Is Distance/space Lorentz contracted?

[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 don't consider proper acceleration, just assume identical ships and propulsion systems 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?

 

[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.

 

[Doc Al]

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 don't consider proper acceleration, just assume identical ships and propulsion systems 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.

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.

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?

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

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

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

 

[Austin0]

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.

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 don't really see it.

Thanks

 

[Doc Al]

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.

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

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?

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.

 

[JesseM]

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?

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:

...

We can imagine a flotilla of spaceships, each remaining at a fixed value of s by accelerating at 1/s. In principle, these ships could be physically connected together by ladders, allowing passengers to move between them. Although each ship would have a different proper acceleration, the spacing between them would remain constant as far as each of them was concerned.

 

[Austin0]

What do you mean by "the ship frame"? ). For ships undergoing constant proper acceleration, the most common choice of coordinate system for them would be Rindler coordinates, :

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

 

[JesseM]

I meant the ship reference frame in its generic sense with no implication that it was an inertial frame.

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.

I am curious about the genesis of this system with its included gravitational dilation.

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).

Also the Theorem of anisotropic dilation within accerating systems due to relative velocity resulting from length contraction.

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?

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.

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).

Given ideally identical mechanisms and uniform spacing between the stations:

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

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.

 

[Austin0]

=JesseM;2333518]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,

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.

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).

I got that from DrGreg in a discussion including Rindler coordinates. I was referring 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.
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Originally Posted by Austin0
Also the Theorem of anisotropic dilation within accerating systems due to relative velocity resulting from length contraction.

Don't understand this sentence at all. How does relative velocity result 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 traveled 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.

What do you mean "exactly the same procedure"?

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

 

[JesseM]

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.

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.

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'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.

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 traveled due to contraction.

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).

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.

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.

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.

OK, you didn't specify that before. In this case the times will be the same.

Whether that acceleration is "proper" or not , is a separate question that is not relevant here.

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.

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.

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).

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?

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 spaceship paradox[/url].

 

[Austin0]

=JesseM;2333573]
"--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).

I think there may be some question regarding this proposition.

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.

This also is uncertain.

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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.
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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

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

 

[JesseM]

--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).

I think there may be some question regarding this proposition.

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, 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.

This also is uncertain.

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?

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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.
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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).

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

Yes, as seen in the inertial frame where the stations are at rest (and the proper times will be equal too).

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??

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).

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

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.

and the spatial distance between those locations was the same

Only in the inertial frame where the stations were at rest, not necessarily in an accelerating coordinate system of the type I described above.

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.

No idea what you mean by that.

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.

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 ). 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.

 

[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.)

 

[Austin0]

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Originally Posted by JesseM
--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).

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

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.

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.

Originally Posted by JesseM
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.

No time dilation?

Originally Posted by Austin0

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

No idea what you mean by that.

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?

Yes, as seen in the inertial frame where the stations are at rest (and the proper times will be equal too).

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??

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, #1 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). #2 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).

#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

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.

What is the basis for an assumption of loss of synchronization between the two ships?
Or that they don't 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.

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 relativistic rocket page). 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.

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?

 

[Austin0]

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.)

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 can't place any comprehensible interpretation on the lines of simultaneity at all.
Thanks

 

[JesseM]

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.

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?

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.

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.

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.

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.

That it is somehow possible to accelerate a frame or an object and avoid the effects of relative velocity.

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).

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.

No time dilation?

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 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.

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

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?

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

Yes, as seen in the inertial frame where the stations are at rest (and the proper times will be equal too).

No I was talking about in the ships.

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.

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??

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, #1 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). #2 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).

#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.

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.

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?

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.

#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]

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.

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.

What is the basis for an assumption of loss of synchronization between the two ships?

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 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?

And if the rear ship's clock did read 4000 seconds when the lead ship's read 5000

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.

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

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).

do you have any particular explanation for how this could take place?

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.

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?

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.

 

[JesseM]

(continued from previous post because it was too long)

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 relativistic rocket page). 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.

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.

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 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).

Wouldnt it be expected then that if their clocks read the same proper time and they arrived simultaneously in their frame,

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.

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 ??

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.

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 ??

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?

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?

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).

 

[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 traveled due to contraction.

=JesseM;2333573]
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).

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

 

[JesseM]

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 traveled 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

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.

 

[Austin0]

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.

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

 

[JesseM]

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

You can't assume the time dilation will work the same way as for a ship at rest in Rindler coordinates, though, because you want the ship (or at least part of the ship, perhaps the center) to have a constant coordinate acceleration as seen in an inertial frame, which means a non-constant proper acceleration--for this to happen the proper acceleration (and thrust) would have to be continually increasing, whereas for a ship at rest in Rindler coordinates, every point is experiencing a constant proper acceleration. I'm also not sure if it's really correct to call the acceleration "Born rigid" if the proper acceleration isn't constant, but I suppose we could at least have something analogous if we just drew in surfaces of simultaneity for the instantaneous co-moving frame of the center of the ship at each point on its worldline, then drew the worldlines of the front and back such that they were always the same distance from the center in each of these surfaces (this is assuming the acceleration is such that the surfaces of simultaneity never cross one another at a distance closer than the distance from the center to either end).

Anyway, given these conditions, if the acceleration in the inertial frame is a constant a for the center of the ship, then v(t) = at + v0, and x(t) = (a/2)t^2 + tv0 + x0. If we want the initial velocity to be 0.6c at t=0, then v0=0.6c. It will be easier to work in units where c=1, so if a = 10 km/s^2, that equals 10/299792.458 = 0.0000333564 light-seconds/s^2. So v(t) = 0.0000333564*t + 0.6, which means if we want to find the time to reach 0.7c, we can just solve for t in the equation 0.7 = 0.0000333564*t + 0.6, which gives t = 2997.9 seconds, which is about what you got (you rounded off the speed of light a little).

With the length contraction equation, you have to keep in mind that it's only exact for objects moving inertially, with the quasi-Born-rigid condition above the length at any given instant in the inertial observer's frame would depend on the exact shape of the worldline of the front and back and would not be exactly equal to the length you get if you plug the instantaneous velocity of the center into the length contraction equation. That said, I think it would be pretty close to that, since at any given time T in the inertial frame I don't think there'd be a huge difference between the instantaneous velocity of the center in the the tilted surface of simultaneity that contains the back end at time T (this would be a point on the center's worldline at a time slightly later than T when its instantaneous velocity was slightly higher), and the instantaneous velocity of the center in the tilted surface of simultaneity that contains the front end at time T (this would be a point on the center's worldline at a time slightly earlier than T when its instantaneous velocity was slightly lower).

So, if we just use the regular length contraction equation, at 0.6c the ship should be 0.8 km long in the inertial frame, and at 0.7c the ship should be 0.714 km long. Again this agrees with your calculation.

How did you calculate the relative velocity between front and back? Just based on the amount the ship shrunk over the course of the acceleration? Again, an exact answer would depend on knowing the function for the worldline of the front and back, but that method would probably give an approximately correct answer.

 

[Austin0]

You can't assume the time dilation will work the same way as for a ship at rest in Rindler coordinates, though, because you want the ship (or at least part of the ship, perhaps the center) to have a constant coordinate acceleration as seen in an inertial frame, which means a non-constant proper acceleration--for this to happen the proper acceleration (and thrust) would have to be continually increasing, whereas for a ship at rest in Rindler coordinates, every point is experiencing a constant proper acceleration. I'm also not sure if it's really correct to call the acceleration "Born rigid" if the proper acceleration isn't constant, but I suppose we could at least have something analogous if we just drew in surfaces of simultaneity for the instantaneous co-moving frame of the center of the ship at each point on its worldline, then drew the worldlines of the front and back such that they were always the same distance from the center in each of these surfaces (this is assuming the acceleration is such that the surfaces of simultaneity never cross one another at a distance closer than the distance from the center to either end).

Anyway, given these conditions, if the acceleration in the inertial frame is a constant a for the center of the ship, then v(t) = at + v0, and x(t) = (a/2)t^2 + tv0 + x0. If we want the initial velocity to be 0.6c at t=0, then v0=0.6c. It will be easier to work in units where c=1, so if a = 10 km/s^2, that equals 10/299792.458 = 0.0000333564 light-seconds/s^2. So v(t) = 0.0000333564*t + 0.6, which means if we want to find the time to reach 0.7c, we can just solve for t in the equation 0.7 = 0.0000333564*t + 0.6, which gives t = 2997.9 seconds, which is about what you got (you rounded off the speed of light a little).

With the length contraction equation, you have to keep in mind that it's only exact for objects moving inertially, with the quasi-Born-rigid condition above the length at any given instant in the inertial observer's frame would depend on the exact shape of the worldline of the front and back and would not be exactly equal to the length you get if you plug the instantaneous velocity of the center into the length contraction equation. That said, I think it would be pretty close to that, since at any given time T in the inertial frame I don't think there'd be a huge difference between the instantaneous velocity of the center in the the tilted surface of simultaneity that contains the back end at time T (this would be a point on the center's worldline at a time slightly later than T when its instantaneous velocity was slightly higher), and the instantaneous velocity of the center in the tilted surface of simultaneity that contains the front end at time T (this would be a point on the center's worldline at a time slightly earlier than T when its instantaneous velocity was slightly lower).

So, if we just use the regular length contraction equation, at 0.6c the ship should be 0.8 km long in the inertial frame, and at 0.7c the ship should be 0.714 km long. Again this agrees with your calculation.

How did you calculate the relative velocity between front and back? Just based on the amount the ship shrunk over the course of the acceleration? Again, an exact answer would depend on knowing the function for the worldline of the front and back, but that method would probably give an approximately correct answer.

The premise of this calculation ,which stems from the hypothesis I mentioned earlier, is that the velocity differential is calculated from the inertial observation frame.
SO I considered calculating overall distance traveled for the front and the back as a function of time , but then decided that it was equivalent to simply calculating based on the difference itself.
So yes , I simply took the distance that the back moved toward the front ,the relative velocity as a function of total elapsed time. And then divided that by c to derive a normalized relative velocity.
I then assumed that a simple, median average ,overall system velocity (.6.5c) would be close enough for a ballpark given constant acceleration.
I then added the back/front relative velocity to one end and the raw velocity for the other end and used the additive velocity formula to get an internal relative velocity between the two.
Entered this figure into the gamma function to get a gamma factor and then applied this to the gamma corrected dt' for the difference between the back clock and the front
over the total course of acceleration. The accrued time dilation.
I realize I may have written this misleadingly regarding the additive velocity results but I think the actual figure for relative velocity 1.7316 e^{-10} c is accurate.

So there you have it. As I said, I understand that this approach may only provide an approximation as compared to a more detailed analysis but is there reason to think there should be significant difference?
Thanks for your help

 

[Austin0]

Hi JesseM I am somewhat confused by the turn this discussion has taken and hope that this can be dispelled somehow.
I have been learning and thinking about this topic for some time . The Bell question, Born acceleration, the explanation of Rindler dilation on the basis of relative veocity . In the process I am having to learn the hyperbolic trig h functions , not to mention relearning a large part of forgotten basic trig. SO I can not yet apply the Rindler coordinates to derive quantitative values for specific hypothetical situations or follow the derivation of the proper velocity formulas. I certainly have not arrived at any kind of conclusions regarding these questions. In the meantime I am pursuing it in the context of thought experiment and logical enquiry. So that is the spirit with which I posed this thread. Not that I have an answer of any kind but because I have questions. The only thing I have any confident certainty of either knowing or of being known in this context ,are the fundamental postulates of SR which I take as a firm foundation unless empirically proven otherwise . And the known physics of acceleration. I have no conclusions

I have a high degree of respect for the breadth of your knowledge and even more for your desire and willingness to share that ,along with your time, with those who want to learn.
I know in the past you have helped me and I am grateful. So it bothers me that I have somehow provoked an appararently adversarial responce that is beyond simple rational debate. So if I have offended you in some way please let me know.
I have taken it as a tacit understanding that anything I might have to say is simply a logical thread or an opinion and I have no delusions of having any knowledge of the actuality of these hypothetical realities.

Originally Posted by Austin0
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.

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.

Of course you are right. That is the explicit area of applicability of SR and it was with that in mind that I was speaking. But of course to get from one inertial state to another neccessatates a period of acceleration, yes?

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?

No. Do you have knowledge regarding the known physics of acceleration, Propagation of momentum, Intermolecular tensile forces etc. that would dictate that constant acceleration,given that it was within the range of the materials ability to propogate it ,would result in sustained compression or contraction?


Originally Posted by Austin0
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.

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?

AS it happens I am not an advocate of any matrix theory but neither do I rule them out. I just don't know.

But where did this come from?? I certaily didnt bring it into the discussion and fail to see how it relates to the question of contraction or the relative speed of light.
What is the connection you see??

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.

I was under the impression that none of the relativistic effects. Time dilation, contraction,inertial mass increase or desynchronization, had any theory that addressed the physics, the possible mechanism for their actualization. Have I missed something?
It would seem that time dilation is something that happens naturally , simply as a result of velocity. So do you see a logical neccessity to assume that the other effects would be fundamentally different?

Originally Posted by Austin0
The Born hypothesis seems to question this. It proposes that length contraction is not an inherent, inevitable consequence of velocity but rather, is an essentially physical phenomenon that requires specific, carefully contrived, application of force to take effect.

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

.

SO then,, would this mean that a relatively inelastic object, say a steel rod that was kept in a transverse position during Born acceleration, would then be longer relative to that frame when rotated longitudinally wrt the motion. Or would be stressed and break, being suddenly subjected to the cumulative acceleration of the entire journey?

Originally Posted by Austin0
That it is somehow possible to accelerate a frame or an object and avoid the effects of relative velocity.

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. # 1 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).

#1 No in this case I would expect that its unrolled length, transverse relative to longitudinal ,would be related by the Lorentz contraction as measured from a different inertial frame.
Actually I wasn't thinking about silly putty much, I was thinking more in terms of electromagnetically accelerated steel rods or tubes. We are approaching the point technologically where this will be a real possibility. I.e. significant velocities.
As I understand it there are a number of different possibilities for acceleration. Fields pushing , pulling, combinations and uniform.
DO you think that contraction would only occur if the force was greater in the rear?
That uniform acceleration would result in the breakdown of a stretched rod?

#1 Actually you have brought up an interesting question. Suppose a ball of string is on a Born ship that has accelerated . Is it now longer relative to that frame if it is unrolled?


[ Originally Posted by Austin0
#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.

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.

I am not following you here. Refering to DrGregs drawing of this situation posted earlier.
We agree that there is no difference between two separate ships and one extended ship, right??
Looking at his diagram for any given instant of the initial frames timeline:
How is the picture regarding lines of simultaneity any different from a similar picture of an inertial frame?
Doesn't it have the same slope relative to the worldline slope of that instantaneous velocity??


Originally Posted by Austin0
What is the basis for an assumption of loss of synchronization between the two ships?

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. #2 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 relativistic rocket page), or are you #3 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?

#2 I have no need to look at the rocket page [which I actually already have, along with many other forms and derivations]. Your logic is self evidently valid. Its not like I am unaware of this or haven't considered it. Equal proper acceleration must result in equal distance and time. But exactly the same obvious logic is applicable from the ships frame.
So it is not such a simple ,clearcut situation.

Originally Posted by JesseM
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.

My understanding is that lines of simultaneity relate the ships frame to the intial frame ,,,do you have some other interpretation I am not aware of ?
That within frames, the functional definition of simultaneity is having the same proper time reading ,,,is this somehow in error??

Thanks Oh you didnt get back on the time calculations , does this mean you found no serious errors ?

 

[PeterDonis]

Equal proper acceleration must result in equal distance and time. But exactly the same obvious logic is applicable from the ships frame.

Austin0, I've been watching this thread for a few days now but haven't jumped in because I've been waiting for you to say something short and to the point that gets at the heart of the point of disagreement between you and JesseM. I think the above quote is it. PMFJI, but I'd like to give my take on why the above quote doesn't apply in this situation.

I'll give a short version of my counter-argument. If it doesn't convince you, I have a longer one in reserve. :-)

The proposition which you say must apply equally in the ships' frame is the following:

"Equal proper acceleration must result in equal distance and time."

Now "proper acceleration" is an invariant--it's the acceleration which is actually felt by observers in each of the ships. So all observers in all frames can agree that the proper acceleration is equal.

But distance and time are frame-dependent. So if equal proper acceleration, which is the same in all frames, leads to equal distance and time in one frame, it *cannot* lead to equal distance and time in any other frame in relative motion to the first. Therefore, the "same obvious logic" *cannot* apply in the ships' frame if it applies (as it does, by hypothesis) in the station frame.

Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth. It's simply an observation about how the space and time coordinates assigned to given events on given worldlines in different reference frames will be related. And its logic appears to me to be valid.

If the above convinces you, well and good. If not (and I understand if the above argument, even if it's logically valid, doesn't quiet all the intuitions that are raising flags in your mind) then I can follow up by laying out more explicitly how things look from each viewpoint.

 

[Austin0]

Originally Posted by Austin0
Equal proper acceleration must result in equal distance and time. But exactly the same obvious logic is applicable from the ships frame.

=PeterDonis;2344191]Austin0, I've been watching this thread for a few days now but haven't jumped in because I've been waiting for you to say something short and to the point that gets at the heart of the point of disagreement between you and JesseM. I think the above quote is it. PMFJI, but I'd like to give my take on why the above quote doesn't apply in this situation.

Hi PeterDonis Glad to have you jumping in. I don't particularly like looking stupid. But I like even less wasting my own time working with false premises, so if you can point out an error in my perspective it would be welcome.

I'll give a short version of my counter-argument. If it doesn't convince you, I have a longer one in reserve. :-)

I have no doubt you have all sorts of arguments in reserve.

The proposition which you say must apply equally in the ships' frame is the following:

"Equal proper acceleration must result in equal distance and time."

Now "proper acceleration" is an invariant--it's the acceleration which is actually felt by observers in each of the ships. So all observers in all frames can agree that the proper acceleration is equal.

But distance and time are frame-dependent. So if equal proper acceleration, which is the same in all frames, leads to equal distance and time in one frame, it *cannot* lead to equal distance and time in any other frame in relative motion to the first. Therefore, the "same obvious logic" *cannot* apply in the ships' frame if it applies (as it does, by hypothesis) in the station frame.

There is no argument here. Every point is beyond question.
But I think it may be misdirected. It appears there is one of those common semantic misunderstandings here. When I said equal distance and time I meant equal between the ships , not between the frames, where self evidently there would not be agreement on either the distance covered or the time elapsed. Earlier I tried to make it clear that from the ships perspective, the stations were only map markers for the establishment of equality of distance traveled between the ships, with no quantitative evaluation whatsoever.
My train of logic is this:
If the rear ship is R, the lead ship is L ,starting from Stn 1 and 2 respectively and ending up at STn's 101 and 102
If a first trial is conducted, where both R and L start simultaneously from Stn 1 --->101,,
it would be expected tha, not only would they arrive at 101 simultaneously with the same elapsed proper time but that at every point on the way they shared a common simultaneity.
Agreed ??
So if this is repeated, with the sole difference being that L now starts from 2, is there any principle of SR that would either explain or neccessitate a loss of simultaneity within this system??
As JesseM pointed out, we could replace the two ships with a single long frame, so this would apply there also.
Obviously this system could not agree on simultaneity with the initial frame, but is there any reason to assume a different definition of simultaneity within the ships frame itself?, I.e. Equal proper time readings are assumed to be simultaneous within that frame.
Or any loss of simultaneity between ships?
So both JesseM and I, as well as the logic of the situation, agree that:
1) the proper time readings at stations 101 and 102 at the arrivals should be the same.
2) the proper time readings in L and R at arrival should be the same.

Think about this . If you consider that the ship frame at any moment is a system at a relative velocity then it is impossible that two frames could have this kind of agreement.
That two clocks in one frame that are spatially separated and have the same readings , could each be colocated with clocks in another frame that also have the same readings [equal to each other], albeit quantitatively different from the first frames.
Am I missing something here?
So it seems to me that right here is cause to question the sound logic that brings the above conclusion. That maybe in actual practice, the ships would not arrive at the same times in the station frame or vice versa.
After all, the obviously valid logic, pre Micholson-Morley and SR,,,, that (c-v) could not possibly be equal to (c+v) didn't work out too well in the real world did it?
It appears to me that if length contraction was an inherent effect of relative velocity then this problem goes away. If you consider a single long ship instead of two ships and the length contracted, then of course the two ends could not arrive anywhere simultaneously in the station frame with identical proper times at opposite ends of the ship.

Going beyond that: If you assume the stations simultaneity where does that lead?
From this perspective it appears to me that the normal assumption would be that wrt the stations clocks that the rear ship would be running ahead of the lead ship.
Am I incorrect in this?
That within the station frame ,whatever time was observed at a given moment on the clock of the rear ship , it would be assumed that the lead ships clock had not yet reached that same proper time.
Correct??
From this I can see a case for the rear ship reaching 101 before L reaches 102 as calculated from the station frame.
And comparably, from the ships frame, 102 reaching L before 101 reaches R.
So if you have an analysis of simultaneity that is different or sheds light on this it would be great.

Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth. It's simply an observation about how the space and time coordinates assigned to given events on given worldlines in different reference frames will be related. And its logic appears to me to be valid.

No question of the validity of your logic. Although I am not quite sure of exactly what it is that you think I need to be convinced of or what you think I am already convinced of, as so far I am not sure of anything.
I originally tried to keep the question purely kinematic but I think JesseM is right as far as it inevitably broadens to include Born acceleration etc.

If the above convinces you, well and good. If not (and I understand if the above argument, even if it's logically valid, doesn't quiet all the intuitions that are raising flags in your mind) then I can follow up by laying out more explicitly how things look from each viewpoint

I don't think it has much to do with intuitions. I don't think you can get very far with SR or QM if you can't check your intuitions at the door. I know you have a much greater command of math than I do from other threads , so if you could possibly take a quick peek at the calculation of dilation I posted ,i think #16 it would be great , Just to see if the basis is correct and the figures seem in the ballpark. It seems like it could also have some bearing on this question.
It would be appreciated but if not I understand. Thanks for your input , it is most welcome.

 

[PeterDonis]

No question of the validity of your logic. Although I am not quite sure of exactly what it is that you think I need to be convinced of or what you think I am already convinced of, as so far I am not sure of anything.
I originally tried to keep the question purely kinematic but I think JesseM is right as far as it inevitably broadens to include Born acceleration etc.

Ok, let me lay things out more explicitly. I think that both you and JesseM have been conflating two different questions:

(Q1) If you have specified a particular set of events in a particular spacetime, and you know the coordinates of those events in one reference frame, what can you say about the coordinates of those same events in another reference frame?

(Q2) If you have a physical scenario in mind, how can you translate that scenario into a particular set of events in a particular spacetime, and how does that translation depend on the particular physical assumptions you make?

In your original problem specification, as I understand it, you make a set of assumptions that were sufficient to pin down a particular set of events in a particular spacetime. Given that, the question of how those events appear to different observers in relative motion only involves Q1, and is therefore a purely kinematic one (or maybe "geometric" would be a better word). That is, if you have a particular set of events with coordinates in one frame, it's a simple matter of mathematics--applying the equations of relativity--to calculate how those events will appear in any other frame. It does *not* depend on what physical assumptions you make.

(If you have *not* yet made enough assumptions that are sufficient to pin down a particular set of events, then of course the physical assumptions you make *can* affect which particular events you end up specifying; that's where Q2 comes in, and where Born rigidity, etc., can make a difference. But once you've specified a particular set of events, if you then make a physical assumption that would lead to a different set of events, you've made an inconsistent specification of the problem. I'll give an example later on.)

Let me lay out explicitly the set of assumptions I take you to have made in your original problem specification:

Let there be a family of space stations, with the following properties: they are all in inertial motion (i.e., they feel no acceleration), they are all at rest relative to each other, and each adjacent pair is separated by a spatial distance S in the common rest frame of all the stations, which we'll call the "station frame".

Let there be two rocket ships, the rear ship R and the front ship F (I will also use those letters to refer to the observers inside the ships). At time t = 0 in the station frame, R is next to station #1 and F is next to station #2. At time t = 0 in the station frame, both R and F turn on their rocket engines and start accelerating. It is stipulated (this is the key physical assumption) that the proper acceleration is the same for R and F: that is, they both feel the same acceleration (if accelerometers are mounted in both ships, they both give the same reading). R and F continue to accelerate at least until R has passed station #99 and F has passed station #100.

The above is sufficient to pick out a particular set of events in a particular spacetime. The spacetime is simply flat Minkowski spacetime, and the "station frame" is simply a global inertial frame using standard Minkowski coordinates t, x. The particular events are as follows:

Event R1: The event at which R, next to station #1, turns on his rocket engines and starts accelerating.

Event F2: The event at which F, next to station #2, turns on his rocket engines and starts accelerating.

Event R99: The event at which R passes station #99.

Event F100: The event at which F passes station #100.

What I mean by a "particular set of events" is not just that we can *define* the above events (of course we can always do that), but that the coordinates of all four of the above events, in the station frame, are *fixed* by the above specifications. That is, we have picked out four specific individual points in Minkowski spacetime. (If it helps, imagine that we have a big spacetime diagram on the wall, using the coordinates of the station frame; the above specifications are sufficient to let us take four push pins and pin them at four specific points on that diagram.)

Next, we know that the following propositions concerning the above events are true:

(P1) In the station frame, the spatial separation, in the station frame, between R1 and F2 is S.

(P2) In the station frame, R1 and F2 are simultaneous (i.e., they both happen at the same time, t = 0).

(P3) R and F both experience the same proper acceleration between R1-R99 and F2-F100, respectively.

(P4) Given P1 and P3, the spatial separation, in the station frame, between R99 and F100 is S.

(P5) Given P2 and P3, in the station frame, R99 and F100 are simultaneous (i.e., they both happen at the same time, which we'll call t = T).

However, we can also show, using just the above and the equations of relativity (in this case, the Lorentz transformation and the relativistic rocket equations), that the following propositions are true:

(P6) At R99, R is moving relative to the station frame.

(P7) At F100, F is moving relative to the station frame.

(P8) Given P4, P5, P6, and the relativity of simultaneity, R99 and F100 are *not* simultaneous in R's frame.

(P9) Given P4, P5, P7, and the relativity of simultaneity, R99 and F100 are *not* simultaneous in F's frame.

(P10) Given P4, P5, P7, P8, and P9, the question, "what is the spatial separation between events R99 and F100" in either R's frame or F's frame, has no meaning; the events are not simultaneous.

(P11) Given the above, if we want to ask the question, "when R passes station #99, how far in front of it is F?", we have to pick out that event on F's worldline that *is* simultaneous with R99. That event will be somewhere *further* along F's worldline than F100 (i.e., F will reach it *after* it passes station #100, by its own clock--and also by the station frame's clock). Therefore, R will conclude that, when he passes station #99, F has *already* passed station #100. In other words, F is "pulling away" from R, from R's point of view.

(P12) Given the above, if we want to ask the question, "when F passes station #100, how far behind it is R?", we have to pick out that event on R's worldline that *is* simultaneous with F100. That event will be somewhere *before* R99 on R's worldline (i.e., R will reach it *before* it reaches R99, by its own clock--and also by the station frame's clock). Therefore, F will conclude that, when he passes station #100, R has *not yet* passed station #99. In other words, R is "falling behind" F, from F's point of view.

You will have noticed that I did not use the term "ship frame" above. That is because there is no one single "ship frame" that you can use at both R99 and F100--you have to specify which ship. This may be confusing because, at R1 and F2, you *can* specify a single "ship frame" that you can use at those two different events--this is, of course, the station frame, since both ships are (momentarily) at rest relative to the station frame at those events. But because the ships are accelerating, their two frames "separate" after events R1 and F2. If that seems counterintuitive, well, welcome to the world of accelerating observers.

You may already understand and agree with all the above--I suspect you do. But now suppose we consider the following:

What if we *specify*, in addition to all the above, that the spatial separation between the two ships remains constant "in the ship frame"? (For example, I might assume that the two ships undergo "Born acceleration" relative to each other.)

Let's try to construct the scenario along those lines. That will be no problem for three of our four events: R1, F2, and either R99 or F100. I'll pick R99 for concreteness. In other words, I know that I can add the stipulation that the spatial separation between the two ships remains constant as seen by the ships, without any contradiction with the coordinates I have already found for events R1, F2, and R99.

The problem comes with F100, because, by P10, it simply makes no sense to ask what the "spatial separation" is, in the "ship frame" (of either ship), between events R99 and F100, because those events are not simultaneous in the "ship frame" (of either ship). In other words: in order to meet the specification that the spatial separation between the ships remains constant, as seen by the ships, I must pick out a *different* event F100 than the one that is picked out by the assumption that both ships experience equal proper acceleration.

This means that there is simply no way to construct a single consistent scenario that has *both* of the following properties:

* Both ships, R and F, experience equal proper acceleration.

* The spatial separation between the two ships remains constant *as seen by observers on both ships*.

It's geometrically impossible. It's like asking for a plane figure that has both diagonals equal but has two opposite sides not parallel. It can't be done. (In the analogy I gave above, I simply can't put a single push pin in my map of Minkowski spacetime for event F100 that will satisfy both properties.) It doesn't matter what other physical assumptions you make; it doesn't matter what the ships are made of, or how they respond to the stresses of acceleration, or whatever. It's simply a question of geometry.

So in order to discuss anything else about the physics of situations like these, you have to first make a choice. Which of these two physical assumptions do you want to satisfy?

* Do you want both ships to experience equal proper acceleration?

* Or do you want both ships to remain at a constant spatial separation, as seen by the ships?

You *can't* satisfy both. Which one you pick will determine what specific scenario you're talking about, and until you do that, it's hard to have a discussion because you may be talking about different scenarios.

(NOTE: There is one other physical assumption you may have noticed. I kept specifying "Minkowski spacetime". But that was already specified when you said there was a family of space stations with the properties given above (inertial motion, all at rest relative to each other, constant spatial separation between adjacent stations), because only in flat Minkowski spacetime can there exist a family of inertial observers with those properties. In fact, specifying that there are such a family of observers is one way of saying what it means for a spacetime to be flat, Minkowski spacetime.

That may make you wonder whether we could find some other spacetime (perhaps some wacky curved spacetime from General Relativity) in which we *could* construct a single consistent scenario that satisfied both of the above physical assumptions. Unfortunately, I think the answer is still no, at least not if you accept the Einstein field equation of GR. I believe there is actually a theorem in GR that shows this, but I can't remember the reference; if I find it I'll post info about it here.)

 

[Austin0]

=PeterDonis;2345239]
Could we for the purposes of this discussion also consider R and F as being a single extended ship RF with initial length L= Station 1 <--> Station 2 ?
Looking at the geometry of a spacetime diagram at any given point in time,,would the graph for this system RF be distinguishable from a similar diagram for an inertial frame ?
regarding lines of simultaneity etc.
If so in what way?


Event R1: The event at which R, next to station #1, turns on his rocket engines and starts accelerating.

Event F2: The event at which F, next to station #2, turns on his rocket engines and starts accelerating.

Next, we know that the following propositions concerning the above events are true:

(P1) In the station frame, the spatial separation, in the station frame, between R1 and F2 is S.

(P2) In the station frame, R1 and F2 are simultaneous (i.e., they both happen at the same time, t = 0). Also in the R and F t'=0
(P3) R and F both experience the same proper acceleration between R1-R99 and F2-F100, respectively.

(P4) Given P1 and P3, the spatial separation, in the station frame, between R99 and F100 is S.

(P5) Given P2 and P3, in the station frame, R99 and F100 are simultaneous (i.e., they both happen at the same time, which we'll call t = T). IN R and F the same applies t'= T'
However, we can also show, using just the above and the equations of relativity (in this case, the Lorentz transformation and the relativistic rocket equations), that the following propositions are true:

(P6) At R99, R is moving relative to the station frame. CHeck

(P7) At F100, F is moving relative to the station frame. CHeck

(P8) Given P4, P5, P6, and the relativity of simultaneity, R99 and F100 are *not* simultaneous in R's frame.

In the normal interpretation this would mean that (R99) T' could NOT equal (F100) T',,,
true?
Are you saying that the proper time on their clocks would not be the same?
This ,at least , seemed to be one place where JM and I agreed.

What is the meaning and interpretation of lines of simultaneity ,beyond the relative clock desynchronization between frames?

(P9) Given P4, P5, P7, and the relativity of simultaneity, R99 and F100 are *not* simultaneous in F's frame.

(P10) Given P4, P5, P7, P8, and P9, the question, "what is the spatial separation between events R99 and F100" in either R's frame or F's frame, has no meaning; the events are not simultaneous.

IMHO it appears that you are taking one frames simultaneity and applying it in an absolute way to another frame.
For instance: What if the ships sent out radar ranging signals while cojacent with the stations and they also had equal proper time readings. Wouldnt the stations agree that the signals were sent simultaneously?
Wouldn't local observers in the station frame see the signals reflect and return [given receivers of course] at specific points in their frame?
Is there any reason to assume that the covered distance R-F-R and F-R-F in the station frame would be different??
Or that the times in the ships would be different??
Or do you think that if one ship sent out a signal cojacent to a station the other ship would emit its signal some completely different location??
If so why would this be so if their clocks read the same time at the stations??

(P11) Given the above, if we want to ask the question, "when R passes station #99, how far in front of it is F?", we have to pick out that event on F's worldline that *is* simultaneous with R99. That event will be somewhere *further* along F's worldline than F100 (i.e., F will reach it *after* it passes station #100, by its own clock--and also by the station frame's clock). Therefore, R will conclude that, when he passes station #99, F has *already* passed station #100. In other words, F is "pulling away" from R, from R's point of view.

Correct me if you think I am wrong but: the Lorentz math tells us from the perspective of one frame what the relative clock readings will be in another frame at specific locations at anyone point in time.
SO # 1 at R99 [ t=T and t'=T' (R)] it will tell you that a ship clock at F100 [t=T] in the station frame will be running behind a certain degree [ T' (F) < T' (R)].

Or #2 comparably it will tell you that at the location in the station frame where [ T' (F) = T' (R)] F must be farther away than100 and that T (K) station time at that location must > that R99 T.

original post austin0
From this I can see a case for the rear ship R reaching 99 before F reaches 100 as calculated from the station frame.And comparably, from the ships frame, 102 reaching L before 101 reaches R.

So it appears that you chose #2 while I chose #1 But neither one really works because T' 99 = T'100 in this case.

(P12) Given the above, if we want to ask the question, "when F passes station #100, how far behind it is R?", we have to pick out that event on R's worldline that *is* simultaneous with F100. That event will be somewhere *before* R99 on R's worldline (i.e., R will reach it *before* it reaches R99, by its own clock--and also by the station frame's clock). #3Therefore, F will conclude that, when he passes station #100, R has *not yet* passed station #99. In other words, R is "falling behind" F, from F's point of view.

original post austin0
From this I can see a case for the rear ship R reaching 99 before L reaches 100 as calculated from the station frame.
#3 And comparably, from the ships frame, 100 reaching F before 99 reaches R

If you look at #3's I think you will agree they describe the same actual situation.
I just used the ships frame. And from this perspective R is not falling behind.
The distance between 99 and 100 has contracted. In the ships,, falling behind would mean increasing radar range. Do you have any reason to expect that to be the case?

You will have noticed that I did not use the term "ship frame" above. That is because there is no one single "ship frame" that you can use at both R99 and F100--you have to specify which ship. This may be confusing because, at R1 and F2, you *can* specify a single "ship frame" that you can use at those two different events--this is, of course, the station frame, since both ships are (momentarily) at rest relative to the station frame at those events. But because the ships are accelerating, their two frames "separate" after events R1 and F2. If that seems counterintuitive, well, welcome to the world of accelerating observers.

It is not confusing because I am aware of the convention that says an accelerating frame is not a valid inertial frame. And the definitions pertaining . I don't disagree with the reasoning but I do think it is somewhat arbitrary and if my memory serves me Einstein analyzed the twins question from the perspective of the traveling twin being a rest frame and the Earth accelerating [curved world line]. I would suggest it might be useful to not be limited by predetermined definitions in the exploration of a purely hypothetical situation.IMHO
I mean that in both senses. The whole field of acceleration at relativistic velocities is totally without empirical input. The physics of dilation, contraction,and clock desynchronization is a total unknown.
SO I ask ; just looking at a Minkowski diagram of the FR system at a given moment do you see any geometric difference between it and an inertial frame?
Is there any reason to assume that the desynchronization as determined from the station frame [which is of course, always the case] should be imposed on the ship?
Thanks for your very clear rendering of the situation

 

[PeterDonis]

Austin0:

In response to this from my last post:

You will have noticed that I did not use the term "ship frame" above. That is because there is no one single "ship frame" that you can use at both R99 and F100--you have to specify which ship. This may be confusing because, at R1 and F2, you *can* specify a single "ship frame" that you can use at those two different events--this is, of course, the station frame, since both ships are (momentarily) at rest relative to the station frame at those events. But because the ships are accelerating, their two frames "separate" after events R1 and F2. If that seems counterintuitive, well, welcome to the world of accelerating observers.

you said:

It is not confusing because I am aware of the convention that says an accelerating frame is not a valid inertial frame. And the definitions pertaining . I don't disagree with the reasoning but I do think it is somewhat arbitrary and if my memory serves me Einstein analyzed the twins question from the perspective of the traveling twin being a rest frame and the Earth accelerating [curved world line]. I would suggest it might be useful to not be limited by predetermined definitions in the exploration of a purely hypothetical situation.IMHO
I mean that in both senses. The whole field of acceleration at relativistic velocities is totally without empirical input. The physics of dilation, contraction,and clock desynchronization is a total unknown.

There are several points here:

(1) Yes, Einstein did discuss the "twin paradox" from the point of view of the traveling twin, but his conclusion was the same: the stay-at-home twin (the one that remains on Earth) ages more than the traveling twin. The fact that, according to the traveling twin, the Earth was "accelerating", did *not* change the conclusion. In relativity, you can analyze a situation from whatever frame you want, but answers about invariants (for example, which twin ages more when they come back together) are the same no matter what frame you use to obtain them.

(2) If you want to discuss what relativity says about a particular situation, then you *do* need to be "limited by predetermined definitions" as far as the laws of relativity and the geometry of Minkowski spacetime are concerned. If you want to consider alternatives to those definitions, then you're considering alternative theories to relativity--or you're considering spacetimes other than Minkowski spacetime. That's not what I thought this thread was about; I thought you were asking what relativity, the theory as we have it, says about the situation of two accelerating spaceships in Minkowski spacetime. And in any case I think that situation, the one I've just described, is one we should all agree on before we start discussing more complicated ones.

(3) Acceleration to relativistic velocities is *not* "without empirical input". For example, every particle accelerator that's ever been built has subjected huge numbers of subatomic particles to huge accelerations and relativistic velocities, and those particles have exhibited *every* phenomenon that relativity says they should: time dilation, length contraction, clock desynchronization, the whole enchilada. We also have tested relativity with macroscopic objects: the GPS system, for example, wouldn't work if our understanding of both special and general relativistic effects on orbiting objects in the Earth's gravity well were not correct. http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html" .

In other words: I believe that the laws of relativity are valid when applied to the scenario we're talking about (the one I defined in detail in my last post). If you don't, or if you even want to consider the possibility that they're not, then there's not much point in having a discussion unless you can come up with some alternative theory that matches all the experimental evidence but gives different results when applied to the situation we're talking about. Trying to speculate about what might happen if the physical laws we know about are violated, without substituting some other set of laws to guide speculation, is, IMHO, not very useful.

Btw, there is one other postulate I haven't mentioned, which is part of the "laws of relativity" but which deserves to be mentioned separately. This is http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html" . It says that the "timing rate" of an accelerating clock, relative to some inertial clock, at any given event depends only on its instantaneous velocity relative to the inertial clock at that event. This postulate is logically independent of the other postulates of relativity, which is why it deserves separate mention; but it has also been experimentally verified to very high precision, so I'm including it in the "laws" that I'm accepting as true. It's important because it's what we use to relate the lines of simultaneity of the station frame with those of R and F, at events R99 and F100.

So I think this is the first point we need to come to agreement on, if we're going to have any kind of productive discussion about the scenario I've described: (A) are you, for this discussion, accepting the laws of relativity, but just questioning whether I (and JesseM) have correctly described their implications for the scenario? Or (B) are you agreeing that we've described the implications of the laws of relativity correctly, but wondering if there could be some *different* set of laws that would give different implications?

If it's (A), I can proceed to try to convince you that we've described the implications of relativity correctly, even though some of them are highly counterintuitive. If it's (B), though, I don't know what else I can say unless, as I said above, you can present some specific alternative to discuss--for example, give me some different set of laws for spacetime geometry that will enable you to construct an "accelerated frame" that includes both R and F and meets *all* of your requirements (i.e., it has both R and F feeling the same proper acceleration, *and* has the spatial separation between them, as seen in the "ship frame", remain constant--remember that, if you accept the laws of relativity as they stand, this is logically impossible).

So please tell me: is it (A) or (B)? Until I know that, I can't really continue the discussion.

 

[PeterDonis]

SO I ask ; just looking at a Minkowski diagram of the FR system at a given moment do you see any geometric difference between it and an inertial frame?

I saw this on reading through your last post again, and I wanted to comment on it, because there might be another point of misunderstanding lurking here. You can't change the geometry by changing reference frames. The "geometry" is a fixed, four-dimensional object. For example, in the scenario we've been discussing, the "geometry" is Minkowski spacetime. This is a definite four-dimensional object, which is the same regardless of what coordinates we use ("station frame" or "ship frame") to label points on it, just as the Euclidean plane is a definite two-dimensional object, regardless of what coordinates we use ("Cartesian" or "polar") to label points on it.

So when you ask if there is any "geometric difference" between the "ship frame" and an inertial frame, the answer is that of course there isn't: the geometry is the same no matter what frame you choose, just as the distance between New York and Chicago is the same no matter what coordinate system we use for the surface of the Earth. The geometry may "look different" with different coordinates, just as the surface of the Earth "looks different" in a Mercator projection than it does on a globe. But that's not because of any change in the geometry; any invariant, such as the distance between two points on the Earth's surface, or whether two events are simultaneous to a given observer in Minkowski spacetime, will be the same no matter what coordinate system we are using. It is the invariants that really constitute "the geometry", not the particular coordinates we assign in particular reference frames to events.

How does all that relate to what we've been discussing? Well, consider the statement we're having trouble agreeing on:

* Events R99 and F100 are simultaneous in the station frame, but they are *not* simultaneous in R's frame at R99, or in F's frame at F100.

The reason they are not simultaneous to R at R99 or to F at F100 is simple geometry: if we draw R's line of simultaneity through event R99, that line will *not* pass through event F100, and if we draw F's line of simultaneity through event F100, that line will *not* pass through event R99. Both of these facts follow from the fact that R and F are moving relative to the station frame at events R99 and F100; their motion means that their lines of simultaneity through those events are "tilted" up and to the right compared to the station frame's line of simultaneity, which is horizontal and passes through both R99 and F100 (because those events *are* simultaneous in the station frame).

In short, R's line of simultaneity through event R99, and F's line of simultaneity through event F100, are *different lines*. That is a geometric fact, an invariant, and no amount of jiggering with coordinate systems can change it, just as no amount of jiggering with coordinates on the Earth's surface can change the fact that the Equator and the Tropic of Cancer are different lines. So it doesn't matter how you try and construct a "ship frame" for R and F; as long as the geometry of Minkowski spacetime is what it is, you can't make R99 and F100 simultaneous to R and F, because you can't make two different and distinct lines coincide by changing reference frames.

 

[Austin0]

=PeterDonis;2347051]
You will have noticed that I did not use the term "ship frame" above. That is because there is no one single "ship frame" that you can use at both R99 and F100--you have to specify
There are several points here:

(1) Yes, Einstein did discuss the "twin paradox" from the point of view of the traveling twin, but his conclusion was the same: the stay-at-home twin (the one that remains on Earth) ages more than the traveling twin. The fact that, according to the traveling twin, the Earth was "accelerating", did *not* change the conclusion. In relativity, you can analyze a situation from whatever frame you want, but answers about invariants (for example, which twin ages more when they come back together) are the same no matter what frame you use to obtain them.

I did not suggest that Einsteins use of an accelerated frame produced different results.
I was simply suggesting that he was willing to consider an accelerating system as a frame.
AS I mentioned before and JesseM also pointed out, the two ships could just as well be a single long ship. You seemed to be saying that it was necessary to consider the two ships or comparably both ends of one ship as two distnctly separate frames.

(2) If you want to discuss what relativity says about a particular situation, then you *do* need to be "limited by predetermined definitions" as far as the laws of relativity and the geometry of Minkowski spacetime are concerned. If you want to consider alternatives to those definitions, then you're considering alternative theories to relativity--or you're considering spacetimes other than Minkowski spacetime. That's not what I thought this thread was about; I thought you were asking what relativity, the theory as we have it, says about the situation of two accelerating spaceships in Minkowski spacetime. And in any case I think that situation, the one I've just described, is one we should all agree on before we start discussing more complicated ones.

Absolutely. If you see some specific point where I have deviated from the laws or procedures of relativity, please point them out. If you feel that my suggestion of considering the system as a single frame is going outside the laws of relativity then by all means we will drop it.
Would you disagree if I said that this question goes beyond the explicit parameters of fundamental postulates and maths of SR and extends into the realm of speculative physics??
Specifically:
1) Length contraction. The math gives a complete description of the quantitative effect between inertial frames but does not provide predictions regarding accelerating frames.
Eg. JesseM's assertion that it is not an inherent result of motion and would not occur in a uniformly accelerated system. I am not making any judgement regarding the reality of this , only pointing out that this is an area of speculation, not merely applying known principles of SR
Eg. JM's assertion that acceleration would have no effect on clock synchronization.
Would you disagree that this is a speculation on the mechanics of clock desynchronization?? Physics.
If this is the case is it not possible that the Lorentz math does not apply as usual with regard to relative clock desynchronization between frames?
I.e. It would appear that this then describes a situation where there is a system moving at relative velocity but maintaining its original synchronization , the same synchronization as the inertial refernce frame.
Perhaps we have different interpretations and understanding of synchronization and simultaneity?

(3) Acceleration to relativistic velocities is *not* "without empirical input". For example, every particle accelerator that's ever been built has subjected huge numbers of subatomic particles to huge accelerations and relativistic velocities, and those particles have exhibited *every* phenomenon that relativity says they should: time dilation, length contraction, clock desynchronization, the whole enchilada. We also have tested relativity with macroscopic objects: the GPS system, for example, wouldn't work if our understanding of both special and general relativistic effects on orbiting objects in the Earth's gravity well were not correct.

I was aware of particle acceleration and the effects regarding dilation. But would be very interested to learn of the results regarding clock desynchronization as it relates to particles. Actually I have searched for information about particle bunching on the net but have found only references but no explicit explanation regarding how it relates to Lorentz contraction. Any links?
If there is observable clock desynchronization does that indicate that it is not a matter of clock convention but is an intrinsic result of acceleration?

In other words: I believe that the laws of relativity are valid when applied to the scenario we're talking about (the one I defined in detail in my last post). If you don't, or if you even want to consider the possibility that they're not, then there's not much point in having a discussion unless you can come up with some alternative theory that matches all the experimental evidence but gives different results when applied to the situation we're talking about. Trying to speculate about what might happen if the physical laws we know about are violated, without substituting some other set of laws to guide speculation, is, IMHO, not very useful.

100% agreement

Btw, there is one other postulate I haven't mentioned, which is part of the "laws of relativity" but which deserves to be mentioned separately. This is http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html" .
I am very familiar with the hypothesis and assumed it was a given not neccessitating mention.

Given such a limited selection I would of course choose A I am definitely not working with any premises outside SR ,of that I am sure.

 

[Austin0]

Originally Posted by Austin0
SO I ask ; just looking at a Minkowski diagram of the FR system at a given moment do you see any geometric difference between it and an inertial frame?

I saw

I saw this on reading through your last post again, and I wanted to comment on it, because there might be another point of misunderstanding lurking here. You can't change the geometry by changing reference frames. The "geometry" is a fixed, four-dimensional object. For example, in the scenario we've been discussing, the "geometry" is Minkowski spacetime. This is a definite four-dimensional object, which is the same regardless of what coordinates we use ("station frame" or "ship frame") to label points on it, just as the Euclidean plane is a definite two-dimensional object, regardless of what coordinates we use ("Cartesian" or "polar") to label points on it.

So when you ask if there is any "geometric difference" between the "ship frame" and an inertial frame, the answer is that of course there isn't: the geometry is the same no matter what frame you choose, just as the distance between New York and Chicago is the same no matter what coordinate system we use for the surface of the Earth. The geometry may "look different" with different coordinates, just as the surface of the Earth "looks different" in a Mercator projection than it does on a globe. But that's not because of any change in the geometry; any invariant, such as the distance between two points on the Earth's surface, or whether two events are simultaneous to a given observer in Minkowski spacetime, will be the same no matter what coordinate system we are using. It is the invariants that really constitute "the geometry", not the particular coordinates we assign in particular reference frames to events.

How does all that relate to what we've been discussing? Well, consider the statement we're having trouble agreeing on:

* Events R99 and F100 are simultaneous in the station frame, but they are *not* simultaneous in R's frame at R99, or in F's frame at F100.
The reason they are not simultaneous to R at R99 or to F at F100 is simple geometry: if we draw R's line of simultaneity through event R99, that line will *not* pass through event F100, and if we draw F's line of simultaneity through event F100, that line will *not* pass through event R99. Both of these facts follow from the fact that R and F are moving relative to the station frame at events R99 and F100; their motion means that their lines of simultaneity through those events are "tilted" up and to the right compared to the station frame's line of simultaneity, which is horizontal and passes through both R99 and F100 (because those events *are* simultaneous in the station frame).

In short, R's line of simultaneity through event R99, and F's line of simultaneity through event F100, are *different lines*. That is a geometric fact, an invariant, and no amount of jiggering with coordinate systems can change it, just as no amount of jiggering with coordinates on the Earth's surface can change the fact that the Equator and the Tropic of Cancer are different lines. So it doesn't matter how you try and construct a "ship frame" for R and F; as long as the geometry of Minkowski spacetime is what it is, you can't make R99 and F100 simultaneous to R and F, because you can't make two different and distinct lines coincide by changing reference frames.

Hi PeterDonis There is no question that the scope of my ignorance is vast.
On the other hand I have some grasp of area of discussion, .
SO I am afraid you have been flogging a dead horse in this case.
Here I asked a very simple and specific question and it appears that somehow you have interpretated it into indicating having no idea whatsoever of the function of coordinate systems or Minkowski space in particular. In the same post you are referring to here, I gave specific instances of the application of Lorentz desynchronization. Specific responces to your earlier post regarding this. SO if you think my application is wrong that is fine , simply show me. But it is not that we disagree that it would occur.
It seems to me that we need to know our respective interpretations of the meaning of lines of simultaneity or we are going to have confusion as to what we are talking about?
What do you think? thanks

 

[PeterDonis]

Austin0:

I apologize if I've given a wrong impression; I'm sure I've been saying much more than I needed to about how the mechanics of SR work. But some of the things you've been saying have suggested to me that you either don't really know or aren't ready to accept just how much those mechanics constrain the situation we're talking about. That's why I wanted to make absolutely sure (a) that I was stating everything explicitly and correctly, in a way you agreed with, even the simple stuff, so we'd be certain to have a common understanding of the scenario; and (b) that you were willing to accept SR as a valid theory. You've said you are; that makes everything a lot easier.

You said:

1) Length contraction. The math gives a complete description of the quantitative effect between inertial frames but does not provide predictions regarding accelerating frames.

and you later on made similar comments about other aspects of SR, such as clock desynchronization. All of these comments are incorrect. The math *does* provide predictions regarding accelerating frames, because of the clock postulate. At any event on an accelerating worldline, we can define a "momentarily comoving inertial frame", or MCIF; this is just an inertial frame in which the accelerating object, at that event, is momentarily at rest. The clock postulate then tells us that *all predictions about what would be observed in the MCIF at a given event, are also predictions about what will be observed by the accelerating observer at that event.*

By the above, since events R99 and F100 are not simultaneous in R's MCIF at R99, or in F's MCIF at F100, they are not simultaneous in R's and F's "accelerating frame" either. That's all there is to it.

(Btw, when I said in my last post that the geometry doesn't change with a change of reference frame, that it's a fixed, four-dimensional object, and that if the two lines of simultaneity are distinct in one frame, they are distinct in all frames, I was trying to say the same thing I just said above in different words. All these things tie together in a logical whole.)

Before I go any further, I'd like to have your input on what I've just said. Bear in mind where I'm coming from: what I've just said is a logical consequence of SR, which you've said you accept. The general principle I gave, on how to deal with accelerating observers, is not something that's just added into the rest of SR, and could be changed while holding the rest of the theory fixed; it's logically linked to a lot of other things, and all those other things would have to change if the principle changed. (For example, I don't see how you could possibly change the rules for accelerated observers to make events R99 and F100 simultaneous in the "ship frame" without also making them no longer simultaneous in the station frame--this is another way of saying what I was saying above about the geometry being fixed.)

That principle is also essential for constructing the general theory of relativity--without it, GR would not work. It also has been experimentally confirmed to a high degree of precision; check out the two links I gave in post #28, which I'll give here again (I admit that in post #28 I didn't really make them stand out):

http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html"

http://relativity.livingreviews.org/Articles/lrr-2006-3/"

Also, the Usenet Physics FAQ has a short page on http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html" .

 

[Austin0]

Re: Is Distance/space Lorentz contracted?

Austin0:

1) Length contraction. The math gives a complete description of the quantitative effect between inertial frames but does not provide predictions regarding accelerating frames.
and you later on made similar comments about other aspects of SR, such as clock desynchronization. All of these comments are incorrect. The math *does* provide predictions regarding accelerating frames, because of the clock postulate.

This iseems like a case of pulling statements out of context , reinterpreting them and then declaring them wrong.

The complete context was:
Originally Posted by Austin0
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.

Originally posted by Austin0 :
1) Length contraction. The math gives a complete description of the quantitative effect between inertial frames but does not provide predictions regarding accelerating frames.
Eg. JesseM's assertion that it is not an inherent result of motion and would not occur in a uniformly accelerated system. I am not making any judgement regarding the reality of this , only pointing out that this is an area of speculation, not merely applying known principles of SR
Eg. JM's assertion that acceleration would have no effect on clock synchronization.
Would you disagree that this is a speculation on the mechanics of clock desynchronization?? Physics.
____________________________________________________________________________
So maybe you would like it better in the form.
The Lorentz math provides a complete description of the quantitative effect of contraction which would seem to apply through the clock hypothesis but JesseM has asserted it does not apply
The math provides a complete description of relative clock desynchronization but it has been asserted that this does not apply..
It was not my assertion or my belief that they would not apply. I made that even clearer in a previous post..

(3) Acceleration to relativistic velocities is *not* "without empirical input". For example, every particle accelerator that's ever been built has subjected huge numbers of subatomic particles to huge accelerations and relativistic velocities, and those particles have exhibited *every* phenomenon that relativity says they should: time dilation, length contraction, clock desynchronization, the whole enchilada. We also have tested relativity with macroscopic objects: the GPS system, for example, wouldn't work if our understanding of both special and general relativistic effects on orbiting objects in the Earth's gravity well were not correct.

Response Austin0 I was aware of particle acceleration and the effects regarding dilation. But would be very interested to learn of the results regarding clock desynchronization as it relates to particles. Actually I have searched for information about particle bunching on the net but have found only references but no explicit explanation regarding how it relates to Lorentz contraction. Any links?
If there is observable clock desynchronization does that indicate that it is not a matter of clock convention but is an intrinsic result of acceleration?
____________________________________________________________________

Again i made that statement in the context of this discussion. I had made specific references before to electromagnetic acceleration of macro rods and tubes, we are talking about systems of clocks and rulers. DO you really think I could not be aware of the muon results etc etc etc etc or that I could possibly doubt their validity?

At any event on an accelerating worldline, we can define a "momentarily comoving inertial frame", or MCIF; this is just an inertial frame in which the accelerating object, at that event, is momentarily at rest. The clock postulate then tells us that *all predictions about what would be observed in the MCIF at a given event, are also predictions about what will be observed by the accelerating observer at that event.*

____________________________________OK certainly no disagreement over ICMIF's. But my memory is that the postulate goes far beyond that. Besides asserting that there would not be any dilation due to acceleration itself, it basically said that the Lorentz maths would apply on an instantaneous basis as usual.

By the above, since events R99 and F100 are not simultaneous in R's MCIF at R99, or in F's MCIF at F100, they are not simultaneous in R's and F's "accelerating frame" either. That's all there is to it.

Well you seem quite sure but I don't get how you jumped from the simple assumption of an MCIF to events not being simultaneous..
My understanding of lines of simultaneity is: They accurately graph relative clock times and locations between frames. That the intersections with the rest frames coordinate locations have direct spatial and temporal interpretation for both frames. And that observers will always agree on these events.
Exactly as if observers with clocks were actually present and cojacent in both frames. That multiple lines of simultaneity for disparate locations in a frame are simply different segments of a single continuous line which is actually congruent with the x' axis The slope being merely a result of convention. The Minkowski version of JesseM's clocks and rulers. So I don't quite see why you seem to think that they would in some way indicate loss of simultaneity within the frame itself but maybe i am not understanding what you are saying.
________________________________________________________________
(Btw, when I said in my last post that the geometry doesn't change with a change of reference frame, that it's a fixed, four-dimensional object, and that if the two lines of simultaneity are distinct in one frame, they are distinct in all frames, I was trying to say the same thing I just said above in different words. All these things tie together in a logical whole.)
This is exactly the reason I specifically asked you if you agreed that the instantaeous diagram at the moment of colocation with the stations was geometrically identical to, [indistinguishablefrom] a diagram of an inertial frame and you agreed that it was. In that case it is axiomatic that from the stations perspective the ships [or ship] are going to be out of synch.[not simultaneous] and that the ships are going to disagree. It is also axiomatic ,explicitly stated in the fundamental postulates that there is no real or absolute temporal meaning to either view. That there is no real meaning to any evaluations of simultaneity regarding spatially separated events between two frames.
I also fail to understand how if the geometry is the same you could from this geometry derive completely different conclusions from the normal ones?
So if you want me to agree [as it appears] that the simultaneity of the stations is the real true, simultaneity and that therefore it is obvious that the ships simultaneity is false ,well I have a real problem with that as it goes completely against my understanding of the basic postulates.
So if you would care to let me know what you mean by simultaneity and how you interpret lines of same it might be helpful. Or how you see my understanding is in error.

In my mind this complete question is much more complex and this query was only an attempt to get new insight into a limited aspect of it.
It involves fundamental physics as it applies to the Lorentz effects ,acceleration ,clock theorem, EP , Born acceleration .
And the question of explaining the anisotropic time dilation as described by the Rindler implementation of Born acceleration as being due to relative velocity between the front and the back. An explanation apparently in response to the conflict between the implicit dilation due to acceleration, and the clock postulate.
So I have a simple question for you. Do you know that the physics as we know it supports the idea that a constantly accelerated rod or spring would not only sustain significant compression or expansion, but that this would continue to increase over time without an increase in acceleration rate. And further would be a permenant condition after the termination of acceleration?
I do know that I have read both engineers and physicists who did not agree.

What do you think about the apparent conflict between the Rindler dilation and EP ?
If it is assumed that Einsteins elevator has relatively dilated clocks at the floor then, over time, the proper time discrepancy wrt the ceiling clocks would have to increase and the measurement of light speed would become increasingly anisotropic. Registering greater and greater speeds from the ceiling to the floor and lesser the other direction.

Thanks

 

[PeterDonis]

Austin0:

I do agree with you that there are plenty of complexities in how to handle accelerated objects in relativity--but the scenario I've been discussing doesn't address them. For example: it does not address Born rigidity or Born acceleration; it does not address the complexities involved with Rindler coordinates or Rindler observers, or what happens to the speed of light in accelerated frames; it does not address any of the actual physics of how objects respond to stresses, such as the stresses imposed by acceleration. All of that stuff is irrelevant in this particular scenario, because those complexities involve how we determine the worldlines of the observers we're interested in, and in the above scenario, we've already specified those worldlines (they're specified by the condition that both observers experience the same constant proper acceleration, but are separated in space, in the station frame, by one "station separation"). And once you've done that, in any frame, in Minkowski spacetime, you have already implicitly answered every question that can be asked about how any observer would observe events on those worldlines.

I'm not unwilling to discuss all the other complexities; I'm just not ready to discuss them until we have agreement on the simple scenario I specified above, and in particular on the proposition that we currently don't seem to agree on. The simple scenario involves nothing except relativistic kinematics, and without agreement on that, the complexities will never fall into place.

Let me briefly restate the scenario: we have a line of stations, numbered 1, 2, 3, etc., and we have two observers, R and F, who start out at adjacent stations (#1 and #2) and both experience the same constant proper acceleration. Define Rn as the event at which R passes station #n (n >= 1), and Fn as the event at which F passes station #n (n >= 2). We are discussing the proposition:

* In R's frame, or in F's frame, events Rn and Fn+1 are *not* simultaneous, for all n > 1. (They *are* simultaneous for n = 1--i.e., R1 and F2 are simultaneous--because those events happen at t = 0, when R and F are at rest in the station frame. But this is the *only* such pair for which that is true.)

This proposition is the one we seem to disagree about. You said:

Well you seem quite sure but I don't get how you jumped from the simple assumption of an MCIF to events not being simultaneous.

I'll explain how in a moment.

That multiple lines of simultaneity for disparate locations in a frame are simply different segments of a single continuous line which is actually congruent with the x' axis.

This is *not* correct. Distinct lines of simultaneity indicate distinct, different clock readings. Consider the two lines of simultaneity we've been talking about, which I'll define names for as follows:

S-Rn: R's line of simultaneity through event Rn, with n > 1.

S-Fn+1: F's line of simultaneity through event Fn+1, with n > 1.

These two lines of simultaneity are parallel--they have the same slope--but they are not the same line. You appear to agree on that. But that means that, in either R's MCIF or F's MCIF, the clock readings associated with S-Rn and S-Fn+1 are *different*. They have to be, because they intersect the t-prime axis (the "time" axis of either MCIF) at different events. If we did a Lorentz transformation into either R's MCIF at Rn (making Rn the new origin), or F's MCIF at Fn+1 (making Fn+1 the new origin), only one of the two lines of simultaneity, S-Rn (for R's MCIF) or S-Fn+1 (for F's MCIF), would coincide with the new x-axis; the other would *not*. It would be parallel to the new x-axis, yes (indicating that it was a line of constant time, as expected), but it would be either *above* the new x-axis (if we transformed into F's MCIF at Fn+1) or *below* the new x-axis (if we transformed into R's MCIF at Rn).

So, by the above, the events Rn and Fn+1, with n > 1, *cannot* be simultaneous in R's or F's MCIF, because in either MCIF they lie on different lines of constant time, and will be assigned different time coordinates. Therefore, by the clock postulate, they can't be simultaneous in R's or F's accelerated frame either.

You observed that I haven't yet described how things look from the accelerated frame; that's true. But I don't need to in order to prove that events Rn and Fn+1, n > 1, can't be simultaneous in that frame (for either R or F). And if we're not in agreement on that proposition, any description I give of how the accelerated frame looks will seem wrong to you, because I'll be basing it on the truth of that proposition. I have no choice, because the proposition I've stated, as I've said, is a logical consequence of relativistic kinematics, so it has to figure in any description of how things look from an accelerated frame.

 

[Austin0]

Original post Austin0
That multiple lines of simultaneity for disparate locations in a frame are simply different segments of a single continuous line which is actually congruent with the x' axis.

This is *not* correct. Distinct lines of simultaneity indicate distinct, different clock readings.

Hi PeterDonis Well here seems to be an area where I may be wrong in my understanding of lines of simultaneity.
Before going into how I am misinterpreting them I would like a little clarification.
I made that statement in the general context of standard interpretation regarding inertial frames. Persuant to the apparent agreement as to the identicallity of the geometry under consideration and an inertial frame.
So is your quote here applicable to inertial frames ?
Thanks

 

[PeterDonis]

Before going into how I am misinterpreting them I would like a little clarification.
I made that statement in the general context of standard interpretation regarding inertial frames. Persuant to the apparent agreement as to the identicallity of the geometry under consideration and an inertial frame.
So is your quote here applicable to inertial frames ?
Thanks

Yes, it is. Each line of simultaneity, in a given inertial frame, is just a line parallel to the x-axis of that inertial frame. Each such line has its own time coordinate in that frame--which is just the time coordinate of the event where that line intersects the t-axis of that frame. All events that lie on a given line of simultaneity in a given frame have the time coordinate of that line. So events that lie on different lines of simultaneity have different time coordinates.

(Edit: The x-axis itself, for a given inertial frame, is just the line of simultaneity with the time coordinate t = 0 in that frame.)

 

[Austin0]

--------------------------------------------------------------------------------

Yes, it is. Each line of simultaneity, in a given inertial frame, is just a line parallel to the x-axis of that inertial frame. Each such line has its own time coordinate in that frame--which is just the time coordinate of the event where that line intersects the t-axis of that frame. All events that lie on a given line of simultaneity in a given frame have the time coordinate of that line. So events that lie on different lines of simultaneity have different time coordinates.

(Edit: The x-axis itself, for a given inertial frame, is just the line of simultaneity with the time coordinate t = 0 in that frame.)

Having thought it over I see that I was wrong in the statement as I meant it.
I was forgetting that a ICMIF was not a true inertial frame.
But I did not mean it in the sense that you seem to talking about here either.

Original post Austin0
That multiple lines of simultaneity for disparate locations in a frame are simply different segments of a single continuous line which is actually congruent with the x' axis.

This is *not* correct. Distinct lines of simultaneity indicate distinct, different clock readings.

I didn't mean that the particular lines at a given moment indicated the same clock readings which is self-evidently false. I meant that from the F frames perspective they were different segments [i.e. different time slices] but that every event [colocation of positions and clocks] that appears on one line will also, at some point in time, appear on the other line.
That over time the set of these events is singular and represents a single set of clocks ,congruent to the x' axis in the F' frame.
But I can see that this does not pertain to an accelerating system because such a system has a dynamic metric. So one line does in fact represent a unique set of clock times and locations due to a change of clock rate during the interval separating them.
If I am still incorrect in my interpretation above let me know.

 

[Austin0]

=PeterDonis;2353089]Austin0:

I do agree with you that there are plenty of complexities in how to handle accelerated objects in relativity--

I'm not unwilling to discuss all the other complexities; I'm just not ready to discuss them until we have agreement on the simple scenario I specified above, and in particular on the proposition that we currently don't seem to agree on. The simple scenario involves nothing except relativistic kinematics, and without agreement on that, the complexities will never fall into place.

Well PeterDonis, at the prospect that you might discuss all the other complexities, I would probably agree to being the real shooter of John Lennon. :-}
Aside from great knowledge, you seem to have a rare degree of objectivity. The ability to play the "devils advocate" as you put it.
Although in my case so far, there seems to be a tinge of the devils prosecutor. ;-(
So I am a little befuddled by the course of this discussion and still unclear on what it is I am supposed to agree to.
I have tried to respond to you point by point, but so far I have no response from you, aside from the few points that have been somewhat taken out of context, reinterpreted or scanned for subtext indicating doubt of SR or wild speculative alternate theories. I am not quibbling over this and understand your desire to have a firm base of understanding but it leaves me in doubt as to whether you actually read the rest.


Originally Posted by Austin0
Equal proper acceleration must result in equal distance and time. But exactly the same obvious logic is applicable from the ships frame.

I've been watching this thread for a few days now but haven't jumped in because I've been waiting for you to say something short and to the point that gets at the heart of the point of disagreement between you and JesseM. I think the above quote is it. PMFJI, but I'd like to give my take on why the above quote doesn't apply in this situation.

Austin0 Post 9
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?

Originally Posted by Austin0
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.

So if you are saying that logic and physics does not lead to productive insights within the parameters I wanted to examine the situation from, then at this point, I can't argue. It has certainly not led me to any useful conclusions and it is even more certain that it is not a productive format for discussion with those of this forum.

#1 Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth. It's

#2 * Do you want both ships to experience equal proper acceleration?* Or do you want both ships to remain at a constant spatial separation, as seen by the ships?

You *can't* satisfy both. Which one you pick will determine what specific scenario you're talking about, and until you do that, it's hard to have a discussion because you may be talking about different scenarios.

It seems to me there is a bit of contradiction here.

#1 States there is no assumption of physics involved.

#2 Makes all kinds of physical assumptions. Born acceleration seems to be based on several physical assumptions . Non-Born acceleration, as outlined by JesseM, also makes several physical assumptions. And the assumption that there is no other possibility other than these two is also a prediction of physics , no?

SO it looks like; having, a priori , made all possible physical assumptions regarding the situation, it resolves to a purely kinematic problem.
Well it certainly leads to definitive results of a sort.
But, as the original question was an attempt to circumvent physical assumptions beyond the known physics of acceleration, it pretty much negates that pursuit.
Or at least, certainly redirects it into an examination of these physical assumptions themselves.

So if all the above assumptions are to be taken as known facts and in addition the stations frames simultaneity is to be taken as the only relevant criteria; then the transformations can, of course, only produce one set of figures. There is no logic involved there, it is only a set of functions.
So if this is the agreement that you are asking for then you have it.

I do have a question though: The Lorentz math itself makes physical predictions which are integrated into the transform. Newton and Galilleo in one.
So the Born hypothesis says that the Lorentz math is insufficient to accurately describe the physics of accelerated motion without additional assumptions. JesseM has said it is not an accurate description of the physics of non-Born accelerated motion.
So how is expecting the math [as a transform] to make accurate coordinate predictions in this context , any different than expecting Galillean transforms to produce accurate predictions regarding relativistic motion between frames where Newtonian kinematics doesn't apply?

Thanks

 

[PeterDonis]

Austin0:

I'm going to respond to things in both of your last two posts here. You said:

I have tried to respond to you point by point, but so far I have no response from you, aside from the few points that have been somewhat taken out of context, reinterpreted or scanned for subtext indicating doubt of SR or wild speculative alternate theories. I am not quibbling over this and understand your desire to have a firm base of understanding but it leaves me in doubt as to whether you actually read the rest.

I did, but I've already explained why I haven't commented on most of what you said. Once again, let me reiterate that there are at least two separate questions involved here:

(1) Given a specific worldline (or worldlines) in Minkowski spacetime, how will observers traveling along that worldline (or worldlines) "see" other events?

(2) Given a description of a physical situation, how do we translate that description into a specification for one or more worldlines to which we can then apply our knowledge with regard to question #1 above?

So far I have only been talking about question #1, because your original specification of the scenario was enough to determine a pair of specific worldlines in Minkowski spacetime. Question #1 is what I'm talking about when I say that a question is "purely kinematic": given a specific worldline in Minkowski spacetime, the question of how observers traveling along that worldline will perceive events is *entirely* determined by the specification of the worldline and the laws of SR. No "physical assumptions" other than those are involved.

In the course of this thread, you, and JesseM, and you in response to my posts, have also talked about a lot of things that *only* become involved when you're talking about question #2--in other words, when you haven't yet obtained a specification of a particular worldline or worldlines that an object or parts of an object will follow. In particular, the statements JesseM was making about whether or how acceleration would affect the length of bodies, how bodies would respond to stresses, whether length contraction would apply, etc., were talking about question #2, *not* question #1. All of those things are necessary to consider if you want to figure out, for example, what specific worldlines each end of a rod made of steel, let's say, would follow if subjected to a certain acceleration. But if you have already specified (as your original scenario did) the worldlines you want to consider, all that other stuff is irrelevant.

Right now, as I've said, I am *only* talking about question #1. So I'm simply not going to respond to things that don't involve question #1, or that mix in question #1 with other stuff. There's no point in my doing so until we have a clear understanding of question #1, by itself, with the specific worldlines specified in the scenario we're discussing.

In response to my statement that:

#1 Does the above argument convince you? I should add that this argument is purely kinematic; it doesn't make any assumptions about how objects respond to stresses under acceleration, and so forth.

...

#2 * Do you want both ships to experience equal proper acceleration?* Or do you want both ships to remain at a constant spatial separation, as seen by the ships?

You *can't* satisfy both. Which one you pick will determine what specific scenario you're talking about, and until you do that, it's hard to have a discussion because you may be talking about different scenarios.

You said:

It seems to me there is a bit of contradiction here.

#1 States there is no assumption of physics involved.

#2 Makes all kinds of physical assumptions. Born acceleration seems to be based on several physical assumptions . Non-Born acceleration, as outlined by JesseM, also makes several physical assumptions. And the assumption that there is no other possibility other than these two is also a prediction of physics , no?

There's no contradiction. My statement #1 was in reference to a specific scenario, the one in which you specified that both ships experience equal (constant) proper acceleration--which is enough to specify a worldline for each ship. In other words, #1 only involved question #1 above, and so, as I've said, is purely a matter of kinematics--the specific worldlines and the laws of SR.

My statement #2 was simply pointing out that the specification that both ships experience equal proper acceleration, and the specification that both ships will remain at constant spatial separation, as seen by the ships, are two *different* specifications; they result in two *different* pairs of worldlines. It is true that, in order to determine that, other "physical assumptions" may be involved, but that's part of question #2, which I'm not yet discussing. Even if it were to turn out (which it won't) that those two specifications, equal proper acceleration vs. constant spatial separation, *did* result in the same pair of worldlines, I would still not want to discuss that until we have agreement on question #1 by itself, dealing with the pair of worldlines that are specified by equal (constant) proper acceleration. If it helps, just suppose that I've withdrawn my statement #2 above from discussion until we are finished with question #1.

With regard to question #1, I want to first make sure that we are using the same definitions for terms. First, you said:

I was forgetting that a ICMIF was not a true inertial frame.

Yes, it is. An MCIF *is* an inertial frame, by definition; it's just that the accerating object is only at rest in that frame momentarily, because it's accelerating. But considered just as an inertial frame, an MCIF is no different from any other.

Also, you said:

I didn't mean that the particular lines at a given moment indicated the same clock readings which is self-evidently false. I meant that from the F frames perspective they were different segments [i.e. different time slices] but that every event [colocation of positions and clocks] that appears on one line will also, at some point in time, appear on the other line.

An "event", as I've been using the term (and as it's standardly used in relativity) means a single *point* in spacetime. In any given inertial frame, an event has specific coordinates t, x (and y, z too if we're using all three dimensions of space--we've just been using x in this thread to keep things simple). For a given inertial frame, no event can possibly appear on more than one line of simultaneity--the one that has the t-coordinate assigned to that event.

If you meant in your quote above the same thing by "event" that I mean, then your statement is simply false: events that appear on one line of simultaneity *cannot* appear on any other. If you were using "event" to mean something else, then please give an explicit definition.

Next, you said:

That over time the set of these events is singular and represents a single set of clocks ,congruent to the x' axis in the F' frame.
But I can see that this does not pertain to an accelerating system because such a system has a dynamic metric. So one line does in fact represent a unique set of clock times and locations due to a change of clock rate during the interval separating them.

I'm not sure exactly what you mean by this, given my uncertainty about how you're using the word "event" above. But let me state what I think you may be saying about accelerated frames in my own words:

* Given a worldline along which an observer experiences constant proper acceleration, each individual event on that worldline has a *different* MCIF. In other words: the MCIF at any given event on that worldline will be in motion relative to the MCIF at any other event on that worldline--no two MCIFs at different events on that worldline can be at rest relative to one another.

And finally, you said:

So if all the above assumptions are to be taken as known facts and in addition the stations frames simultaneity is to be taken as the only relevant criteria; then the transformations can, of course, only produce one set of figures. There is no logic involved there, it is only a set of functions.
So if this is the agreement that you are asking for then you have it.

First, I never said that "the stations frames simultaneity is to be taken as the only relevant criteria". I simply said that the station frame is the only frame in which events Rn and Fn+1 are simultaneous, for n > 1. But let me split that claim up, and take it in two stages. Let's first consider just inertial frames alone, without making any claims as to how the "accelerating frame" relates to them. Let's just consider the claim that the station frame is the only *inertial* frame in which events Rn and Fn+1 are simultaneous, for n > 1. In other words, do you agree to the following propositions? (I'm using the same names for events that I used in my last post.)

* Events Rn and Fn+1 are simultaneous in the station frame for all n > 1.

* Events Rn and Fn+1 are *not* simultaneous in R's MCIF at Rn, or in F's MCIF at Fn+1, for all n > 1.

I don't know that I'd say there's no "logic" involved, although maybe "mathematics" would be a better word--but mathematics is just logic using equations and symbols as tools. And again, please note that those two propositions don't talk about the "accelerating frame" itself--just the station frame and the two MCIFs, R's at Rn and F's at Fn+1--all inertial frames. I'm not asking here if you agree that the "accelerating frame" itself *has* to see things the same way at each event as the MCIF; I want to make sure first that you agree on how things will look in the MCIFs themselves. Since those are just inertial frames, there shouldn't be an issue, since you've already said you agree that relativistic kinematics gives correct predictions for how things look in inertial frames in relative motion. But I want to be sure before going on.

 

[PeterDonis]

So I am a little befuddled by the course of this discussion and still unclear on what it is I am supposed to agree to.

Maybe it would help if I responded to this statement as well. Here's why I came into this thread:

You (Austin0) raised the issue of an apparent logical contradiction between two propositions (to save time I'll state them using my more recent specification of the scenario):

(1) According to SR, observers R and F will *not* observe events Rn and Fn+1 to be simultaneous for all n > 1; those respective pairs of events will only be simultaneous in the station frame.

(2) However, the proposition that "equal acceleration leads to equal distances and times" would seem to imply that events Rn and Fn+1 *would* be simultaneous to R and F, for n > 1; and this appears to be confirmed by the fact that the proper time on R's clock for event Rn will be the same as the proper time on F's clock for event Fn+1, for n > 1.

(I know the above is not the only issue you raised; but as I've said, I believe that until the above issue is resolved, any discussion of the other issues you raise is premature.)

Now I believe SR to be a correct theory (within its domain of applicability--that is, as long as the effects of spacetime curvature are negligible, so that we can assume spacetime to be globally flat). That means that anything which is a logical consequence (or mathematical consequence, if you like) of the laws of SR along with whatever specific premises are assumed for a given scenario, I believe to be true. Given that, I'm quite sure proposition #1 above is not in error; the situation you posed is not a new one, and I've worked through it myself in detail in the past, and I've found plenty of references to confirm my solution. I'm also quite sure that the proper time on R's clock for event Rn *is* the same as the proper time on F's clock for event Fn+1; that's a simple consequence of the solution for proposition #1 (and anyway, you agree that it is, so it hasn't been in question).

So given the above, I believe that there has to be something wrong with the proposition that "equal acceleration leads to equal distances and times". I already know what *I* believe is wrong with that proposition: I stated it in my first post in this thread. "Equal acceleration" is an invariant (provided we mean "proper acceleration", the acceleration that would be measured by an accelerometer moving with the given observer, but we've already agreed on that), but distances and times are frame-dependent, so there can only be *one* frame in which equal accelerations can lead to equal distances and times, and we already know the station frame is that frame.

You weren't satisfied with that resolution of the issue, so I'm now trying to pinpoint exactly where our understanding diverges. Every question I've asked and you've answered has narrowed it down.

 

[Saw]

Please ignore this if it distracts you from your discussion. It’s just that I have followed it and I feel as if the discussion were incomplete without a practical background, i.e., a problem to solve. Trying to think of one, the example has reminded me of a similar discussion we’ve had.

Imagine that the ships (we keep calling them R and F) depart from the same place and fly in opposite directions. After some “whiles” with equal proper accelerations, each ship reaches its respective finish line. Both finish lines are, in the station frame, equidistant from the origin.

So a referee in the station declares that the arrivals are simultaneous and the race to be a draw.

However, in R’s MCIF, R has arrived at its finish line earlier than F at its own. In other words, when it reaches its finish line, a longer patch of the station’s frame has passed by R itself than by F, as per R's simultaneity line.

But F’s MCIF measures that F has arrived earlier! At that time, a longer patch of the station’s frame has passed by F than by R.

In spite of that, R’s and F’s MCIFs read the same proper time intervals when they reach their respective finish lines.

PeterDonis, would you agree that the solution of the problem (who wins the race) is a draw?

Austin0, if we all agreed that the solution, in accordance with orthodox SR, is a draw would you not mind that SR holds that in R’s and F’s MCIFs there is discrepancy with the station frame and between themselves as to simultaneity/distances traversed?

 

[PeterDonis]

PeterDonis, would you agree that the solution of the problem (who wins the race) is a draw?

If the definition of "who wins the race" is "who reaches their respective finish line first, as judged from the station frame", then yes; your statement of the scenario stipulates that they arrive at their finish lines simultaneously in the station frame.

However, you can't say categorically that that is "the" solution of the problem precisely because simultaneity is frame dependent. Any condition on what happens "first" when the respective events are spacelike separated has to include a specification of whose frame "first" is going to be judged from. Otherwise the condition is ambiguous.

 

[atyy]

If the definition of "who wins the race" is "who reaches their respective finish line first, as judged from the station frame", then yes; your statement of the scenario stipulates that they arrive at their finish lines simultaneously in the station frame.

However, you can't say categorically that that is "the" solution of the problem precisely because simultaneity is frame dependent. Any condition on what happens "first" when the respective events are spacelike separated has to include a specification of whose frame "first" is going to be judged from. Otherwise the condition is ambiguous.

I think this is the same scenario that Saw and I discussed a little while ago. If it is, I think Saw is right. Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

The coordinate independent way of selecting the station frame is to make R and F both carry clocks, and declare the winner the one who reaches his end point in least own proper time.

 

[PeterDonis]

The coordinate independent way of selecting the station frame is to make R and F both carry clocks, and declare the winner the one who reaches his end point in least own proper time.

Yes, this would be a good way of avoiding a frame-dependent specification, and under this definition, the race would be a draw.

Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

 

[atyy]

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

Indeed. R has the temerity to say that he won the race, even though he was at rest!

 

[Saw]

Yes, this would be a good way of avoiding a frame-dependent specification, and under this definition, the race would be a draw.

This is a way to phrase it with which I disagree. In this particular respect, at this more fundamental level, I think Atyy and I fully agree. I’ll explain myself. This is something which is at the core of SR. Someone might say it’s philosophical, but it seems to me totally physical. After all, physics is about solving problems, isn’t it?

With your statement, you seem to be implying that SR means that the “specification” of the race (the rule for determining who wins) can be arbitrary: if you choose a coordinate-independent quantity, you’re doomed to disagreement; if instead you manage to specify an invariant, then you’ll get agreement; but the choice is whimsical, there is no physical reason to prefer one over the other. If this were true, the significance of SR would be that a whole bunch of problems (those where the relevant events are non-colocal) have been shifted to the realm of problems without an objective solution.

To take an extreme example: this would amount to the same as saying that, in the classical scenario, the police has no objective reason for arresting you for driving faster than the speed limit… wrt the ground! You could always say that in your brother’s frame, who was co-moving with you, you were at rest! And you could challenge the traffic regulations on the grounds that they are arbitrary and discriminatory! That would not be right: it’s obvious that, in this context, the ground frame is the relevant reference. That does not mean that any other frame is not apt for predicting what is wrong and what is right. A spaceship flying by just has to measure the speed of the ground wrt it and your speed wrt it, apply the relativistic rule for combining velocities and thus conclude that you were creating danger on the road, which is the outcome that the rule tries to avoid, i.e., the spirit of the problem.

The same applies to this race example. You have to find the spirit of the problem, the practical objective: what do we want to reward by saying R or F won the race? Technology of the spaceship, ability in driving it? It seems to me that both proper acceleration and proper time are good indications thereof. So the choice of these indicators is not arbitrary but a must.

Of the 3 frames, the station frame is the only inertial frame, because the inertial frames associated with R and F are ambiguous, since R and F both accelerated. A frame in which R is always at rest is non-inertial.

This is true, but I don't think it disqualifies R or F from having a legitimate "point of view" (if you don't like the word "frame") according to which they are at rest and other objects are moving. R's or F's point of view will not "work the same" in all respects as that of an inertial observer, but that's OK; as long as spacetime is flat, SR can be used to work out how R's or F's point of view will work.

Here instead I tend to rather agree with PeterDonis. If I have understood PeterDonis well, with the concept of MCIF you can always convert R and F frames into an inertial frame at a specific event. Once this is done, you find that they disagree with the station frame and between themselves in terms of simultaneity and hence in terms of distance traversed by R and F. However, they do not disagree in terms of who won the race. They understand the spirit of the problem and so do not trust their simultaneity judgments on face value and work a little further. For example, what R should do is finding F’s proper time = dt^2 – dx^2. If the result is the same as the reading of his own clock, he agrees that the race was a draw.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

 

[Austin0]

Austin0:

Yes, it is. An MCIF *is* an inertial frame, by definition; it's just that the accerating object is only at rest in that frame momentarily, because it's accelerating. But considered just as an inertial frame, an MCIF is no different from any other.

OK it was simply to this, not inconsequential difference ,that I was refering. That regarding the frame it was only transitory and was in effect a series of MCIF's with differing metrics

#1* Events Rn and Fn+1 are simultaneous in the station frame for all n > 1.

#2* Events Rn and Fn+1 are *not* simultaneous in R's MCIF at Rn, or in F's MCIF at Fn+1, for all n > 1.

reduced to this,disregarding all other considerations, there has never been any disagreement. I am sure I have explicitly recognized this more than once.Two frames cannot agree on simultaneity regarding any two events
On #1 alone,, #2 is an automatic conclusion.

I don't know that I'd say there's no "logic" involved, although maybe "mathematics" would be a better word--but mathematics is just logic using equations and symbols as tools.

Perhaps I phrased that badly. Obviously the math was derived with impeccable logic, both mathematical and mental. I meant that having reached the point of applying the math, you are already beyond the point of exercising your own logical mind toward the question.

And again, please note that those two propositions don't talk about the "accelerating frame" itself--just the station frame and the two MCIFs, R's at Rn and F's at Fn+1--all inertial frames. I'm not asking here if you agree that the "accelerating frame" itself *has* to see things the same way at each event as the MCIF; I want to make sure first that you agree on how things will look in the MCIFs themselves. Since those are just inertial frames, there shouldn't be an issue, since you've already said you agree that relativistic kinematics gives correct predictions for how things look in inertial frames in relative motion. But I want to be sure before going on

Approached from the point of #'s 1 and 2 above, it appears to me to be straight forward application. Just looking at it mentally, it seems to me , we have R now behind ,so its clock is running behind ,exactly opposite the normal situation. F is ahead, with clocks running ahead.
They turn off the engines and resynch their clocks so that R is running ahead and F running behind and we have an inertial frame like any other ,except that it is exactly the same size as it was at rest in its original frame as observed in that original frame. And will continue to be so through any period of extended acceleration. As this is assumed to apply to a single ship as well , this presents an extraordinary picture. A frame that continues to attain greater and greater instantaneous velocities but remains the same length as observed from an inertial frame.
If you have done the math I am sure it all works out fine and I would like to hear any points I am missing.
So what is your interpretation of this picture?? Consider a single long ship.
What immediately occurs to me : The ship is extended beyond its rest length and Lorentz contracted back to exactly the same length?
or
The ship is not extended because Lorentz contraction is a result of specific application of force? But simultaneity, in effect , extends it so the stations are relatively contracted?
Or there is a flaw in the original logic [my own] and this does not describe an actual possible reality
All of these seem to have real problems so what is a better one??
And the real questions behind the query: Does the Lorentz contraction apply simply to material bodies or does it apply to extended frames and the space between bodies?
Is the phenomenon a purely physical one of forces, with implied stresses etc. or is it simply a stressfree result of the geometry of spacetime?
Is it fundamentally a temporal effect that is perceived as a spatial effect??
This one I have spent a lot of time playing with.
These questions do not seem to be clarified by this exercise. Maybe I should have left out the stations and simply gone directly to the physics??
thanks

 

[PeterDonis]

With your statement, you seem to be implying that SR means that the “specification” of the race (the rule for determining who wins) can be arbitrary: if you choose a coordinate-independent quantity, you’re doomed to disagreement; if instead you manage to specify an invariant, then you’ll get agreement; but the choice is whimsical, there is no physical reason to prefer one over the other.

I didn't mean to imply that there was no reason to prefer the invariant way of specifying who wins the race (i.e., preferring "lower elapsed proper time" as the criterion). Of course there is: invariants are the things everybody can agree on, so you avoid the ambiguity of specifying things in frame-dependent ways. I was just pointing out that you do, in fact, have to specify the invariant; you can't just say "whoever reaches the finish line first wins", without qualification, and expect everybody to agree, because those words are ambiguous. Saying "whoever has the least elapsed proper time when they cross their designated finish line, wins" is unambiguous.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

Of course R and F can each calculate what the other's proper time elapsed would be, given the appropriate data. Anyone can. That's not in question.

 

[atyy]

Here instead I tend to rather agree with PeterDonis. If I have understood PeterDonis well, with the concept of MCIF you can always convert R and F frames into an inertial frame at a specific event. Once this is done, you find that they disagree with the station frame and between themselves in terms of simultaneity and hence in terms of distance traversed by R and F. However, they do not disagree in terms of who won the race. They understand the spirit of the problem and so do not trust their simultaneity judgments on face value and work a little further. For example, what R should do is finding F’s proper time = dt^2 – dx^2. If the result is the same as the reading of his own clock, he agrees that the race was a draw.

Or did I catch it wrong? Would R and F, in spite of being non-inertial, make those calculations? Well, that was your discussion, which I don’t want to interrupt. Just thought that having this practical background might be helpful.

I think what I said agrees with all of your points and PeterDonis's too.

 

[Saw]

I didn't mean to imply that there was no reason to prefer the invariant way of specifying who wins the race (i.e., preferring "lower elapsed proper time" as the criterion). Of course there is: invariants are the things everybody can agree on, so you avoid the ambiguity of specifying things in frame-dependent ways. I was just pointing out that you do, in fact, have to specify the invariant; you can't just say "whoever reaches the finish line first wins", without qualification, and expect everybody to agree, because those words are ambiguous. Saying "whoever has the least elapsed proper time when they cross their designated finish line, wins" is unambiguous.

No, that's not the point. You're saying that the reason to prefer one specification over the other is just unambiguity. Unambiguity is not sufficient. With this rationale we could also say that the race is won by the ship whose driver is a lady. That's also unambiguous. (Well, not always..., but you catch the point). And unambiguity is necessary at the level of the ultimate meaning of the chosen concept, but only at that level. In fact, the specification might perfectly be: "whoever arrives earlier in the station frame". A different thing is that the ship frames may reach the same conclusion (not as to simultaneity, but as to the solution of the problem) through their own measurements of proper time. But as I said I don´t want to interrupt the discussion. If it's to going to be controversial, let's leave it aside. If you want to contest this, you can do it in the thread Elegant Universe Example, where this subject is already advanced.

Of course R and F can each calculate what the other's proper time elapsed would be, given the appropriate data. Anyone can. That's not in question.

ok, please answer Austin0.