# Graphical Derivation of the CADO Equation

And with a very brief (non-instant) turnaround: explain to me how it is possible that her clock's tickrate "suddenly accelerates" instantly due to Charlie making his turnaround many many lightyears away.

Here is the explanation you keep asking for, and I keep giving you:

Charlie's plane of simultaneity shifts when he turns around. This causes Charlie himself (not Alice) to conclude that Alice's time is currently at a different time than it was before.

The calculations are correct, but the interpretation that Alice jumps forward in time from t=10 to t=70 [or that her tickrate accelerates] due to Charlie's turnaround is wrong.

This is your strawman. I have never said that Alice herself experiences anything due to Charlie's turnaround. It is Charlie who experiences the change in Alice's clock position & rate etc.

(Needy Bate: not Alice.)

And if Charlie understands what his jumps does, neither does Charlie.

Then what time do you think Charlie would say is on Alice's clock, if it is not 10 or 70? Is it is 40 or what?

Here is the basic "twin paradox" with an outbound leg of uniform motion, a very brief turnaround, and an inbound leg of uniform motion:

ACCORDING TO THE STAY-HOME TWIN:
t=0 t'=0
t=20 t'=10
t=40 t'=20
(Very brief turnaround)
t=40 t'=20
t=60 t'=30
t=80 t'=40
(Twins re-united, both agree on their ages)

ACCORDING TO THE TRAVELING-TWIN:
t'=0, t=0
t'=20 t=10
(Very brief turnaround)
t'=20 t=?????????????
t'=40 t=80
(Twins re-united, both agree on their ages)

I will leave it up to the nay-sayers to take a wild guess what ????????????? should be.

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I hope I'm wrong, but I think Not Einstein would say ?????????????=40 and I think Q-reeus would say ?????????????=10, lol.

The correct answer is 70 of course.

Later guys! I will not be checking in this thread any more. Have a great life.

Risking the accusation of an argument from authority, here's Brian Cox on the matter:

Look again at what he says at the 2:15 point.

One last thing that just occurred to me...

'Actual' time dilation rates are always relative to a given proper reference frame. Uniform circular motion of traveling twin as per #30. The instantaneous relative motions are always reciprocal, so your argument implies clocks for both traveling twin B and home twin A tick at equal retarded rates 1/gamma, as measured from each other's frame. But it's not so. In fact that retarded rate 1/gamma only applies to B's clock as measured from A's frame. In B's frame, A's measured clock rate is increased by factor gamma.
If you care to work it through, this is entirely consistent with Doppler shift analysis. Whereas you will have a very hard time reconciling what I quoted of you above, with circular motion case.

For the case of the traveling twin in uniform circular motion, the Doppler effect data for a stationary observer would depend on their location. For example, one gets different Doppler effect data if the stationary observer is located at the center of the circle, compared to if the stationary observer is located outside the circle. Yet both of those stationary observers are in the same proper reference frame. This disproves your underlined claim, above.

Of course once the Doppler effect is factored out, then both such stationary observers would agree on the time dilation rate, which would be 1/gamma. And the only reason the traveling twin finds it to be gamma instead of 1/gamma is because he is turning around constantly, which causes his simultaneity to shift so that the other twin's clock goes ahead, just as it does in the purely linear case.

If you choose a polygon path instead of a circle, the time dilation rate would be 1/gamma for both twins (reciprocal), everywhere except for the turns at the corners. In those locations, the same effect happens, ensuring the stationary twin ends up older, just as it does in the purely linear case.

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The actual time dilation rate is 1/gamma on both the outward leg, and the inward leg of the journey. We are not talking about a circular journey, this is a straight path outward, a turnaround, and a straight path back.

The Doppler shifted rates do not represent the actual time dilation rates, as they need to be converted as described in your own link. Notice also that your link never says that the Doppler approach supercedes the Minkowski diagrams in any way. The Minkowski diagrams all show the stay-home-twin's time changing when the traveling-twin turns around, the very thing you are denying.
It's all rather moot, but I will take back my claim in #271 that Doppler rates always reflect 'actual' aging rates of the other observer. It does for circular motion as per #30, but not generally, and not for straight-line in-out journeys that CADO caters for. The main point I emphasized back in #260 & again in #265 remains - what matters is the accumulated age difference when reunited. Anything inferred in between - especially any hint of 'backwards time travel', is a suspect artifact of a particular method of analysis and particular interpretation thereof.

If you and MF are comfortable with 'actual' backwards time jumps occuring, go ahead. Doppler analysis has no such artifact, and has the distinct advantage that each observer always has an ever accumulating record of the other observer's clock ticks, with no room for 'missing' ticks. Hence there is no room for error when comparing each other's net elapsed proper time upon reunion. Doppler rates are always positive thus 'backwards time jumps' are a foreign notion.

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For the case of the traveling twin in uniform circular motion, the Doppler effect data for a stationary observer would depend on their location. For example, one gets different Doppler effect data if the stationary observer is located at the center of the circle, compared to if the stationary observer is located outside the circle. Yet both of those stationary observers are in the same proper reference frame. This disproves your underlined claim, above.
No it doesn't, for two reasons.
Firstly, because it was *specified* clearly in #30:
"...In the case of constant circular motion about an axis intersecting the home twin,..."

Secondly, even for off-axis location of home twin, the *net* result will have the same 1/gamma vs gamma rates asymmetry over each complete 2pi radians elapsed cycle. All that changes is there will be a superposed quasi-harmonic modulation of Doppler rates. Which modulation averages to zero overall.
Of course once the Doppler effect is factored out, then both such stationary observers would agree on the time dilation rate, which would be 1/gamma. And the only reason the traveling twin finds it to be gamma instead of 1/gamma is because he is turning around constantly, which causes his simultaneity to shift so that the other twin's clock goes ahead, just as it does in the purely linear case.
But 'factoring out the Doppler shift' is tantamount to rejecting SR entirely. Pointless notion. It's necessarily always there as an intrinsic feature. And does there reflect the true relative rates of aging.
If you choose a polygon path instead of a circle, the time dilation rate would be 1/gamma for both twins (reciprocal), everywhere except for the turns at the corners. In those locations, the same effect happens, ensuring the stationary twin ends up older, just as it does in the purely linear case.
Above a certain level of polygon subdivision, the distinction between true circular and polygon will blur into imperceptibility and at no time would the traveling twin detect 'straight line 1/gamma dilation rate' for the home twin's clock (home twin always detects 1/gamma rate for traveling twin's clock). Nor is there then any room for sudden backwards time jumps.
Best to just accept it's all a consequence of integrating over the standard SR spacetime infinitesimal interval ds^2 = c^2dt^2 - (dx^2 + dy^2 + dz^2) in both frames properly.

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Here is the explanation you keep asking for, and I keep giving you:

Charlie's plane of simultaneity shifts when he turns around. This causes Charlie himself (not Alice) to conclude that Alice's time is currently at a different time than it was before.
And here is my response that I keep giving you: if the effect only happens with and around Charlie, and has no effect on Alice('s time), and Alice's worldline is thus unaffected, she didn't travel back in time.

It's not a strawman: it's the conflict that you (apparently) still don't see clearly.

I have never said that Alice herself experiences anything due to Charlie's turnaround.
True, but you are talking about her travelling back in time. If that's not something that she experiences, if it's not something that's "real to her", how can it be said to have happened to her? It can't, and therefore Charlie concluding that Alice travelled back in time is incorrect.

It is Charlie who experiences the change in Alice's clock position & rate etc.
Right, and since Alice doesn't travel back in time (just look at her worldline in the Minkowski diagram), Charlie concluding that she did is wrong.

Then what time do you think Charlie would say is on Alice's clock, if it is not 10 or 70? Is it is 40 or what?
As long as the event in question is in Charlie's "elsewhere"-section in his lightcone, he cannot say, as events in that section can still shift around if Charlie accelerates. When it's in his part-section, he'll (in principle) have information about it, and if it's in his future-section, well, it's the future, it's not yet written.

Basic GR, really.

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I hope I'm wrong, but I think Not Einstein would say ?????????????=40 and I think Q-reeus would say ?????????????=10, lol.
Please stop putting words in my mouth.

The correct answer is 70 of course.
Could very well be; I haven't bothered checking, because it's really not that important to what we are discussing.

Later guys! I will not be checking in this thread any more. Have a great life.
Aww, you were getting closer and closer to figuring out the conflict! Ah well, your loss. If you don't want to learn, I can't make you.

Look again at what he says at the 2:15 point.
All right, lets!

"...protects that. Even though you can, sort of, slow time down and speed it up, and you can even reverse the order of certain things happening, you can't reverse the order of things that cause other things to happen, and it's all down to the minus sign. ..."
Let's break that down.

"Even though you can, sort of, slow time down"
Note that this isn't a reversal of the flow of time.

I've explicitly stated agreement with this happening in this thread.

"and speed it up,"
This isn't about a reversal of the flow of time either.

I've expressed agreement with this happening in this thread too.

"and you can even reverse the order of certain things happening,"
Yes, this is something that I brought up: the elsewhere-section of the lightcone. Note that this isn't a reversal of the flow of time: it's explicitly about reordering events.

"you can't reverse the order of things that cause other things to happen,"
Yes, this is something I explicitly mentioned when bringing up the lightcone.

"and it's all down to the minus sign."
I indeed didn't brought this up; it has little added value to this conversation, and I doubt all participants would manage GR, seeing how many trouble they are having with SR already.

So Mike_Fontenot, what did you want to point out? That Prof Brian Cox agrees with everything I've been saying in this thread? Or that you maybe misinterpreted what he said?

I will leave it up to the nay-sayers to take a wild guess what ????????????? should be.

Could very well be; I haven't bothered checking, because it's really not that important to what we are discussing.
I should probably answer that anyway, lest I be accused of dodging the question. I agree with whatever SR says the ????????????? should be, because I think SR is correct (as correct as any scientific theory can be, of course) in the domain in which it is applicable.

[...]

So Mike_Fontenot, what did you want to point out? That Prof Brian Cox agrees with everything I've been saying in this thread? Or that you maybe misinterpreted what he said?

I'm saying that he clearly disagrees with you. You and I will just have to agree to disagree about that.

I have another comment. If I understand you correctly, you're OK with sudden increases of the home twin's age (according to the traveler), but that you're not OK with sudden decreases in her age (according to the traveler). If so, I think you've got a big logical problem with that view. If the traveler suddenly changes his velocity from +v to -v (from moving away from her to moving toward her), then he will conclude that she suddenly gets a lot older. But if he then changes his velocity from -v back to +v, he will conclude that she suddenly gets a lot younger. Those two back-to-back velocity changes are equivalent to no change in his velocity at all. So if her first sudden ageing (according to him) is valid, then her immediately following sudden decrease in age must also be valid.

Here's another issue I want to address: the question of whether the home twin ever perceives that her time is "running backward". The shortest answer is a question: "Compared to WHAT?". I.e.., to talk about how her time is "running", some other time reference is needed to quantify what "running" means. What is that other time reference? A lengthier answer is the following:

At each instant of the home twin's life, her "state" at that instant is defined by the the arrangements and current motions of all the molecules in her body at that instant. Most important is that her perception of herself and her life at that instant is "recorded" in the state of her brain cells at that instant. At that instant in her life, that state, whatever it is, is fixed and unchangeable for all eternity. It can't change, just like the image on any particular frame of a movie reel is fixed and can't change, regardless of whether some projectionist is running his projector forwards or backwards (and at different speeds), or stopped completely. Any perception of the progression of those images, from one image to the next, is purely in the eyes of the viewer watching what the projectionist is projecting. Similarly, any progression of the home twin's age depends on whose "perspective" we're referring to ... each "observer" in general will come to a different conclusion about the progression of the home twin's age, and each is right. It is possible for an arbitrarily large number of inertial observers to be momentarily co-located wrt the traveling twin at the instant of his turnaround, each moving at a different velocity wrt the home twin, and they will all come to different conclusions about her current age at that instant. They are all equally right. And the home twin herself is also equally right about her own conclusion about her age progression during his turnaround.

As to why we each perceive that our own age is progressing at a constant "rate" (whatever that is) toward the future is an open question that has never been convincingly answered by physics. The differential equations that govern physics run equally well in both directions of time. The only compelling description I've seen is that our universe for some reason (perhaps by pure chance) just happened to begin in an extremely highly-ordered (low entropy) state, which makes almost all possible trajectories progress to less-ordered states. Progression from highly disordered states to an extremely highly ordered state is not impossible, but is much less likely than the huge number of progressions in the opposite direction. But that still doesn't really explain why we all perceive a flow of time from the past toward the future. Perhaps the people who argue that the progression of time is just an illusion are correct. But even if they are correct, it would be hard or maybe impossible to do everyday physics work without assuming a progression of time from past to future.

I'm saying that he clearly disagrees with you. You and I will just have to agree to disagree about that.
Interesting that you are unwilling to actually talk about that disagreement though; you only assert its existence by pointing out a timestamp, and then refuse to even explain your position.

I have another comment. If I understand you correctly, you're OK with sudden increases of the home twin's age (according to the traveler), but that you're not OK with sudden decreases in her age (according to the traveler). If so, I think you've got a big logical problem with that view. If the traveler suddenly changes his velocity from +v to -v (from moving away from her to moving toward her), then he will conclude that she suddenly gets a lot older. But if he then changes his velocity from -v back to +v, he will conclude that she suddenly gets a lot younger. Those two back-to-back velocity changes are equivalent to no change in his velocity at all. So if her first sudden ageing (according to him) is valid, then her immediately following sudden decrease in age must also be valid.
No, I'm not OK with sudden increases as described either. Neddy Bate also came up with this misrepresentation of my position. I have already addressed it pages ago in my post #233:
I fully agree that if one is a problem, the other one is too. However, one doesn't stand out like a sour thumb, while the other one does. Thus my concerns only seem arbitrary to people that fail to understand the underlying basis of those concerns. Basic things like the arrow of time and causality.
Here's another issue I want to address: the question of whether the home twin ever perceives that her time is "running backward". The shortest answer is a question: "Compared to WHAT?". I.e.., to talk about how her time is "running", some other time reference is needed to quantify what "running" means. What is that other time reference?
In the context of the discussion we were having, this question has been explicitly answered: compared to Charlie's clock. In fact, this has been implicitly stated by Neddy Bate and me over and over again.
But I think you'll find this question in this context is actually irrelevant: in SR, it is with respect to any clock!

A lengthier answer is the following:

At each instant of the home twin's life, her "state" at that instant is defined by the the arrangements and current motions of all the molecules in her body at that instant.
Wonderful point! Let's say Charlie's frame switching is near instantly, and his distance to Alice is large. This means the "jump backwards in time" is large, and happens in a small timespan for Charlie. Can you please describe to me what the molecules in Alice's body are doing during that time? For example, what's her temperature? What's the relativistic mass of her molecules? What's her internal pressure?

Most important is that her perception of herself and her life at that instant is "recorded" in the state of her brain cells at that instant. At that instant in her life, that state, whatever it is, is fixed and unchangeable for all eternity. It can't change, just like the image on any particular frame of a movie reel is fixed and can't change, regardless of whether some projectionist is running his projector forwards or backwards (and at different speeds), or stopped completely. Any perception of the progression of those images, from one image to the next, is purely in the eyes of the viewer watching what the projectionist is projecting. Similarly, any progression of the home twin's age depends on whose "perspective" we're referring to ... each "observer" in general will come to a different conclusion about the progression of the home twin's age, and each is right. It is possible for an arbitrarily large number of inertial observers to be momentarily co-located wrt the traveling twin at the instant of his turnaround, each moving at a different velocity wrt the home twin, and they will all come to different conclusions about her current age at that instant. They are all equally right. And the home twin herself is also equally right about her own conclusion about her age progression during his turnaround.

As to why we each perceive that our own age is progressing at a constant "rate" (whatever that is) toward the future is an open question that has never been convincingly answered by physics. The differential equations that govern physics run equally well in both directions of time.
Note that while the equations do work fine in both directions, T-symmetry is proven to be broken: the two directions are not symmetric.

The only compelling description I've seen is that our universe for some reason (perhaps by pure chance) just happened to begin in an extremely highly-ordered (low entropy) state, which makes almost all possible trajectories progress to less-ordered states. Progression from highly disordered states to an extremely highly ordered state is not impossible, but is much less likely than the huge number of progressions in the opposite direction.

Now, can you please explain to me how Charlie changing velocity leads superluminally to a closed system next to Alice suddenly decreasing in entropy?

But that still doesn't really explain why we all perceive a flow of time from the past toward the future. Perhaps the people who argue that the progression of time is just an illusion are correct. But even if they are correct, it would be hard or maybe impossible to do everyday physics work without assuming a progression of time from past to future.
So... are you suggesting that your resolution to the conflict is "time is an illusion"?

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No, I'm not OK with sudden increases as described either.
[...]

Over the years, I've noticed that many people, including some professional physicists, find the idea of the home twin suddenly getting younger (according to the traveling twin) to be so abhorrent that they will embrace ANY alternative explanation, no matter how absurd that alternative explanation may be. Some just argue that the whole concept of time at a distance is meaningless, and should never even be considered. Others embrace alternative schemes for calculating the time at a distance. An alternative that is embraced by most of the powers-that-be on the "Physics Forums" forum is the Dolby and Gull explanation, called the "Radar Method". It avoids any possibility of negative ageing by the home twin (according to the traveling twin), but at the expense of violating causality: the current age of the home twin, during an instantaneous (or short) turnaround, depends on how the traveling twin will choose to accelerate well AFTER the given turnaround. That means that the home twin's current age at the instant of the turnaround is not merely unknown then, but isn't even defined then. That seems like pure desperation to me.

The article alluded to but not customarily given a useful link to in #296: https://arxiv.org/abs/gr-qc/0104077
Seems rigorous enough, to the extent any 'resolution' of a traveling twin 'paradox' can be. No evident hint of desperation there.

More relevantly imo, after more than twenty years plugging away with promoting CADO, there is even ONE identifiable physically consequential implication? Say, a modified/improved GPS, or modified/improved resolution of current Hubble 'constant' discrepancy? Or anything else that can be measured and/or used? Or is it actually nothing more when all boiled down, philosophical distraction?

And btw MF you are STILL evading. #258 -> #91 (not #90) -> #30. Supposedly generally applicable CADO formula manifestly fails for circular motion case. Yes or no?

Over the years, I've noticed that many people, including some professional physicists, find the idea of the home twin suddenly getting younger (according to the traveling twin) to be so abhorrent that they will embrace ANY alternative explanation, no matter how absurd that alternative explanation may be. Some just argue that the whole concept of time at a distance is meaningless, and should never even be considered. Others embrace alternative schemes for calculating the time at a distance. An alternative that is embraced by most of the powers-that-be on the "Physics Forums" forum is the Dolby and Gull explanation, called the "Radar Method". It avoids any possibility of negative ageing by the home twin (according to the traveling twin), but at the expense of violating causality: the current age of the home twin, during an instantaneous (or short) turnaround, depends on how the traveling twin will choose to accelerate well AFTER the given turnaround. That means that the home twin's current age at the instant of the turnaround is not merely unknown then, but isn't even defined then. That seems like pure desperation to me.
Erm, that's nice and all, but that has nothing to do with me, my position, or the part of my post you quoted. Care to actually address what I said?

'Sudden time jumps' is so reminiscent of e.g. 'superluminal light spots': https://en.wikipedia.org/wiki/Lighthouse_paradox
And likewise physically inconsequential.

The difference is that with the "moving" light spots, the spots from one instant to the next are different spots ... they are different photons that are arriving. So there isn't a single material object that is moving. In the case of the home twin suddenly getting older or younger during the traveler's sudden velocity change, it is always one single twin (the home twin) whose ageing is being determined by the traveler.

[...]
More relevantly imo, after more than twenty years plugging away with promoting CADO, there is even ONE identifiable physically consequential implication?
[...]

Yes. It explains why the home twin is the older when the traveler returns, instead of being the younger, as the traveler would expect if he only knows about the famous time-dilation result, and doesn't know that the home twin suddenly ages by a very large amount during his sudden turnaround. The home twin is not surprised by the outcome, because all she needs to know is the famous time-dilation result, because she never accelerates.