# A New Simultaneity Method for Accelerated Observers in Special Relativity

Okay, I think I understand better now. In your method, the outbound leg is actually nothing like CMIF at all.

I see no difference on the outbound leg (between the CMIF method and my method) at all. The outbound leg will be exactly the same for ANY simultaneity method that is causal (and the CMIF method and my method are both causal). Any causal simultaneity method must simply use the time-dilation equation (TDE) to determine the age correspondence between the two twins on the outbound leg, because the traveler MAY decide to not change his velocity at the "turnaround". The age correspondence has to be the same on the outbound leg outbound leg as for the case where the twins aren't real twins, but where both were born at the same instant when their perpetually inertial (different) mothers happened to momentarily pass each other. The instantaneous velocity change by the traveler at the origin in the "real twins" case has no effect on the correspondence between their ages, because the distance between the twins is zero when the velocity change occurs. There is no age-correspondence effect for a co-located velocity change, in both the CMIF method and in my method. (The "disagreement interval" (DI) in my method, when their separation is zero, has zero length, so he agrees with his PIO on the entire outbound leg). (Recall that the amount of his ageing during the DI is just how much he ages during the transit of a pulse from her to him; when they are co-located, that time interval is zero.) The age correspondence IS different on the outbound leg for the Dolby&Gull method and for the Minguzzi method, but they are both non-causal: they both say that the EFFECT of the instantaneous velocity change happens well BEFORE the velocity changes. In fact, in the Minguzzi method, the effect happens as soon as the traveling twin leaves the origin ... his ACD on the outbound leg has a slope of 1.0, NOT 1/gamma. In the D&G method, the ACD starts out obeying the TDE (and has an initial slope of 1/gamma), but suddenly increases the slope well before the velocity change occurs.

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I see no difference on the outbound leg (between the CMIF method and my method) at all.

I think you are only considering the ages of two twins, "He" and "She." My post mentions a clock at rest in the stay-home frame which is synchronised to Her clock, and located at x=23.09. That is the place where He will eventually come to a stop (in the simplest scenario). I am talking about the time displayed on that clock, at the instants of time before and after He accelerates away from Her. If he is using the CMIF method, he says it changes from t=0.00 (before) to t=13.33 (after). (See Minkowski diagram below.) Your method says that that clock displays t=0.00 (before) and t=0.00 (after) because you say that your method does not change the time on a distant clock just because someone accelerates. Correct?

If you still do not agree, then I will show you another Minkowski diagram of the outbound leg using the CMIF method, with a clock at rest in the stay-home frame which is synchronised to Her clock, and located at x=-23.09. For the time displayed on that clock, at the instants of time before and after He accelerates away from Her, if he is using the CMIF method, he says it changes from t=0.00 (before) to t=-13.33 (after). You surely do not think that your method has clocks going backwards, now do you?

I think you are only considering the ages of two twins, "He" and "She."

That's all that NEEDS to be considered. A simultaneity method's SOLE job is to produce an age correspondence diagram (ACD): a plot of the home twin's age, according to the traveling twin (him), at each instant of his life. And for ANY scenario that begins with the twins together at the origin, with the traveler (he) changing instantaneously after birth to a positive velocity (moving away from the home twin (her) in the positive X direction on the Minkowski diagram), the CMIF method and my method give EXACTLY the same results on the entire outbound leg (before he changes their relative velocity at the velocity change point "T"). BOTH methods say that her current age according to him anywhere on the outbound leg (before the instantaneous velocity change at T) is always just his age divided by gamma.

My post mentions a clock at rest in the stay-home frame which is synchronised to Her clock, and located at x=23.09. That is the place where He will eventually come to a stop (in the simplest scenario). I am talking about the time displayed on that clock, at the instants of time before and after He accelerates away from Her. If he is using the CMIF method, he says it changes from t=0.00 (before) to t=13.33 (after). (See Minkowski diagram below.) Your method says that that clock displays t=0.00 (before) and t=0.00 (after) because you say that your method does not change the time on a distant clock just because someone accelerates. Correct?

Yes. On the outbound leg, my method doesn't agree with the CMIF method about the current reading on THAT CLOCK, but my method DOES agree with the CMIF method about the current age of the home twin (before the velocity change at "T").

Rather than talking about clocks, it's much more enlightening to do another scenario where the two twins aren't really twins, but who are a distance 23.09 apart at their births, and mutually stationary then. Position him (the traveler) at his birth at the origin of the diagram as usual, but position her at birth at the position 23.09 on the positive X axis. Let her send him a pulse when they are born. That pulse travels downward to the right. While that pulse is in transit, they agree with each other that they always have the same age. As the instant that he receives that pulse, she transmits a second pulse to him, and he instantaneously changes his velocity to 0.57735. As soon as he accelerates, the disagreement interval (DI) begins ... i.e, he doesn't agree with his new PIO. And the DI ends when he receives her second pulse. So just as in the standard scenario (with them co-located at birth), he concludes that her current age doesn't instantaneously change when he accelerates at a distance from her, but that her current age DOES immediately start to linearly increase faster than his rate of ageing.

I know that you badly want to find an inconsistency in my method, but there aren't any inconsistencies in it (just as their aren't any inconsistencies in the CMIF method). They are both consistent, and we can't tell which one gives the true simultaneity. Which one we use depends on which one we each think has the more desirable characteristics.

On the outbound leg, my method doesn't agree with the CMIF method about the current reading on THAT CLOCK, but my method DOES agree with the CMIF method about the current age of the home twin (before the velocity change at "T").

Yes I know, that is what I was saying in my post #200. Your method has the traveler using something that superficially looks like the CMIF method on the outbound leg, when considering only the stay-home twin's age. But if he considers all of the other synchronised clocks which can be at rest in the perpetually inertial frame of the stay-home twin, (other than the stay-home twin's clock at the origin), it becomes clear that your method is not the same as the CMIF method at all.

I know that you badly want to find an inconsistency in my method, but there aren't any inconsistencies in it (just as their aren't any inconsistencies in the CMIF method). They are both consistent, and we can't tell which one gives the true simultaneity. Which one we use depends on which one we each think has the more desirable characteristics.

On the contrary, when I mistakenly thought you were using the exact CMIF method for the outbound leg, but not for the inbound leg, it was then that I thought your method was inconsistent. But now that I understand that your method is never exactly the same as CMIF method, I don't think it is inconsistent any more. It is consistent in that the traveler's first acceleration has the same features that his second acceleration has, namely, that distant clocks do not change the time they are currently displaying just because someone accelerates. That is why I brought up the clock (or, if you prefer, another twin who is not really a twin) located at x=23.09.

Sounds like we might be in a little better agreement now than we've been in recently. I remember we were actually confederates for years before I discovered my method!

I'll respond to Halc and Neddy as soon as I finish the analysis of the multiple velocity changes scenario. I worked on it most of the day yesterday. Almost done (I think).
I've been staying out of it for a while since I've been waiting for the dust to settle. This post from 2 weeks ago seemed to indicate that the method proposed at the time needed work, but it seems unchanged in the document in the areas where I find problems. My post 164 addressed some problems with the method described in section 10 of the paper, notably that Bob can arrive home and tell Alice that she's far younger than she really is. If there's an update to the method that solves this problem, then it hasn't made its way to your page that you evangelize so heavily. If there isn't a fix to the problem, then my post 164 has not been addressed.

My post 164 addressed some problems with the method described in section 10 of the paper, notably that Bob can arrive home and tell Alice that she's far younger than she really is. If there's an update to the method that solves this problem, then it hasn't made its way to your page that you evangelize so heavily. If there isn't a fix to the problem, then my post 164 has not been addressed.

I've never analyzed a case (where the traveler IS reunited with the home twin) for which the two twins FAIL to agree about the correspondence between their two ages at the reunion. If that DID happen, then clearly that would indicate that something is wrong with my method. But it hasn't ever happened.

Somewhere buried in all my recent postings, I DID resolve the issue of when the S equation can be used. Recall that the basic method for constructing the age correspondence diagram (ACD) is to do it via the Minkowski diagram, without using the S equation. (It is often even easier to do it that way anyway). But some of Neddy's questions revealed that for multiple velocity changes that are too close together, the S equation doesn't work. What I found was this: the S equation for the subsequent velocity change can't be used when that velocity change occurs while in the "disagreement interval" (DI) of the previous velocity change. In that case, the Minkowski diagram must be used to determine the slope.

I DO need to add that information to my webpage ... I didn't realize that I hadn't done that yet. I HAVE added some stuff to my webpage in various places recently, but I forgot to add the above conclusions about when the S equation can't be used.

I've never analyzed a case (where the traveler IS reunited with the home twin) for which the two twins FAIL to agree about the correspondence between their two ages at the reunion. If that DID happen, then clearly that would indicate that something is wrong with my method. But it hasn't ever happened.
That's fine, but then you need to point out where to find the new method. I simply utilized the 'S = '... equation in section 10, and as quoted in post 164, and it yields a slope that isn't always steep enough to get Alice's age up to what it needs to be in time for the reunion.

Somewhere buried in all my recent postings, I DID resolve the issue of when the S equation can be used.
...
What I found was this: the S equation for the subsequent velocity change can't be used when that velocity change occurs while in the "disagreement interval" (DI) of the previous velocity change. In that case, the Minkowski diagram must be used to determine the slope.
So DI is anytime the previous S = slope hasn't run to completion? What does the diagram tell you to do? I mean, Minkowski says the frame to use is arbitrary, so are we using Bob's frame here? That would be CMIF, which is known to exhibit discontinuities and backwards aging and such. So it's something else if you feel that these things are to be avoided.

Sorry for the stupid questions, but I got behind when you indicated that it wasn't all yet worked out, and the paper is damn near impossible to read with the total lack of diagrams and all the preamble about signals being passed back and forth and such, none of which seems critical to actually using the method. Anyway, I didn't find the section that mentioned when to use the S equation and what to do when you decide not to. It seems a pretty shaky method when the rule is to use this computation, except when it doesn't work, in which case use something else.

I DO need to add that information to my webpage ... I didn't realize that I hadn't done that yet. I HAVE added some stuff to my webpage in various places recently, but I forgot to add the above conclusions about when the S equation can't be used.
Oh good, so it's not just me not finding it. This thread is way to long to try to glean the new rules out of it, especially when I'm never sure if it's a work in progress still. Anyhoo, kindly point to where you put it when you got it stable.

So DI is anytime the previous S = slope hasn't run to completion? What does the diagram tell you to do?

What you do is, when the previous velocity change happens, you draw a pulse from her (on the horizontal axis) to him (on his worldline, where that pulse intersects his worldline). Once he has received that pulse, he again agrees with the PIO, and the disagreement interval (DI) is over. At that point, the S equation works if there is another velocity change then, or later. But if a velocity change happens before the DI ends, the S equation can't be used then. But that doesn't mean we can't analyze that situation ... in that case, we just need to use the Minkowski diagram instead of the S equation to get the slope of the segment beginning with that velocity change.

What you do is, when the previous velocity change happens, you draw a pulse from her (on the horizontal axis) to him (on his worldline, where that pulse intersects his worldline).
Horizontal axis? What determines that? Why don't we just use that the whole trip? Keep in mind that using the phrase "when X happens to Bob, Alice does Y" necessarily leverages a method for simultaneity at a distance, and seemingly not your method.

I've just added the new material, about when the S equation can't be used to determine the slope of the section of the ACD following a velocity change, to my webpage:

The new section is now Section 12, and I incremented the section numbers of the following sections accordingly.

Horizontal axis? What determines that? Why don't we just use that the whole trip?

It sure seems like it would be easier to do that, rather than to do Mike's method. Also, Mike's method seems arbitrary. We can have a perpetually inertial frame with a set of synchronised clocks, and we can also have a recently accelerated observer who is stationary with respect to that perpetually inertial frame. This newcomer proclaims to the perpetually inertial people that their clocks are not really synchornised. Naturally they ask him why not. He can either tell them it is because the distances between the clocks are not what they think they are, although the speed of light is what they think it is. Or, he can tell them it is because the speed of light is not what they think it is, although the distances between the clocks are what they think they are. Or, he can tell them it is because the speed of light is not what they think it is AND the distances between the clocks are not what they think they are. It is arbitrary, and I don't know which Mike actually chose, because I am not interested enough to try to figure it out.

And besides, the perpetually inertial people would be completely justified to think the newcomer is just wrong, mistaken, or insane. They know how they manufactured their measuring rods, and they know how many fit in the distance between their clocks. Who is this newcomer to say that some of those rods are different lengths than others? The perpetually inertial people also know the speed of light, as they have performed physics experiments, and studied relativity. Who is this newcomer to say that the speed of light varies? They would have every reason to just ignore him, or even require him to take a mental health evaluation. This newcomer is certainly not doing any physics, he is just stating opinions while claiming they are facts, without any justification.

Who is this newcomer to say that some of those rods are different lengths than others? *

It would have been even better if I had said, "Who is this newcomer to say that some of those rods are different lengths than others, or even more bizarrely, that the lengths of the rods are changing over time?" This also applies to the speed of light, "Who is this newcomer to say that the speed of light is changing over time?"

After all, the newcomer will eventually tell the perpetually inertial people that their clocks are synchronised now, but they weren't earlier. Meanwhile, all of the perpetually inertial people would be scratching their heads, knowing that nothing has changed with any of their clocks. I'm telling you, they would be justified in putting the newcomer into psychiatric observation.

( * Too late to edit my post, so I added it here.)

Concerning this attitude of there being a 'correct' method, I wonder what you mean by 'correct'. The only interpretation of that assertion is that the method corresponds to some metaphysical truth, but the method chosen is not consistent with any suggested metaphysics in any known philosophical interpretation except possibly solipsistic-idealism where Alice in fact doesn't exist at all when not directly experienced, and not even then in the same way you exist. This method you've created is not consistent with your claims in your paper and posts, and that makes it irrational.
section 13 said:
I don't believe that my home twin ceases to exist whenever we are separated.
OK, that sort of eliminates the solipsistic idealism. That fit well with the CMIF method, but not the others. The universe is all about me.

she must be a specific age right now. So I believe she must have some specific current age. Her current age is not just one of a set of equally good "conventions" of simultaneity, as many physicists believe. Therefore there must be a single, correct simultaneity method.
All this adds up to an assertion about some kind of objective (independent of viewpoint) ordering of all events in the universe. This is an absolutist stance, and the view says that there is an objective ordering, even if the ordering cannot be experimentally determined. This is a rational stance, and one held by a different set of physicists than the ones you mention that take a relative view, which you call a 'set of equally good "conventions" of simultaneity', which seems to be a strawman way of putting it because there seem to be only three conventions used in practice, and neither is on your list.

So, given your chosen premise that she must be a specific age right now, you've effectively asserted an objective ordering, and then you create a method that isn't objective. That's the part that isn't rational. You get the absurdities that Neddy and yourself point out: Two comoving observers using the same method and getting different answers. You call it a disagreement interval, which effectively admits that your assertion that Alice being a specific age right now to be a false assertion. You're contradicting your own premises.
If she's a specific age right now, then nothing you are doing right now (like moving this way or that) should have any effect at all on her age.

I now suspect that it may be impossible to prove which simultaneity method is the correct one. I DO believe that there IS one and only one correct simultaneity method in special relativity, but we probably just can't know what that correct method is. All we can do is choose which of the known methods we each want to use, based on which one we believe has the most desirable characteristics. I think my method's characteristics are the most desirable among the four simultaneity methods that I'm aware of (CMIF, Dolby&Gull, Minguizzi, and mine): mine is (1) causal, (2) it produces an ACD that is always continuous and piecewise-linear, and (3) it never produces negative ageing of the home twin
You forgot 4) Doesn't violate your stated premise that Alice has an actual age right now. Your method violates that, and Minguzzi does not.
But remember that I said that the physicists (both absolutists and relativists) tend to use only one of three methods for any practical purpose? All three of those methods satisfy your 3 points, and one satisfies the 4th one that includes your absolute premise.

Method 1: (the most commonly used method): Her age is as we see it. If I move away quickly, her age rate drops significantly, mostly due to Doppler effect.
I pointed out this method in an earlier post. Everybody is looking at Betelgeuse right now because it suddenly got uncharacteristically dimmer and they're wondering if it's about to go supernova. I found 50 link to articles about it and not one of them says that it might have gone supernova 650 years ago. They all say it might go any time now (It probably isn't). That's all using method 1. Ditto for all the news about the black hole mergers.
This method fails the test of commutivity. If I am 16 and she appears 8, then she, at 8, will see me as perhaps 4. This criteria was not one of your 3+1 points. Your method fails it as well. The method doesn't work over large distances.

Method 2: Frame of home base, the place where the reunion takes place. That's almost always Earth mean frame. This method is commutative, and works well locally but not over large distances. More precisely, all computations are done relative to the frame in which the departure event and the reunion event are at the same spatial location.

Method 3: Comoving coordinate system: The age of every event is the spacetime interval between the big bang and that event. This is a favored method for very large scale, and because it is the only method that foliates all of spacetime, it is favored by many absolutists. It is commutative. Drawback is that it isn't inertial. Two stationary objects tend to separate over time. An inertial object with no forces acting on it tends to slow over time, losing kinetic energy. All this is pretty non-Newtonian. The method is rarely used for local computations like trips to nearby galaxies and such.

The first two methods cannot be 'the correct' absolute method since neither foliates all of spacetime.

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This paper by Ovyind Gron provides some interesting observations that may be useful.
https://arxiv.org/abs/1002.4154
The twin paradox and the principle of relativity, Oyvind Gron, (Submitted on 22 Feb 2010)
In the standard formulation of the twin paradox an accelerated twin considers himself as at rest and his brother as moving. Hence, when formulating the twin paradox, one uses the general principle of relativity, i.e. that accelerated and rotational motion is relative. The significance of perfect inertial dragging for the validity of the principle of relativity is made clear. Three new results are reviewed in the discussion. A cosmic time effect which cannot be reduced to the gravitational or the kinematical time dilation. Perfect dragging in an exact solution of Einsteins field equations describing flat spacetime inside a shell with Kerr spacetime outside it. An extended model of Minkowski spacetime in order to avoid introducing absolute acceleration and rotation through the asymptotic emptiness of the Kerr spacetime.

Hello.

I finally had the opportunity to really sit down and work through Mike's paper.

I have confirmed the coordinates of tau1-tau10, but I find I have issues with the method of adding together the periods when the photon goes from T7 to R according to the OCMIO and when the photon goes from R to Q according to the ICMIO.

I agree with Neddy Bate that Mike's CADO method, from 1999, was superior and more representative of Special Relativity.

I'm very impressed with your diligence, Jonathan. I will take your analysis seriously, and will respond asap. (It's actually good timing, as I have just wrapped up doing the nasty job of taxes.)

You DID correctly confirm the result that, for her pulse that crosses point R, she ages 4.88 years during the transit of that pulse before it gets to R (according to the OCMIO), and that she ages 11.54 years between R and when he receives the pulse (according to the ICMIO). So he then adds 4.88 + 11.54 = 16.42 years to determine how much she ages during the complete transit of that pulse (according to him). Since she was 28.45 years old when she transmitted that pulse, he concludes that she is 44.87 years old when he received the pulse at age 38.64.

So that gives us ONE point on the age correspondence diagram (ACD) curve: when he is 38.64 years old, she is 44.87 years old (according to him).

How do we get the whole ACD curve? If getting every point on the curve (or getting enough points that we are confident that we know what the curve looks like) were each as hard as the above calculations, my method wouldn't be very practical. (That is the case with the Minguizzi method: each point on his ACD (except for the first section) is hard to determine). But fortunately, the complete ACD for my method is very easy to determine. All you really need to do is determine only two points on the ACD, each of which is easier than in the above process.

To get the first point, we look at the pulse he receives from her immediately before he changes his velocity from +0.57735 to -0.57735. It is entirely in the left half plane (to the left of the vertical line passing through the velocity-change point (T) and intersecting the horizontal axis (her worldline). So during that entire pulse, he agrees with the OCMIF. It is easy to determine that she is 26.67 years old, and he is 32.66 years old, when he receives that pulse (all according to him).

The second easy point is determined by considering the pulse that she transmits when he changes his velocity. That pulse is entirely in the right half plane. He receives that pulse exactly at the point where he begins to agree with the ICMIO. It is easy to determine that, when he receives that pulse, she is 63.09 and he is 44.61 years old (both according to him).

So with those two points, we know where the first section of the ACD intersects the middle section of the ACD, and we know where the middle section of the ACD intersects the third section of the ACD. Since he agrees with the OCMIO during the whole first section, we know that that section is just a straight line of slope 1/gamma = 0.817, starting from the origin. And since he agrees with the ICMIF during the whole final section, we know that section is just a straight line of that same slope 1/gamma = 0.817. And we already have the location of the beginning of the third section, and we also know the point where the third section ends (from her simple analysis of the trip), so we can plot the whole third section.

That leaves only the middle section to determine. IF that section were curved, we would have to do a lot of work to determine that curve ... each point we determine in that section would need to be determined using the process we used to determine their ages for the pulse that was partly in the left half plane and partly in the right half plane (and which intersected the vertical line coming down from the velocity-change point halfway to the horizontal axis). We would need to do that same process for many other pulses that intersect that vertical line at other points.

Fortunately, we don't need to do ANY of those calculations where the pulse is partly in both half planes, because the middle section turns out to be a straight line! That is VERY fortuitous! ALL we have to do draw a straight line between the two points we've already got: the point where the first section ends, and the point where the third section begins.

In you response, you wrote:

However, now “Her” world-line is broken up into three segments, (T7-T8), (T8-T10), (T10-T9), and “He” is adding together the times from T7 to T8, and the times from T10 to T9, but still skipping the middle time.

Nothing that you've identified as "being skipped" is needed, or relevant. ALL that's needed is that we have been able to easily produce the ACD.

You also wrote in your response:

Again what’s missing here is the fact that these events occurred as the ICMIO traversed
the space between event u1 and Q. The events from u1 to TA did not happen to “Him” at all.

That causes no problem at all.

At the end of your post above, you wrote:

I agree with Neddy Bate that Mike's CADO method, from 1999, was superior and more representative of Special Relativity.

I don't agree that the CMIF simultaneity method is any more representative of Special Relativity than my new simultaneity method.

Special Relativity is SILENT on simultaneity for an observer who accelerates.

CMIF ASSUMES that an observer who is not currently accelerating always agrees with the PIO riding along with him. Special relativity makes no such assumption.

My method assumes that pulses that are partly in each half plane should be handled using the process I've described in (the first part of ) my webpage, and which you have correctly duplicated. Using my assumption produces a result in which there are no discontinuities in the ACD. And it produces a result in which the observer who accelerates will disagree with the PIO riding along with him for some period of time after the velocity change has occurred (the "disagreement interval" (DI)). Eventually, the observer who accelerated in the past will agree with the PIO, and will continue to agree with the PIO for the remainder of the trip (if there are no additional accelerations at a distance).

My method assumes that pulses that are partly in each half plane should be handled using the process I've described in (the first part of ) my webpage, and which you have correctly duplicated. Using my assumption produces a result in which there are no discontinuities in the ACD. And it produces a result in which the observer who accelerates will disagree with the PIO riding along with him for some period of time after the velocity change has occurred (the "disagreement interval" (DI)). Eventually, the observer who accelerated in the past will agree with the PIO, and will continue to agree with the PIO for the remainder of the trip (if there are no additional accelerations at a distance).

In regards to the Disagreement Interval (DI), do you believe that the speeds of the clocks held by the traveling twin ("Him") and the comoving Perpetually Inertial Observer (PIO) would naturally be ticking at different speeds, or they would they be at the same speed, unless carefully calibrated to use your technique to achieve a continuous Age Correspondence Diagram, with the stay-at-home twin, "Her"?

In regards to the Disagreement Interval (DI), do you believe that the speeds of the clocks held by the traveling twin ("Him") and the comoving Perpetually Inertial Observer (PIO) would naturally be ticking at different speeds, or they would they be at the same speed, unless carefully calibrated to use your technique to achieve a continuous Age Correspondence Diagram, with the stay-at-home twin, "Her"?

I'm not sure I follow all your wording in the above quote. But I can say that the ICMIO is the same age as the traveling twin (he) on the entire return leg. They don't disagree on the entire return leg about ANYTHING that is happening locally ... but during the DI, they disagree about what is happening at a distance, and in particular, they disagree about the current age of the home twin (her).