Graphical Derivation of the CADO Equation

For the dynamic duo;

The aos is a mathematical device for the purpose of local clock synchronization. It is logistically impossible for distant clocks. The converging frames/observers have their x axes aligned parallel.

 

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Not to answer for Mike, but I had the same question earlier in the thread.

We start with the inverse Lorentz transformation:
t = γ(t' + (vx' / c²))

We give a new name to t:
cado_t = γ(t' + (vx' / c²))

We distribute the gamma term:
cado_t = γt' + (vγx' / c²))

Now substituting t=γt' for such a case:
cado_t = t + (vγx' / c²))

Now substituting x=-γx' for such a case:
cado_t = t - (vx / c²))

And that is the cado equation. Just rename t to cado_h, and rename x to L.
cado_t = cado_h - (vL / c²))
cado_t = cado_h - (v * L)

 

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I think I've figured out a very concise and general rule to decide if the CADO equation can be used by one accelerating twin when the other twin can also be accelerating at the same or at some other time (with all accelerations being restricted to instantaneous velocity changes).

Here is the "quick and dirty" explanation:

First, we draw a Minkowski diagram whose horizontal axis is the twins' mother's age. (Their mother never accelerates.) Then, we draw the world lines of the two twins on that diagram. In the following rule, I will call the twin whose opinion we want "the observer", and the other twin "the observed".

Then the rule is this:

We can use the CADO equation to determine the observed's current age for that portion of the the observer's life that lies to the left on the diagram of any kinks in the observed's world line.

It should be noted that the above rule gives a sufficient condition to be able to use the CADO equation. It is not a necessary condition to be able to use the CADO equation. There is a necessary and sufficient rule that would allow additional scenarios to use the CADO equation, but it becomes so complicated to actually use the CADO equation in those scenarios that the general non-CADO-equation method (which must always be used for finite accelerations) is easier and simpler to use.

 

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Mike;

Regarding the example in the section "Idealized Instantaneous Velocity Changes"

Neither the rest observer A or the moving observer B, need calculations to determine their own time, they have clocks. The rest observer A is only concerned with motion induced effects of time dilation and length contraction when transforming his description to that of B.

From section 1:
"2. It must specify how the traveler, at each instant of his life, is to determine the current age and the current position of each and every object (or person) in the (assumed flat) universe."

Since the discovery of a finite speed of light, the current description of the universe is not possible for any observer. They can only observe how things were. All perception is historical. In SR it can be done with the 'radar' method using light to measure distances, for local (on an astronomical scale) events.

In this spacetime drawing, the time and space axes are prefixed with the observer id. The red aos for B is assumed to match A times with simultaneous B times, thus A10 corresponds to B20. The green line transforms time dilated B times to A times.
The problem is, the signal left B while moving outbound and returned while B is moving inbound. B intercepts the signal early at D', instead of at D. Now A10 corresponds to B12. The last signal that maintains the aos is A5.36.
This is more of the distorted conclusions resulting from an instantaneous change in velocity. It might be applicable for fundamental particles, but not in the world of macro objects.

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I needed to add a sentence to the rule I gave in my last posting, but I'm no longer allowed to edit that posting, so I'll re-post it here, with the additional sentence added to the end of the rule.

_____________________

I think I've figured out a very concise and general rule to decide if the CADO equation can be used by one accelerating twin when the other twin can also be accelerating at the same or at some other time (with all accelerations being restricted to instantaneous velocity changes).

Here is the "quick and dirty" explanation:

First, we draw a Minkowski diagram whose horizontal axis is the twins' mother's age. (Their mother never accelerates.) Then, we draw the world lines of the two twins on that diagram. In the following rule, I will call the twin whose opinion we want "the observer", and the other twin "the observed".

Then the rule is this:

We can use the CADO equation to determine the observed's current age for that portion of the the observer's life that lies to the left on the diagram of any kinks in the observed's world line. That "CADO-correct" region includes the vertical line that passes through the leftmost kink in in the observed's world line.

It should be noted that the above rule gives a sufficient condition to be able to use the CADO equation. It is not a necessary condition to be able to use the CADO equation. There is a necessary and sufficient rule that would allow additional scenarios to use the CADO equation, but it becomes so complicated to actually use the CADO equation in those scenarios that the general non-CADO-equation method (which must always be used for finite accelerations) is easier and simpler to use.

 

Mike,

May I ask your opinion of NotEinstein's claim that whenever SR calculates a distant clock running backwards, that SR is giving incorrect results, and that GR must be used instead, and that GR will calculate that the distant clock does not run backward?

When I speak of a distant clock running backward, I mean as calculated by an accelerating frame. Using the example on your web page, before the turnaround-acceleration, the traveling twin calculates cado_t=40-(0.866*34.64))=10, and after the turnaround-acceleration he calculates cado_t=40-(-0.866*34.64))=70, but if he were to do another turnaround-acceleration, he would calculate cado_t=40-(0.866*34.64))=10 again. The distant clock going from 70 to 10 would be an example of a clock running backward, and NotEinstein claims that is a violation of causality. Your opinion?

 

phyti,

What does aos mean?

In SR there is an assumption that all clocks at rest in one inertial reference frame can be synchronised to each other.

The calculations which show a "jump" in time are using instantaneous accelerations. For finite accelerations, there would be continuous change instead of a jump. But there could still be situations where distant clocks run backwards.

 

When the traveling twin instantaneously changes his velocity, so that he is then moving away from the home twin (or moving away from her faster than he was previously), then according to the traveler, her age will instantaneously decrease (she will instantaneously get younger), provided that their separation is sufficiently great.
That is a correct result. It does not violate causality. And GR is never needed in an SR scenario. SR assumes that there are no significant masses involved, and thus no gravitational fields. The equivalence principle is of value only as a way to use SR to show how GR has to be ... namely, that its geometry can't be Euclidean. It was invaluable to Einstein in 1905 in that role ... it guided him to his GR theory that he didn't complete until 10 years later.

 

I agree. Thank you for confirming. I had never heard of NotEinstein's claim before, and just wanted to make sure I wasn't missing something.

 

From https://en.wikipedia.org/wiki/Twin_paradox#Viewpoint_of_the_traveling_twin :

And that ref A 11 is indeed a paper by A. Einstein from 1918 discussing the Twin Paradox. But yeah, what did he know about relativity?

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Oh, and it turns out I was right that it is not possible to act on the clocks jumping forward/backwards in a way that allows one to send information back into the past:
https://en.wikipedia.org/wiki/Rietdijk–Putnam_argument#Andromeda_paradox

 

I need to make one correction in my last post.

I said:

"When the traveling twin instantaneously changes his velocity, so that he is then moving away from the home twin (or moving away from her faster than he was previously), then according to the traveler, her age will instantaneously decrease (she will instantaneously get younger), provided that their separation is sufficiently great."
(color emphasis added)

That qualification should have been:

" provided that their separation is non-zero".

For instantaneous velocity changes (in the direction described above), she will instantaneously get younger no matter how close they are, provided their separation is non-zero. Her age just won't decrease by very much if they are very close together. In the case of finite accelerations, there may not be any decrease in her age if they are not sufficiently far apart.

 

And I think that's also the resolution to the causality discussion: all of this "clocks ticking backwards"-stuff is happening in the space-like region of the light-cone (as I mentioned earlier). You can change the order of events in that region by changing frame. However, causality doesn't kick in until the events in question become light-like (and later time-like). The clocks aren't really ticking backwards, and they can never be observed as doing such; you're just re-ordering events in the space-like region. But since we're talking about a single clock, when the events come out of the space-like region, they have to be in order: that's where causality kicks in.

So the "backwards ticking", while a genuine result from the theory, has no physical meaning in the sense that it has zero effect on anything. As soon as information (light) from the clock reaches Alice, she will conclude that there was no real jump back in time after all.

In conclusion: we were all a bit right. Causality is indeed not violated, but nothing is jumping back in time in any measurable or meaningful way.

 

Your quote from Wiki said:

"According to the equivalence principle, the traveling twin may analyze the turnaround phase as if the stay-at-home twin were freely falling in a gravitational field and as if the traveling twin were stationary." (color emphasis added by me)

An SR scenario can be cast into GR via the equivalence principle, as an aid to understanding GR, not as an aid to understanding SR. Formulating the equivalent GR scenario requires postulating fictitious gravitational fields in the GR scenario. Those fields don't exist in the SR scenario. Read Einstein's 1905 paper. He doesn't use gravitational fields in it. In fact, he excludes them.

 

Now you say, "it is not possible to act on the clocks jumping," as if this is your own idea. But that was exactly the idea I was using when I invited you to show how the alien could possibly make any "effect" on earth precede its "cause". I was hoping that you'd realize that if the alien can't do that, then causality is not violated.

At the time, your definition of "violation of causality" was simply that if the alien calculated a different ordering of events, then causality was automatically violated. I tried to get you to consider that that no effect ever preceded its cause in the earth's own frame, and you accused me of choosing a preferred frame. I'm glad you are now having a change of heart.

The "backwards ticking" has as much physical meaning as any other effect of SR, such as length contraction, and time dilation. SR provides a definition of simultaneity for inertial frames, including how they assess relatively moving inertial frames. The backwards ticking is required under certain circumstances.

And Alice is not the one who ever thought there was a jump in time. She never even accelerated.

Anyhow, I am glad we are all of the same page now, (hopefully). I apologize if my "Amy and Bikey" post seemed over the top. I thought it was pretty funny though!

 

False; re-read the quote. There's nothing about "an aid to understanding GR" in there.

Also, "may" does not exclude "must". If it turns out this gravitational field is so strong that GR must be used, then it must be.

It's not a "GR scenario".

Except that the two interpretations of the scenario are equivalent: that's why it's called the "equivalence principle".

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Only if you read the 1918 paper.

But he does in the 1918 paper, and the two interpretations are equivalent: that's why it's called the "equivalence principle".

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So you are disagreeing with Einstein, are claiming that the Einstein Equivalence Principle is wrong, and thus that his 1918 paper is nonsense?

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First things first.

Funny? You were shouting (all caps), and it was an almost endless stream of misrepresentations of what I said. Misrepresentations I corrected you on before. To me, it appear you either got really upset and lost it, or you were massively trolling. Since the second is against the rules, I gave you the benefit of the doubt, and chose to respond as if it was the first. If you really did mean it as a joke, please never try to joke again.

I accept your apology though.

​

I completely missed that; in which post did you make that argument?

But if time is measured to be flowing in reverse, causality is still violated, as effects are preceding causes. I tried to explain this to you at the end of post #114, but you continued to fail to grasp that all reference frames are equally valid.

No, not calculated. Measured. If I said "calculated" anywhere, I apologize; I didn't mean to use that word. Now, I know I didn't explain the different anywhere, because I didn't realize there was any. Turns out, I was more right that I though in my trying to use careful wording. If you use the following definitions:
"observe": light of the event has reached you;
"measure": subtract the light-travel from an observation;
"calculate": light of the event hasn't reached you yet;
then why arguments still work. Because I've been used "measured" instead of "calculated", I've excluded the space-like region.

And you are still doing that, even in this very sentence. You haven't once made an argument that causality isn't violated from the viewpoint of the alien which calculates time flowing backwards, which was the entire point.

First of all, it's more a change of brain, not heart.

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And second, I still stand by most of my arguments. I've just realized that they are more nuanced than I thought.

False; length contraction and time dilation are measurable; this "backwards ticking" is not.

I agreed that it's something that necessarily pops up in the model. But it has no physical impact/meaning.

Sorry, I messed up the names; I indeed meant the traveler.

Well, obviously we're still not, but I did manage to answer my own questions, so I guess we can move on.

 

The SR clock synch convention is a method to simulate clock synchronization for the local observer Ann, who can assume a pseudo rest frame if moving at a constant velocity. Bob can also apply the same method in his pseudo rest frame. Each has a personal synch system established. Per SR there is no universal system of time. Via the rules of physics, the clock rates depend on the velocity of the frame where each clock is located.

The CADO theory states:

The age of Ann 'jumps' forward from 3 to 7 at the instant reversal at B4, because the red axis of simultaneity now points in that direction.

Yet the next clock image detected by B is A3.

This contradicts the 'jump' statement.

The red aos is part of the simulation mentioned above. The physical x axis does not point in time, but in the direction of motion, from B to A.

The space time drawings cannot be interpreted in the manner of road maps.

If the velocity profiles/(time lines) are predetermined, numerically or graphically, time dilation for each segment can be found, and aging compared.

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I have found a necessary and sufficient condition for determining whether or not the CADO equation can be used, when both twins sometimes instantaneously change their velocities. And I have added that new information to the end of Section 14 of my webpage. Here is an excerpt from that section, giving the new information:
_________________

It should be noted that the above rule gives a sufficient condition to be able to use the CADO equation. It is not a necessary condition to be able to use the CADO equation. It is possible to give a necessary and sufficient condition that allows us to know whether the CADO equation can be used or not. If there is an instant t in the observer's life when the intersection of his line of simultaneity with the observed's world line forms the hypotenuse of a right triangle whose base is a horizontal section of the observed's world line, and whose vertical side is formed by the vertical line through the point t on the observer's world line, then the observer can use the CADO equation at that instant t in his life. That very long and complicated-sounding condition is actually simple to visualize on a sketch of the Minkowski diagram: what the condition defines is a right triangle which is similar to the right triangle you get in the Minkowski diagram when the home twin is perpetually inertial, and from which the CADO equation is derived. The only difference in the new triangle is that its base can be above or below the horizontal axis. The new triangle defines a "generalized" CADO equation which looks just like the regular CADO equation, except the variable name used for the first term on the right-hand-side of the equation should be changed from "CADO_H" to "CADO_M", because it now represents the opinion of the mother, not the opinion of the (sometimes accelerating) home twin. (That first term on the right-hand-side of the CADO equation always needs to refer to the opinion of a perpetually-inertial person, whose world line is the horizontal axis of the Minkowski diagram.) So you can easily spot potential regions where the (generalized) CADO equation might be usable by looking for portions of the home twin's world line that are horizontal. If those horizontal portions are long enough to include the entire base of the right triangle, then the generalized CADO equation can be used.

 

I am the originator of the CADO equation, but I'm not the originator of the example with "Ann" that you are referring to, so I can't help you there.