# The Time Dilation Equation (TDE) of Special Relativity

#### Mike_Fontenot

Registered Senior Member
The time dilation equation (TDE) is one of the most important results in all of special relativity. It can single-handedly resolve the Twin "Paradox". Special relativity says that the TDE specifies what ANY particular inertial observer MUST conclude about the rate of ageing (compared to their own rate of ageing) of anyone moving with respect to them. In particular, it says that the traveling twin (he) MUST use the TDE to determine how much SLOWER the home twin is ageing compared to his own rate of ageing whenever he is inertial. And likewise, SHE must use the TDE to determine how much slower HE is ageing than she is, according to HER, during her entire life (since she is ALWAYS inertial). They each MUST conclude that, whenever they are inertial, the other twin is ageing gamma times slower than their own rate of ageing ... they have no other choice: their own experimental measurements CONFIRM the correctness of the TDE. Whenever SHE is unaccelerated (which is ALL the time), she says he is ageing gamma times slower than she is, and that is a totally REAL conclusion for her. And whenever HE is unaccelerated (which is ALL the time EXCEPT for the one instant where he is reversing his velocity), he MUST conclude that she is ageing gamma times slower than he is. And that is a totally REAL conclusion for him. And for HIM, those results then REQUIRE that, according to HIM, SHE instantaneously gets older during HIS instantaneous turnaround. That HAS to happen ... otherwise he won't agree with their respective ages at the reunion, and they obviously MUST agree at the reunion (when they are eye-to-eye).

The time dilation equation (TDE) is one of the most important results in all of special relativity. It can single-handedly resolve the Twin "Paradox". Special relativity says that the TDE specifies what ANY particular inertial observer MUST conclude about the rate of ageing (compared to their own rate of ageing) of anyone moving with respect to them. In particular, it says that the traveling twin (he) MUST use the TDE to determine how much SLOWER the home twin is ageing compared to his own rate of ageing whenever he is inertial. And likewise, SHE must use the TDE to determine how much slower HE is ageing than she is, according to HER, during her entire life (since she is ALWAYS inertial). They each MUST conclude that, whenever they are inertial, the other twin is ageing gamma times slower than their own rate of ageing ... they have no other choice: their own experimental measurements CONFIRM the correctness of the TDE. Whenever SHE is unaccelerated (which is ALL the time), she says he is ageing gamma times slower than she is, and that is a totally REAL conclusion for her. And whenever HE is unaccelerated (which is ALL the time EXCEPT for the one instant where he is reversing his velocity), he MUST conclude that she is ageing gamma times slower than he is. And that is a totally REAL conclusion for him. And for HIM, those results then REQUIRE that, according to HIM, SHE instantaneously gets older during HIS instantaneous turnaround. That HAS to happen ... otherwise he won't agree with their respective ages at the reunion, and they obviously MUST agree at the reunion (when they are eye-to-eye).

There are three distinctive results of SR, and only one of them is time dilation (TD). In the above paragraph, you discuss time dilation, but you also say, "according to HIM, SHE instantaneously gets older during HIS instantaneous turnaround." That is because of relativity of simultaneity (RoS) and nothing to do with time dilation. I think you have a pretty firm grip on those two aspects of SR because you have studied the twin scenario so much.

However, there is a third result of SR which you seem to have misunderstood, probably because it does not play a very important role in the twin scenario. It is length contraction (LC). That became very important when you switched from the twin scenario to Bell's scenario, and you dropped the ball badly there. Having spent much less time studying length contraction, you jumped to your own conclusions and published them prematurely, without regard for how they did not comport with the specifics of what SR actually says. Sorry, but it is true.

Then, once you were so deeply invested in your ideas, you felt you had to steadfastly adhere to them as if they were something taught to you in a text book, rather than something you came up with yourself. You became a student who learned a subject from a textbook you wrote yourself, errors and all. That lead to the dead end where you had no choice but to defend something incorrect for the rest of your life.

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The time dilation equation (TDE) is one of the most important results in all of special relativity.
It's important to bear in mind that the equation you're referring to is just one special case of the much more important and more general Lorentz tranformation equations.
It can single-handedly resolve the Twin "Paradox".
No, it can't, because the equation only compares two inertial frames at a time. During the turnaround in the twin paradox, somebody's frame is non-inertial.
Special relativity says that the TDE specifies what ANY particular inertial observer MUST conclude about the rate of ageing (compared to their own rate of ageing) of anyone moving with respect to them.
The term "rate of ageing" is a very strange term to use, and I don't know quite what you mean by it.

A "rate", in physics, usually refers to the the derivative of something with respect to a time variable. In the case of a "rate of ageing", that would imply a unit of measurement equivalent to a time over a time, or a dimensionless quantity. Maybe you could argue for the Lorentz factor as the sort of thing you're really talking about there, but in my opinion that's an unnecessarily tortuous way of talking about it.
They each MUST conclude that, whenever they are inertial, the other twin is ageing gamma times slower than their own rate of ageing ... they have no other choice: their own experimental measurements CONFIRM the correctness of the TDE.
The twin paradox is a thought experiment, remember. I don't know what "experimental measurements" you're referring to. That particular experiment has never been done. Of course, time dilation has been confirmed by other experiments.
Whenever SHE is unaccelerated (which is ALL the time), she says he is ageing gamma times slower than she is, and that is a totally REAL conclusion for her.
Not exactly. While HE is turning around, SHE says that he is aging even more slowly. However, in the twin paradox, that turnaround takes zero time for HER, so that observation is irrelevant for HER.
And whenever HE is unaccelerated (which is ALL the time EXCEPT for the one instant where he is reversing his velocity), he MUST conclude that she is ageing gamma times slower than he is.
Yes. The time dilation equation of SR applies when there are two inertial frames and one proper time. It's a special case.
And for HIM, those results then REQUIRE that, according to HIM, SHE instantaneously gets older during HIS instantaneous turnaround.
Not those results. You have to jump out of the "system" of the time dilation equation of SR to draw any conclusions about who is older at the end of the trip, because that turnaround creates its own set of problems, concerned with HIM moving non-inertially for a time. The solution to that is not to be found anywhere in the time dilation equation. As Neddy said, a solution can be arrived at by considering, instead, the separate issue of the relativity of simultaneity.
That HAS to happen ... otherwise he won't agree with their respective ages at the reunion, and they obviously MUST agree at the reunion (when they are eye-to-eye).
Yes, but the time dilation equation won't provide the explanation you need, on its own.

(My [Mike Fontenot's] changes and additions are in red):

There are three distinctive results of SR, and only one of them is the time dilation equation (TD). In the above paragraph, you discuss the time dilation equation, but you also say, "according to HIM, SHE instantaneously gets older during HIS instantaneous turnaround." That is because of relativity of simultaneity (RoS) and nothing to do with time dilation.

In this simple scenario, I DON'T get that result (about her instantaneously ageing during his instantaneous turnaround, according to him) from my very simple equation that gives that result. Instead, I just get it by applying the TDE to BOTH inertial segments of his trip, giving her ageing (according to him) during those two segments of his life. THEN, I subtract that amount from her TOTAL ageing during his trip. The difference is the amount of her instantaneous ageing during his turnaround (according to HIM). So I conclude that, according to him, she instantaneously gets 60 years older during his instantaneous turnaround. Before that, he uses the TDE to tell him that she ages by 10 years during his outbound leg, and 10 years during his inbound leg. So she is 80 years old, and he is 40 years old, at their reunion, as required by both HER analysis and HIS analysis. In more complicated scenarios (where he undergoes more than one instantaneous velocity change during his trip), it is necessary to make use of my simple equation that directly computes her instantaneous ageing (according to him). Or, her instantaneous ageing can be determined graphically (although that requires more work).

I think you have a pretty firm grip on those two aspects of SR because you have studied the twin scenario so much.

However, there is a third result of SR which you seem to have misunderstood, probably because it does not play a very important role in the twin scenario. It is length contraction (LC). That became very important when you switched from the twin scenario to Bell's scenario, and you dropped the ball badly there. Having spent much less time studying length contraction, you jumped to your own conclusions and published them prematurely, without regard for how they did not comport with the specifics of what SR actually says. Sorry, but it is true.

I completely disagree. I believe my understanding of the length contraction equation is just fine. Bell's scenario is fundamentally different from the twin paradox (beyond the fact that it concerns finite accelerations, not instantaneous velocity changes). The accelerations they specify are different for the two separated rockets: the front rocket has a greater acceleration than the rear rocket, and consequently the separation between the two rockets increases, according to the people on the rear rocket. That is why the initial inertial observers (IIO's) see no length contraction between the two rockets ... they say the separation between the rockets is constant.

The example I gave (using an additional rocket located with the home twin, and stationary with respect to her immediately before the traveling twin [in his rocket] instantaneously changes his velocity) showed that the distance between the two rockets (when they instantaneously and equally change their velocity) does NOT instantaneously increase, as some people incorrectly believe.
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I (Mike Fontenot] said:

It [the Time Dilation Equation (TDE) ] can single-handedly resolve the Twin "Paradox".

No, it can't, because the equation only compares two inertial frames at a time. During the turnaround in the twin paradox, somebody's frame is non-inertial.

I'll explicitly show you how the TDE does resolve the twin paradox:

In the twin paradox, the traveler (he) first uses the TDE to tell him that, while he ages 20 years on his inertial OUTBOUND leg, she ages by 10 years (according to him). Then, he uses the TDE to tell him that, while he ages 20 years on his inertial INBOUND leg, she ages by 10 years (according to him). So her total ageing (according to him) on both of his inertial legs is 20 years. But he finds, at their reunion, that she is 80 years old (and he is 40 years old then). So he then knows that she HAD to have instantaneously aged by 60 years during his instantaneous turnaround ... there was no other time when she could have done that extra ageing.

(In a more complicated scenario, where there are multiple instantaneous velocity changes by the traveling twin at multiple times during his trip, it IS necessary to be able to directly compute how her age instantaneously changes during EACH of his instantaneous velocity changes. I've previously given the simple equation that does that. It can also be determined graphically.)

The whole issue here is in the insistence of a "instantaneous turnaround", which in of itself, is not physically realistic. You are always going to have some finite, non-zero time in which the turnaround is achieved. During which time the TDE for inertial frames is not valid for the observer doing the turnaround, and you have to use another formulation. One that accounts for the distance and direction of the clock in consideration, and the (finite) magnitude of the acceleration.

This is similar To the "ideal battery" problem in electronics. It is common when analyzing a circuit (especially when new to the subject) to treat batteries as being ideal and conductors having no resistance. With an ideal battery, the source voltage does not change with the load, and there is no voltage drop across an conductor with no resistance.

This creates a problem if you short an ideal battery with such a conductor. You end up with a scenario where you should both have no voltage difference between the ends of the conductor and also have the full source voltage difference between the ends, which is contradictory.
The solution is that ideal batteries and conductors are not physically possible. Both the conductor and battery have some non-zero resistance, and in real life you get a voltage divider, with the both the battery and conductor having a voltage drop across them.

I'll explicitly show you how the TDE does resolve the twin paradox:

In the twin paradox, the traveler (he) first uses the TDE to tell him that, while he ages 20 years on his inertial OUTBOUND leg, she ages by 10 years (according to him). Then, he uses the TDE to tell him that, while he ages 20 years on his inertial INBOUND leg, she ages by 10 years (according to him). So her total ageing (according to him) on both of his inertial legs is 20 years. But he finds, at their reunion, that she is 80 years old (and he is 40 years old then). So he then knows that she HAD to have instantaneously aged by 60 years during his instantaneous turnaround ... there was no other time when she could have done that extra ageing.
This is all well and good, but the TDE does not provide any reason for her aging 60 years during the turnaround. HE can deduce that something must have happened to allow HER to age 60 years while he was turning around, but he has no explanation of what that something might be, if he's only working with the TDE.

This is all well and good, but the TDE does not provide any reason for her aging 60 years during the turnaround.

The REASON is that Special Relativity REQUIRES it!

An addendum to my post immediately above: Here is an elaboration:

In the standard scenario where the traveling twin (he) only changes his velocity ONCE during his trip, ALL of his life during his trip is inertial except for that single instant in his life when he reverses course. So, at all other times during his trip, he is inertial, and can therefore use the time dilation equation (TDE) to tell him that his home twin is currently ageing gamma times slower that he himself is ageing. For the speed 0.866 ly/y, gamma equals 2.0, so he says she is ageing half as fast as he is, during both of his inertial segments.

From the specification of the scenario, for each of the two segments of his trip, he ages 20 years, and (by the time dilation equation) he says she ages 10 years during each of those two segments. So he says he ages a total of 40 years during the trip (because he doesn't age at all during his instantaneous turnaround [and she agrees]). And he says that SHE ages a total of 20 years during the two inertial legs of his trip. But he knows that SHE has correctly used the time dilation equation (TDE) to conclude that (according to her), while he ages a total of 40 years during his trip, she ages a total of 80 years. And he MUST agree with that (because they are eye-to-eye and looking at each other at their reunion).

So, he knows that her total ageing during his trip is 80 years, but that her total ageing during his two inertial legs is only 20 years. Where did she age the missing 60 years (80 - 20)? The ONLY place that could have happened is during his instantaneous turnaround ... there is no other alternative. It couldn't have happened during his instantaneous departure when they were born, and it couldn't have happened during their instantaneous reunion (because in both of those instants, they are face-to-face, looking at each other, and they would have seen any instantaneous ageing if it had occurred). That proves that, ACCORDING TO HIM, she instantaneously MUST have aged by 60 years during his turnaround. She certainly didn't FEEL that happening, and from her point of view, it didn't happen ... according to her, his turnaround took place during a single instant of her life. But to him, that instantaneous ageing HAD to have happened to her ... there is no other way they could agree about their respective ages at their reunion.

Mike: you just repeated what you said before. My point about the insufficiency of the TDE on its own stands.

The whole issue here is in the insistence of a "instantaneous turnaround", which in of itself, is not physically realistic. You are always going to have some finite, non-zero time in which the turnaround is achieved. During which time the TDE for inertial frames is not valid for the observer doing the turnaround, and you have to use another formulation.
[...]

It's not difficult to handle the case where the acceleration is finite and the duration of the acceleration is not infinitesimal. I did that in a paper I published in "Physics Essays" in late 1999 (about 25 years ago): it's in Volume 12, Number 4, December 1999. I've scanned-in a diagram from that paper below. It shows the age of the home twin (she) (plotted vertically), according to the traveling twin (he), versus his age, plotted horizontally. The first half of the plot shows what happens when the he is accelerating in the direction TOWARD her: he says her age rapidly INCREASES (by about 60 years during only two years of his ageing). And then the second half of the plot shows what happens when the he is accelerating in the direction AWAY FROM her: he says her age rapidly DECREASES (by slightly less than 60 years during two years of his ageing).

So what happens when the acceleration is finite is qualitatively similar to what happens when there is an instantaneous change in his velocity.

Mike: you just repeated what you said before. My point about the insufficiency of the TDE on its own stands.

My post #9 above is quite detailed. Feel free to point out an error in that post.

It shows the age of the home twin (she) (plotted vertically), according to the traveling twin (he), versus his age, plotted horizontally.
The diagram you posted mixes frames pretty badly, but seems to correctly illustrate the concept you were trying to convey, which is what happens IF he chooses to use his CMIF at all times.

As for the frame mixing, the lower axis could be either. The vertical axis is according to him.
Just taking the top middle data point, it says:
CADO(v) = 81.2 y <-- This is what gets plotted
v = ~0.774c
(moving closer)
L = 40.0 ly

So it mixes values from two frames, which the text explains as far as I remember. Most of the values are as measured in her frame despite the chart claiming to be things 'according to him', and the method only works if she is inertial all the time.

All that said, the numeric values are accurate. It seems you knew how to do kinematics back 25 years ago and lost it more recently.
The graph is basically a higher energy (years of acceleration) and lower distance (L) illustration of the Andromeda paradox, where, when choosing to use a CMIF coordinate system, the age of a distant object goes up and down as you accelerate back and forth. You seem to round your numbers by truncation rather than rounding to nearest.

But almost nobody makes the choice to use these accelerating CMIF frames. For instance, I've never found an astronomer or any physicist at all that suggests the age of stars and other objects in Andromeda wobbles back and forth over 750 years every day, and collectively lost well over 10000 years last month. The only times anyone chooses to use such a coordinate system are ones that find pragmatic utility in doing so, which are pretty much confined to discussions of the Andromeda paradox and other educational illustrations of how SR works. I've never seen such a method used for instance outside of a thought experiment.

And that's been my point, that real 'observers' tend to choose a coordinate system that yields the best pragmatic utility to the situation at hand. My example of the speeding ticket illustrated that, something to which you had no reply.

My point about the insufficiency of the TDE on its own stands.
This was confirmed by Mike's post. He needed to reach for a relativity of simultaneity equation to make that chart, the TDE not coming into play.

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Before I respond to Halc's last post, I need to correct an error in my above post.

I said: "The first half of the plot shows what happens when the he is accelerating in the direction TOWARD her".

That isn't correct. His acceleration TOWARDS her only lasts from when he is 26 until he is 27. (That's when the curve is increasing it's slope.) That's the only part of this diagram that is needed to show how his instantaneous turnaround in the standard twin paradox is a limiting case of a finite acceleration existing for a finite duration.

In this finite acceleration case, when he accelerates toward her (at 1 g), he says that results in her quickly ageing from 17 years old (when he is 26) to her being 49.1 years old (when he is 27). This is analogous to the simple instantaneous turnaround case, when he instantaneously changes his his velocity from 0.866 (away from her) to -0.866 (toward her), and concludes that her age instantaneously increases by 2*0.866*d, where "d" is their distance apart, according to her.

[...]
But almost nobody makes the choice to use these accelerating CMIF frames. For instance, I've never found an astronomer or any physicist at all that suggests the age of stars and other objects in Andromeda wobbles back and forth over 750 years every day
[...]

Brian Greene did that in his NOVA program several years ago ... it's still available (I give the link below). He picks an alien about as far from us as is possible, and has that alien riding a bicycle back and forth in a small circle, toward us and away from us. Because of that vast separation from us, the alien's motion causes the alien to conclude that the current time on earth is swinging back and forth over several hundred years.

https://www.pbs.org/wgbh/nova/video/the-fabric-of-the-cosmos-the-illusion-of-time/

Scan forward to the 21:39 point. The important stuff just lasts for several minutes past that point.

He picks an alien about as far from us as is possible, and has that alien riding a bicycle back and forth in a small circle, toward us and away from us.
Seen it quite some time ago. I approve of the unusual take that it does it from the alien perspective, not ours.
Brian chooses that simultaneity method precisely for the reasons I gave in my post: It serves Brian's pragmatic purpose of illustrating the Andromeda effect. It serves no pragmatic purpose to the alien at all, so there's no reason to suppose the alien on the bike is using such a method.

While you're confirming my points, you confirmed this one:
It seems you knew how to do kinematics back 25 years ago and lost it more recently.
when you corrected something that wasn't wrong, into something that was:
His acceleration TOWARDS her only lasts from when he is 26 until he is 27.
The labels on the picture correctly say that at age 27 he is stationary relative to the distant object, and at 28, his velocity relative to that inertial object has become negative (moving closer). Now you apparently suggest that going from stationary to 0.774c towards it constitutes accelerating away. This is not the first time you suggest that positive acceleration from stationary results in negative motion, but 24 years ago you would not have suggested that.

Anyway, I agreed with the picture and the numbers (before your 'correction'), despite the mix of values as measured from different frames. The slope of the lines really does decrease over time as coordinate acceleration (relative to the magenta inertial frame) goes down. That coordinate acceleration maxes out at velocity zero when proper and coordinate acceleration match.

My post #9 above is quite detailed. Feel free to point out an error in that post.
I already did, in posts #3 and #7. They were quite detailed, too.

Halc said:

"The labels on the picture correctly say that at age 27 he is stationary relative to the distant object,

I (Mike) say: True.

Halc said:

"and at 28, his velocity relative to that inertial object has become negative (moving closer)."

I (Mike) say: True, but his velocity has been negative (moving closer) ever since he was infinitesimally older than 27.

Halc said:

"Now you apparently suggest that going from stationary to 0.774c towards
(i.e., -0.774 with my convention), it constitutes accelerating away."

I (Mike) say: I don't know what you mean by "accelerating away". I'll elaborate, and maybe that will clarify things, and eliminate our misunderstandings. When he is 26, according to him, she is 17, they are 40 ly apart, and he is moving away from her at +0.774. Between his age of 26 to 27-, they are continuing to move apart, but he is slowing down (because he is accelerating in a direction OPPOSITE to his motion relative to her. Then, when he is 27, they are momentarily stationary and 40.5 ly apart (but he is still accelerating at -1g, so when he is 27+, he starts to move TOWARD her). By the time he is 28, she is 81.2, they are 40 ly apart, and moving at a relative speed of -0.774 (moving closer together). (That's the end of that story, according to our needs here).

The material below is older, but some of it may still be helpful:

Halc said:

"For instance, I've never found an astronomer or any physicist at all that suggests the age of stars and other objects in Andromeda wobbles back and forth over 750 years every day."

But then (after I gave the Brian Greene NOVA reference), Halc said:

"Seen it quite some time ago. I approve of the unusual take that it does it from the alien perspective, not ours."

Now, I (Mike) respond: Why does it matter whether it's our perspective or the alien's perspective, that time goes back and forth at a distance when someone (human or otherwise) accelerates, sometimes toward a distant person, and sometimes away from a distant person?

Then Halc says:

"Brian chooses that simultaneity method precisely for the reasons I gave in my post: It serves Brian's pragmatic purpose of illustrating the Andromeda effect. It serves no pragmatic purpose to the alien at all, so there's no reason to suppose the alien on the bike is using such a method."

I (Mike) respond:

What a bizarre statement! It is exactly the same point that Penrose was making in the Andromeda scenario: whenever someone (he) accelerates back and forth, he will conclude that a very faraway person's (her) age rapidly changes, both positively and negatively. And that is also what happens in the (expanded) Twin Paradox, with multiple velocity changes by the traveler (in opposite directions): (1) positive ageing by the home twin (she) (according to the accelerating twin (he)) when he accelerates toward her, and then (2) negative ageing by her when he accelerates in the direction away from her.

It's important to bear in mind that the equation you're referring to [the time dilation equation (TDE)] is just one special case of the much more important and more general Lorentz tranformation equations.

The original source of the TDE and the LCE is indeed the Lorentz equations. The Lorentz equations are used to derive the time dilation equation (TDE) and the length contraction equation (LCE). But once you have those, they are the simplest way to answer almost all questions in Special Relativity, and the Lorentz equations just muddy the water.

I said: "It (the TDE) can single-handedly resolve the Twin "Paradox"."

And you said:

No, it can't, because the equation only compares two inertial frames at a time. During the turnaround in the twin paradox, somebody's frame is non-inertial.

OK ... I'll explicitly SHOW how to do it!

Take the case where the traveling twin (he) leaves home, right after his birth, at a speed of o.866 ly/y, and gets to his turnaround point when he is 20. Everyone agrees about that. And everyone agrees that he ages another 20 years on the return trip, so everyone agrees that he is 40 when he returns (because his turnaround is instantaneous and he doesn't age at all during that). By using the TDE, he concludes that she ages 10 years on his outbound leg. And SHE uses the TDE to conclude that when he is 20 at the turnaround, she is 40. They disagree. They are BOTH right. That is just the way Special Relativity IS. Get used to it.

Then, he reverses course, and returns to her at the same speed (0.866 ly/y) as on his outbound leg. He gets 20 years older during his return leg (everyone agrees about that), and he says she ages 10 years on his return leg (she doesn't agree ... she says she gets 40 years older during his return leg). So he EXPECTS to find that she is 20 years old at their reunion. But SHE knows that he will be 40 years old at their reunion (THAT'S part of the scenario description, and she knows (from the time dilation equation) that she will be 80 years old then. They MUST agree about their respective ages at the reunion, because they are colocated and face-to-face there. He is surprised to see that she is 80 years old at the reunion, because the TDE told him that she would age half as fast as he does on each of his inertial legs. Where did he go wrong in his expectations? He THOUGHT she'd be 20 at the reunion, but he found her to be 80. When did that additional 60 years of ageing by her occur during his trip, according to him? It couldn't have occurred during EITHER of the two inertial legs of his trip, because the TDE rules those times with an iron fist. The ONLY place that extra 60 years of ageing by her (according to him) could have occurred is during his instantaneous turnaround ... there is no other possible alternative.

In the simple scenario above, there is only one non-inertial instant during his trip. That is what has allowed us to solve the "paradox" ... there is only ONE instant when she could have undergone that instantaneous ageing (according to him). But in more complicated scenarios, there can be multiple such times, and we need to be able to directly compute all of her instantaneous ageings that he says happen. That can be done using Minkowski diagrams, OR easier by using a very simple equation I derived long ago, called "the delta_CADO" equation. "delta_CADO" is the amount of the instantaneous ageing by her, according to him, when he instantaneously changes his velocity wrt her. If the separation between the two twins at some instant is "L" lightyears, and if the instantaneous change in his velocity at that instant is denoted delta_v, where

delta_v = v_after_velocity_change - v_before_velocity_change,

then

delta_CADO = -L * delta_v .

Velocities are POSITIVE when directed AWAY from her, and NEGATIVE when directed TOWARD her.

Very simple, and it works for arbitrarily many velocity changes by the traveler.

This is all well and good, but the TDE does not provide any reason for her aging 60 years during the turnaround. HE can deduce that something must have happened to allow HER to age 60 years while he was turning around, but he has no explanation of what that something might be, if he's only working with the TDE.

The REASON for her ageing instantaneously (either positively or negatively) when he instantaneously changes his velocity is that the TDE of Special Relativity DEMANDS it (when it is applied THREE times: twice by him [once for each of his inertial legs] and once by her [for her single inertial "leg", just staying at home]). That's the ONLY reason needed. From the SINGLE assumption that a light pulse in ANY inertial frame will ALWAYS travel at 186,000 miles per second, as measured in that frame by its occupants, EVERYTHING ELSE in Special Relativity follows!! So if that one assumption is NOT TRUE, then Special Relativity is not true, because that is the ONLY assumption that Special Relativity makes.