# Is Distance/space Lorentz contracted?

1. Sep 3, 2009

### Austin0

Hi A simple thought experiment:

A line of inertial space stations at 10 ls intervals stretching indefinitely into the distance.
Two spaceships A and B located at adjacent stations.
For simplicity dont consider proper acceleration, just assume identical ships and propulsion sytems and mechanically identical fuel feed etc. In this question it is only important that the accelerations be exactly equal.
Disregard contraction of the ships themselves, as relative to the spatial separation this is irrelevant, vanishingly small. SO distance between will be measured from the center of the ships.
Simultaneity of blastoff is not an issue as they are both starting out in the same inertial frame with conventionally synched clocks all around.

So the question is simple:
1) In the frame of the stations does the distance between the ships remain the same or does it contract???

2) In the frame of the ships does the distance remain the same or does it expand???

2. Sep 3, 2009

### Mr Whippy

1) As the ships accelerate closer to the speed of light, they shrink in length as does the distance between them.

2) No one on board notices this, as since time is slowing down it takes longer for the light to move from one place to another along the direction of travel. So to the crew the ship appears to remain undistorted.

Finally if they reached the speed of light, all the ships and the distance between them would have zero length, fortunately that speeds prohibited.

3. Sep 3, 2009

### Staff: Mentor

This is the same setup as in Bell's infamous Spaceship Paradox. The way the ships are accelerated really does matter. Let's assume that the accelerations are such that according to the original inertial frame the speeds of the ships are always the same.

Since the ships always have the same speed, the distance between them cannot change.

From the view of a co-moving inertial frame (which constantly changes, since the ships accelerate) the distance between the ships grows larger.

4. Sep 3, 2009

### Austin0

Hi Doc As I understood the Bell scenario the question related to the contraction of the ships themselves and whether that contraction would increase the separation between them.

Is there some explanation for the assumption that in the ship frame the distance would increase given that; in that frame also, the expectation would be ; equal acceleration , equal instantaneous velocity , equivalent distance covered per time???

In the Born rigidity question there is some rationale for an assumption of unequal acceleration relative to physical location, but in this case I dont really see it.

Thanks

5. Sep 3, 2009

### Staff: Mentor

That's not my understanding. What matters in that scenario is the distance between the ships.

I would not agree with those expectations. Since the ship's view of simultaneity will be different than that of the station frame, the ships see themselves as not maintaining the same speed at all times.

The way I visualize this is by thinking in terms of small bursts of velocity change. At any given time, let's say that the ships are moving at a speed v with respect to the station. If another burst of speed is given to the ships, those bursts will be (by stipulation) simultaneous in the station frame but at different times in the ship frame.

6. Sep 3, 2009

### JesseM

What do you mean by "the ship frame"? Unlike with inertial frames, there is no unique way to construct a coordinate system for an accelerating object. Also note that if the ship is accelerating at a constant rate from the perspective of some inertial observer, then it is not experiencing constant "proper acceleration" (constant proper acceleration meaning the G-force experienced on board the ship is constant, and the instantaneous rate of acceleration in the ship's own instantaneous inertial frame from one moment to the next is constant...this is also the type of acceleration that would be produced if the ship's thrust was constant in its own instantaneous inertial rest frame from one moment to the next). For ships undergoing constant proper acceleration, the most common choice of coordinate system for them would be Rindler coordinates, where the time coordinate along the ship's worldline matches its proper time, and the Rindler coordinate system's definition of simultaneity and distance always matches that of the ship's instantaneous inertial rest frame (I suppose you could also construct a coordinate system with these properties for a ship which was not undergoing constant proper acceleration). If you had two ships undergoing constant proper acceleration in such a way that their acceleration at any given time was the same from the perspective of a fixed inertial frame, then in each ship's own Rindler frame the other ship would not be at rest; in order for one ship to be at rest in another ship's Rindler frame, the ships must have different accelerations as seen in an inertial frame, and different proper accelerations too, with ships closer to the "Rindler horizon" accelerating more quickly--the two ships would then be undergoing Born rigid acceleration. This page on the Rindler horizon has a diagram and a little discussion:

7. Sep 4, 2009

### Austin0

I meant the ship reference frame in its generic sense with no implication that it was an inertial frame.
I am somewhat familiar with the Rindler system and am studying it more right now.
I am curious about the genesis of this system with its included gravitational dilation.
Did this all originate with Born and the rigidity hypothesis???
Also the Theorem of anisotropic dilation within accerating systems due to relative velocity resulting from length contraction. I have searched the net without result so if you happen to know where ,when and who originated this it would be appreciated.

Regarding this question I tried to keep it as simple as possible.
So I have a couple of questions for you.
Without considering what observers in the station frame might observe, but only regarding the stations themselves as milage markers:
Assume ship A takes off by itself. Accelerating as stipulated ,[constant as regards propulsion mechanism].
At some point in time it reaches ,say, the 100th station from it's origen and notes its own elapsed proper time.
Then ship B does exactly the same procedure from the adjacent station.

Given ideally identical mechanisms and uniform spacing between the stations:

# 1) the proper elapsed times would be the same?

# 2) the times would be different.?

If # 2) then what possible principle of physics or SR would explain this???

If #1) then what possible reason would there be to expect anything different to occur simply because they happened to take off simultaneously as determined within a single inertial frame???
Thanks

8. Sep 4, 2009

### JesseM

I don't understand what you mean by "in its generic sense". For any given non-inertial object, there are an infinity of distinct ways to construct a non-inertial coordinate system where it is at rest, with these coordinate systems disagreeing about things like simultaneity and distances. Again, you can probably get a unique system by adding the stipulation that 1) coordinate time along its worldline matches proper time, and 2) at any point on its worldline, the set of events that are defined to be simultaneous with that point are the same that would be simultaneous with that point in the object's instantaneous inertial frame at that moment, and distances in this plane of simultaneity would match the instantaneous inertial frame as well. I believe the Rindler coordinate system for an object undergoing uniform proper acceleration has both these properties.
What do you mean? There's no true gravitational time dilation in this system, although one can talk about pseudo-gravitational time dilation in a pseudo-gravitational field (see here).
Don't understand this sentence at all. How does relative velocity result from length contraction? Does anisotropic dilation just mean that different members of the "flotilla of spaceships" described in the quote at the end of my last post would experience different time dilation factors, whether we're talking about dilation in an inertial frame (where you can see from the diagram that ships closer to the Rindler horizon have a greater velocity at any given moment) or in the Rindler coordinate system?
What do you mean "exactly the same procedure"? Do you want them to both have the same proper acceleration? Note that in the "flotilla of spaceships", ships at different positions must have different proper accelerations in order to have them keep a constant distance from one another in Rindler coordinates (which is the same as saying they have a constant distance from one another in their instantaneous inertial rest frame at each moment, because of property 2) I mentioned above...this is also the same as saying they are undergoing Born rigid acceleration).
The answer would obviously be yes if they both had the same proper acceleration and started from rest in the frame of the inertial observer, but I think it would be no if we were talking about different members of the flotilla that are at rest relative to one another in Rindler coordinates/are accelerating in a Born rigid way.

9. Sep 4, 2009

### Austin0

That is exactly what I meant by in its generic sense. Without any specific coordinate definitions but merely as a perspective with it being at rest.

I got that from DrGreg in a discussion including Rindler coordinates. I was refering to it as "a non-uniform time metric" and he suggested it would be more conventional if I used "gravitaional time dilation" also specifically stating that SR didnt make a distinction between pseudo-gravitational and true gravitational.
___________________________________________________________________________
Originally Posted by Austin0
Also the Theorem of anisotropic dilation within accerating systems due to relative velocity resulting from length contraction.

I am sorry. I assumed you were familiar with the Theorem. It doesn't specifically relate to
different ships but proposes that within a single ship there would be greater dilation at the back because, as viewed fronm an inertial frame the back must have a greater instantaneous velocity than the front to account for the extra distance travelled due to contraction. I guess Sylas is the one I should ask. He made it sound like this was a well known and established Theorem.

Exactly that. The second ship, as an independant sequence of events, notes the proper time as they initiate acceleration , sits back and counts stations as they go by and then notes the time when they reach the 100th from their starting point.
This is a sequence of events. There is no need for a coordinate system. No need to calculate acceleration rates or instantaneous velocities or relative time dilation or anything.
This is purely a matter of logic and known physics. They have the same acceleration ,same instantaneous thrust , as determined by the identicallity of their mechanisms of propulsion. Whether that acceleration is "proper" or not , is a separate question that is not relevant here.
It appears to me that, by your definition of proper acceleration within the Rindler construct ; if they had equal acceleration as determined by equal mechanism in their own frame [as stipulated], that this would not be "proper" within Rindler where the term has been defined to mean unequal acceleration , with greater acceleration being required at the rear.

So for the moment, if you could consider this question solely within the given parameters , with no reference to how things appear in the station frame or whether the acceleration is proper or not, what does your logic predict would be the outcome of the events???

thanks

10. Sep 4, 2009

### JesseM

But then there can be no definite answer to questions like whether the distance between two accelerating ships remains constant in their own rest frame; it will depend on how you define the rest frame.
There's no meaningful difference between considering a pair of ships undergoing Born rigid acceleration and considering the front and back of a single ship undergoing Born rigid acceleration.
OK, that's a decent conceptual argument for why the back must have greater velocity, but I don't think it's really rigorous and it doesn't sound like a "theorem"--to figure out exactly how the velocity of the front compares with the velocity of the back as seen in an inertial frame, I think you'd have to do a detailed analysis involving the assumption of Born rigid acceleration (and if you don't assume Born rigidity, there's no particular reason to assume the ship will length contract as its velocity increases).
Again, unless you say how the two ships accelerate the question is not well-defined. If you assume they both have the same constant proper acceleration, then sure, they will both find the same proper time elapses between the start and passing the 100th, and the time elapsed in an inertial frame observing them will be the same too. But if you want them to be undergoing Born rigid acceleration so a taut string between the two ships wouldn't break or become slack as they accelerate, then one ship will have a greater proper acceleration than the other, and in this case the proper time will presumably be different.
OK, you didn't specify that before. In this case the times will be the same.
If they have the same instantaneous thrust, then obviously they have the same proper acceleration! Proper acceleration just means the acceleration in their instantaneous inertial rest frame (which also determines the G-forces experienced on board due to acceleration), and if they have the same thrust in their instantaneous inertial rest frame, then they must have the same acceleration in this frame if no other forces are acting on them.
I think you're confusing the concept of "proper acceleration" with "Born rigid acceleration". Every accelerating object has some value for its proper acceleration at a given moment, which again is defined just as the acceleration in its instantaneous inertial rest frame at that moment. But ships (or ends of a single ship) which are at rest in the same Rindler coordinate system are undergoing Born rigid acceleration, which means each ship individually has a constant proper acceleration, but the proper acceleration of the one at the rear is greater than the proper acceleration of the one at the front (the G-force experienced on the one at the rear would be higher than the G-force experienced on the one at the front).
Assuming you mean "whether the acceleration is Born rigid or not", the answer is that it's impossible to answer the question without saying something about the acceleration. But you earlier said both ships used the same thrust--do you understand that this automatically implies that the acceleration is not Born rigid, and that it implies both ships must experience the same proper acceleration? In this case, as I said, the times will be identical for both ships. But this also implies the distance between the ships in their own instantaneous inertial rest frames is changing from one moment to the next, so a string between them which was initially taut would break as in the [URL [Broken] spaceship paradox[/url].

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11. Sep 4, 2009

### Austin0

I think there may be some question regarding this proposition.

This also is uncertain.

___________________________________________________________________________
Originally Posted by Austin0
It appears to me that, by your definition of proper acceleration within the Rindler construct ; if they had equal acceleration as determined by equal mechanism in their own frame [as stipulated], that this would not be "proper" within Rindler where the term has been defined to mean unequal acceleration , with greater acceleration being required at the rear.
_______________________________________________________________________

It appears we have agreed that under the stipulated condition of equal thrust, the times and distances would in fact, be equal????

SO if the ships agree that they have traveled an equal distance in an equal time [as transmitted with EM coms] how does it follow that the distance between the ships in their instantaneous co-moving frame is changing from moment to moment??

If by their frames simultaneity they arrived at spatial locations at the same proper time and the spatial distance between those locations was the same as it was when they were at rest how does it follow that a ruler stretched between them would or could be stretched???

If we consider the situation from the perspective of the station frame there are some logical inferences to be drawn from normal SR principles:

The fact that the ships are accelerating does not negate the effects of average instantaneous velocities. SO by loss of simultaneity we would assume that the trailing ship would reach its 100th station before the leading ship reached its 100th station if the two ships read equal proper times at their arrivals.

Based on this they would conclude that :
1) the distance between the ships was contracted relative to when they were at rest wrt the station frame.
2) the rear ship would neccessarily, have to have had a greater acceleration [or relative velocity] to have traveled a greater distance relative to the lead ship to effect this contraction.
3) the ships clocks were desynchronized relative to the stations clocks.

Do you see a problem with any of this???

They question from the ship perspective is somewhat more problematic.
They would agree that the stations clocks were desynchronized.
It would appear that the only way the distances between the stations could appear contracted is as a function of time. Or rather as a function of dilated time [relative to their own clocks when they were at rest in the station frame].
As for the stations themselves, they could appear contracted due to loss of simultaneity.

So does this make sense???
Thanks

12. Sep 4, 2009

### JesseM

Why do you say that? If you place no conditions on the way the front and back of the ship accelerate, then anything is possible with regards to how the distance between front and back behaves. For example, if both front and back have the same acceleration at each moment in some inertial frame, then the ship's length will stay constant in that frame, provided the ship is made of elastic material and can physically stretch in its own instantaneous rest frame without breaking apart. You could even accelerate the front more than the back, so the ship would elongate as seen by an inertial observer.
Again, why? The same constant proper acceleration implies that [coordinate distance traveled] as a function of [coordinate time since beginning to accelerate from rest] as seen in an inertial frame will be the same--do you disagree?
The phrase "proper acceleration within the Rindler construct" appears to be meaningless, although I suspect that you are just trying to talk about Born rigid acceleration. Again, any object whatsoever always has a well-defined "proper acceleration" regardless of how it is accelerating or what coordinate system you choose to use (proper acceleration is by definition a coordinate-invariant quantity, just like proper time and proper distance). If you are given the coordinate position of a function of time for some pair of objects in Rindler coordinates, you can calculate those object's proper acceleration, even if they not at rest in these coordinates and are not accelerating in a Born rigid way (and if they are accelerating in a Born rigid way and are at rest in Rindler coordinates, then note that their proper acceleration is different from their coordinate acceleration in Rindler coordinates, which is naturally zero since they are at rest in this system).
Yes, as seen in the inertial frame where the stations are at rest (and the proper times will be equal too).
They only agree that they each take the same proper time to travel between 100 stations. But if you define an accelerating coordinate system for each ship which has the properties I mentioned earlier [namely 1) coordinate time along its worldline matches proper time, and 2) at any point on its worldline, the set of events that are defined to be simultaneous with that point are the same that would be simultaneous with that point in the object's instantaneous inertial frame at that moment, and distances in this plane of simultaneity would match the instantaneous inertial frame as well], then it is not going to be true that both ships took the same coordinate time to travel past 100 stations (suppose both ships start accelerating at the same moment in their initial rest frame...then although in this same inertial frame they'll both pass 100 stations at the same moment, it will also be true when the front ship reaches the 100th station, his plane of simultaneity for his co-moving frame is tilted relative to this first frame, so according to his definition of simultaneity the rear ship has not yet passed 100 stations). It's also not necessarily going to be true that they traveled the same coordinate distance (since the distance between stations is continually changing in this type of coordinate system).
Huh? Reaching the spatial locations at the same proper time has nothing to do with reaching them simultaneously in their co-moving instantaneous rest frame. If the front ship passes 100 stations at a proper time of 5000 seconds according to his own clock, then he'll certainly agree that the rear ship passes 100 stations at a time of 5000 seconds according to its own clock, but he'll say that in his own co-moving instantaneous rest frame at the moment he passes the 100th station, the rear ship's clock has not yet reached 5000 seconds, maybe it only reads 1000 seconds at that moment or something.
Only in the inertial frame where the stations were at rest, not necessarily in an accelerating coordinate system of the type I described above.
No idea what you mean by that.
Nonsense, if they both start accelerating at the same moment in the inertial frame, and they both use the same constant proper acceleration, then the situation is totally symmetrical and they will both have the same function for displacement as a function of time in this frame (the function for displacement as a function of time as seen in a fixed inertial frame for an object undergoing constant proper acceleration is given on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]). If you think otherwise, then you probably have some confused notion of "loss of simultaneity"--I don't really understand what you mean by that phrase in this context, since here we are only talking about what is seen in the inertial station frame, not about what is going on in other frames with different definitions of simultaneity.

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13. Sep 4, 2009

### DrGreg

Austin0, maybe the attached spacetime diagrams may help.

The top diagram shows what happens in your scenario. The red and blue lines represent two spaceships. The dots along each line are ticks at one-second intervals (say) of each ship's own clock. As both ships accelerate in the same way, for each ship after $\tau$ ticks have passed the velocity of each must be the same relative to the initial inertial frame. The distance between the ships as measured in the initial frame remains constant; it's the horizontal separation in the diagram. But in each ship's own frame (if we assume for the sake of argument that we use co-moving inertial observers to define the frame) the separation must increase, so that the Lorentz-contracted distance in the initial frame can be constant. The grey lines in the diagram are the lines of simultaneity for each of the ships.

For contrast, the bottom diagram shows two ships accelerating under Born-rigid acceleration, so that the distance between ships is constant in the ship's own Rindler frame, and the back ship is accelerating more than the front ship. The distance between the ships contracts in the original inertial frame.

(For what it's worth, the red and blue lines aren't just sketched, they're plotted accurately using Excel.)

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14. Sep 7, 2009

### Austin0

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Originally Posted by Austin0
I think there may be some question regarding this proposition.

I say that because my operative assumption is:
That the Lorentz math, the gamma function in it's various applications ,is a valid description of the physics of motion as it occurs in the real world.
That, along with the constancy of c , it is intrinsically frame invariant. Not as a result of any coordinate system or convention but as an inherant property of the physics of that real world.
You can create any coordinate system and clock conventions you care to, but length contraction and clock desynchronization are still going to occur witnin frames at relative velocities. No matter how you decide to set your clocks, you are not going to be able to measure the actual anisotropic relative speed of light.
The Born hypothesis seems to question this. It proposes that length contraction is not an inherant, inevitable consequence of velocity but rather, is an essentially physical phenomenon that requires specific, carefully contrived, application of force to take effect.
That it is somehow possible to accelerate a frame or an object and avoid the effects of relative velocity.

No time dilation???

Originally Posted by Austin0

The fact that the ships are accelerating does not negate the effects of average instantaneous velocities.

This is what I mean. You are not considering the Lorentz effects that would result from the relative velocity during the course of acceleration.

Originally Posted by Austin0
It appears we have agreed that under the stipulated condition of equal thrust, the times and distances would in fact, be equal????

No I was talking about in the ships.

Originally Posted by Austin0
SO if the ships agree that they have traveled an equal distance in an equal time [as transmitted with EM coms] how does it follow that the distance between the ships in their instantaneous co-moving frame is changing from moment to moment??

#1 What is the significance of his simultaneity wrt the initial frame??? Both ships are now traveling at relative velocity wrt that frame and all that is important is the relationship between the two ships at this point in time.
And there is no question of the simultaneity of the takeoff's so what meaning could it have that the ship is no longer simultaneous wrt the initial frame and any assumption derived through that regarding their takeoffs or their current simultaneity???

#2 Of course the distance is dynamically changing during the course of acceleration but what reason is there to assume that it is not changing equally for both ships and what is the significance of the transient changes regarding the total distance covered at a single moment in spacetime [when they arrive at 100]

Originally Posted by Austin0
If by their frames simultaneity they arrived at spatial locations at the same proper time

What is the basis for an assumption of loss of synchronization between the two ships???
Or that they dont occupy a single instantaneous co-moving reference frame??
And if the rear ship's clock did read 4000 seconds when the lead ship's read 5000 that would have to mean that the spatial separation had actually increased by a large percentage and the rear ship was far from it's 100th station, do you have any particular explanation for how this could take place???

On the concept of equidistance maintained in the Born fleet:
What is the meaning of a single instantaneous co-moving reference frame within which they maintain their distance???
Is this some kind of magical instantantaneous determination of position at all points in the frame or is there a hypothetical observer at some location within this frame?
Certainly it would appear that from within the ships they could not even agree on how far apart any two were. They would get different radar measurements from A to B than from B to A. And with a dynamic increase in relative time dilation , the distance between them seems like it must also change with time.

Originally Posted by Austin0
If we consider the situation from the perspective of the station frame there are some logical inferences to be drawn from normal SR principles:

Originally Posted by Austin0
SO by loss of simultaneity we would assume that the trailing ship would reach its 100th station before the leading ship reached its 100th station if the two ships read equal proper times at their arrivals.

As I said ,you are disregarding the effects of relative velocity. Just imagine that immediately before hitting stations 100, they turned off their engines. SO when actually colocated with the stations they were totally valid inertial frames.
Wouldnt it be expected then that if their clocks read the same proper time and they arrived simultaneously in their frame, that they could not arrive simultaneously at the stations in the stations frame???
And doesn't it follow that stopping acceleration wouldn't provoke some instantaneous changes in their clocks,, that it must be assumed that this loss of simultaneity wrt the station frame was happening gradually all along the course , resulting from instantaneous relative velocity at every point ??
I guess you could assume that they would remain in synch and that a conventional synchronization procedure would then make them out of synch with the station ??
What is the assumption for the Born fleet? DO they remain in synch except for the unequal dilation or go out of synch through velocity plus the dilation effect???

15. Sep 7, 2009

### Austin0

Hi DrGreg Thanks for the drawings ,,I 've got to get a copy of EXCEL

It appears from the drawing of the equal acceleration system that the lines of simultaneity are parallel at any given time. Would this imply normal simultaneity in the system?
In fact it looks like a picture of an inertial frame that is simply getting faster over time.
The only problem with the picture is that the distance between does not diminish comparably.
On the Born ships I cant place any comprehensible interpretation on the lines of simultaneity at all.
Thanks

16. Sep 7, 2009

### JesseM

The constancy of c is not frame invariant if you allow for non-inertial frames! The speed of light is certainly not always c in non-inertial frames like Rindler coordinates. All the equations derived from the Lorentz transformation--time dilation and length contraction and velocity addition and so forth--are only intended to apply in the system of inertial frames which are related to one another by the Lorentz transformation.

Also, even in inertial frames, the length contraction equation is intended to apply to objects that move in some "rigid" way--if you move an object from one rest frame to another, the length contraction equation only gives you the right answer for its initial length vs. its final length (both as measured in the initial rest frame) if the internal stresses in the object don't change when it's at rest in one frame vs. at rest in the other, as measured by observers in each frame. If the internal stresses do change then there's no reason to expect it to apply--do you imagine the Lorentz contraction equation would still work if you compared a scrunched spring in one frame to the same type of string that had been stretched nearly to its breaking point in another frame?
What is the "actual" anisotropic relative speed? Are you suggesting that only in one frame is the speed of light "really" isotropic, and that other frames that measure it as being anisotropic are just making an error because they are using a "wrong" definition of simultaneity? Perhaps you are an advocate of the Lorentz ether theory?

In any case, clock desynchronization is not something that happens "naturally" when you accelerate a pair of clocks to change their rest frame, it's based on manually resetting the clocks in their new rest frame according to the Einstein synchronization convention.
Length contraction is only intended to apply when the object in one frame appears physically identical to how it looked in the other frame, as measured by observers at rest in each frame--that includes measurements of internal stresses at each point along the object, how far apart the atoms making it up are at each point as measured by observers at rest relative to the object. Again, if you take a spring which is scrunched up in one frame and then accelerate its ends in a way that causes the spring to look stretched in its final rest frame, do you expect the length contraction equation to apply when comparing its initial length in the first frame to its final length after acceleration? If you take some silly putty that's been pushed together into a little ball and then pull on its ends such that it looks like a long snake once you get done accelerating it (as seen by observers in its new rest frame), do you expect the length contraction to apply to its initial vs. final length? If you do, you have just badly misunderstood what the length contraction is meant to do--normally it's just meant to tell you about the length of a single rigid inertial object as measured in two different frames, but if you want to use it to deal with the case of an object accelerated from one frame to another (in order to predict how much the length of the object will change as seen in the initial rest frame), it only works if the object appears physically identical in its new rest frame after the acceleration to how it looked in its original rest frame before the acceleration, including the fact that the internal stresses at each point along its length should remain the same.
You aren't "avoiding the effects of relative velocity", you're physically changing the object itself as seen in its own rest frame. It's still true that if you take a point along an object which is experiencing a stress S as measured by observers at rest relative to it, then if the object is moving relative to you, the distance between the atoms at that point is shrunk relative to the distance between atoms you would see if that point along the object was at rest relative to you and experiencing the same stress S as measured in your frame. But if you accelerate the object in a non-rigid way that changes the physical stresses at different points along the object as measured in its rest frame, it's just absurd to think that the Lorentz contraction equation will relate its length before vs. after the acceleration as measured in your frame. That would be like like thinking that if you took a rolled-up ball of string at rest in your frame, then accelerated it while also unrolling the string, that you should expect the length of the unrolled string after acceleration to be related by the Lorentz transformation to the length before the acceleration (both as measured in your frame).
Of course there is time dilation. Both ship's clocks measure less time between the start point and passing the 100th station than the time elapsed between these events in the station rest frame. But both ship's clocks are slowed down by exactly the same amount at every moment in the station rest frame, since their velocity at every instant is the same in the station rest frame (since they both are undergoing the same proper acceleration, their velocity as a function of time function will be identical--again, just look at the equations on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] page if you don't believe this). So, the proper time for each ship is the same as the other ship, even though it's less than the coordinate time in the station rest frame.
Of course I am, you just don't understand them. Look, multiple people with expertise in relativity have told you you're wrong about this, instead of confidently asserting that they are all wrong and you are right, could you consider approaching this discussion in the spirit of asking questions about things you don't understand rather than making assertions?
And I just said the proper times will be equal, i.e. the times as measured by the ship clocks. I don't know what it means to say the distances will be equal from the perspective of the ships, since you haven't specified the details of what sort of non-inertial frames you want each ship to use.
The significance is that we already know that the two ships will reach the 100th station simultaneously in the initial frame, so this tells us that if we are interested in what happens in the instantaneous co-moving frame at the moment the lead ship passes the 100th station, it must be true that the back ship has not yet passed its own 100th station in this frame. The initial frame is just being used as a tool to help figure out how things would look in the co-moving frame, since the Lorentz transformation tells us how one frame's surface of simultaneity looks tilted when graphed in another frame.
This is a pretty vague question, but as a quick summary, I'd say that although the assumption of equal thrust implies the matched velocities in the inertial station frame, it does not imply matched velocities in a non-inertial frame for each ship whose definition of simultaneity always matches up with that of their own instantaneous co-moving inertial frame. If you have a row of ships which all take off simultaneously in the station frame and all experience the same constant thrust, then in each ship's own non-inertial frame, it will see that ships farther and farther behind him in the row are accelerating slower and slower, while ships farther and farther ahead of him in the row are accelerating faster and faster.
Each ship's perspective on their own journey will be symmetrical--each ship get the same answer for the distance from the first station they passed at the moment they pass their 100th station. But each ship has a different starting station and a different 100th station, and we already know (by considering their surfaces of simultaneity as seen in the station rest frame) that their perspectives on the other ship's journies are not symmetrical--the lead ship sees that the rear ship has not yet reached its 100th station at the moment the lead ship does, while the rear ship sees that the lead ship reaches its 100th station well before the rear ship does. The fact that both non-inertial frames agree the times for each ship to pass 100 stations are different, and that both non-inertial frames see the distance between stations continually changing over time, is good reason to suspect both non-inertial frames will say that the distance between a ship's first station and its 100th station at the moments the ship is passing each one will differ for the two ships, although I haven't actually done the calculation.
Again, it's just based on drawing a spacetime diagram of the two ship's worldlines from the perspective of the inertial station frame, where we know how the surface of simultaneity for a ship's instantaneous co-moving inertial frame should look. Do you have any experience in drawing spacetime diagrams with multiple frames' surfaces of simultaneity? If not you really need to learn this stuff before getting into advanced discussions of acceleration. And if so, are you disagreeing that under conditions of thrust the two ships will reach their 100th station simultaneously in the station rest frame (if so then you really need to familiarize yourself with the equations on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] page), or are you disagreeing that the fact that if they reach the 100th stations simultaneously in the station frame this implies that the rear ship has not yet reached its 100th station in the instantaneous co-moving inertial frame of the front ship as it is passing its own 100th station?
Only in the co-moving inertial frame of the lead ship (or the non-inertial frame where the lead ship is at rest). In the station frame, both ships' clocks always show the same time.
Remember that you're talking about the non-inertial frame of the lead ship here. I wouldn't jump to any solid conclusions about what's happening to the distances in this frame, since in this frame the distance between stations is continually decreasing (provided we're using a non-inertial coordinate system where distances and simultaneity at each moment match up with the ship's instantaneous co-moving inertial frame at that moment).
Sure, the explanation for any weird stuff that happens in the ship's non-inertial frame is just based on how this non-inertial frame is constructed, how distances and simultaneity in this frame are always supposed to match up with the ship's co-moving instantaneous inertial rest frame at each moment. So if you want to figure out the distance to the rear ship in the front ship's non-inertial frame at a particular point on the front ship's worldline, or the time on the rear ship in the front ship's frame at that point, all you have to do is figure out their positions and proper times in the inertial station frame, then draw in the surface of simultaneity for the instantaneous co-moving inertial frame at that point using the usual rules for drawing one inertial frames' surface of simultaneity in a diagram drawn from the perspective of another inertial frame.
At any point on an accelerating object's worldline, different inertial frames will have different values for the object's instantaneous velocity at that point. Naturally there will be one inertial frame where the object's instantaneous velocity is exactly zero at that point, and that is all that is meant by the "instantaneous co-moving inertial frame". If you had an inertial observer at rest in that frame who happened to be next to the accelerating object at that instant, then all other frames would agree that the two had matched velocities at that instant.

Last edited by a moderator: May 4, 2017
17. Sep 7, 2009

### JesseM

(continued from previous post because it was too long)
If they just turn off their engines without decelerating to come to rest in the station frame, then they will not pass their 100th stations simultaneously in their new inertial rest frame. Their velocity as a function of time was identical in the station rest frame so they must pass their 100th stations simultaneously in the station rest frame, which means in any other inertial frame they do not pass them simultaneously. Again, if you disagree that they have identical velocities as a function of time in the station rest frame, you really need to read up on the http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken] which deals precisely with the question of a ship's velocity as a function of time in the inertial frame it is initially at rest in, under conditions of constant thrust (constant proper acceleration).
But they don't arrive simultaneously in this frame, and you have presented no argument as to why you think they do. I have already explained why I know that their velocity as a function of time will be identical in the station rest frame if they are both undergoing the same proper acceleration (and they take off simultaneously in this frame), because this is what the relativistic rocket equations tell us.
The two clocks always remain synchronized with one another in the station frame, although of course they are both falling behind coordinate time in that frame. Again just look at the relativistic rocket page, which shows the relation between clock time on an accelerating rocket and coordinate time in the frame it took off from, under the assumption of constant proper acceleration.
Remain in sync from when to when? A convention synchronization procedure applied when? I am assuming that they were both synchronized in the station frame before they started accelerating in the station frame, and that they both began accelerating simultaneously in the station frame. Are you assuming something different?
Clocks at different positions in the Born fleet are ticking at different rates both in Rindler coordinates and from the perspective of some inertial frame (remember that different clocks in the fleet have different proper accelerations). In Rindler coordinates, I believe the way clocks at different positions tick at different rates can be seen as equivalent to gravitational time dilation under the equivalence principle (I'm not totally sure of this, but my understanding from some previous physicsforums discussions is that a small room whose top and bottom are both sitting at constant Schwarzschild radius in the gravitational field of a spherical planet, small enough that tidal forces are negligible, can be treated as equivalent to a similar room accelerating in a Born rigid way in flat spacetime with the same G-forces at the top and bottom as with the room in the gravitational field).

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18. Sep 8, 2009

### Austin0

Originally Posted by Austin0
there would be greater dilation at the back because, as viewed fronm an inertial frame the back must have a greater instantaneous velocity than the front to account for the extra distance travelled due to contraction.

Hi JesseM Maybe you could take a quick look at this?? This for a single ship as measured from an inertial frame. Considered to have constant acceleration .
I did a work up of a hypothetical case but my math is rusty so I thought I would run it by you.
Inertial frame F
Accelerating System S'
rest L'= 1 km
a= 1000g= 10km /s$$^{2}$$

Range .6c ===> .7c
.7c-.6c =.1c = 3 x 10$$^{4}$$km/s

Time dt= (3 x 10$$^{4}$$km/s)/(10km/s) =3000 s

Contraction v$$_{i}$$=.6c ------- $$\gamma$$=1.25 --- = L'$$_{0}$$=.8 km
v$$_{f}$$ =.7 -------- $$\gamma$$= 1.4 --- =L'$$_{1}$$ =.71km

Difference in length over course of acceleration = .09 km
.09km/ 3000s = 3 x 10 $$^{-5}$$ km /s

relative velocity between front and back v$$_{fb}$$= (3 x 10 $$^{-5}$$ km /s) /(3 x 10 $$^{5}$$km /s ) = 10$$^{-10}$$c
Additive average relative velocity between front and back = (.65+10$$^{-10}$$)+ .65c = .65+ (1.7316 e$$^{-10}$$ )

average velocity difference v$$_{d}$$= 1.7316 e$$^{-10}$$ c
avg $$\gamma$$= 1 +( 2.9484 x 10 $$^{-20}$$ ) between front and back

Relative to inertial frame F ,,, S' avg v=.65c $$\gamma$$= 1.32
dt/1.32 = 3000/1.32 = 2,273 s = overall elapsed time on S'

2,273 x (1 +( 2.9484 x 10 $$^{-20}$$ )) = 6.782 x 10 $$^{-17}$$ s
elapsed time difference between back and front.

As I said I am rusty and could have easily dropped an exponent or counted all the zeros or 9's on the calculator screen wrong but does this seem in the ballpark???
Or is there some other fundamentally different way to calculate that would derive a significantly different result?
I assumed constant acceleration as observed in the inertial frame , of course the calculated (a) factor wouldn't neccessarily be healthy for humans but it made for smaller numbers
Thanks

19. Sep 8, 2009

### JesseM

When you say you want a constant acceleration as observed in an inertial frame, is that for both the front and the back? In that case, both will necessarily have the same function for distance as a function of time in that inertial frame, which means there will be no change in distance between the front and back from one moment to the next (and as in the Bell spaceship paradox, this will cause the ship to become physically stretched, increasing the stress until the ship breaks apart). Again, the length contraction equation only relates length before and after acceleration if the object is physically unchanged as seen in its own rest frame before and after acceleration, including no changes in the internal stresses at various points along the object.

20. Sep 8, 2009

### Austin0

No this is assuming length contraction , Born rigidity and Rindler relative time dilation btween the back and the front.
Thanks