The case for True Length = Rest Length

[GrayGhost]

Please understand, I am not debating you, I'm trying to understand you, ...

OK, I appreciate that, thanx.

Back in post #271, you defined your use of POV to be a specific FOR in which the observer is at the origin:

Now you are saying that it's still a POV even if there is no observer located at the origin:

And then you ask me how I define POV:

Indeed.

I said:

What any observer measures of space and time has no relation to any frame of reference. They don't need to be thinking about a frame of reference or have defined a frame of reference to make a measurement or to make an observation.​

Well, I agree with your second sentence here. I disagree with the first sentence ... The measurement made by one's own ruler is related to the ruler itself, and the units-of-measure wrt said ruler is related to the units-of-measure wrt one's own assigned coordinate system axes. The frame-of-reference assigned to oneself is the (say) cartesian cooridnate-system assigned to oneself. So I see them all as related. Now, I do realize that a measurement may be made w/o the consideration of mapping said measurements into a coordinate system. Normally you would want to map it, but maybe you only wish to know (say) the separation and do not care about the mapping within the system. Yet, this does not mean the coordinate system and the measurements made by rulers are not related, IMO.

You are the one who insists on putting a non-inertial observer at the origin of some kind of non-inertial FOR or at the origins of a series of inertial FOR's or whatever it is you have in mind that you don't seem to be able to communicate precisely.

I'm not sure how else to communicate it. I tweeked an old illustration and posted it, and provided a narrative description of it. The debates here have been over side-issues, mainly semantics, and not my specific point at hand.

At one point or another here, I think everyone has agreed that I consider ... twin B's experience during non-inertial motion is the "collective equivalent" of an infinite number of contiguous corresponding inertial frames-of-reference of which twin B momentarily occupies. I do not see this as the frankenstein-force-fit description, as DaleSpam suggested prior. IMO, twin B's POV "actually is" the very same as said collective equivalent. However, twin B must sum the LT solutions for each of those infitesimal segments considered (over the interval), and this summing is what allows the LTs that were designed for the all-inertial case to apply to the non-inertial POV.

So that is why I'm asking you to consider the observer (twin B) after he has come to rest with respect to his twin at the halfway point of his trip. Please describe his POV that you think is natural for him and decribe what he sees of his twin, the twin's clock, the heavens, and all the other descriptions that you gave earlier when the traveling twin did not stop but instead reversed direction.

When twin B comes to rest with A at the turnabout point, his POV is the same as A's in these respects ...

wrt A (and thus B), the following apply ... Bodies in motion are length-contracted and clocks in motion tick slower than his own. No doppler effects are existent wrt EM radiated by inertial sources stationary wrt A (and thus B). Otherwise, the EM is doppler shifted per the doppler eqn of speical relativity. The effects of gravitation are "considered ignorable" in my responses here.​

GrayGhost

 

[ghwellsjr]

The measurement made by one's own ruler is related to the ruler itself, and the units-of-measure wrt said ruler is related to the units-of-measure wrt one's own assigned coordinate system axes. The frame-of-reference assigned to oneself is the (say) cartesian cooridnate-system assigned to oneself. So I see them all as related. Now, I do realize that a measurement may be made w/o the consideration of mapping said measurements into a coordinate system. Normally you would want to map it, but maybe you only wish to know (say) the separation and do not care about the mapping within the system. Yet, this does not mean the coordinate system and the measurements made by rulers are not related, IMO.

OK, you're saying that after an observer makes measurements, he has the option of mapping them to a coordinate system. That part I get and agree with. I'm still unclear about which coordinate system you think is assigned to the observer.

At one point or another here, I think everyone has agreed that I consider ... twin B's experience during non-inertial motion is the "collective equivalent" of an infinite number of contiguous corresponding inertial frames-of-reference of which twin B momentarily occupies. I do not see this as the frankenstein-force-fit description, as DaleSpam suggested prior. IMO, twin B's POV "actually is" the very same as said collective equivalent. However, twin B must sum the LT solutions for each of those infitesimal segments considered (over the interval), and this summing is what allows the LTs that were designed for the all-inertial case to apply to the non-inertial POV.

(It was JesseM, not DaleSpam, that made that suggestion. See post #236.)
Here's another thing I don't understand. You keep talking about an observer using Lorentz Transforms to solve for something involving summing but you have not made it clear what the starting inertial frame is that he is working with, nor the set of events (1 time and 3 spatial coordinates) in that frame, nor the relative speed between that first FOR and the FOR he wants to convert the events in to. And after doing that for one FOR and he does it again for the next FOR, what is it that he sums and what is the significance of the sum? I just have no idea what you are thinking. Please elaborate instead of just repeating the same general recipe.

When twin B comes to rest with A at the turnabout point, his POV is the same as A's in these respects ...

wrt A (and thus B), the following apply ... Bodies in motion are length-contracted and clocks in motion tick slower than his own. No doppler effects are existent wrt EM radiated by inertial sources stationary wrt A (and thus B). Otherwise, the EM is doppler shifted per the doppler eqn of speical relativity. The effects of gravitation are "considered ignorable" in my responses here.​

GrayGhost

Well, since we only have twin A and twin B with no relative motion, then the only thing in your list that might apply is "No doppler effects are existent wrt EM radiated by inertial sources stationary wrt A (and thus B)", but I'm not sure if you meant that to also apply to both A and B or if you meant to only apply it to other potential objects/observers and specifically exclude twin A and twin B.

But what I need to know is the differences between the POV for the two twins. Do they share any of the coordinates between their two FOR's? Do they have the same time coordinate? Do they share any of their spatial axes? Do all their axes point in the same direction?

Can you please fill in these details even if you think they are obvious, because they are not obvious to me.

Thanks.

 

[Dale]

twin B's experience during non-inertial motion is the "collective equivalent" of an infinite number of contiguous corresponding inertial frames-of-reference of which twin B momentarily occupies. I do not see this as the frankenstein-force-fit description, as DaleSpam suggested prior. IMO, twin B's POV "actually is" the very same as said collective equivalent.

You certainly can arbitrarily adopt that convention and define B's POV in that way. But that is merely a personal choice and is not a standard convention. That convention has some problems, such as the fact that the one way speed of light is not c in it, and that it assigns multiple coordinates to the same event. But you are certainly free to use it anyway.

The point is that the phrase "B's POV" unambiguously refers to the reference frame where B is at rest if B is inertial, but if B is non-inertial it is ambiguous unless you specify what convention you are using.

 

[GrayGhost]

You certainly can arbitrarily adopt that convention and define B's POV in that way. But that is merely a personal choice and is not a standard convention. That convention has some problems, such as the fact that the one way speed of light is not c in it, and that it assigns multiple coordinates to the same event. But you are certainly free to use it anyway.

Well, the twin B POV is not an always-inertial POV. Your 2 points here are valid, however whether that's any real problem depends on how you look at the overall-picture there. Where you figure the 1-way speed of light cannot be c, I figure it differently ... that events move in spacetime (per B) while he himself undergoes proper acceleration. They must move in a way that ensures the 1-way speed of light is always c, during-any-single-moment-considered anywhere in space (per B). The LTs define the spacetime relation between the twin A and B POVs, and since the LTs require the 1-way speed of light to be c, then never can the speed of light "not be c" for-any-moment-considered at-any-location-in-space.

During twin B's own proper acceleration, events move, and bodies move thru space and time non-linearly, IOWs differently than what would be expected in an all-inertial case (in which case events don't even move). Twin A is inertial, and at any point during twin B's traversal, B exists at some specific location in space and time (per A). If twin A is diligent enough, then at any said moment, twin A can determine (by summing LTs solns for infitesimal segments as he goes) how A himself must exist in space and time per twin B, and also what twin B's clock should then read. When we later look at twin B's clock, nav data, and track data, twin B should hold twin A precisely where twin A predicted he would. Both observers are bound by the LTs, and the spacetime coordinates are invariant under rotation.

The point is that the phrase "B's POV" unambiguously refers to the reference frame where B is at rest if B is inertial, but if B is non-inertial it is ambiguous unless you specify what convention you are using.

Well, therein lies the problem IMO. I'd say by Einstein's convention, in the sense that the 1-way speed of light is c across the all-of-space in any single moment considered (per B). However, the Einstein convention is ... T₁ = 1/2*(T₀+T₂). This convention cannot apply (per B) in-the-usual-all-inertial-way during B's own proper acceleration, because events, including emission, reflection, and reception move as his own acceleration continues. Yet, what is upheld is this ... light's speed is always c in-any-moment-considered at-any-point-in-space. Also, add that there is no convention that can consistently correctly guess where anyone is in space (now), given said-other has the ability to undergo proper acceleration and his flight is not preplanned. However, each twin does have the ability to determine where-the-other-twin-was when his own radar's reflection event occurred (off the other twin), although that would be a more difficult thing for twin B to determine compared to the ease of the all-inertial observer (ie A) or the all-inertial scenario. The fact is, each observer exists at some specific location in spacetime, and if the convention used does not accurately figure it, then the convention is somewhere between less-than-perfect and unsatisfactory IMO.

GrayGhost

 

[Mike_Fontenot]

[...]The LTs define the spacetime relation between the twin A and B POVs, and since the LTs require the 1-way speed of light to be c, then never can the speed of light "not be c" for-any-moment-considered at-any-location-in-space.
[...]

Just FYI:

When I wrote my original relativity paper (which derives and explains the CADO equation), I referred to the quantity "v" (or its nondimensional version beta) as "the velocity of the traveler, relative to the home twin", without specifying "according to WHOM?". It is well known that any two inertial frames will always agree about their relative velocity ... it is not necessary to specify "according to WHOM?" in that case.

The CADO reference frame is DEFINED by requiring that the accelerating traveler, at each instant of his life, always agree (about remote distances and times) with the inertial frame with which he is momentarily stationary at that instant. I call that the "MSIRF(t)", for "Momentarily Stationary Inertial Reference Frame" at the instant "t" in the traveler's life.

And since the home twin's inertial frame always agrees about relative velocity with each of those MSIRF's, I thought that the accelerating traveler must also agree. That's why I didn't feel the need to specify "according to WHOM?" when I referred to the quantity "v" or "beta" in my paper.

But I later realized that I was incorrect: the accelerating traveler does NOT agree about velocities with his current MSIRF. They agree about distances and times at that instant, but not about velocities at that instant. The reason lies in the fact that velocities always by definition are based on changes of a distance during a very small change in time. I.e., the velocity of the home twin, relative to the traveler, according to the traveler, at some instant t of his life, refers to how that distance changes for two very closely-spaced times of his life very near the instant t. And the MSIRF's at those two different times in his life AREN'T the SAME inertial reference frame. THAT'S why the traveler generally won't agree about relative velocities with his current MSIRF.

Fortunately, this omission on my part didn't affect the results in my paper, because all the results were correct when the quantity "v" and "beta" in my equations referred to the velocity of the traveler, relative to the home twin, ACCORDING TO THE HOME TWIN. I.e., it WAS necessary for me to specify "according to WHOM" when I referred to a relative velocity.

Mike Fontenot

 

[GrayGhost]

Just FYI:

But I later realized that I was incorrect: the accelerating traveler does NOT agree about velocities with his current MSIRF. They agree about distances and times at that instant, but not about velocities at that instant. The reason lies in the fact that velocities always by definition are based on changes of a distance during a very small change in time. I.e., the velocity of the home twin, relative to the traveler, according to the traveler, at some instant t of his life, refers to how that distance changes for two very closely-spaced times of his life very near the instant t. And the MSIRF's at those two different times in his life AREN'T the SAME inertial reference frame. THAT'S why the traveler generally won't agree about relative velocities with his current MSIRF.

Indeed. Thanx for the correction there Mike. I've been thru this before, but it has been a very long time. So, while it makes the overall spacetime predictions more complex, it in no way makes it undo'able. Each twin has the added burden of predicting what the other holds as "the current" instantaneous-relative-velocity, which of course must be done to determine the gamma factor at that instant. So during periods of relative acceleration, the current "instantaneous velocity and gamma factor" are personal per POV. By "per POV", I mean "per twin A and per twin B".

GrayGhost

 

[Dale]

Where you figure the 1-way speed of light cannot be c, I figure it differently ... that events move in spacetime (per B) while he himself undergoes proper acceleration. They must move in a way that ensures the 1-way speed of light is always c, during-any-single-moment-considered anywhere in space (per B).

I am not following your description, I don't think that it is possible to avoid both of the problems that I mentioned above this way. Could you put it mathematically for clarity?

Say that you have an inertial unprimed frame:
(t,x)

And in that inertial frame there is an observer B with a timelike worldline:
(t_B(\lambda),x_B(\lambda))

What is the expression or operation to determine B's coordinates:
(t',x')

 

[GrayGhost]

Just FYI:

But I later realized that I was incorrect: the accelerating traveler does NOT agree about velocities with his current MSIRF (Momentarily Stationary Inertial Reference Frame). They agree about distances and times at that instant, but not about velocities at that instant. The reason lies in the fact that velocities always by definition are based on changes of a distance during a very small change in time. I.e., the velocity of the home twin, relative to the traveler, according to the traveler, at some instant t of his life, refers to how that distance changes for two very closely-spaced times of his life very near the instant t. And the MSIRF's at those two different times in his life AREN'T the SAME inertial reference frame. THAT'S why the traveler generally won't agree about relative velocities with his current MSIRF.

Mike,

I'll have to rethink my last response on this matter. I figure your paper may have had it right in the first place. I'll repost a new response on this soon.

GrayGhost

 

[GrayGhost]

I am not following your description, I don't think that it is possible to avoid both of the problems that I mentioned above this way. Could you put it mathematically for clarity?

I was wondering when you were going to get around to asking that question. Patience is a virtue :)

GrayGhost

 

[ghwellsjr]

I was wondering when you were going to get around to asking that question. Patience is a virtue :)

GrayGhoat

I was wondering when you are going to get around to answering my questions from post #302. I only have so much patience :)

 

[Mike_Fontenot]

[...]
So, while it makes the overall spacetime predictions more complex, it in no way makes it undo'able.
[...]

Actually, the calculations required to determine simultaneity according to the accelerating traveler, beyond those required to determine simultaneity according to the home-twin, are very simple and easy: basically, only one multiplication and one addition (or subtraction) are needed. Those simple arithmetic operations are performed on three quantities, all of which are as concluded by the (unaccelerated) home-twin. So those three quantities must be determined (for each instant of the accelerated traveler's life) regardless of whether you want to determine the traveler's "point-of-view", or the home-twin's "point-of-view".

The two additional simple arithmetic operations, that are required to determine the traveler's conclusions about simultaneity, are just those specified in the CADO equation, which is given here:

https://www.physicsforums.com/showpost.php?p=2934906&postcount=7 .

The three quantities that are needed in the CADO equation are defined in the above posting. All three quantities are computed using a single inertial frame: the frame of the home-twin.

In the idealized cases of instantaneous velocity changes, the determination of those three quantities is trivial.

For the more realistic cases of accelerations which are "piece-wise constant", the determination of the three quantities is more complicated, but they all still DO have closed-form analytic solutions, which are fairly widely known and will probably be familiar to almost everyone with a thorough understanding of special relativity (at the level of Taylor & Wheeler, for example).

For completely general acceleration profiles, the three quantities can still be determined, but numerical integration is required. Fortunately, those cases are rarely needed.

The CADO equation is valid for ALL situations where the "home-twin" is perpetually unaccelerated. It IS possible, if necessary, to determine simultaneity, according to each of two observers, who are BOTH accelerating, in completely arbitrary ways. But that again requires iterative numerical methods, and the simple CADO equation is not generally applicable in those cases.

Mike Fontenot

 

[GrayGhost]

I was wondering when you are going to get around to answering my questions from post #302. I only have so much patience :)

Touche^'

 

[GrayGhost]

Mike Fontenot,

My position is that the LTs must apply per twin B, even during his proper acceleration. If twin B and the momentarily colocated MSIRF-observer disagree on the instantaneous relative velocity of twin A (as you contend), then a problem arises. I've determined the root of this problem, and how to resolve it. Your contention that they must disagree, is half correct but incomplete IMO. I'll need some time to figure how to articulate this in a "cut to the chase" manner. That said, I am not sure as yet whether this has any impact on your paper.

GrayGhost

 

[Mike_Fontenot]

[...]
My position is that the LTs must apply per twin B, even during his proper acceleration.
[...]

The Lorentz equations DO apply. But the quantity "v" that appears in the Lorentz equations needs to be "the relative velocity between the home-twin and the traveler, according to the home-twin", NOT according to the traveler. Or, since the MSIRF(t) at any given instant "t" in the traveler's life, always agrees with the home-twin about their relative velocity, you can equally well specify the velocity "v" in the Lorentz equations as "the relative velocity between the home-twin and the MSIRF(t), according to the MSIRF(t)" ... it's the same number in either case.

Here's something for you to think about, while you are mulling all this over:

Take the standard twin "paradox" scenario, with gamma = 2. Suppose that immediately before the turnaround, their separation according to the home-twin, is L lightyears. The traveler says their separation is L/2 lightyears then.

Half way through the turnaround (when the home-twin says their relative velocity is zero), the home twin says their separation is still L, and the traveler NOW also says their separation is L lightyears. So the traveler says that their separation has changed by L/2 lightyears, during an infinitesimal amount of his ageing, so he says that their relative velocity during that first half of the turnaround has been infinitely large.

Denote the age of the traveler at the beginning of the turnaround as t1, and the age of the traveler at the midpoint of the turnaround as t1+delta, where delta is infinitesimally small, but non-zero). Denote the MSIRF at the beginning of the turnaround as MSIRF(t1), and the MSIRF at the midpoint of the turnaround as MSIRF(t1+delta) ... they are DIFFERENT inertial frames.

Ask yourself this: what do MSIRF(t1) and MSIRF(t1+delta) say about THEIR own separation (with respect to the home-twin) during the first half of the turnaround? Do either of them agree with the traveler, that the separation changes by L/2 during the infinitesimal time delta, and thus that the velocity during the time delta is infinitely large?

Mike Fontenot

 

[GrayGhost]

...

For reference ...

https://www.physicsforums.com/attachment.php?attachmentid=32982&d=1299884366"​

I've already thought thru all the points you mentioned in your last post here Mike, and I do not disagree with them. It does not change my opinion that your assumption of the instantaneous twin A velocity (per B) is incomplete.

(1) There is the relative twin B velocity recorded by twin A thru A-space over A-time.
(2) There is the relative twin A velocity recorded by twin B thru B-space over B-time.​

Wrt (2), there are 2 components of the relative velocity. Over a twin B virtually-instant-acceleration ...

component one ... is the relative velocity between B and an observer of the A-frame momentarily colocated with B. B will (virtually) agree with said inertial observer's assessment. This is the very same relative velocity that twin A holds of the remotely located luminal twin B.

component two ... is the added relative velocity (over and above component one) which results from the angular rotation of B's own sense-of-simultaneity, and this velocity component increases in conjunction with increased range (eg twin A range). This superpositional effect is why the inertial twin A can move superluminally wrt B per B, during B's rapid enough proper acceleration. It is not the result of any energy expenditure by (or upon) the always-inertial twin A. The further away twin A is, the faster twin A must traverse B-space per B, which in theory (unlikely in practice) could exceed speed c and approach infinite. However, this velocity component does not affect the slope of the twin A worldline, no matter how far away A might be from B.​

I submit that the instantaneous velocity of twin A per B (at some B-time) "is equivalent to" the slope of the B-worldline per A (at that same B-time).

The instantaneous velocity of twin A (from component one) is what must be used by twin B when B applies the LTs to any instant of his own time. He cannot use the collective velocity (both components added) within the LTs.

You said ...

Mike_Fontenot: The Lorentz equations DO apply. But the quantity "v" that appears in the Lorentz equations needs to be "the relative velocity between the home-twin and the traveler, according to the home-twin", NOT according to the traveler.​

So while you agree with me in that v must be that as recorded by twin A, you disagreed (prior) that "the instantaneous twin A velocity" as recorded by B can agree with the MSIRF-observer colocated with B. On the one hand I agree with you, given B is calculating twin A velocity in the classical way, ie change in recorded position over time. However on the other hand, the slope of the worldline is indicative of a velocity different from the classical calculation ... because classically, events (eg. the location of the A clock per B upon commencement of B acceleration) do not move in space and time with accelerated POVs.

So are we (maybe) saying the very same thing and I do not quite realize it, or is there anything in what I say that you disagree with?

GrayGhost

 

[Mike_Fontenot]

[...]

Velocity is velocity. It's silly to re-define it.

Mike Fontenot

 

[Dale]

So GrayGhost are you going to answer the question I asked above? I don't think that what you described above is logically possible, but it is hard to tell without a mathematical formulation.

 

[GrayGhost]

Velocity is velocity. It's silly to re-define it.

It's not a matter of redefining velocity Mike. It's the matter of how the LTs may be applied by B from his accelerating POV, and in a way that matches the twin A experience. I was merely pointing out the problems at hand. In the LTs, velocity is < c, and light's speed is c. You have stated that twin B can use the LTs, however that he must use the velocity that A records of B, not that which B records of A. That's what I said too, I made the attempt to justify why twin B can (and should) use that velocity ... my position being that there exists a 1:1 mapping of like-A/B-worldline-slopes, the slope of the A-worldline per B matching the slope of the B-worldline per A.

The problems are these ...

(1) twin A can move thru B-space superluminally when B undergoes proper acceleration. Obviously, that velocity should not be used in the LTs by twin B.

(2) light's speed varies from c because B's own sense-of-simultaneity dynamically rotates while the light travels. However, the LTs require an invariant c.​

IMO, these problems are resolved by using the instantaneous slope of the A-worldline for v in the LTs (where v is always < c), as opposed to B determining the A-velocity from change in position over duration (which can be superluminal).

GrayGhost

 

[Dale]

these problems are resolved by using the instantaneous slope of the A-worldline for v in the LTs (where v is always < c), as opposed to B determining the A-velocity from change in position over duration (which can be superluminal).

What is "instantaneous slope" if not "change in position over duration". You are contradicting yourself. This is why working out the math is so important.

 

[ghwellsjr]

Here's something for you to think about, while you are mulling all this over:

Take the standard twin "paradox" scenario, with gamma = 2. Suppose that immediately before the turnaround, their separation according to the home-twin, is L lightyears. The traveler says their separation is L/2 lightyears then.

Half way through the turnaround (when the home-twin says their relative velocity is zero), the home twin says their separation is still L, and the traveler NOW also says their separation is L lightyears. So the traveler says that their separation has changed by L/2 lightyears, during an infinitesimal amount of his ageing, so he says that their relative velocity during that first half of the turnaround has been infinitely large.

Denote the age of the traveler at the beginning of the turnaround as t1, and the age of the traveler at the midpoint of the turnaround as t1+delta, where delta is infinitesimally small, but non-zero). Denote the MSIRF at the beginning of the turnaround as MSIRF(t1), and the MSIRF at the midpoint of the turnaround as MSIRF(t1+delta) ... they are DIFFERENT inertial frames.

Ask yourself this: what do MSIRF(t1) and MSIRF(t1+delta) say about THEIR own separation (with respect to the home-twin) during the first half of the turnaround? Do either of them agree with the traveler, that the separation changes by L/2 during the infinitesimal time delta, and thus that the velocity during the time delta is infinitely large?

Mike Fontenot

When the traveler decelerates and becomes at rest in the frame in which the home-twin has always been at rest, the home-twin has no awareness of this event until long after it has occurred. You cheat when you give him knowledge from the frame of reference that we are aware of.

The traveler also does not experience an infinite velocity when he decelerates. He does not see the home-twin suddenly fly away from him. If he remained stationary in his original rest frame for a long time, instead of accelerating back toward home, he would gradually see the home-twin moving away from him, just as he would see all other objects in the sky (at rest in his original rest frame), both in front of him and behind him start to move away from him.

What if at the "moment" (as you define NOW for both twins) when the traveler decelerates, the home-twin were to also accelerate away from the traveler with the exact same acceleration profile, would you conclude that in the MSIRF of the home-twin, the traveler has not accelerated at all but continued on his steady speed away from the home-twin? Would this actual increase in speed in the home-twin negate your conclusion that the relative infinite velocity between the two twins has gone away? And would the traveler also agree from his MSIRF that the relative velocity has not changed at all and that the separation between the two of them remains constant at L/2?

 

[GrayGhost]

this post posted by accident.

 

[GrayGhost]

What is "instantaneous slope" if not "change in position over duration". You are contradicting yourself.

For reference ...

https://www.physicsforums.com/attachm...2&d=1299884366"​

DaleSpam,

Look at the illustration in the link above. If we are to assume that velocity is nothing more than the change of position over time, then twin A moves superluminally from point 1 to point 3 thru B-space during B's own virtually-instant proper acceleration. Clearly, this velocity cannot be used in LTs, and for good reason. I don't see this as overly complicated, personally.

My position is that the slope of the A-worldline dictates A's current velocity wrt B per B, far as "what velocity B should use in the LTs" goes. If you imagine a uniform almost-virtually-instant twin B proper acceleration, it's quite easy to envision how the A-worldline progresses from vertical to the slope of 0.866c, never exceeding 0.866c, let alone c.

Now you assume I am contradicting myself. However, there are 2 processes occurring wrt the A-worldline (per B) as B accelerates ...

(1) The A-worldline rotates steadily from vertical to a slope indicative of 0.866c, never exceeding 0.866c.

(2) The intersection of "the A-worldine and B-line-of-simultaneity" moves thru B-space, and can be superluminal.​

Now if the velocity is to be determined by the receipt of light signals, and doppler shifts converted into their appropriate dilation equivalent, then superluminal motion is never detected. The established velocity will match the current slope of the A-worldline per B. The reason this works out as such, is because events move in space and time per B during his acceleration, something that does not happen classically. One such event would be the location of the A-clock upon commencement of B acceleration, ie twin B's departure event from twin A. As twin B accelerates, said event drops further and further back in B time, and further and futher away in B-space, per B. This keeps the worldline slope at sub-c, which in my illustration never exceeds 0.866c even though A must move thru B-space superluminally (if velocity is determined in the classical way change in position over time).

GrayGhost

 

[Mike_Fontenot]

[...]

In the twin "paradox" example I gave previously (with gamma = 2), IF the traveler concludes that his distance to the home twin increases by L/2 lightyears during his infinitesimal ageing during the first half of his turnaround, then he MUST conclude that their relative velocity was infinite (on average) during that section of his life. Period. Non-negotiable.

And IF (as in the CADO reference frame) the traveler's conclusions about those two distances (at the beginning and at the midpoint of the turnaround) AGREE with the respective MSIRFs' conclusions about those distances at those two instants, then those two distances ARE L/2 and L, respectively. So the traveler MUST conclude that the distance to his home twin increases by L/2 during the first half of the his turnaround. Period. Non-negotiable.

IF the traveler ALWAYS agrees, about the instantaneous distance to his home twin, at each instant of his life, with his MSIRF at that instant, then he WILL disagree with that MSIRF about the relative velocity of the home twin at any instant during the first half of his instantaneous turnaround. Each MSIRF is an inertial frame, and NO inertial frame will EVER conclude that the home twin has an infinite relative velocity with respect to the traveler.

If you want to use a reference frame for the traveler, for which the relative velocity of the home twin during the first half of the turnaround ISN'T infinite, then that reference frame CAN'T agree with the conclusion of the traveler's MSIRF about his current distance to the home twin, at each instant during that first half of the turnaround. You just CAN'T have it both ways.

Mike Fontenot

 

[Dale]

For reference ...

https://www.physicsforums.com/attachm...2&d=1299884366"​

DaleSpam,

Look at the illustration in the link above.

The link doesn't work.

If we are to assume that velocity is nothing more than the change of position over time

It is not an assumption, it is a definition.

If you want to re-define velocity that is OK, but you will have to be very very clear and precise. No handwaving, just precise mathematical definitions. You are using non-standard terms and you are re-defining standard terms, so you cannot assume that I will understand what you mean without a rigorous treatment.

Again, say that you have an inertial unprimed frame:
(t,x)

And in that inertial frame there is an observer B with a timelike worldline:
(t_B(\lambda),x_B(\lambda))

What is the expression or operation to determine B's coordinates:
(t&#039;,x&#039;)

And now the operation to determine the velocity.
v_A(\lambda)=f(?)

 

[GrayGhost]

The link doesn't work.

Ahhh, a Cut-N-Paste problem there. Here's a repost with th correct web address ...

****************************************************************

For reference ...

https://www.physicsforums.com/attachment.php?attachmentid=32982&d=1299884366"​

DaleSpam,

Look at the illustration in the link above. If we are to assume that velocity is nothing more than the change of position over time, then twin A moves superluminally from point 1 to point 3 thru B-space during B's own virtually-instant proper acceleration. Clearly, this velocity cannot be used in LTs, and for good reason. I don't see this as overly complicated, personally.

My position is that the slope of the A-worldline dictates A's current velocity wrt B per B, far as "what velocity B should use in the LTs" goes. If you imagine a uniform almost-virtually-instant twin B proper acceleration, it's quite easy to envision how the A-worldline progresses from vertical to the slope of 0.866c, never exceeding 0.866c, let alone c.

Now you assume I am contradicting myself. However, there are 2 processes occurring wrt the A-worldline (per B) as B accelerates ...

(1) The A-worldline rotates steadily from vertical to a slope indicative of 0.866c, never exceeding 0.866c.

(2) The intersection of "the A-worldine and B-line-of-simultaneity" moves thru B-space, and can be superluminal.​

Now if the velocity is to be determined by the receipt of light signals, and doppler shifts converted into their appropriate dilation equivalent, then superluminal motion is never detected. The established velocity will match the current slope of the A-worldline per B. The reason this works out as such, is because events move in space and time per B during his acceleration, something that does not happen classically. One such event would be the location of the A-clock upon commencement of B acceleration, ie twin B's departure event from twin A. As twin B accelerates, said event drops further and further back in B time, and further and futher away in B-space, per B. This keeps the worldline slope at sub-c, which in my illustration never exceeds 0.866c even though A must move thru B-space superluminally (if velocity is determined in the classical way change in position over time).

GrayGhost

 

[GrayGhost]

In the twin "paradox" example I gave previously (with gamma = 2), IF the traveler concludes that his distance to the home twin increases by L/2 lightyears during his infinitesimal ageing during the first half of his turnaround, then he MUST conclude that their relative velocity was infinite (on average) during that section of his life. Period. Non-negotiable.

Let's see how twin B plugs that infinite twin A velocity value into the LTs, see where twin A is placed by B into twin B's own system.

And IF (as in the CADO reference frame) the traveler's conclusions about those two distances (at the beginning and at the midpoint of the turnaround) AGREE with the respective MSIRFs' conclusions about those distances at those two instants, then those two distances ARE L/2 and L, respectively. So the traveler MUST conclude that the distance to his home twin increases by L/2 during the first half of the his turnaround. Period. Non-negotiable.

In this, we agree.

IF the traveler ALWAYS agrees, about the instantaneous distance to his home twin, at each instant of his life, with his MSIRF at that instant, then he WILL disagree with that MSIRF about the relative velocity of the home twin at any instant during the first half of his instantaneous turnaround. Each MSIRF is an inertial frame, and NO inertial frame will EVER conclude that the home twin has an infinite relative velocity with respect to the traveler.

Seems to me that when twin B and the MSIRF observer are colocated, twin A must exist somewhere in spacetime and both those fellows must agree (even though they will disagree as to how twin A got there over time) ... assuming they have diligently and correctly maintained the current location of twin A. The reason they must agree Mike, is because they are momentarily of the very same frame-of-reference. All observers who are stationary wry one another, even if momentarily, must agree on where twin A then is. Maintaining the current twin A location via continuous radar tracking (and processing) is more laborious for twin B than for the momentarily colocated MSIRF-observer, but it doesn't change the fact that at that instant ... they must map where twin A is (identically) within their own (overlaid) system(s).

If you want to use a reference frame for the traveler, for which the relative velocity of the home twin during the first half of the turnaround ISN'T infinite, then that reference frame CAN'T agree with the conclusion of the traveler's MSIRF about his current distance to the home twin, at each instant during that first half of the turnaround. You just CAN'T have it both ways.

The fact that it is more difficult for twin B to (keep track and) determine the location of twin A at any instant during his own acceleration, does not lead that he should disagree with the momentarily colocated MSIRF-observer. All observers at rest with each other agree on the location and clock readout of a moving observer.

GrayGhost

 

[Dale]

If we are to assume that velocity is nothing more than the change of position over time, then twin A moves superluminally from point 1 to point 3 thru B-space during B's own virtually-instant proper acceleration.

Exactly. And again, it is not an assumption, it is a definition. If you wish to re-define velocity then you may, but you need to be completely specific about it.

If you imagine a uniform almost-virtually-instant twin B proper acceleration, it's quite easy to envision how the A-worldline progresses from vertical to the slope of 0.866c, never exceeding 0.866c, let alone c.

Now you assume I am contradicting myself. However, there are 2 processes occurring wrt the A-worldline (per B) as B accelerates ...

(1) The A-worldline rotates steadily from vertical to a slope indicative of 0.866c, never exceeding 0.866c.

(2) The intersection of "the A-worldine and B-line-of-simultaneity" moves thru B-space, and can be superluminal.​

Your point (1) is clearly not true in your diagram. The slope of the A worldline clearly exceeds 0.866c, or even c. Regarding (2) "the intersection of the A-worldline and the B-line-of-simultaneity" is just a long-winded way of saying the position of A in the B frame. The velocity of A in the B frame is by definition the derivative of this. Again, if you wish to redefine velocity you will have to be very specific. More math less english. Even if B's acceleration is finite this can lead to v>c.

Now if the velocity is to be determined by the receipt of light signals, and doppler shifts converted into their appropriate dilation equivalent, then superluminal motion is never detected.

Yes, that is the Dolby and Gull approach, not the naive/CADO approach.

The established velocity will match the current slope of the A-worldline per B. The reason this works out as such, is because events move in space and time per B during his acceleration, something that does not happen classically.

What does this mean? What is the formula that describes this "motion of events"?

I think perhaps this is the key thing that you need to define, then you could possibly define your new concept of velocity as some sort of motion in addition to or relative to this motion of events. But you really need to be clear and mathematically precise here if you want to ensure a self-consistent outcome.

Honestly, rather than patching up such a strange ad-hoc concept, I think you would be much better served actually learning some differential geometry. But if you do want to pursue this the place to start seems to be this concept of the motion of events. Start by expressing that mathematically.

 

[Mike_Fontenot]

Let's see how twin B plugs that infinite twin A velocity value into the LTs, see where twin A is placed by B into twin B's own system.
[...]

That's the whole point: the traveler must NOT use HIS value of the relative velocity in the Lorentz equations, he MUST use the HOME-TWIN'S value of the relative velocity in the Lorentz equations. Or, since the MSIRF always agrees with the home-twin about their relative velocity, the traveler can use the MSIRF's value of the velocity ... it's the same number.

I think I stated that very clearly in the first part of this posting:

https://www.physicsforums.com/showpost.php?p=3223917&postcount=314 .

Here's an excerpt from that posting:

[BEGIN EXCERPT]:

"Originally Posted by GrayGhost

[...]
My position is that the LTs must apply per twin B, even during his proper acceleration.
[...]

The Lorentz equations DO apply. But the quantity "v" that appears in the Lorentz equations needs to be "the relative velocity between the home-twin and the traveler, according to the home-twin", NOT according to the traveler. Or, since the MSIRF(t) at any given instant "t" in the traveler's life, always agrees with the home-twin about their relative velocity, you can equally well specify the velocity "v" in the Lorentz equations as "the relative velocity between the home-twin and the MSIRF(t), according to the MSIRF(t)" ... it's the same number in either case."

[END EXCERPT]

I think you are perhaps overloaded, and are trying to do so many things so fast that you are missing some important things in some of the previous posts.

[...]
Seems to me that when twin B and the MSIRF observer are colocated, twin A must exist somewhere in spacetime and both those fellows must agree (even though they will disagree as to how twin A got there over time) ...
[...]
The fact that it is more difficult for twin B to (keep track and) determine the location of twin A at any instant during his own acceleration, does not lead that he should disagree with the momentarily colocated MSIRF-observer. All observers at rest with each other agree on the location and clock readout of a moving observer.

They DO agree about the current distance to the home-twin, and about the current age of the home-twin. But they DON'T agree about the home-twin's current relative velocity.

In another previous posting, I described WHY they disagree about the relative velocity:

https://www.physicsforums.com/showpost.php?p=3217917&postcount=305 .

Here's an excerpt from that posting:

[BEGIN EXCERPT]:

And since the home twin's inertial frame always agrees about relative velocity with each of those MSIRF's, I thought that the accelerating traveler must also agree. That's why I didn't feel the need to specify "according to WHOM?" when I referred to the quantity "v" or "beta" in my paper.

But I later realized that I was incorrect: the accelerating traveler does NOT agree about velocities with his current MSIRF. They agree about distances and times at that instant, but not about velocities at that instant. The reason lies in the fact that velocities always by definition are based on changes of a distance during a very small change in time. I.e., the velocity of the home twin, relative to the traveler, according to the traveler, at some instant t of his life, refers to how that distance changes for two very closely-spaced times of his life very near the instant t. And the MSIRF's at those two different times in his life AREN'T the SAME inertial reference frame. THAT'S why the traveler generally won't agree about relative velocities with his current MSIRF.

Fortunately, this omission on my part didn't affect the results in my paper, because all the results were correct when the quantity "v" and "beta" in my equations referred to the velocity of the traveler, relative to the home twin, ACCORDING TO THE HOME TWIN. I.e., it WAS necessary for me to specify "according to WHOM" when I referred to a relative velocity.

[END EXCERPT]

I KNOW you can understand the above stuff ... just slow down a little bit.

Mike Fontenot

 

[Mike_Fontenot]

Just FYI:

In a follow-up paper to my original paper on accelerating observers, I derived "the velocity of the traveler, relative to the home-twin, according to the traveler". The result is (for units where c has unity magnitude, so I'll leave c out of the equation, for simplicity):

V = v - (L*v*a)/gamma ,

where BOTH V and v are "the velocity of the traveler, with respect to the home twin", but V is "according to the traveler", and v is "according to the home-twin" (or, equivalently, "according to the MSIRF"). v is positive when the twins are moving apart.

L is the (positive) separation between traveler and home-twin, according to the home-twin.

"a" is the traveler's acceleration, as read on an accelerometer he carries. "a" is positive when in the direction of positive v.

All of the quantities in the equation are for some arbitrary, but given, instant of the traveler's life.

Mike Fontenot

 

[GrayGhost]

I think perhaps this is the key thing that you need to define, then you could possibly define your new concept of velocity as some sort of motion in addition to or relative to this motion of events. But you really need to be clear and mathematically precise here if you want to ensure a self-consistent outcome.

Honestly, rather than patching up such a strange ad-hoc concept, I think you would be much better served actually learning some differential geometry. But if you do want to pursue this the place to start seems to be this concept of the motion of events. Start by expressing that mathematically.

Dale Dale DaleSpam. It's quite interesting you know. The theory demands that no material body can accelerate to speed c, because of energy considerations. Add, speed c represents a cosmic speed limit. Yet per twin B, twin A could move superluminally thru B-space. So what does it all mean? Hmmm.

Conceptually, it's quite clear in my mind, personally. Events move in B-space per B while B continues his own proper acceleration. Such an event is the B-departure-from-A. This is what keeps the velocity < 0.866c in my example (let alone < c), even though A moves superluminally per B from a standpoint of the-change-in-position-over-time.

You know ... this reminds me of the problem whereby many folks remain stuck, because they generally tend to ignore the time dilation while focusing only on spatial changes. Neither can be ignored, and they always exist in unison. In this case, IMO, twin A moves superluminally (per B) only if B focuses only on the spatial contraction while ignoring the time dilation component altogether. If the time dilation component is not ignored, then the cosmic speed limit is not violated. Add, it's referred to as time dilation ... however really, its a dilation of spacetime.

Indeed, what needs to be done is to math-model the twin B experience as the-collection-of-momentary-colocated-inertial-frames-of-reference that B co-occupies. Yes, differential geometry should get the job done. I haven't seen it anywhere, but I find it very difficult to believe no one has done this as yet. Now, Mike Fontenot believes he has done this. Where Mike simply states that twin B cannot use his own recorded A-velocity in the LTs (he must use an A-framer's recorded v), I merely made the attempt to explain WHY twin B must do so, and why it works. Wrt that matter, I figure I'm saying the same fundamental thing as Mike, but I'm not sure he realizes it yet.

Also, I never looked at it as though I was changing the definition of relative velocity. However, maybe I am now that you mention it? I suppose it'd be a rather presumptuous thing to do, assuming it hasn't yet been done. As I said though, I doubt I'm the first to suggest this, and I'd be very very surprised if no one has modeled it as yet. It's not a change to the classical velocity definition far as non-luminal speeds go ... it applies only to the case of luminal (and superluminal) motion from a non-inertial POV, in the relativistic case.

Wrt another comment you made ... it seems to me that if the Dolby and Gull approach converts doppler shift to the appropriate spatial offset, it should be equivalent to what I've been saying. Yes? I mean, the relativistic doppler shift is the result of dilation in space and time both, so said conversion must account for both the time component and the space component.

GrayGhost