How are the twins distinguished?

[Mike_Fontenot]

What are "his own potential measurements", if not measurements on some set of rulers and clocks?

His measurements use only his own wristwatch.

[...] Also, for an event far away from him, even if the event was simultaneous with an event on his worldline when he was moving at speed v₁, he may not actually get light from the event until his is moving at v₂,
[...]

That's why I used the term "potential measurements". The proof involves showing what he WOULD measure IF he chose to permanently remain inertial at the speed v₁ ... there never would BE a different speed v₂ in that case. If the home twin constantly sends messages reporting her age, the traveler will be receiving messages from the instant he stops accelerating (and also before then, of course, but those aren't involved in the calculations). The traveler can take all the time he needs to make his measurements and do his calculations.

Mike Fontenot

 

[JesseM]

His measurements use only his own wristwatch.

Well, how does an accelerating observer define the time of distant events using his own wristwatch? Does look at the distance on some ruler, and then subtract (distance)/c from the time he got a signal from the event? If so, if there are multiple inertial rulers moving at different velocities, which does he choose? The one that's at rest relative to him when he receives the signal, or the one which has the property that when you subtract (distance)/c from the time he received it, it was at rest relative to him at that time on his watch? Again it seems like a matter of arbitrary convention.

That's why I used the term "potential measurements". The proof involves showing what he WOULD measure IF he chose to permanently remain inertial at the speed v₁ ... there never would BE a different speed v₂ in that case.

So if he receives a signal at time T on his watch when he is moving at v₁, does he calculate the time the event would have occurred in his frame if he had been moving at v₁ forever, or something else?

 

[Mike_Fontenot]

Well, how does an accelerating observer define the time of distant events using his own wristwatch? Does he look at the distance on some ruler, [...]
[...]

He doesn't use ANY ruler.

[...]
So if he receives a signal at time T on his watch when he is moving at v₁, does he calculate the time the event would have occurred in his frame if he had been moving at v₁ forever, or something else?

To understand exactly what calculations and observations the (formerly) accelerating observer must carry out, it's first necessary to answer the question "After the accelerating observer stops accelerating, how soon does he become 'an inertial observer', and exactly how soon can he legitimately perform the same calculations that a perpetually inertial observer can perform?". To do that requires a fairly lengthy argument that isn't appropriate or practical in this venue ... but it's all in the paper, for anyone who really wants to know.

Mike Fontenot

 

[stevmg]

You have missed out one of the 6 phases in the above list. It should be more like this:

1) acceleration from the Earth (let's say it is constant acceleration d²s/dt² = a)
2) trip speed at a constant rate: ds/dt is constant (call it v)
3) deceleration at constant rate from trip velocity to 0 (-d²s/dt² = -a)
4) continuing the negative acceleration in 3) above until a negative constant trip velocity is achieved -ds/dt = -v
5) trip speed at a constant rate: ds/dt is constant (-v)
6) deceleration or positive acceleration) from the trip velocity d²s/dt² = a, until the actual final speed of B wrt A is zero.

This makes it consistent with the scenario described in this Wikipedia page:

http://en.wikipedia.org/wiki/Twin_p...lt_of_differences_in_twins.27_spacetime_paths

The Wiki page gives a formula for the 2 constant velocity phases and the 4 accelerating phases.

d \tau=2T_c/ \sqrt{1+(aT_a/c)^2}+4\frac{c}{a}\, asinh(aT_a/c)

where T_a is the coordinate time of a single accelerating phase and T_a is the coordinate time of a single cruising phase.

The whole thing can be expressed in terms of velocity as:

d \tau=2T_c/\gamma+4\frac{cT_a}{v \gamma}\, asinh(v\gamma/c)

where v is the cruising velocity.

Both the equations above assume all the acceleration phases last the same time and the cruising phases last the same time.

We fixed that error (the returning constant velocity phase missing) in a prior post. We downloaded the Wikipedia article you referred to and we haven't got a clue on how to interpret it.

But, things being as they are, we will let that slide and just know that the "moving" traveler ages slower. That's good enough for me for now.

Thanks for all your help.

 

[Passionflower]

But your method also requires the onboard computer to be constantly figuring out the clock's position in some frame (which necessarily has requires judgments about simultaneity in that frame) and use that to adjust the rate of ticking, it can't just directly adjust the rate of ticking based on the measurements of the accelerometer, since as I pointed out your method requires that the clock's rate is continually increasing relative to proper time during the entirely inertial inbound leg.

I you read the article you would understand that the 'inertial clock' is not based on a frame of simultaneity. Why do you think it prompted me to attend members of this possibility if not for the difficulties in using frames of simultaneity in accelerating situations? I am just trying to help and thought it would be interesting for my fellow members to know about this alternative. Perhaps it is wasted on you, as you seem to be rather negative about it. No offense you can always ignore it!

Yes an on-board computer would be necessary, but so what? What's your point? You think it is too advanced for a spaceship that travels relativistically to have an on-board computer?

 

[Passionflower]

Notice the loop. I never suggested that, considering the loop.[ i.e. the round trip], that this would not be the case.

Ok so then what do you mean by differential time dilation if there is no loop?

Alright let's talk about acceleration and your potential misconceptions.

This is what you initially wrote:

The initial acceleration could in fact be a decceleration so if the home clock increased in rate wrt the ship proper time htis would not reflect reality as the ship clock could in fact be ticking faster than the Earth clock.

And then you wrote:

This also requires an assumption of absolute acceleration.
Given that the initial starting velocity is purely relative with no actual value this implies that that the resulting change in clock rate could be an increase or a decrease.

So consider two inertial twins T_home and T_travel at the same location. Explain to me how if T_travel undergoes proper acceleration and T_home stays inertial T_travel's clock rate can increase with respect to T_home's clock rate?

 

[Austin0]

Originally Posted by Austin0

Within the accelerating frame how would observers be able to tell at what point they were at rest with home and had started on the return trip to increase the home clock rate?

As the captain of the accelerating ship ( one that travels in the x-direction, stops and returns ) I can monitor my acceleration on the outgoing leg and calculate my relative velocity v wrt to home. At some point I give the order to reverse engines, and apply acceleration until I calculate that v = 0.

If you take my quote in context it was in responce to the idea that simply through accelerometer readings the clock could be computerized to calibrate the onboard "home" clock.
I didn't suggest that there was no way to calculate when the ship reached a state of rest wrt earth. I in fact explicitly described how this could be done through interferometer Doppler readings of a continuous signal from earth.
Can I assume you do understand that the decceleration/acceleration phase at turn around is actually one continuous acceleration in the direction of earth?
Not two discrete periods as far as onboard thrust and continuous accelerometer readings is concerned. That to determine the abstract relative point of rest wrt Earth requires additional reference whether it is calculation , DOppler or some other outside reference.
That the accelerometer will by itself give no clue as to this point.

This was the point of contention and misunderstanding here.
Would you disagree with any of this?

 

[stevmg]

Again to Passionflower: I have apologized in a prior post (#87) and via private message my apology for flippant statements earlier. As I stated to you (as well as the monitors of PF, if they are watching) that this form of communication will not be repeated to either you or anyone else in the future.

All I understand from all this is that once there is motion, there is a loss of time (time dilation) to the "moving" object in relation to the stationary object. When it is a simple trip at constant velocity, one can postulate that looking from the FR of the moving object, the "stationary" object is moving so the time dilattion occurs to that object (we have been referring to "that object" as "A" while the original moving object was designated "B.")

JesseM has shown that when you have a constant motion (velocity) away from an onject A followed by a constant velocity motion back to A, whether it be from two different objects (call the B and B') traveling at constant velocity but in opposite directions, or the same object B which turns around, with relation to A, the combined proper times of B and B' journey is less than the proper time of the interval that A experiences. This is a consequence of the hyperbolic relation of distance and time given by the Minkowski equations. Even if the same object experiences
1) acceleration
2) constant velocity
3) deceleration to zero
4) acceleration (or continuation, if you will, of the deceleration in 3)
5) constant velocity back to A
6) deceleration to zero so B rejoins A
B experiences less proper time than A. Even with the application of those complicated hyperbolic arcsine equations for proper time in aceleration/deceleration (which I, as yet, cannot do) you still wind up with less time evolved.

JesseM went on to show that choosing any other FR and sticking with it as the "stationary" FR that A still experienced more proper time than B.

Postulating Passionflower's statement (I hope I am correct here) that if you look strictly from B, A would "slow down." I don't know how one uses an FR which itself has acceleration in it, which would be the case if you chose B's FR. Einstein gave the example of the falling elevator and that to the man in it, he doesn't experience any difference than a man sitting still and not moving (no gravity, of course.) One thing is for sure - if you could choose B's FR as your central FR, A would accelerate... and A would be the one who slowed down but I don't know if you can do that. Can one choose an accelerating/decelerating FR and make that one central and make the universe move about it? As it has been stated, in a theoretically gravity free universe, one can do anything as Einstein talked about with his elevator man.

One interesting thing, though, if the man and elevator were accelerating together they would be under the influence of a gravity. A second person in an FR which was stationary would experience the pull of this gravity and the equal and opposite resitance or force as he/she stands on his stationary platfrom. Meanwhile, the elevator man is experiencing no gravity as he is falling and feels weightless. If he were chosen as the central FR, the guy on the platform would appear to accelerate upward and experience the force produced by that upward accelerating motion.

This is weird, though.

So, can one use an FR which is accelerating/decelerating as a central FR?

 

[Passionflower]

To stevemg: I received your message and I replied. As far as I am concerned this mater is resolved and is behind us.

Postulating Passionflower's statement (I hope I am correct here) that if you look strictly from B, A would "slow down."

No.

Assume A and B are inertial at event X.

Then if B undergoes proper acceleration and A stays inertial then it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from B to X. And a smaller path length means less elapsed time.

 

[granpa]

this has gone on for 8 pages now.
Guys, the twins paradox is not THAT hard.

you've been given good answers and you keep rejecting them and then you just keep asking the same questions over and over. Maybe its time that you started questioning your assumptions.

As I said before, its all about relativity of simultaneity. That is how how each can see the other as aging more slowly when they are moving with respect to each other.
And its also about how relativity of simultaneity changes when one accelerates.
you don't even need to know how it changes WHILE it is accelerating. you can just calculate its state BEFORE and AFTER the acceleration and from that you can see what must have happened over the course of the acceleration.

 

[stevmg]

To stevemg: I received your message and I replied. As far as I am concerned this mater is resolved and is behind us.


No.

Assume A and B are inertial at event X.

Then if B undergoes proper acceleration and A stays inertial then it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from B to X. And a smaller path length means less elapsed time.

I stand corrected.

Did you mean "it is always true that the spacetime distance between A (at any event on his worldline) and X is larger than the path length from B to X." or

"it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from A to X?"

Or, is there a difference between path length and spacetime distance for the same object (in this case B) which leaves your original statement as posted, "it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from B to X." - in which event it is back to the books for me.

 

[yuiop]

If you take my quote in context it was in responce to the idea that simply through accelerometer readings the clock could be computerized to calibrate the onboard "home" clock.
I didn't suggest that there was no way to calculate when the ship reached a state of rest wrt earth. I in fact explicitly described how this could be done through interferometer Doppler readings of a continuous signal from earth.
Can I assume you do understand that the decceleration/acceleration phase at turn around is actually one continuous acceleration in the direction of earth?
Not two discrete periods as far as onboard thrust and continuous accelerometer readings is concerned. That to determine the abstract relative point of rest wrt Earth requires additional reference whether it is calculation , DOppler or some other outside reference.
That the accelerometer will by itself give no clue as to this point.

This was the point of contention and misunderstanding here.
Would you disagree with any of this?

Let us say the traveling twins accelerates for time \tau_{a1} as measured by his own clock and with constant acceleration (a₁).

He can then calculate that the time that elapsed (T_a1) in the Earth frame during the acceleration phase is:

T_{a1} =\frac{c}{a_1}\, sinh(a_{a1} t_{a1}/c)

and that his terminal velocity (v) after the acceleration phase is:

v = \frac{a_1 T_{a1}}{\sqrt{1+(a_1 T_{a1}/c)^2}}

He now cruises for time \tau_{c} as measured by his own clock and can calculate the time elapsed T_c on the Earth clock as

T_c = \frac{\tau_c}{\sqrt{1-v^2/c^2}}

He now decelerates with constant acceleration -a₂. He can calculate the time \tau_{a2} (as measured by his own clock) it will take to come to rest with respect to the Earth as:

t_{a2}=\frac{c}{a_2}\, artanh(v/c)

The above equation is valid even if the rocket does not actually stop but continues reversing direction.

The time that elapses on the Earth clock during the deceleration phase is:

T_{a2} = \frac{c}{a_2}\, sinh(a_2 \tau_{a2}/c)

None of the above calculations require contact with the Earth after take off. All that is required is an onboard accelerometer, a clock and a calculator for the rocket captain to know when he has come to rest with respect to the Earth and how much time has elapsed on the Earth relative to his own clock in the Earth rest frame.

 

[stevmg]

Let us say the traveling twins accelerates for time \tau_{a1} as measured by his own clock and with constant acceleration (a₁).

He can then calculate that the time that elapsed (T_a1) in the Earth frame during the acceleration phase is:

T_{a1} =\frac{c}{a_1}\, sinh(a_{a1} t_{a1}/c)

and that his terminal velocity (v) after the acceleration phase is:

v = \frac{a_1 T_{a1}}{\sqrt{1+(a_1 T_{a1}/c)^2}}

He now cruises for time \tau_{c} as measured by his own clock and can calculate the time elapsed T_c on the Earth clock as

T_c = \frac{\tau_c}{\sqrt{1-v^2/c^2}}

He now decelerates with constant acceleration -a₂. He can calculate the time \tau_{a2} (as measured by his own clock) it will take to come to rest with respect to the Earth as:

t_{a2}=\frac{c}{a_2}\, artanh(v/c)

The above equation is valid even if the rocket does not actually stop but continues reversing direction.

The time that elapses on the Earth clock during the deceleration phase is:

T_{a2} = \frac{c}{a_2}\, sinh(a_2 \tau_{a2}/c)

None of the above calculations require contact with the Earth after take off. All that is required is an onboard accelerometer and clock and a calculator for the rocket captain to know when he has come to rest with respect to the Earth and how much time has elapsed on the Earth relative to his own clock in the Earth rest frame.

Outstanding!

stevmg

 

[Passionflower]

Did you mean "it is always true that the spacetime distance between A (at any event on his worldline) and X is larger than the path length from B to X." or

"it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from A to X?"

No, neither.

Or, is there a difference between path length and spacetime distance for the same object (in this case B) which leaves your original statement as posted, "it is always true that the spacetime distance between B (at any event on his worldline) and X is larger than the path length from B to X." - in which event it is back to the books for me.

Indeed, the same object.

Consider the above mentioned scenario on a spacetime diagram:
Attach a cord from event X to B. When B starts to accelerate and perhaps later goes inertial or what ever it likes, at each time the cord between X and his current position represents the spacetime distance, which is always the longest possible time, between the two events. However the curve on which B travels is the path that represents B's proper time. Now if you draw it you will see that the curve is at all times longer than the straight line but in relativity we do not use an Euclidean metric but a Minkowski metric and in this metric the line that shows longest is in fact the shortest.

Now since we always can have a hypothetical twin travel inertially on X,B we have a closed loop (e.g. a twin experiment) and thus this proves that at all times we can assert that B's clock must be going slower than than A's clock. B could even have a special clock on-board that at each instant of his trip calculates the proper time of this hypothetical twin, this clock would simply measure the distance between X,B.

For those who are interested: imagine B traveling with various rates of acceleration and various cruising periods, what could you say about the total [strike]volume[/strike] area between X,B and the B's path, what does it represent?

 

[yuiop]

Postulating Passionflower's statement (I hope I am correct here) that if you look strictly from B, A would "slow down." I don't know how one uses an FR which itself has acceleration in it, which would be the case if you chose B's FR. Einstein gave the example of the falling elevator and that to the man in it, he doesn't experience any difference than a man sitting still and not moving (no gravity, of course.) One thing is for sure - if you could choose B's FR as your central FR, A would accelerate... and A would be the one who slowed down but I don't know if you can do that. Can one choose an accelerating/decelerating FR and make that one central and make the universe move about it? As it has been stated, in a theoretically gravity free universe, one can do anything as Einstein talked about with his elevator man.

One interesting thing, though, if the man and elevator were accelerating together they would be under the influence of a gravity. A second person in an FR which was stationary would experience the pull of this gravity and the equal and opposite resistance or force as he/she stands on his stationary platform. Meanwhile, the elevator man is experiencing no gravity as he is falling and feels weightless. If he were chosen as the central FR, the guy on the platform would appear to accelerate upward and experience the force produced by that upward accelerating motion.

This is weird, though.

I think you have it about right. Just to clarify. In the accelerating rocket scenario, the accelerating rocket measures less time between two spatially separated events than the inertial observers. In the elevator scenario, the observers in the falling elevator do not experience proper acceleration. Locally they can treat the falling elevator as an inertial RF. They can place clocks at the top and bottom of the elevator and synchronise them. The "stationary" observer on the platform (who is experiencing proper acceleration) should measure less elapsed time between the events (bottom of the elevator passing the platform) and (top of the platform passing the platform) than the observers in the elevator. Hopefully I have applied the equivalence principle correctly there. The basic rule is that the observers in the RF that have to use two spatially separated clocks to measure the elapsed time will always measure more elapsed time than the observer in the RF that uses a single clock and is present at both events.This is true even if there is no acceleration involved.

 

[yuiop]

For those who are interested: imagine B traveling with various rates of acceleration and various cruising periods, what could you say about the total volume between X,B and the B's path, what does it represent?

I'm interested, but I am too lazy to work it out for myself Presumably you have already worked it out?

 

[Austin0]

Well, you're both right, depending on how you look at it.


But it is also the case that there is a single non-inertial frame in which B is at rest throughout. In that sense B has one frame.

So really it depends what you mean by "frame".

I would also comment that non-inertial frames aren't as easy to define as inertial frames: you have some choice in precisely how to define your non-inertial frame, so there is really more than one non-inertial frame in which B is at rest. For simple linear motion in the absence of gravity, as in this example, however, there is a "natural" choice of frame (where frame simultaneity coincides with co-moving inertial simultaneity). But in other cases, such as rotating frames, problems arise if you want to extend a local frame to be shared by multiple rotating observers. But that's another story...

If the inertial observer on Earth is E and the accelerating traveler is ET
I think we can define a singular reference frame for ET as an elemental system of clocks and ruler.
Given the requisite breakthoughs in energy and propusion needed for an actual enactment of this scenario, it is not a stretch to imagine a fleet of computerized drones arranged along the path of ET that co-accelerate according to a preprogrammed schedule for the entire trip.
This could either be Born rigid or uniform through the fleet.
This kind of singular frame is perhaps the only kind that would conform to reality in the same way that is applicable wrt inertial frames. Providing frame agreed events and rational temporality.
So in this context ET's frame is initially, simply a clone of E's ruler and clocks.
At initiation of acceleration, ET's clock would be passing by E's position at an increasing rate to reach a point of contant rate, and then diminishing rate approaching turn around, when they would be momentarily at rest only to repeat the same sequence in the return phase of the trip.
Now regarding the interim relationship between E and ET's clocks as observed by E , this is subject to various assumptions:

1)We can assume ET's clocks local to E might at points run backwards. Succeeding passing clocks displaying decreasing proper time readings.
But we cannot assume ET's local clock running backwards, so in this case there is no meaningful information regarding the instantaneous relationship of E and ET's proper times possible through the local relationship at E's position.

2) We can assume that at some point or points eg. at the end of the acceleration phases , that the ET clocks are resynchronized by convention.
Assuming magical instantaneous resynch, this could result in significant local changes in ET's clocks. If the procedure is conducted from ET's end this would mean sudden adjustments to the clocks proximate to E at that time. As this is self evidently a mechanical adjustment of the clocks it has no actual temporal meaning wrt the relationship of E and ET at that moment.
Any naive calculations based on this conventional synchronization can also have zero temporal meaning regarding the two spatially separated points.

This highlights a problem with the mathematical structure wrt accelerated systems.
The math provides a temporal interval of the relative local clock readings, but makes no distiction between the inertial and the accelerated system.
Specifically it allows the interpretation that this interval could imply a local change in the proper time of E simply by the assumption of simultaneity in ET.
Clearly this is not consistent with physical reality in this circumstance, as the only change possible is in the dynamic system.
It leads to possible conditions and conclusions that ET's local clocks could be be suddenly colocated, not only with previous points of E's worldline but with previous points of ET's frame's own worldlines. I.e. Two spatial points and clocks from ET's frame simultaneously colocated at some previous point in E's history.

It seems clear that the application of the math based on these assumptions, not only does not contain any meaningful temporal information re: the relationship between E and ET, but is inconsistent, not only with SR, but also with our fundamental understanding of spacetime.
I.e All points of both frames would move forward though time even if at different rates.
To attribute any meaning to these calculations is to divorce them from any possible real world consistent singular coordinate system or any actually implemented system of clocks and rulers.

Now to seriously consider the requirements of a non-inertial frame consistent with reality:
1) The assumption that the clocks in all cases would move forward in time
2) A system of synchronization.
Perhaps a continuous synch signal to update local clocks.
This would obviously require differential calculations to compensate for the difference in velocity at the point of transmission and the point of reception.
The difference in distance between transmission and reception resulting from this velocity difference.
The gamma difference due to the above.
Accelerometers. Etc

Highly complex but not impossible in princple I wouldn't imagine.

If this was either theoretically or practically [in the future] implemented it would seem to be a system where [to an approximation] c would be constant.
It would eliminate all unrealistic jumps of temporality, as even with reversal of direction all changes would be incremental and local and the relationship of simultaneity between frames, that is consistent with reality wrt inertial frames would apply here as well.
Not time traveling clocks etc.

You will notice, I am sure, that I neglected Born rigid acceleration in this picture as well as Rindler coordinates.
Re: Rindler. WHether or not this is accurate within a limited domain it does not seem to lend itself to extended frames. So I am sure it could be included with proper care and complication as could Born acceleration but I left them both out for simplicity.

 

[Austin0]

I think you have it about right. Just to clarify. In the accelerating rocket scenario, the accelerating rocket measures less time between two spatially separated events than the inertial observers. In the elevator scenario, the observers in the falling elevator do not experience proper acceleration. Locally they can treat the falling elevator as an inertial RF. They can place clocks at the top and bottom of the elevator and synchronise them. The "stationary" observer on the platform (who is experiencing proper acceleration) should measure less elapsed time between the events (bottom of the elevator passing the platform) and (top of the platform passing the platform) than the observers in the elevator. Hopefully I have applied the equivalence principle correctly there. The basic rule is that the observers in the RF that have to use two spatially separated clocks to measure the elapsed time will always measure more elapsed time than the observer in the RF that uses a single clock and is present at both events.This is true even if there is no acceleration involved.

I think you are right, in principle the clock at the top would be running faster than the clock at the bottom of the elevator although it would have to be a pretty tall elevator to possibly measure any difference I'm sure

 

[yuiop]

But in other cases, such as rotating frames, problems arise if you want to extend a local frame to be shared by multiple rotating observers. But that's another story...

The other story and I will keep it brief because it slightly off topic, is this.

Consider a rotating ring. Observers on the ring want to synchronise clocks. They assume the speed of light is constant and isotropic and use Einstein's clock synchronisation method to synchronize clocks all the way around the ring. If they start with a master clock at point A and start synchronising clocks in one direction around the ring, they find when they get back to A, the last synchronised clock is not synchronised with A! No matter what they do they can not synchronise all clocks with each around the perimeter. One way they can resolve this issue is to use an alternative synchronise method of using a clock at at the centre of the ring and synchronise all clocks relative to the central master clock. When they do this they find the speed of light is not isotropic. This indicates that in certain accelerating scenarios, the speed of light can not be considered constant and isotropic over extended distances.

 

[Austin0]

The other story and I will keep it brief because it slightly off topic, is this.

Consider a rotating ring. Observers on the ring want to synchronise clocks. They assume the speed of light is constant and isotropic and use Einstein's clock synchronisation method to synchronize clocks all the way around the ring. If they start with a master clock at point A and start synchronising clocks in one direction around the ring, they find when they get back to A, the last synchronised clock is not synchronised with A! No matter what they do they can not synchronise all clocks with each around the perimeter. One way they can resolve this issue is to use an alternative synchronise method of using a clock at at the centre of the ring and synchronise all clocks relative to the central master clock. When they do this they find the speed of light is not isotropic. This indicates that in certain accelerating scenarios, the speed of light can not be considered constant and isotropic over extended distances.

I assume you mean not isotropic wrt measurement with or against direction of rotation?

 

[starthaus]

The other story and I will keep it brief because it slightly off topic, is this.

Consider a rotating ring. Observers on the ring want to synchronise clocks. They assume the speed of light is constant and isotropic and use Einstein's clock synchronisation method to synchronize clocks all the way around the ring. If they start with a master clock at point A and start synchronising clocks in one direction around the ring, they find when they get back to A, the last synchronised clock is not synchronised with A! No matter what they do they can not synchronise all clocks with each around the perimeter. One way they can resolve this issue is to use an alternative synchronise method of using a clock at at the centre of the ring and synchronise all clocks relative to the central master clock. When they do this they find the speed of light is not isotropic. This indicates that in certain accelerating scenarios, the speed of light can not be considered constant and isotropic over extended distances.

Not exactly. The coordinate light speed is a function of the clock synchronization method, this is a well known fact. This is why coordinate light speed is not considered meaningful either in SR or in GR. By contrast, local light speed, by not being dependent on the clock synchronization method, is meaningful and it is isotropic in both SR and GR.

 

[JesseM]

I you read the article you would understand that the 'inertial clock' is not based on a frame of simultaneity.

I didn't say it was! I have always been talking about the non-inertial clock, my point is that both the method I suggest and the method you suggest require calculations of the clock's position in some reference frame, a frame which naturally does have some particular definition of simultaneity.

Why do you think it prompted me to attend members of this possibility if not for the difficulties in using frames of simultaneity in accelerating situations? I am just trying to help and thought it would be interesting for my fellow members to know about this alternative. Perhaps it is wasted on you, as you seem to be rather negative about it.

I'm not trying to be negative, just pointing out that your method doesn't actually completely avoid the need to think about simultaneity, though it only requires us to think about the definition of simultaneity in some inertial frame, not a more complicated definition in a non-inertial frame (but then the same thing is true of the method I suggested where the onboard clock is programmed to maintain a constant rate of ticking in an inertial frame...there is a difference though, in my method the final answer for the rate of speedup at any given point on the clock's worldline depends on what frame you chose, while in your method the final answer would not depend on the choice of frame).

Yes an on-board computer would be necessary, but so what? What's your point? You think it is too advanced for a spaceship that travels relativistically to have an on-board computer?

The point was not the computer itself, the point was that the computer's calculations would necessarily involve figuring out the ship's position some reference frame which had its own definition of simultaneity, so you aren't fully dispensing with issues of simultaneity (though as I said the final answer for the rate of speedup is frame-independent in your method). The bolded part of my comment was the important part:

But your method also requires the onboard computer to be constantly figuring out the clock's position in some frame (which necessarily has requires judgments about simultaneity in that frame)

 

[Austin0]

Notice the loop. I never suggested that, considering the loop.[ i.e. the round trip], that this would not be the case.

Ok so then what do you mean by differential time dilation if there is no loop?

I was never talking about a situation where there was no loop.
The only mention I have made to that case in this thread is that I don't think any definitive statement can be made regarding total elapsed time or differential dilation if there are two systems with no other referents which do not become recolocated. I.e. Loop

Alright let's talk about acceleration and your potential misconceptions.

This is what you initially wrote:

The initial acceleration could in fact be a decceleration so if the home clock increased in rate wrt the ship proper time this would not reflect reality as the ship clock could in fact be ticking faster than the Earth clock.

And then you wrote:

This also requires an assumption of absolute acceleration.
Given that the initial starting velocity is purely relative with no actual value this implies that that the resulting change in clock rate could be an increase or a decrease.

So consider two inertial twins T_home and T_travel at the same location. Explain to me how if T_travel undergoes proper acceleration and T_home stays inertial T_travel's clock rate can increase with respect to T_home's clock rate?

Assume two travelers starting out at home and accelerating to 0.9c wrt home when they both go inertial.
One traveler then accelerates back to home.
DO you think that this traveler's clock would then tick slower than the 0.9c traveler's clock as measured by home observers ? DO you think that observed between two sets of points equidistant in home frame, that the returning traveler who is accelerating would be observed to have less elapsed proper time between its two points than the 0.9 traveler between an rqual two points?

 

[JesseM]

He doesn't use ANY ruler.

Well, can you give some overview about how he decides which tick of his own clock is simultaneous with some distant event, given that there is some gap between the event and the light from the event reaching him (at least in any inertial frame)? If he doesn't know the distance between him and the event, how does he figure out the difference between the time he saw the signal and the time it "really" occurred according to the simultaneity scheme you are proposing?

To understand exactly what calculations and observations the (formerly) accelerating observer must carry out, it's first necessary to answer the question "After the accelerating observer stops accelerating, how soon does he become 'an inertial observer', and exactly how soon can he legitimately perform the same calculations that a perpetually inertial observer can perform?". To do that requires a fairly lengthy argument that isn't appropriate or practical in this venue ... but it's all in the paper, for anyone who really wants to know.

Is the paper available anywhere online? Also, is the paper just making an argument about a useful way to define simultaneity for an accelerating observer within the context of regular SR, or is it challenging some aspect of SR? If the latter you shouldn't really be discussing it in this forum but rather in the Independent Research forum...I notice that "Physics Essays" does apparently sometimes publish papers which disagree with mainstream conclusions from SR, like http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PHESEM000019000001000006000001&idtype=cvips&gifs=yes&ref=no .

 

[stevmg]

Consider the above mentioned scenario on a spacetime diagram:
Attach a cord from event X to B. When B starts to accelerate and perhaps later goes inertial or what ever it likes, at each time the cord between X and his current position represents the spacetime distance, which is always the longest possible time, between the two events. However the curve on which B travels is the path that represents B's proper time. Now if you draw it you will see that the curve is at all times longer than the straight line but in relativity we do not use an Euclidean metric but a Minkowski metric and in this metric the line that shows longest is in fact the shortest.

I hate to overuse this word, but another "outsanding" explanation of why twin B has experienced less of an interval than twin A. Yes, the Minkowski Geometry is "opposite" to Euclidean geometry in that the longest arc is the shortest time.

Now since we always can have a hypothetical twin travel inertially on X,B we have a closed loop (e.g. a twin experiment) and thus this proves that at all times we can assert that B's clock must be going slower than than A's clock. B could even have a special clock on-board that at each instant of his trip calculates the proper time of this hypothetical twin, this clock would simply measure the distance between X,B.

For those who are interested: imagine B traveling with various rates of acceleration and various cruising periods, what could you say about the total area between X,B and the B's path, what does it represent?

Does it represent "gravity" or the effect of acceleration/deceleration (whatever you want to call it?) In effect, the curving of spacetime in the t dimension produced by applying a force.

I don't know, just guessing, but a scientific (I know a smidgeon about this) guess as opposed to a wild guess.

 

[stevmg]

Assume two travelers starting out at home and accelerating to 0.9c wrt home when they both go inertial.
One traveler then accelerates back to home.
DO you think that this traveler's clock would then tick slower than the 0.9c traveler's clock as measured by home observers ? DO you think that observed between two sets of points equidistant in home frame, that the returning traveler who is accelerating would be observed to have less elapsed proper time between its two points than the 0.9 traveler between an rqual two points?

Call one traveler A, the other B (man, that is secret-coded, isn't it?)
Let's assume they travel the same path from initial before B turns around after acheiving the 0.9c and accelerates back to the initial statring point. A stays at 0.9c.

I DO believe that B's clock, while he is accelerating back, will tick faster than A's clock, while A is inertial. Why? Because at inertial speed for A, time dilation has a slower tick rate than home base. While B is accelerating back to home base, his velocity is getting smaller and time dilation, though present, is decreasing. Until B reaches home base, his clock will be continually slower than home base (but "speeding up" and catching up to home base.) A's clock will remain the slowest because he is at max time dilation for this experiment.

 

[stevmg]

Let us say the traveling twins accelerates for time \tau_{a1} as measured by his own clock and with constant acceleration (a₁).

He can then calculate that the time that elapsed (T_a1) in the Earth frame during the acceleration phase is:

T_{a1} =\frac{c}{a_1}\, sinh(a_{a1} t_{a1}/c)

and that his terminal velocity (v) after the acceleration phase is:

v = \frac{a_1 T_{a1}}{\sqrt{1+(a_1 T_{a1}/c)^2}}

He now cruises for time \tau_{c} as measured by his own clock and can calculate the time elapsed T_c on the Earth clock as

T_c = \frac{\tau_c}{\sqrt{1-v^2/c^2}}

He now decelerates with constant acceleration -a₂. He can calculate the time \tau_{a2} (as measured by his own clock) it will take to come to rest with respect to the Earth as:

t_{a2}=\frac{c}{a_2}\, artanh(v/c)

The above equation is valid even if the rocket does not actually stop but continues reversing direction.

The time that elapses on the Earth clock during the deceleration phase is:

T_{a2} = \frac{c}{a_2}\, sinh(a_2 \tau_{a2}/c)

None of the above calculations require contact with the Earth after take off. All that is required is an onboard accelerometer, a clock and a calculator for the rocket captain to know when he has come to rest with respect to the Earth and how much time has elapsed on the Earth relative to his own clock in the Earth rest frame.

yuiop -

May we start another thread for discussion of the derivation of the above equations? I will call it just that - "Derivation of proper time in acceleration in SR"
Go to:
https://www.physicsforums.com/showpost.php?p=2865296&postcount=1

 

[Mike_Fontenot]

[...]
If he doesn't know the distance between him and the event, how does he figure out the difference between the time he saw the signal and the time it "really" occurred according to the simultaneity scheme you are proposing?

He DOES determine the distance. But he doesn't use any rulers to do it. He does it by timing a round-trip light signal. His measurements and calculations only involve first-principle notions of velocity, distance, and time.

Is the paper available anywhere online?

No, unfortunately it's not. But any university library should either have it, or else be able to get it via inter-library loan.

Also, is the paper just making an argument about a useful way to define simultaneity for an accelerating observer within the context of regular SR, or is it challenging some aspect of SR?

No, nothing about it challenges Einstein's SR.

But it is NOT just describing a useful way to define simultaneity for an accelerating observer. The paper proves that there is only ONE way to define simultaneity for the accelerating traveler, if you want to insure that his definition of simultaneity never disagrees with his own elementary measurements and first-principle calculations.

And once you have defined, for the accelerating traveler, both simultaneity AND stationarity (i.e., his definition of distance), in a such a way that neither definition conflicts with his own measurements and calculations, then you have defined HIS reference frame. That reference frame is unique...there are no choices to be made, given the above requirement of avoiding any conflicts with his own measurements and calculations.

The above definition of a frame for the accelerating traveler shares a trait in common with Einstein's definition of the inertial frame used in the Lorentz equations: BOTH frames are constructed so that they never conflict with the elementary measurements (using the observer's own wristwatch) and first-principle calculations of the observer stationary at the origin of those frames.

For an inertial observer, it's possible (in the spirit of GR) to define the coordinates in an almost arbitrary way. But if you do that, you decrease he meaningfulness of the coordinates, to the observer. In GR, you don't have a choice: you CAN'T use Lorentz coordinates over the whole universe. But in SR, for an inertial observer, you CAN use Lorentz coordinates. The fact that the Lorentz equations are used so widely in SR testifies to the utility of that choice. As is the fact that time-dilation and length-contraction (which both follow directly from the Lorentz equations) are so widely used in SR.

I'm basically making the same kind of choice, for the accelerating observer: choose based on what is MEANINGFUL to the observer. Nothing is more meaningful to the observer than what the observer measures (with his own wristwatch) and calculates (using only first-principles) on his own.

Mike Fontenot

 

[Mike_Fontenot]

[...]
And its also about how relativity of simultaneity changes when one accelerates.
you don't even need to know how it changes WHILE it is accelerating. you can just calculate its state BEFORE and AFTER the acceleration and from that you can see what must have happened over the course of the acceleration.

I'll just add this:

In the simplest version of the traveling twin example (with a single instantaneous speed change at the midpoint), it IS possible to INFER the change in the home twin's age (according to the traveler), during the instantaneous turnaround: you know that both twins obviously must agree about their respective ages when they are reunited, and you know how much the traveler says his twin ages during the two inertial portions of his trip. So the home twin's ageing during the turnaround (according to the traveler) must be just enough to make the totals agree at the reunion.

But in only slightly more complicated examples (e.g., with multiple instantaneous velocity changes), it becomes important to KNOW how to directly calculate the amount of the home twin's ageing during each of the velocity changes. And in the case of FINITE accelerations, being able to directly calculate the home twin's ageing during each of the traveler's segments of finite acceleration is indispensable.

It turns out to be easy to do that. For the cases of instantaneous velocity changes, the required calculations are almost trivial to carry out. For piecewise-constant accelerations, the calculations are a bit more complex, but they can still be done, if necessary, with a good calculator. And with a computer program, they are very easy.

Both of the above types of problems can be handled with a simple equation that I derived many years ago, which I call "the CADO equation". The CADO equation follows directly from the Lorentz equations ... the CADO equation really just automates what you can deduce from the geometry of the Minkowski diagram.

I also, many years ago, implemented the CADO equation in a computer program I call "the CADO program".

It's also possible to use the CADO equation to do completely general acceleration profiles, but you (usually) can't do it in a closed-form way...it requires some numerical integrations. Fortunately, piecewise-constant accelerations are usually all you really need to be able to handle.

Here is a description of the CADO equation that I've posted previously, in other threads:
____________________________________________________________Years ago, I derived a simple equation that relates the current ages of the twins, ACCORDING TO EACH TWIN. Over the years, I have found it to be very useful. To save writing, I write "the current age of a distant object", where the "distant object" is the stay-at-home twin, as the "CADO". The CADO has a value for each age t of the traveling twin, written CADO(t). The traveler and the stay-at-home twin come to DIFFERENT conclusions about CADO(t), at any given age t of the traveler. Denote the traveler's conclusion as CADO_T(t), and the stay-at-home twin's conclusion as CADO_H(t). (And in both cases, remember that CADO(t) is the age of the home twin, and t is the age of the traveler).

My simple equation says that

CADO_T(t) = CADO_H(t) - L*v/(c*c),

where

L is their current distance apart, in lightyears,
according to the home twin,

and

v is their current relative speed, in lightyears/year,
according to the home twin. v is positive
when the twins are moving apart.

(Although the dependence is not shown explicitly in the above equation, the quantities L and v are themselves functions of t, the age of the traveler).

The factor (c*c) has value 1 for these units, and is needed only to make the dimensionality correct. For simplicity, you can generally just ignore the c*c factor when using the equation.

The equation explicitly shows how the home twin's age will change abruptly (according to the traveler, not the home twin), whenever the relative speed changes abruptly.

For example, suppose the home twin believes that she is 40 when the traveler is 20, immediately before he turns around. So CADO_H(20-) = 40. (Denote his age immediately before the turnaround as t = 20-, and immediately after the turnaround as t = 20+.)

Suppose they are 30 ly apart (according to the home twin), and that their relative speed is +0.9 ly/y (i.e., 0.9c), when the traveler's age is 20-. Then the traveler will conclude that the home twin is

CADO_T(20-) = 40 - 0.9*30 = 13

years old immediately before his turnaround. Immediately after his turnaround (assumed here to occur in zero time), their relative speed is -0.9 ly/y. The home twin concludes that their distance apart doesn't change during the turnaround: it's still 30 ly. And the home twin concludes that neither of them ages during the turnaround, so that CADO_H(20+) is still 40.

But according to the traveler,

CADO_T(20+) = 40 - (-0.9)*30 = 67,

so he concludes that his twin ages 54 years during his instantaneous turnaround.

The equation works for arbitrary accelerations, not just the idealized instantaneous speed change assumed above. I've got an example with +-1g accelerations on my web page:

http://home.comcast.net/~mlfasf

The derivation of the equation is given in my paper

"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.

Mike Fontenot

 

[JesseM]

He DOES determine the distance. But he doesn't use any rulers to do it. He does it by timing a round-trip light signal. His measurements and calculations only involve first-principle notions of velocity, distance, and time.

OK, so basically some variation of radar distance, except instead of just the simple assumption that if a signal is sent at T1 and returns at T2 then the event of it bouncing back at you must have been simultaneous with your clock reading T2 - (T2 - T1)/2, you're presumably doing some more complicated calculations to define simultaneity in terms of the time a signal was sent and the time it returned.

But it is NOT just describing a useful way to define simultaneity for an accelerating observer. The paper proves that there is only ONE way to define simultaneity for the accelerating traveler, if you want to insure that his definition of simultaneity never disagrees with his own elementary measurements and first-principle calculations.

"First-principle calculations" has no well-defined meaning, I don't see why whatever calculations you are doing to determine simultaneity are more "first-principle" then the simple rule of saying that a signal bounced back at T2 - (T2 - T1)/2, perhaps at a distance of c*(T2 - T1)/2. Likewise with "elementary", I don't see why a measurement of distance involving a radar method is more "elementary" than a method involving local measurements on some ruler.

And once you have defined, for the accelerating traveler, both simultaneity AND stationarity (i.e., his definition of distance), in a such a way that neither definition conflicts with his own measurements and calculations

But I don't think there's going to be any fundamental physical reason to prefer your methods of making "measurements and calculations" over some others like the ones I suggested. Certainly the resulting frame you define won't be "physically preferred" in the sense that the laws of physics take some special simple form in that coordinate system that they don't take in other non-inertial coordinate systems (whereas that is true of inertial coordinate systems).

then you have defined HIS reference frame. That reference frame is unique...there are no choices to be made, given the above requirement of avoiding any conflicts with his own measurements and calculations.

But the "choices" are precisely in how to define the observer's "measurements and calculations"! Again, you could pick some different method of measurement, or the same method of measurement but with different method of calculating simultaneity and distance from measurements, and you would get a different frame.

 

[Passionflower]

I was never talking about a situation where there was no loop.
The only mention I have made to that case in this thread is that I don't think any definitive statement can be made regarding total elapsed time or differential dilation if there are two systems with no other referents which do not become recolocated. I.e. Loop

Do you understand that we can always construct a loop between the event distance and path length distance?

Assume two travelers starting out at home and accelerating to 0.9c wrt home when they both go inertial.
One traveler then accelerates back to home.
DO you think that this traveler's clock would then tick slower than the 0.9c traveler's clock as measured by home observers ? DO you think that observed between two sets of points equidistant in home frame, that the returning traveler who is accelerating would be observed to have less elapsed proper time between its two points than the 0.9 traveler between an rqual two points?

Ok, now we have three observers.

Three observers inertial at event E₀, A remains inertial, B and C accelerate and then go inertial after some time B accelerates again while C remains inertial which is event E₁.

Then we can conclude that:

1: The distance between E₀ and E₁ is always longer than both the path length of B and C. Which means that B and C are aging less with respect to A.

2. The distance between E₁ and a time measurement at B is always longer than the path length of B. Which means that B ages less with respect to C.

3. Combining 1 and 2 we get: B ages the least, then C and then A.

Feel free to present a scenario that runs counter to this without changing the goal posts by having A or C accelerate at a later stage.

 

[Al68]

Certainly the resulting frame you define won't be "physically preferred" in the sense that the laws of physics take some special simple form in that coordinate system that they don't take in other non-inertial coordinate systems (whereas that is true of inertial coordinate systems).

I don't really want to get in the middle here, but doesn't that "accelerated frame convention" have the distinct advantage that whatever coordinates are assigned to distant events will be completely independent of the future motion of the accelerated observer?

It would seem that the only way for an accelerated observer to assign coordinates to distant events that would be consistent with the possibility of future inertial motion, and consistent with SR, would be to use the ICMIF coordinates.

 

[JesseM]

I don't really want to get in the middle here, but doesn't that "accelerated frame convention" have the distinct advantage that whatever coordinates are assigned to distant events will be completely independent of the future motion of the accelerated observer?

I think you could come up with other non-inertial systems where this was true. At any event E on the accelerating observer's worldline consider the surface of simultaneity in the observer's instantaneous inertial rest frame at E, then as long as the surface of simultaneity for the non-inertial frame is defined in such a way that every point on the surface of simultaneity for the same event E is guaranteed to be on or "below" (in the past of) that inertial surface of simultaneity, then coordinates assigned to events shouldn't depend on the observer's future motion.

edit: actually on second thought, lying "under" the inertial plane of simultaneity isn't really the issue, all that matters is that the plane of simultaneity for any point on the worldline is defined in such a way that the future behavior of the worldline doesn't matter--for example, at any point we could define a plane of simultaneity by imagining what would happen if we had particles which could move FTL, and imagining sending them out in all directions such that they moved at 200c in the accelerating observer's instantaneous inertial rest frame, so the spacelike paths of all these particles would define a surface of simultaneity.

 

[Austin0]

I was never talking about a situation where there was no loop.
The only mention I have made to that case in this thread is that I don't think any definitive statement can be made regarding total elapsed time or differential dilation if there are two systems with no other referents which do not become recolocated. I.e. Loop

Do you understand that we can always construct a loop between the event distance and path length distance?

Perhaps not?

Are you talking about drawing a hyperbolic curve for an accelerated frame AF in M spacetime ? Starting from x₀ on the origen worldline of inertial reference frame F
and terminating at point x₁ with an instantaeous velocity of say 0.9c
and then drawing an inertial straight line from x₀ to x₁ for frame IF
? Is this what you mean by a loop??

In any case this is a question I have been meaning to bring up someplace:
Is the proper time greater for IF between x₀ and x₁ than the curved worldline between the same two points?
If AF was a Euclidean curve I would naturally assume it would be longer and therefore less proper time than IF, but it's not is it?
SO the question is : Is the integrated proper time derived from infinitesimal line evaluations, which as far as I understand, is a sum of infinitesimal instantaneous v_i*\gamma(v_i) relative to F,,, less than the direct IF line interval * \gamma(v_f) relative to F ?
I don't know,,, the math is beyond me,,, but the clock hypothesis makes me think they might be equivelant.
I asked in another thread about doing a direct type of area calculus between IF and AF and it seemed the answer was no,, but I am not sure.
In any case doing both relative to F should give proper times shouldn't it?

Assume two travelers starting out at home and accelerating to 0.9c wrt home when they both go inertial.
One traveler then accelerates back to home.
DO you think that this traveler's clock would then tick slower than the 0.9c traveler's clock as measured by home observers ? DO you think that observed between two sets of points equidistant in home frame, that the returning traveler who is accelerating would be observed to have less elapsed proper time between its two points than the 0.9 traveler between an rqual two points?

Ok, now we have three observers.

Three observers inertial at event E0, A remains inertial, B and C accelerate and then go inertial after some time B accelerates again while C remains inertial which is event E1.

Then we can conclude that:

1: The distance between E0 and E1 is always longer than both the path length of B and C. Which means that B and C are aging less with respect to A.

2. The distance between E1 and a time measurement at B is always longer than the path length of B. Which means that B ages less with respect to C.

3. Combining 1 and 2 we get: B ages the least, then C and then A.

Feel free to present a scenario that runs counter to this without changing the goal posts by having A or C accelerate at a later stage.

To make the comparison I think you need two more events.
E_C at a point in A where the time of C is observed and E_B in A where the time of B is observed in A E1 --> E_C = E1 --> E_B

WHat you are saying here seems to contradict your accelerometer/clock idea.
If you look at when B begins accelerating back toward earth; it is a decceleration wrt earth.
IN your scenario this meant the computer would slow down the "home " clock. I.e. Meaning the ship clock 's proper rate was speeding up, yes??
So from the time that B begins accelerating towards A until reaching v=0 wrt A , B's clock rate would be speeding up, getting into phase with A's.
Through this whole phase B would be slowing down relative to A while C maintained a constant high velocity. So do you think that B or C traveled more over this time period??

 

[Passionflower]

WHat you are saying here seems to contradict your accelerometer/clock idea.
If you look at when B begins accelerating back toward earth; it is a decceleration wrt earth.
IN your scenario this meant the computer would slow down the "home " clock. I.e. Meaning the ship clock 's proper rate was speeding up, yes??
So from the time that B begins accelerating towards A until reaching v=0 wrt A , B's clock rate would be speeding up, getting into phase with A's.

No, the clock rates getting closer in the end does not influence the already accumulated time difference.

With respect to the spacetime diagram and curves, look at my earlier posting I think it pretty much states what you are looking for.

Consider a traveler accelerating from event E₁ with a constant proper acceleration of \alpha. After some time the traveler takes a measurement on his clock, it states \tau and he marks this event E₂.

Now what is the slowest way of getting from E₁ to E₂?

What we know about the traveler at E₂:

w = \sinh(\alpha \tau)

\gamma = \sqrt{1+w^2}

So the total coordinate distance traveled is:

d = \frac{\gamma -1}{\alpha}

Thus the slowest way of getting from E₁ to E₂ is:

\tau_{min} = \sqrt{\tau^2 - d^2}

So the traveler's relative clock rate is:

\frac {\tau_{trav}}{\tau_{min}}

 

[Austin0]

WHat you are saying here seems to contradict your accelerometer/clock idea.
If you look at when B begins accelerating back toward earth; it is a decceleration wrt earth.
IN your scenario this meant the computer would slow down the "home " clock. I.e. Meaning the ship clock 's proper rate was speeding up, yes??
So from the time that B begins accelerating towards A until reaching v=0 wrt A , B's clock rate would be speeding up, getting into phase with A's.

No, the clock rates getting closer in the end does not influence the already accumulated time difference.

Of course not. But we are not here comparing B and A , at the point of beginning of return acceleration, B and C have the same accumulated proper time. We are just looking at the comparative clock rates between B and C while B is deccelerating wrt home A.
If B's clock is speeding up relative to A it seems clear it would also be speeding up relative to C No?

Are you talking about drawing a hyperbolic curve for an accelerated frame AF in M spacetime ? Starting from x₀ on the origen worldline of inertial reference frame F
and terminating at point x₁ with an instantaeous velocity of say 0.9c
and then drawing an inertial straight line from x₀ to x₁ for frame IF

With respect to the spacetime diagram and curves, look at my earlier posting I think it pretty much states what you are looking for.

Given this diagram with AF and IF as accelerated and inertial world lines between two points with the rest frame being F
I am talking about applying the formula below [which I can't do] and comparing the total proper time of AF with the total proper time of inertial line IF. Directly from this diagram.

Consider a traveler accelerating from event E₁ with a constant proper acceleration of \alpha. After some time the traveler takes a measurement on his clock, it states \tau and he marks this event E₂.

Now what is the slowest way of getting from E₁ to E₂?

What we know about the traveler at E₂:

w = \sinh(\alpha \tau)

\gamma = \sqrt{1+w^2}

So the total coordinate distance traveled is:

d = \frac{\gamma -1}{\alpha}

Thus the slowest way of getting from E₁ to E₂ is:

\tau_{min} = \sqrt{\tau^2 - d^2}

So the traveler's relative clock rate is:

\frac {\tau_{trav}}{\tau_{min}}

Here again we are not interested in the relationship between B and A Both B and C are going to have less elapsed proper time than A.
But after the beginning of return acceleration there is no direct way to compare B and C except over some equal separation of two sets of events . To compare B and A's elapsed time and compare C and A's elapsed time and then be able to derive a comparison.
Unless you have a better idea.

 

[Passionflower]

Here again we are not interested in the relationship between B and A Both B and C are going to have less elapsed proper time than A.
But after the beginning of return acceleration there is no direct way to compare B and C except over some equal separation of two sets of events . To compare B and A's elapsed time and compare C and A's elapsed time and then be able to derive a comparison.
Unless you have a better idea.

Alright you are not interested. Perhaps you should look again and see if you understand that the section you refer to is about an accelerated path versus a geodesic path and not about comparing A, B and C.

It seems we communicate very badly, perhaps it is better if you ask someone else as it seems I am not capable of giving you any help.

 

[Austin0]

Alright you are not interested. Perhaps you should look again and see if you understand that the section you refer to is about an accelerated path versus a geodesic path and not about comparing A, B and C.

It seems we communicate very badly, perhaps it is better if you ask someone else as it seems I am not capable of giving you any help.

I did not say I was not interested but rather that the question was concerning something else.
And yes you are right there are two different questions here. The accelerated path vs geodesic path in a MInklowski diagram and the completely different question with A, B and C

The curved vs straight was prompted by your statement that a curved path could always be turned into a loop and I asked you if that was what you meant.

 

[Passionflower]

The curved vs straight was prompted by your statement that a curved path could always be turned into a loop and I asked you if that was what you meant.

Yes, and I took the trouble explaining it.

But your comment on that explanation "Here again we are not interested in the relationship between B and A" seems to indicate you did not even read it and already concluded it was referring to something else.

 

[Austin0]

Originally Posted by Austin0
The curved vs straight was prompted by your statement that a curved path could always be turned into a loop and I asked you if that was what you meant.

Yes, and I took the trouble explaining it.

But your comment on that explanation "Here again we are not interested in the relationship between B and A" seems to indicate you did not even read it and already concluded it was referring to something else.

WHich was the explanation regarding the loop?
I certainly read everything carefully but didn't see anything as a specific reply to the loop question.

 

[Passionflower]

WHich was the explanation regarding the loop?

The explanation you were not interested in.

 

[Al68]

I think you could come up with other non-inertial systems where this was true. At any event E on the accelerating observer's worldline consider the surface of simultaneity in the observer's instantaneous inertial rest frame at E, then as long as the surface of simultaneity for the non-inertial frame is defined in such a way that every point on the surface of simultaneity for the same event E is guaranteed to be on or "below" (in the past of) that inertial surface of simultaneity, then coordinates assigned to events shouldn't depend on the observer's future motion.

edit: actually on second thought, lying "under" the inertial plane of simultaneity isn't really the issue, all that matters is that the plane of simultaneity for any point on the worldline is defined in such a way that the future behavior of the worldline doesn't matter--for example, at any point we could define a plane of simultaneity by imagining what would happen if we had particles which could move FTL, and imagining sending them out in all directions such that they moved at 200c in the accelerating observer's instantaneous inertial rest frame, so the spacelike paths of all these particles would define a surface of simultaneity.

Maybe I'm missing something here, but if a non-inertial frame is defined so that the coordinates assigned to events are independent of the observer's future motion, then don't the coordinates assigned necessarily have to be the same coordinates assigned for the case of all future motion being inertial motion, ie the same coordinates assigned by a co-moving inertial observer, to be consistent with the SR simultaneity convention?

It seems to me that "independent of future motion" necessarily means "the same as the ICMIF", since the latter is one of the possible future motions of the accelerated observer.

 

[JesseM]

Maybe I'm missing something here, but if a non-inertial frame is defined so that the coordinates assigned to events are independent of the observer's future motion, then don't the coordinates assigned necessarily have to be the same coordinates assigned for the case of all future motion being inertial motion, ie the same coordinates assigned by a co-moving inertial observer, to be consistent with the SR simultaneity convention?

I don't see why--after all, you can design a non-inertial rest frame for an inertial observer, keeping in mind that in SR "non-inertial frame" doesn't necessarily mean that objects at fixed coordinate positions in the frame are moving non-inertially, it just means a frame that doesn't satisfy the two postulates of SR. For a simple example, suppose an inertial observer is moving at 0.8c in the +x direction of an inertial (x,t) frame. Then we can obtain a non-inertial (x',t') frame where that observer is at rest using the Galilei transformation:

x' = x - 0.8c*t
t' = t