How are the twins distinguished?

[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