# A New Simultaneity Method for Accelerated Observers in Special Relativity

#### MikeFontenot

Until recently, I've been an avid proponent of the "co-moving inertial frames" (CMIF) simultaneity method (previously called the "CADO" method by me). I had claimed to have proven that the CMIF method is the ONLY method that agrees with the accelerated observer's own elementary observations and elementary calculations. But I recently concluded that there was a loophole in that proof, and therefore I had failed to prove what I thought I had proven. I decided to take a fresh look at the whole issue of simultaneity for an accelerated observer. In the course of doing that, I discovered a new simultaneity method that shows, with a very simple proof, that the CMIF method isn't correct. My new method says that when the accelerating observer instantaneously changes his velocity, the current age of the home twin DOESN'T instantaneously change. Instead, the slope of the age correspondence curve instantaneously changes its slope from a constant less than one to a constant greater than one. And then after a well-defined passage of time, the slope instantaneously switches back to the same constant less than one that occurs in the first segment. So the "curve" in the age correspondence diagram is always a continuous, piecewise-linear line of three straight line segments. Unlike the Dolby and Gull simultaneity method, and the Minguzzi simultaneity method, my method is causal, i.e., effects are always PRECEDED by causes. My new method is explained in detail on my webpage referenced below (in front of the old information on my webpage, which I now know to be incorrect).

Michael Leon Fontenot

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#### MikeFontenot

I got a new result from my simultaneity method, and have added it near the end of the new material on my webpage. I was able to prove that in my new method, the home twin's current age, according to the traveling twin, can never DECREASE. I.e., she can never get YOUNGER, according to him. That is a nice property for a simultaneity method to have, because the prospect of the home twin getting younger is repugnant to many physicists. In contrast, the well-known co-moving inertial frames (CMIF) simultaneity method implies that she DOES get younger (according to him) when he accelerates in the direction away from her (when their separation is sufficiently great).

A New Simultaneity Method - CADO

#### GatheringKnowledge

As far, as I understand the basic rules of SRT, causality is always maintained, if the timeline for a chain of events doesn't "extend" beyond the light cone on a spacetime diagram (time-like events). In shortcut, there's no way you can observe the aging process being reversed for someone who's in motion relative to you, unless he won't be moving faster than light. Maybe istead pasting a link to your website, you can try to more or less explain, what this new method is all about...

Here you have a cool interactive Minkowski diagram, so you can show a practical example

topsquark

#### MikeFontenot

The most important result I got in my paper (and webpage) "A New Simultaneity Method for Accelerated Observers in Special Relativity" is a proof that the "co-moving inertial frames" (CMIF) simultaneity is incorrect. Although most of the paper is about the new simultaneity method I've discovered (which I'll hereby christen as "Fontenot's method"), I don't make use of that method in my proof that CMIF simultaneity is incorrect.

I recently realized that my original proof wasn't correct. But I was able to correct it, and I believe the new proof is "airtight". Here is the new proof:

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So, when he receives her message immediately before his turnaround, we’ve determined that he knows that she aged 9.76 years during the pulse’s transit, and that she is currently 26.67 years old. Then, he instantaneously changes his velocity to -0.57735 ly/y, now heading back toward her. How old does he say she is now?

In order to answer that question, we need to define some important instants of time. It is important to realize that although the velocity change is instantaneous, there are nevertheless three distinct instants related to the turnaround that must be distinguished. The first instant is when he receives her pulse. The second instant is when his instantaneous velocity change occurs. Denote that instant as time "T" in his life. "T" is a specific, finite number. The third instant is when he first is no longer accelerating, and is moving at the new constant negative velocity. There is no finite amount of time that separates these three instants. But they are separated by an infinitesimal amount of time. We could express this by writing the instant when he receives her message as "T-", and the instant when he begins his constant negative velocity as "T+".

Now, given the above terminology, we can reason as follows. At the instant "T-", he has received her message, and has determined that she has aged by 9.76 years during the complete transit of her message. Her message says that she was 16.91 years old when she transmitted the message. So she is 26.67 years old at the instant "T-".

Next, we need to make use of some definitions and an additional pulse. She transmitted the first pulse when she was 16.91 years old. Call that the instant in her life "TAU". She transmits the additional pulse at the instant "TAU+", which is only infinitesimally later than the instant "TAU". The first pulse was received by him at the instant "T+". The additional pulse is received by him at the instant "T++", which in only infinitesimally different than the instant "T+". So when he receives that additional pulse, he knows that her age only changed by at most an infinitesimal amount during his instantaneous velocity change. CMIF simultaneity says that she aged by a large finite amount during his instantaneous velocity change. Therefore the CMIF simultaneity method is incorrect. That is a very startling fact, given that the CMIF simultaneity results are found in many (maybe most) text books on special relativity.
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The link to my webpage is:

The proof is near the end of Section 7.

#### MikeFontenot

I have just improved the wording of my proof that the co-moving inertial frames simultaneity method (CMIF) is incorrect. If anyone is trying to identify an error in my proof, it should be easier now to localize any such error. Here is the improved wording:
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Now, given the above terminology, we can reason as follows. At the instant "T-", he has received
her message, and has determined that she has aged by 9.76 years during the complete transit of
her message. Her message says that she was 16.91 years old when she transmitted the message.
So she is 26.67 years old at the instant "T-".

Next, we need to make use of some definitions and an additional pulse. She transmitted the first
pulse when she was 16.91 years old. Call that instant in her life "TAU". The first pulse was
received by him at the instant "T-" (immediately before his velocity change). The additional pulse is
received by him at the instant "TA-" (immediately after his velocity change), which is only
infinitesimally different than the instant "T-". Call the time when she transmitted that additional
message "TAU+", which is only infinitesimally later than the instant "TAU". So when he receives that
additional pulse, he knows that her ageing during that message's transit is only infinitesimally
different than her ageing during the transit of the first message. And her age when she transmitted
that additional message was only infinitesimally greater than her age when she transmitted the first
message. For both messages, her age when he receives the message is the sum of her age when
she transmitted the message, plus her ageing during the transit of the message. Therefore her
current age (according to him) when he receives the additional message is at most only infinitesimally
different than her current age when he received the first message. I.e., her current age (according to
him) at the instant "TA-" (after his velocity change) is at most only infinitesimally different than her
current age at the instant "T-" (before his velocity change). But CMIF simultaneity says that she aged
by a large finite amount during his instantaneous velocity change. Therefore the CMIF simultaneity
method is incorrect
. That is a very startling fact, given that the CMIF simultaneity results are found in
many (maybe most) text books on special relativity.
___________________

The proof is near the end of Section 7 of my webpage:

Also, be aware that the proof does NOT make use of my new simultaneity method (which I'm calling "Fontenot's method" or "Fontenot's simultaneity method"). I show that the CMIF simultaneity method is incorrect purely from first principles, combined with the fact that on the outbound leg (up to but not including the instantaneous velocity change) the traveler (he) is equivalent to an inertial observer (because he has never accelerated), and we only need the time-dilation equation to determine his conclusion about his twin's current age on that leg.

#### MikeFontenot

I've just discovered that my proof, that the CMIF simultaneity method is incorrect, is invalid. Here is the previous argument, with the newly discovered error discussed below that:

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The entire lifetime of the first pulse is in his past when he instantaneously changes his velocity. And the lifetime of that pulse is many years. Her ageing during the transit of that pulse is part of history ... no one can change that amount of her ageing. And her age when she transmitted that pulse is also an unchangeable fact. So the sum of those two facts, which is her current age immediately before his velocity change, is also a fact which cannot change.

And my last posting shows that the two pulses differ only infinitesimally. They both exist for many years, and are only infinitesimally different. Up until the instant that he changes his velocity, the amount she has aged during the transit of the pulses is essentially the same ... the difference is only infinitesimal.

But what about the second pulse (which he receives immediately after he changes his velocity)? My last posting showed that when he receives the first pulse, the second pulse is itself almost complete. MOST OF HER AGEING during the entire second pulse has already happened when he receives the first pulse. The only part of the second pulse which hasn't happened yet, when he accelerates, is the last infinitesimal part.
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I now see that I haven't successfully proved the above sentence with the phrase I've now highlighted using ALL CAPITAL LETTERS. I haven't been able to prove that she can't age by a large finite amount during that infinitesimal remaining part of the second pulse. And I'm beginning to suspect now that it CAN'T be proven.

So where does that leave me? It leaves me with having defined a new simultaneity method that has some very nice properties. It is causal (like the CMIF method), it produces an Age Correspondence Diagram (ACD) that has no discontinuities, is piecewise-linear, never decreases (so she never gets younger, according to him), and is easy and quick to obtain. So people who are horrified about instantaneous (and especially negative) ageing will probably like it. But I can't (and probably never will be able to) prove that the CMIF method is incorrect, or that my method is correct.

However, I DO believe that there IS only one correct simultaneity method for the accelerated observer. I DON'T accept that either method is equally good. And I don't accept that simultaneity at a distance is meaningless. That is based on my philosophical belief that she doesn't cease to exist whenever they are separated, and IF she still exists, she MUST be doing something specific "right now". And if so, she MUST have a specific age right now, because her brain at each instant of her life contains a record of what she was doing at that instant. So, I believe that EITHER the CMIF method, OR my method, is correct, but they aren't BOTH correct. We just don't know which one is the correct one. I reject the Dolby&Gull method, and the Minguizzi method, because they are non-causal, and special relativity (at least Einstein's version of it) is causal. So, as for now, the CMIF method and my method are "the only games in town".

#### MikeFontenot

My new simultaneity method, which I call Fontenot's simultaneity method, produces an age correspondence diagram (ACD), which is a plot of the home twin's current age (according to the traveling twin), as a function of the age of the traveling twin. My ACD is piecewise-linear, with no discontinuities. It consists of three connected straight lines. The first portion has a slope of 1/gamma_1. The last portion has a slope of 1/gamma_2. gamma_1 and gamma_2 are the gamma factors corresponding to v_1 and v_2. The middle section of that diagram (starting immediately after the traveler's instantaneous velocity change from v_1 to v_2) has a constant upward slope that, in the standard twin paradox scenario with v_2 = -v_1, is greater than one. That slope can be determined graphically. It can also be computed analytically. Denote the slope of the middle section of the ACD curve as S. Then the S equation is

S = (1 / gamma_2) + gamma_2 * (1 - v_2) * (v_1 - v_2),

where v_1 is the relative velocity before the velocity change, v_2 is the relative velocity after the velocity change, and gamma_2 is the gamma factor corresponding to v_2. Velocities are positive when the twins are diverging, and negative when the twins are converging. It is possible to show, by using the S equation, that my simultaneity method never produces a negative value for S. I.e., in my method, the traveling twin never says that the home twin is getting younger. That is in contrast to the CMIF method, which predicts negative aging of the home twin for some scenarios. My simultaneity method also differs from Dolby and Gulls' simultaneity method, and from Minguzzi's simultaneity method, in that theirs are non-causal. My simultaneity method, and the CMIF simultaneity method, are both causal.

I call the above S equation Fontenot’s equation.

More information, including a plot of the ACD for the twin paradox with v = +-0.866 ly/y and gamma = 2.0, in available on my webpage:

#### MikeFontenot

I'm going to try to upload two jpegs: the ACD and the Minkowski diagram, for the case of two separated velocity changes, starting from v1 = 0.57735 at the birth of the twins, then changing to v2 = 0.0 when he is 32.66 years old, and then later changing to v3 = -0.57735. The three different ages for him on that last velocity change are 36.89, 44.21, and 55.75 years old. I've marked on the ACD where he begins to agree with the perpetually-inertial observer, for each of those three scenarios ... they are the final low-slope lines of slope 1/gamma = 0.817.

And here is the Minkowski diagram:

In the Minkowski diagram above, I failed to write the amount of her ageing during the upper portion of the L0 pulse: it is 4.227 years. That is what the PIO (perpetually-inertial observer) AFTER the velocity change calculates. The PIO BEFORE the velocity change determined that her ageing during the lower portion of the pulse (up to the point P0) is 7.974 years. So the traveler concludes that her aging during the entire pulse is 7.974 + 4.227 = 12.201 years. And she was 21.133 years old when she transmitted the pulse. So he concludes that she was 21.133 + 12.201 = 33.334 years old when he received her pulse. He was 36.887 years old then. The fact that he ADDS the amounts of her ageing during the two portions of the pulse (as determined by the two PIO's), to determine her current age when he receives her pulse, is the HEART of the definition of my simultaneity method ... everything follows from that.

The question has been asked: "Can my equation, for the slope of the ACD after a velocity change, always be used?". The shortest answer is "No". A slightly longer answer is, "The traveler (he) can, after he changes his velocity, use my equation to determine how fast his home twin (she) is ageing (compared to his own rate of ageing) IF he currently agrees (about her current age) with the perpetually-inertial observer (PIO) riding along with him then. If he DOESN'T agree with the PIO, he CAN'T use my equation, and he must use the Minkowski diagram analysis to determine her relative rate of ageing then. Basically, immediately after he changes his velocity, he can use my equation to determine the slope of the ACD which results from that velocity change, PROVIDED he hasn't changed his velocity too recently in the past.

Postscript: Apparently, images aren't allowed. Sorry. I've put both those images on my webpage, so they CAN be seen there. They are at the end of the new section of my webpage (which describes my new simultaneity method), before the old material on my CADO equation that is useful for the co-moving inertial frames (CMIF) simultaneity method. The webpage link is: