Graphical Derivation of the CADO Equation

[NotEinstein]

Yes, I do have two different values of CADO_H and that may well show that the CADO equations are lacking.

But I also have three different values of t' for Charlie, one value in post #94 which you are quoting (t'=60), and two different values in a post just a little further down which was supposed to pertain to the same scenario, post #96 (t'=-60 and t'=0).

But what I am doing with the different values of t' is required in order for the Lorentz equations to give the correct results. And I give the same flexibility to the CADO equations by just copying & pasting the resulting values of γt' and -γx' into the CADO equations.

So, you are ignoring the definition of CADO_H so that it works. May I suggest you thus have proven that the CADO-equation is lacking, if not downright incorrect, as I said earlier? If you have to modify definitions to make it work, then it doesn't work.

So if CADO is lacking in that sense, then so is SR.

No, the variables in SR are clearly defined and they work. If you find them to be ambiguous or need to modify them, that's your lack of understanding, not a lack in SR.

 

[NotEinstein]

The twins do not have to have any mass, and the accelerations do not have to be instantaneous. Shouldn't GR reduce to SR in that case?

Only if those changes cause the gravitational field to be weak enough to be neglected.

I assume you would answer yes, because you call this a "weak spot" in SR, so it must predict the result that you think is wrong,

This jump in logic alludes me. If the acceleration is slow enough, doesn't it just slow down the clocks instead of reversing time?

namely time going backwards as well as forwards for very distant clocks (not local).

Not sure why you included "forwards" there; I never complained about that?

 

[Neddy Bate]

So, you are ignoring the definition of CADO_H so that it works. May I suggest you thus have proven that the CADO-equation is lacking, if not downright incorrect, as I said earlier? If you have to modify definitions to make it work, then it doesn't work.

I already conceded CADO may be lacking, it's right there in what you are quoting. But you are right that I didn't say anything about it being downright incorrect, though I am willing to say now that could be case under strict definitions.

No, the variables in SR are clearly defined and they work. If you find them to be ambiguous or need to modify them, that's your lack of understanding, not a lack in SR.

Okay, I think I see the distinction you are trying to draw. You seem to be saying that the strict definition of CADO_H does not allow it to be as flexible as the strict definition of t'.

I hope you are okay with my use of t'=60, t'=-60, and t'=0 and that you do not think that I made errors there? If you do think I made errors, then I would be grateful if you would show me the Lorentz equations where t' is the same in all of the contexts I spoke of there.

 

[Neddy Bate]

Only if those changes cause the gravitational field to be weak enough to be neglected.

Which I would think would be the case if the masses are nearly zero, and the accelerations are nearly zero.

This jump in logic alludes me. If the acceleration is slow enough, doesn't it just slow down the clocks instead of reversing time?

No, the reversal of time (on a distant clock in a different frame) has to do with the direction of travel being reversed by the traveler. The magnitude of the effect has to do with the distance between the traveler and the distant clock under consideration, and also the speed (not the acceleration).

I don't know if you saw the video Mike_Fontenot posted, but you have to forward it to about 6 to get to the part about the alien on the bicycle.

Not sure why you included "forwards" there; I never complained about that?

Funny you mention that because I did edit that out for clarity. The original intended meaning was that you didn't like time going 'backward' being an equally acceptable result as time going 'forward.'

 

[NotEinstein]

I already conceded CADO may be lacking, it's right there in what you are quoting. But you are right that I didn't say anything about it being downright incorrect, though I am willing to say now that could be case under strict definitions.

OK, I think we are in agreement on that then.

Okay, I think I see the distinction you are trying to draw. You seem to be saying that the strict definition of CADO_H does not allow it to be as flexible as the strict definition of t'.

I hope you are okay with my use of t'=60, t'=-60, and t'=0 and that you do not think that I made errors there? If you do think I made errors, then I would be grateful if you would show me the Lorentz equations where t' is the same in all of the contexts I spoke of there.

I don't see a problem there, but I must admit, I haven't checked in details.

Which I would think would be the case if the masses are nearly zero, and the accelerations are nearly zero.

Actually, I'm not sure if the mass is a factor. In Newtonian physics, it divides out; I wouldn't be surprised if it does that as well to first order in GR in this case.

No, the reversal of time (on a distant clock in a different frame) has to do with the direction of travel being reversed by the traveler. The magnitude of the effect has to do with the distance between the traveler and the distant clock under consideration, and also the speed (not the acceleration).

I don't know if you saw the video Mike_Fontenot posted, but you have to forward it to about 6 to get to the part about the alien on the bicycle.

So you are saying that if the acceleration takes place over a period of ten billion plus years, it would cause a constant reversal of time during that period (for an object at a certain distance, etc.)?

Funny you mention that because I did edit that out for clarity. The original intended meaning was that you didn't like time going 'backward' being an equally acceptable result as time going 'forward.'

That's true. Something with causality and such, as I mentioned before.

 

[phyti]

The graphic shows the popular 'twin paradox' with the (red) axis of simultaneity (aos) for the outbound and inbound paths of B. The aos is the basis for the 'time jump' from the lower to the upper position. Supposedly B observes A age instantly from t=3 to t=7 upon reversal. The error: the (blue) light signals originate at A and are not round trip signals originating at B, which is required for the SR clock synch convention. The aos does not apply to the drawing, which is only a case of doppler effects. Notice that the image received by B at the reversal is A2, and the next to be received is A3. B doesn’t see A7 until B6.5. There is no 'jump'.

 

[Neddy Bate]

So you are saying that if the acceleration takes place over a period of ten billion plus years, it would cause a constant reversal of time during that period (for an object at a certain distance, etc.)?

If the current "now" is relative, as SR says, then however gradually it shifts due to velocity changes, it still shifts nonetheless. The distant alien on a bicycle now might say the current "now" on earth is behind our current time now, (when his is pedaling away from earth). Even if it takes him billions of years to turn his bicycle so that it is eventually pedaling toward earth, there will eventually come a time when the distant alien on a bicycle now might say the current "now" on earth is ahead of our current time now. The acceleration rate does not limit him to only our current "now" or before, as far as I know.

That's true. Something with causality and such, as I mentioned before.

I still invite you to try to demonstrate a causality violation because of this effect. I don't think there are any.

 

[Neddy Bate]

The graphic shows the popular 'twin paradox' with the (red) axis of simultaneity (aos) for the outbound and inbound paths of B. The aos is the basis for the 'time jump' from the lower to the upper position. Supposedly B observes A age instantly from t=3 to t=7 upon reversal. The error: the (blue) light signals originate at A and are not round trip signals originating at B, which is required for the SR clock synch convention. The aos does not apply to the drawing, which is only a case of doppler effects. Notice that the image received by B at the reversal is A2, and the next to be received is A3. B doesn’t see A7 until B6.5. There is no 'jump'.

Interesting, but the 'jump' is not supposed to be about what anyone sees with their eyes. It is about the definition of the current "now" in a given reference frame. Also, instead of a jump, it can shift gradually with gradual acceleration. The 'jump' is caused by the idealized instantaneous acceleration which is not realistic, and only serves to simplify the exercises.

 

[Mike_Fontenot]

You guys have gotten WAY off track. The CADO frame and the CADO equation are about SPECIAL RELATIVITY. Special relativity applies to flat spacetime, where there are no gravitational fields. General relativity is irrelevant to the CADO frame and to the CADO equation.

 

[NotEinstein]

If the current "now" is relative, as SR says, then however gradually it shifts due to velocity changes, it still shifts nonetheless. The distant alien on a bicycle now might say the current "now" on earth is behind our current time now, (when his is pedaling away from earth). Even if it takes him billions of years to turn his bicycle so that it is eventually pedaling toward earth, there will eventually come a time when the distant alien on a bicycle now might say the current "now" on earth is ahead of our current time now. The acceleration rate does not limit him to only our current "now" or before, as far as I know.

So if the alien is pedaling so slowly that its effect on the distant object's time is a backwards ticking of 0.5 (in some unit), while it is also ticking forwards with a nominal 1.0, wouldn't you agree that the distant object's time is going with speed: 1.0 - 0.5 = 0.5, and not some negative number? In other words, if the acceleration is slow enough, it would merely cause a slow-down, not a reversal? Or rephrased: there is always an acceleration slow enough that the "shifting backwards in time" of the plane of simultaneity is slower than time ticking forwards, resulting in a net forward flow of time.

I still invite you to try to demonstrate a causality violation because of this effect. I don't think there are any.

Causality says: first cause, then effect. If you go backwards in time, you'll have effect first, then cause. Seems pretty cut-and-dry to me? At the very least, it's a violation of the second law of thermodynamics. Also, since things don't also mirror or charge-conjugate, it's a violation of CPT-symmetry.

 

[NotEinstein]

You guys have gotten WAY off track. The CADO frame and the CADO equation are about SPECIAL RELATIVITY. Special relativity applies to flat spacetime, where there are no gravitational fields. General relativity is irrelevant to the CADO frame and to the CADO equation.

False; through the Einstein Equivalence Principle, accelerations can be interpreted as gravitational fields, and thus with large enough accelerations, SR isn't the proper framework to handle such cases.

But you're right: we've going a bit off-track. Let's re-focus on the problem that surfaced with the CADO-equation: both Neddy Bate and I agree in post #103 (and earlier) that the CADO-equation seems to be lacking. It appears at least one definition of the terms involved needs to be changed in order for it to work. Any comments on that?

 

[Neddy Bate]

You guys have gotten WAY off track. The CADO frame and the CADO equation are about SPECIAL RELATIVITY. Special relativity applies to flat spacetime, where there are no gravitational fields. General relativity is irrelevant to the CADO frame and to the CADO equation.

Well, in post #94 I did use the CADO equation to do some calculations for the NotEinstein scenario. NotEinstein objected to my results because I had to use two different values for CADO_H as follows:

BOB's calculation of Alice's time:
CADO_T = CADO_H - (v * L)
CADO_T = 0 - (0.866 * 0.00)
CADO_T = 0 - 0.00
CADO_T = 0
(Note: To find Alice's biological age according to Bob, we can add the time that was displayed on her clock just before it was reset to 0.)
(CADO_T = 0 + 40)
(CADO_T = 40)

CHARLIE's calculation of Alice's time:
CADO_T = CADO_H - (v * L)
CADO_T = 120 - (0.866 * 138.56)
CADO_T = 120 - 120.00
CADO_T = 0
(Note: To find Alice's biological age according to Charlie, we can add the time that was displayed on her clock just before it was reset to 0.)
(CADO_T = 0 + 40)
(CADO_T = 40)

NotEinstein said that by definition CADO_H should be 0 in both cases, and that L should be 0 for Bob, but that L should be 34.64, thus disproving the CADO equation. Reminder: NotEinstein has all three clocks reset to 0 just before Bob and Charlie both accelerate to 0.866c simultaneously in the stay-home frame.

Do you have any comments or questions on my method, or NotEinstein's objections?

 

[Neddy Bate]

So if the alien is pedaling so slowly that its effect on the distant object's time is a backwards ticking of 0.5 (in some unit), while it is also ticking forwards with a nominal 1.0, wouldn't you agree that the distant object's time is going with speed: 1.0 - 0.5 = 0.5, and not some negative number? In other words, if the acceleration is slow enough, it would merely cause a slow-down, not a reversal? Or rephrased: there is always an acceleration slow enough that the "shifting backwards in time" of the plane of simultaneity is slower than time ticking forwards, resulting in a net forward flow of time.

Yes, I would agree that for some chosen distance that is the case. But then a larger distance could always be chosen. So if the alien and earth are a distance of 10 apart (in some units) and the earth's time is going with speed 1.0-0.5=0.5 then perhaps tripling the distance to 30 might result in some other distant object's time going with speed 1.0-1.5=-0.5. Wouldn't you say?

Causality says: first cause, then effect. If you go backwards in time, you'll have effect first, then cause. Seems pretty cut-and-dry to me? At the very least, it's a violation of the second law of thermodynamics. Also, since things don't also mirror or charge-conjugate, it's a violation of CPT-symmetry.

You are mixing frames. Time does not go backwards on earth just because a distant alien is pedaling his bike in a circle. The 'current now on earth' according to the alien is what changes. It can change to a time that you and I right now would say is from the past, or it can change to a time that you and I right now would say is from the future. How can the alien (or anyone) use that to violate causality?

 

[NotEinstein]

Yes, I would agree that for some chosen distance that is the case. But then a larger distance could always be chosen.

And a smaller acceleration will make the "backwards in time" go away again. Your point being?

So if the alien and earth are a distance of 10 apart (in some units) and the earth's time is going with speed 1.0-0.5=0.5 then perhaps tripling the distance to 30 might result in some other distant object's time going with speed 1.0-1.5=-0.5. Wouldn't you say?

Because you just needed to modify the set-up in a crucial way to argue against my point, I feel pretty confident you know I'm right.

You are mixing frames.

No, I'm not. You are misinterpreting what I said.

Time does not go backwards on earth just because a distant alien is pedaling his bike in a circle.

That's quite literally the entire point of SR, isn't it? It's not just that clocks "appear" to go slower, they really do. This "backwards in time"-stuff is equally real. The frame of the distant alien is just as valid as Earth's. Or are you now arguing for a preferred frame?

The 'current now on earth' according to the alien is what changes.

Yes, and the "current now on Earth" is going backwards (according to the alien), then time is thus going backwards on Earth (according to the alien), which is a violation of causality.

It can change to a time that you and I right now would say is from the past,

Now *you* are mixing frames! It's the distant alien that sees things go backwards in time. Our observations here on Earth are neither here nor there in this case.

or it can change to a time that you and I right now would say is from the future. How can the alien (or anyone) use that to violate causality?

Let's say there's a closed system on Earth, that (according to us on Earth) is increasing in entropy. The distant alien, measuring time on Earth going backwards, measures the entropy of this closed system to be decreasing, which is a violation of the second law of thermodynamics. Also, the alien measures me having second degree burns, and then cooking water jumps up from the ground, grazing by my arm, healing it. The water then flows neatly into the cup on the table, which similarly formed as shards simultaneously and spontaneously jumped up from the floor. I'd call all of that a pretty big case of "violation of causality", wouldn't you? Yet, it's what the distant alien measures... (according to SR.)

 

[Mike_Fontenot]

The value of the equivalence principle lies only in what special relativity requires general relativity to be like. It has no value in the opposite direction. Einstein used he equivalence principle to show him (in 1905) that in the still unknown general relativity, geometry couldn't be Euclidean: the ratio of the circumference of circle to its diameter couldn't in general be equal to pi, and the sum of the internal angles of a triangle couldn't generally be 180 degrees. It took him 10 more years to finish his general relativity theory, but the equivalence principle told him some very important things about general relativity in 1905.

 

[Neddy Bate]

And a smaller acceleration will make the "backwards in time" go away again. Your point being?

Yes, I already agreed that you can choose an acceleration small enough that time will only slow down, and not go backwards for some given distance. Then I can choose a larger distance, and time does not merely slow down, but goes backwards.

Reminder: You are the one saying it can never go backward. I am showing that it can. Or are you going to say GR kicks in for ridiculously low accelerations and ridiculously large distances?

Because you just needed to modify the set-up in a crucial way to argue against my point, I feel pretty confident you know I'm right.

No, I do not think you are right at all. You could be right, I admit that, and maybe I do't know enough GR to understand why.

But as it stands, I see no reason to conclude that these ridiculously small accelerations you are talking about require GR. To me, GR should reduce to SR in such cases.

That's quite literally the entire point of SR, isn't it? It's not just that clocks "appear" to go slower, they really do. This "backwards in time"-stuff is equally real. The frame of the distant alien is just as valid as Earth's. Or are you now arguing for a preferred frame?

Yes, the frame of the Alien is just as valid as earths. And it is the alien's frame which has the time on earth going backward, not the earth's own frame.

Just like when Charlie determined Alice was 10 years old just before he decelerated. She really was 10 in that frame, even though she was 40 in her own frame. See how that works? Different frames disagree. There is nothing Charlie can do to make her age change from 40 to 10 in her own frame, but he himself can change to any frame he wants, by simply accelerating.

Yes, and the "current now on Earth" is going backwards (according to the alien), then time is thus going backwards on Earth (according to the alien), which is a violation of causality.

So you claim, but you have not shown any example. The first obstacle you need to overcome is that time does not go backwards in the earth's own frame, it goes backward in the alien's accelerating frame. The alien is a huge distance away from the earth, and he would have a very difficult time causing any effect on earth. That is what you need to show if you think causality is violated.

Now *you* are mixing frames! It's the distant alien that sees things go backwards in time. Our observations here on Earth are neither here nor there in this case.

So, for you, "causalty is violated when the alien determines the earth's time went backward" can be a true statement even though time never goes backward in the earth frame itself. For me, causality is not violated in that case, because the earth's time proceeds forward in its own frame. So each effect on earth is preceded by a cause on earth.

Do you also feel that causality is violated when you watch a movie in reverse?

Let's say there's a closed system on Earth, that (according to us on Earth) is increasing in entropy. The distant alien, measuring time on Earth going backwards, measures the entropy of this closed system to be decreasing, which is a violation of the second law of thermodynamics. Also, the alien measures me having second degree burns, and then cooking water jumps up from the ground, grazing by my arm, healing it. The water then flows neatly into the cup on the table, which similarly formed as shards simultaneously and spontaneously jumped up from the floor. I'd call all of that a pretty big case of "violation of causality", wouldn't you? Yet, it's what the distant alien measures... (according to SR.)

No I wouldn't call that "violation of causality," because on earth, the hot water comes before the burn, and the glass drops to the floor before it breaks.

Consider this analogy: Imagine someone claiming that length contraction is a violation of the laws of physics, on the grounds that a steel bar, which is not undergoing any forces, should not become shorter. I reply that the steel bar does not become shorter in its own frame, so the laws of physics are not violated. Do you at least see my consistency there?

 

[Mike_Fontenot]

The CADO equation is correct, because the Lorentz equations are correct, and the CADO equation was derived purely from the Lorentz equations (both analytically, and also from the Minkowski diagram, which itself comes from the Lorentz equations).

You guys are wasting your time. You are making lots of mistakes. Mistakes that the newer variable names used in the webpage version of the CADO equation are designed to make much less likely.

 

[Neddy Bate]

The CADO equation is correct, because the Lorentz equations are correct, and the CADO equation was derived purely from the Lorentz equations (both analytically, and also from the Minkowski diagram, which itself comes from the Lorentz equations).

You guys are wasting your time. You are making lots of mistakes. Mistakes that the newer variable names used in the webpage version of the CADO equation are designed to make much less likely.

Did I make any mistakes here?

BOB's calculation of Alice's time:
CADO_T = CADO_H - (v * L)
CADO_T = 0 - (0.866 * 0.00)
CADO_T = 0 - 0.00
CADO_T = 0
(Note: To find Alice's biological age according to Bob, we can add the time that was displayed on her clock just before it was reset to 0.)
(CADO_T = 0 + 40)
(CADO_T = 40)

CHARLIE's calculation of Alice's time:
CADO_T = CADO_H - (v * L)
CADO_T = 120 - (0.866 * 138.56)
CADO_T = 120 - 120.00
CADO_T = 0
(Note: To find Alice's biological age according to Charlie, we can add the time that was displayed on her clock just before it was reset to 0.)
(CADO_T = 0 + 40)
(CADO_T = 40)

NotEinstein said that by definition I should have used CADO_H=0 in both cases, and that I should have used L=0 for Bob and L=34.64 for Charlie. Your thoughts?

Reminder: NotEinstein has all three clocks reset to 0 just before Bob and Charlie both accelerate to 0.866c, simultaneously according to Alice's stay-home frame.

 

[Mike_Fontenot]

Did I make any mistakes here?

[...]

NotEinstein said that by definition I should have used CADO_H=0 in both cases, and that I should have used L=0 for Bob and L=34.64 for Charlie. Your thoughts?

Reminder: NotEinstein has all three clocks reset to 0 just before Bob and Charlie both accelerate to 0.866c, simultaneously according to Alice's stay-home frame.

I do admire your "Never say die" stamina. It is very admirable. But I still think you're wasting your time. I haven't been willing to spend any time trying to dig through all of NotEinstein's contortions, so I can't comment on your responses to his posts. But I can tell you that spending any time working on any questions about gravitational effects on calculations in special relativity is folly ... there are no gravitational fields in special relativity, by assumption.

 

[Neddy Bate]

I do admire your "Never say die" stamina. It is very admirable. But I still think you're wasting your time. I haven't been willing to spend any time trying to dig through all of NotEinstein's contortions, so I can't comment on your responses to his posts. But I can tell you that spending any time working on any questions about gravitational effects on calculations in special relativity is folly ... there are no gravitational fields in special relativity, by assumption.

Thank you. For the record, I do think the CADO equation is a helpful simplification of the Lorentz equations. I think it helps because it puts all the input variables in one frame, and saves people from having to think about two frames at once.

However, there are times when the CADO equation still requires some careful thought, which is to be expected, because that is the case with the Lorentz equations as well.

For example, the reason I use CADO_H=120 for Charlie's calculation of Alice's time, is because in his own frame the time would be t'=60. Multiplying by the gamma factor (to switch to the other frame) gives 120. It is a bit awkward to have all the input variables in the frame where the events are simultaneous, but to still have to use two different times to describe them.

With the Lorentz equations, I would have used t'=60 for Charlie and t'=0 for Bob. Those are still two different times, but at least they don't come from the frame where the events are defined to be simultaneous.