alice now see a bullet going 1.8c and a rocket going .9c?

I am uncertain on a few question still sorry

 

[Nugatory]

What would Alice see standing on a planet if a ship going .9c relative to Alice flies by parallel to her and shoots a bullet with a muzzle velocity of .9c relative to the ship in the same direction of travel .9c.

(I've tried to guess what you're asking and have clarified accordingly. You really have to get in the habit of never stating a velocity or speed without also considering what it is relative to)

If I have guessed correctly what you're asking, Alice will see the ship moving at a speed of .9c relative to her, and will see the bullet moving at a speed of .994c relative to her.

I got this result from the relativistic velocity addition formula: $w=\frac{u+v}{1+uv}$ (and measuring distance in light-seconds and time in seconds so that $c=1$ and don't need to clutter things up with factors of $c$ and $c^2$)

 

[TheScienceOrca]

(I've tried to guess what you're asking and have clarified accordingly. You really have to get in the habit of never stating a velocity or speed without also considering what it is relative to)

If I have guessed correctly what you're asking, Alice will see the ship moving at a speed of .9c relative to her, and will see the bullet moving at a speed of .994c relative to her.

I got this result from the relativistic velocity addition formula: $w=\frac{u+v}{1+uv}$ (and measuring distance in light-seconds and time in seconds so that $c=1$ and don't need to clutter things up with factors of $c$ and $c^2$)

Thanks, now Let's just imagine a ship traveling at .9c relative to Alice. On that ship a bullet is shot in the same direction of motion at .9c. Let's say the bullet ship stop exactly 1 second relative to Alice.

Would the distance from the bullet to Alice be .994 LS or 1.8 LS If we stop all of the objects 1 second relative to the objects then the bullet would be .994 LS or 1.8 LS away from alice?

 

[pbuk]

(I've tried to guess what you're asking and have clarified accordingly. You really have to get in the habit of never stating a velocity or speed without also considering what it is relative to)

If I have guessed correctly what you're asking, Alice will see the ship moving at a speed of .9c relative to her, and will see the bullet moving at a speed of .994c relative to her.

I got this result from the relativistic velocity addition formula: $w=\frac{u+v}{1+uv}$ (and measuring distance in light-seconds and time in seconds so that $c=1$ and don't need to clutter things up with factors of $c$ and $c^2$)

Alice will see the ship moving at a speed of .9c relative to her, and will see the bullet moving at a speed of .994c relative to her. The ship will appear shortened along the line of flight, and the bullet will appear shortened even more. If Alice can see a clock on the ship it will appear to move slowly, and the spinning bullet will appear to spin even more slowly. However Alice won't actually see the ship at all because any light from the ship will be shifted into the near infrared by the Doppler effect (approximately 4x wavelength distortion), and any light from the bullet even more so (approx 20x wavelength distortion).

Bob who is on the ship will see...

You can't possibly get your head around all this stuff working it out from thinking about the fact that the speed of light in a vacuum is invariant - even Einstein didn't do that, he started with the maths which showed him that some weird things would be observed and his non-mathematical thought experiments to make sense of it came later.

Instead you need to find a book or online resource or tutorial or something (can't anyone recommend one?), learn this stuff and then if you need to come back and ask questions.

 

[Nugatory]

Thanks, now Let's just imagine a ship traveling at .9c relative to Alice. On that ship a bullet is shot in the same direction of motion at .9c. Let's say the bullet ship stop exactly 1 second relative to Alice.

Would the distance from the bullet to Alice be .994 LS or 1.8 LS


If we stop all of the objects 1 second relative to the objects then the bullet would be .994 LS or 1.8 LS away from alice?

In one second of Alice's time the bullet moving at .9c relative to Alice will move a distance of .9 light-seconds according to Alice and therefore be .9 light-seconds away according to Alice.

You on the planet will see the bullet approaching at .994c relative to you, and Alice approaching at .9c relative to you, so the bullet is only gaining on Alice by .094c - for every second of your time that passes after the gun is fired, the bullet will be .094 light-seconds closer to you than Alice is.

Thanks to time dilation, Alice's .436 seconds is one second of your time and her one second is 2.3 seconds of your time. When Alice says that the bullet is .9 light-seconds away from her one second after the gun is fired, you will say that the bullet is $2.3\times{.094}=.22$ light-seconds away from her 2.3 seconds after the gun is fired.

 

[TheScienceOrca]

In one second of Alice's time the bullet moving at .9c relative to Alice will move a distance of .9 light-seconds according to Alice and therefore be .9 light-seconds away according to Alice.

You on the planet will see the bullet approaching at .994c relative to you, and Alice approaching at .9c relative to you, so the bullet is only gaining on Alice by .094c - for every second of your time that passes after the gun is fired, the bullet will be .094 light-seconds closer to you than Alice is.

Thanks to time dilation, Alice's .436 seconds is one second of your time and her one second is 2.3 seconds of your time. When Alice says that the bullet is .9 light-seconds away from her one second after the gun is fired, you will say that the bullet is $2.3\times{.094}=.22$ light-seconds away from her 2.3 seconds after the gun is fired.

I think I am misscommunicated I am not in this scenario, I will try to restate it.

Imagine looking at an x and y grid.

Lets say Alice is floating in the middle of the universe at 0,0

a ship at 5,0 is moving up the y-axis at .9c parallel to alice (x will always stay 5).

Their z is equal.

If this rocket also shoots a bullet in the same direction (up the y-axis at .9c relative to itself).

You stated the bullet would be moving .994c relative to alice and the rocket still .9c.


If the ship fires its bullet right when it's at (5,0).

In 1 second relative to Alice, according to your statements the bullet would be .994c LS away and the rocket .9c LS away. Which means the bullet is only .094c away from the rocket even though the bullet is traveling .9c relative to the rocket.

This is because as 1 second has passed for Alice, but not a full second for the bullet right?




1 second relative to the rocket what would the coordinates be from 5,0 start point?

1 second relative to the bullet what would the coordinates be form 5,0 start point?

I am slowly putting this together, I just don't understand if light travels at the speed of light shouldn't it be instant because time is traveling infinitely slow around it?

 

[PeterDonis]

if light travels at the speed of light shouldn't it be instant because time is traveling infinitely slow around it?

No. Light travels at the speed of light in any frame; that is, it travels at the speed of light relative to any observer (since an observer must have nonzero rest mass and so can't move the way light does). The speed of light is not instantaneous.

 

[phinds]

TheScienceOrca, since you have persistently had some difficulty in understanding what you are being told, let me expand slightly on what Peter has said.

Let's take an observer at, say, the origin of his own grid system. In his grid system, that is, his frame of reference, he looks out and sees numerous spaceships traveling at all different directions relative to him and at all different speeds relative to him. They are all also traveling at different speeds relative to each other and they are all different distances from our observer.

Now, our observer sets of a "light bomb" that shoots light off in all directions at once.

The light will reach all of the above mentioned spaceships at different times.

Our observer AND ALL OF THE SPACESHIPS will see the light traveling at c when it reaches them.

It will be blue-shifted for some of them and red-shifted for some of them but that is irrelevant to speed of the beam of light that reaches them. EVERYONE sees the light as traveling at c.

 

[PeterDonis]

Lets say Alice is floating in the middle of the universe at 0,0

a ship at 5,0 is moving up the y-axis at .9c parallel to alice (x will always stay 5).

The x dimension is superfluous here, and it will make the math easier if we leave it out and only deal with the $t$ and $y$ coordinates. That's what I'll do below; everything I do will be valid regardless of what $x$ coordinates (or $z$ coordinates, for that matter) we assign to Alice and the ship and the bullet, as long as they are all constant.

If this rocket also shoots a bullet in the same direction (up the y-axis at .9c relative to itself).

You stated the bullet would be moving .994c relative to alice and the rocket still .9c.

Yes. (More accurately, 0.9945c, which is the accuracy I'll use in calculations below.)

In 1 second relative to Alice, according to your statements the bullet would be .994c LS away and the rocket .9c LS away.

Yes.

Which means the bullet is only .094c away from the rocket

Relative to Alice; *not* relative to the rocket. Distances get transformed when you change frames, just like velocities do. Also there is relativity of simultaneity to consider. See below.

even though the bullet is traveling .9c relative to the rocket.

But not relative to Alice. You have to be very careful not to switch frames in mid-stream, so to speak, which is what you did in the sentence I just quoted (in two parts so you can see exactly where you switched--in between the two parts I quoted).

This is because as 1 second has passed for Alice, but not a full second for the bullet right?

No, it's more than that. Let's work out the coordinates that you asked for.

We have the following events, given with their coordinates in Alice's frame:

Event A: Alice and the ship start out co-located at (0, 0), and the ship fires the bullet at the same instant. The ship moves at 0.9c relative to Alice, and the bullet moves at 0.9945c relative to Alice.

Event B: The ship is located at (1, 0.9) after 1 second relative to Alice. (Note that we're using coordinates in which time is in seconds and distance is in light-seconds.)

Event C: The bullet is located at (1, 0.9945) after 1 second relative to Alice.

Now what we want are the coordinates of events B and C relative to the ship (note that event A has the same coordinates relative to the ship, since it's the origin of both frames). This is easily obtained via the Lorentz transformation; if $t, y$ are the coordinates relative to Alice, and $t', y'$ are the coordinates relative to the ship, then we have:

$$
t' = \gamma \left( t - \frac{v y}{c^2} \right)
$$
$$
y' = \gamma \left( y - v t \right)
$$

where $\gamma = 1 / \sqrt{1 - v^2 / c^2}$. For $v = 0.9c$, we have $\gamma = 2.294$, and this gives the following event coordinates relative to the ship:

Event B: (0.4359, 0) (note that we expect $y' = 0$ here because the ship is at rest at $y' = 0$ in its own frame)

Event C: (0.2408, 0.2168)

Note carefully several things:

(1) In the ship's frame, event B happens *less* than 1 second after event A. This is an example of time dilation: only 0.4359 seconds elapse on the ship between two events that are 1 second apart for Alice.

(2) In the ship's frame, event C happens *before* event B (whereas in Alice's frame, they happen at the same time). This is an example of relativity of simultaneity: events that are simultaneous in one frame are not simultaneous in another frame. But it also means that, if we want to know how far away the bullet is from the ship at event B, in the ship's frame, event C is the *wrong* event to look at. Instead, we need to look at:

Event D: (0.4359, 0.3923) is the event where the bullet is, in the ship's frame, when the ship is at event B (note that the time of this event, in the ship's frame, is the same as the time of event B). The bullet is 0.3923 light seconds away from the ship, in the ship's frame, when 0.4359 seconds have elapsed, because the bullet is moving at 0.9c relative to the ship. (We can verify this, by the way, by taking the ratio of $y'$ to $t'$ for event C; $0.2168 / 0.2408 = 0.9$, as expected.)

And just for completeness, we can transform the coordinates of event D back to Alice's frame, simply by inverting the sign of $v$ in the equations above; this gives

Event D: (1.81, 1.8) in Alice's frame. Notice that the $y$ coordinate here is 1.8 = 0.9 + 0.9; this is not a coincidence. It has to be that way because of how all the math combines: the way velocities add, and the way coordinates transform. It might be instructive for you to work out, from the various equations already given, how this comes about.

 

[Simon Bridge]

Recap:

A ship moving at 0.9c relative to Alice fires a bullet in the direction of it's motion at 0.9c relative to the ship.
(The bit in italics is what got left out of the original statement and makes all the difference.)

After one second by Alice's clock, Alice acquires a "snapshot" of the situation.
There is no need to stop the ship and bullet - Alice just wants to look at the the two at an instant of time.

After 1s, Alice finds that the ship has traveled a distance of 0.9 light-seconds from before, and the bullet has traveled a distance of 0.994 light-seconds. The distance between the ship and the bullet, as measured by Alice, is 0.094 light-seconds.
Now: Alice could naively deduce that the bullet is traveling 0.094c with respect to the ship but it is more accurate to say that the separation between the ship and the bullet increases at the rate of 0.094c.

The point of this example was to discuss how things could travel FTL with respect to each other without violating special relativity.
This is a common but somewhat misleading way to describe the effect. To see what I mean, we need to look at another situation:

This time, two ships pass each other, and Alice, at t=0, but traveling in opposite directions at 0.9c (wrt Alice).
At t=1s, the first ship has gone 0.9 light-seconds in one direction while the second ship has gone 0.9 light-seconds in the opposite direction.
This means that, in one second, the ships are 1.8 light-seconds apart.
Now: Alice could naively deduce that the second ship is traveling 1.8c with respect to the 1st ship but it is more accurate to say that the separation between the ships increases at the rate of 1.8c.

All this sort of thing is why we have to be careful with our descriptions when relativity is involved.

A similar issue comes up in post #1, where there was an unspoken assumption that absolute velocity can be determined... I'm not sure that idea has quite been shaken but, even so, some future soul may google here...

There is no experiment you can do to determine your constant velocity - but it is not because time dilation and length contraction conspire to prevent it. Rather it is the other way around: time dilation and length contraction are the consequence of there being no experiment to determine you own constant velocity. This means that the concept of velocity is meaningless without also specifying what it is relative to. This is an idea that can take some students quite a long time to wrap their heads around.

It is similar to how an object will have a different height depending on how far away the ruler it - this effect is called "perspective". We define the "proper height" to be that measured by a ruler that is right next to the object ... and we just go around calling it "height". Special relativity shows us there is also a perspective-like effect at different speeds. To get the proper height of an object, we now need to say the rule has to be stationary with respect to the object as well as being right next to it.

Relativity and perspective are both consequences of the laws of geometry that Nature happens to use, it's just that we are not used to needing to use all of them in one go and our everyday language is too vague to cope.