Graphical Derivation of the CADO Equation

Discussion in 'Physics & Math' started by Mike_Fontenot, Sep 5, 2018.

1. First, here is an excerpt, from Section 2 of my webpage, of the definition of the CADO equation:

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Given the above definition of the CADO frame, it is possible to derive a very simple, and very useful equation, called "the CADO equation", which allows the traveler to determine, at each instant t in his life, the current age of any given distant perpetually-inertial object or person (the "home-twin" in the twin paradox scenario).

First, it is important to understand that, for any given instant t in the traveler's life, the home-twin and the traveler will generally disagree with one another about how old the home-twin is at that instant of the traveler's life. There are two quantities in the CADO equation which represent each of the twins' conclusions about the home-twin's age when the traveler's age is t. The quantity CADO_T denotes the traveler's conclusion about the home-twin's age, when the traveler's age is t, whereas the quantity CADO_H denotes the home-twin's conclusion about the home-twin's age, when the traveler's age is t.

The CADO equation can be written (most simply) as

where

v is their current relative speed, according to the home-twin, at the given instant t in
the traveler's life, with v taken as positive when the twins are moving apart,

L is the distance from the home-twin to the traveler, at the given instant t in the
traveler's life, according to the home twin,

and

the asterisk denotes multiplication.

Strictly speaking, the quantity L(t) is the position of the traveler, relative to the home-twin, according to the home-twin, when the traveler's age is t. The distinction will be clarified later (in Section 5), but for now, it's simplest to just think of it as a distance (a number either positive or zero).

The above equation gives the relationship between those four quantities (CADO_T, CADO_H, v, and L), at the given instant t of the traveler's life. I.e., although it is not shown explicitly, each of the four quantities in the equation are functions of t.

What makes the CADO equation especially useful is that it allows the quantity CADO_T, which is a quantity which is otherwise relatively difficult to determine, to be easily calculated from the other three quantities (CADO_H, L, and v ), which are each very easy to determine.

_____________________________________

I originally derived the CADO equation, many years ago, from the Minkowski diagram. I have recently made an addition to the webpage, to explicitly show how to do that derivation. The addition is near the end of Section 11 of the webpage (the section entitled "Graphical Interpretation of the CADO Frame).

Here is a link to the webpage:

Last edited: Sep 5, 2018

3. originIn a democracy you deserve the leaders you elect.Valued Senior Member

Messages:
10,812
I don't understand.

What does, "CADO_T denotes the traveler's conclusion about the home-twin's age" mean? I don't know what he will conclude.
Does he think the home twin and himself are the same age? Does he know the age difference because he has already worked out the time dilation? If that is true what is the point of additional calculations??

Lets put the units of velocity in km/s
Lets say L is in km.

That means the units of the left side of the equation is years and the right side is year - km^2/s. That makes no sense! I have no idea what year - km^2/s is supposed to mean?!!

How do you subtract km^2/s from years????

Last edited: Sep 5, 2018

5. Here is the next paragraph in Section 2 of the webpage, that immediately follows the excerpt. I didn't include it in the excerpt in the interest of brevity:
___________________

In order to make the equation strictly correct, a factor of c*c dividing the last term is required, where c is the constant speed of any light pulse, as determined by any perpetually-inertial observer. If the time and spatial units are chosen so that c has unity value, the factor in that case is required only for dimensional correctness. In this article, units of years and lightyears will be used exclusively (but often abbreviated as y and ly), and the factor of c*c will be suppressed entirely, purely for simplicity and brevity.

____________________________

As for your other misunderstandings, I recommend that you read the complete Section 2 in the webpage. If you still have questions, read the next section (Section 3: "Idealized Instantaneous Velocity Changes"), which explicitly works through a simple example, and shows exactly how to apply the CADO equation.

Last edited: Sep 5, 2018

7. Which means what?

or

or something else?

8. The last term in the equation is v * L.

9. For the record, CADO stands for "current age of a distant object".

What above definition? I understand you copy-pasted this from your own webpage, but this is not how you start a thread. It forces people to visit your website in order to understand what your topic is about. That can be considered a form of advertisement, which is against the forum's rules.

It seems you define c differently than mainstream physics does. Is there a reason for that?

And what's the difference between the CADO frame and the proper frame (or "comoving frame")? The way I read the CADO frame's definition, they appear to be identical?

10. Thanks. So would you say that the formally correct version of your equation would be the following?

If so, then your equation looks much like the Lorentz transformation, only without the gamma factor:
t' = γ(t - (vx / c²))
where:
γ = 1 / √(1 -(v²/c²)) = gamma

What would you say are the differences between the two, and how does your version manage without a gamma factor?

11. originIn a democracy you deserve the leaders you elect.Valued Senior Member

Messages:
10,812
So the equation you gave wasn't correct? Not a good start. What do you mean by strictly correct? Was your original equation wrong, strictly wrong, kinda wrong or sorta right? These are new math terms for me.

Messages:
2,527
Why doesn't he bring religion into the "discussion"?

That seems to be de rigueur for the place these days. 90% bullshit, 10% opposition to bullshit, that just might be moderated, because even the idiots deserve to have a voice on a platform other than farcebook or twatter.

13. I've got another question. Imagine there is a triplet: Alice, Bob, and Charlie. Alice and Bob stay at home, but Charlie takes a rocket into space. But, at a distance D, Charlie comes to a complete stop (with respect to Alice and Bob). In other words, $v=0$. Afterwards, according to your formula, all will agree how old Alice is, because with $v=0$ your equation reduces to: $\text{CADO}_T=\text{CADO}_H$.
But now, both Bob and Charlie start moving (both in the same direction that Charlie was moving in earlier) with the same speed (using the same acceleration profile, if that's relevant). Even though they are in the same reference frame all the time, Bob and Charlie will now disagree on how old Alice is, due to Charlie starting with $L=D$ while Bob starts with $L=0$. That seems weird to me; I'm pretty sure SR predicts that Bob and Charlie would agree on Alice's age, because in SR time dilation is a function only of speed, not distance.

How do you resolve this conflict?

14. I see you have been trying to see if the cado equation 'breaks' under some condition. I have been trying to do that also, and I commend your efforts here.

I think you have a good case, except, I think you have to say in which frame Bob and Charlie start moving simultaneously. I haven't actually checked to see if the cado equation would work for the case where they start to move simultaneously in the moving frame, instead of the stay-home frame. To me, it's strange that it works at all, even in the simpler case.

Last edited: Sep 6, 2018
15. I'm not prepared to plow through all that article linked to in #1, but will take the simple formula presented there as 'universally valid' as claimed.
As NotEinstein has pointed out in #11, Mike Fontenot's CADO formula cannot handle general nonuniform motion of the traveling twin - e.g. an arbitrary period spent in circular motion. Indeed, just by considering the basics of the Doppler shift analysis:
it should be clear the scalar product term v*L will fail to account for the asymmetries in relative aging, depending if v as a velocity not speed is directed away from or towards or perpendicular to the at home twin. Similarly L needs to be a displacement vector not scalar distance. Even with that sorted out, such a minimally modified CADO expression cannot even correctly yield relative aging rates in general, let alone accumulated relative time spans. The case of circular motion of traveling twin should make it obvious why.

Time for a long rethink to ponder what's really required to encapsulate a physically meaningful CADO formula. One that will be very different to what's given in #1.
That Wikipedia article contains all that is necessary. At minimum Mike you will have to have an integral expression involving vector products, and contains the usual Lorentz time dilation gamma factor.

16. You're right, I should have specified it explicitly, although from the context it should be clear what frame I had in mind: it's the proper frame of any of the three people (after Charlie has come to a complete stop), which is indeed the same as the stay-home frame.

I don't find it surprising such a simple equation can give the correct answer under the given conditions. If you look at the Minkowski diagram, the time difference is basically the difference on the y-axis (= of the stay-home frame) between a certain y-coordinate and where the y'-axis of the moving party at that same y-coordinate intersects the y-axis. With a constant $v$, there is an obvious direct linear relation between the time difference and distance travelled. Similarly, there obviously is a direct relation between the value of $v$ and this time difference. Whether this is correctly captured in the CADO-equation I haven't checked at this moment in time.

I don't have access to Mike's paper from 1999, but its abstract strongly suggests the CADO-equation is already given in it. Which means that Mike has had almost 20 years already to rethink... Just sayin'.

Mike_Fontenot: Why so many threads about the same equation? (And I'm not even counting the threads on other fora.) Is it really that special? If so, please enlighten us!

18. Okay I think I figured this out.

On the webpage, the basic twin scenario is set up with v=0.866c so that gamma=2 and the time the traveling twin has been traveling is 40 years as measured by the stay-home twin. So the stay-home twin says the distance traveled is 0.866*40=34.64 lightyears. The stay-home twin also says the age of the travelling twin is 40/2=20 years. This is all basic stuff not really related to the CADO equation.

But at that moment, the turnaround is about to happen. This is where the CADO equation comes in. It is shown in action where the traveling twin calculates the age of the stay-home twin to be 40-(0.866*34.64)=10 years just before the turnaround. Then the sign of the velocity is reversed to represent the turnaround having happened, and the CADO equation is shown in action again. This time the traveling twin calculates the age of the stay-home twin to be 40-(-0.866*34.64)=70 years just after the turnaround.

The same results can be obtained with the inverse Lorentz transformations as follows:
t = γ(t' + (vx' / c²))
t = 2(20 + (0.866 * -17.32))
Note that this equation uses t' as an input, which is only 20 instead of the 40 used in the CADO equation.
Note also that this equation uses x' as an input, which is only -17.32 instead of the 34.64 used in the CADO equation.
But watch what happens if we distribute the gamma term:
t = 2(20 + (0.866 * -17.32))
t = 40 + (0.866 * -34.64))
t = 40 - (0.866 * 34.64))
Boom! We have arrived at the CADO equation.
Before the turnaround:
t = 40 - (0.866 * 34.64))
t = 10
And then reversing the sign of the velocity, to represent the turnaround:
t = 40 - (-0.866 * 34.64))
t = 70
So the CADO equation is just a clever way of using stay-home-twin coordinates instead of traveling-twin-coordinates, which allows it to leave off the gamma term entirely.

Last edited: Sep 6, 2018
19. Nice work! So it is a somewhat trivial result that is at least valid under very specific conditions. Mike_Fontenot: Can you tell us if it's valid under more conditions than only the turnaround point of the twin paradox set-up as Neddy Bate just demonstrated?

20. Yes.

Interesting question. I don't know the answer. Two pieces of information may help your search for an answer to that question. I first derived the CADO graphically, from the Minkowski diagram, many years ago. But when later I published my paper,

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

I included a purely analytical derivation of the CADO equation based only on the Lorentz equations. You might want to take a look at that derivation, to see if it sheds any light on your question. The Physics Essays journal sells electronic copies of all their past articles, so you can get a copy from them. I don't know if they still sell paper copies or not. (I'll see if I can find a link to their webpage, and if so I'll post it below). Also, you may be able to find bound volumes of past issues of that journal in a university library near you ... The University of Colorado (near where I live) has that journal.

Secondly, I've recently added to the end of Section 11 of my webpage (the section entitled "Graphical Interpretation of the CADO Frame") an explicit description of how to derive the CADO equation from the Minkowski diagram. That might also help answer your question.

The Physics Essays website is

http://physicsessays.org

21. (Emphasis mine.)

Wait, it's been more than 20 years ago since you came up with this, and in all that time you haven't even checked how it relates to the most basic things of SR (i.e. the Lorentz transformation)? Why do you expect us to invest even the slightest amount of effort, when you clearly haven't yourself?