# The Time Dilation Equation (TDE) of Special Relativity

From post 18:
I'll elaborate, and maybe that will clarify things, and eliminate our misunderstandings.
...
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)
OK, it is this last statement that seems to contradict what you said in post 14:
His acceleration TOWARDS her only lasts from when he is 26 until he is 27.
You say now (post 18) that he is still accelerating at -1g, so when he is 27+ ...
I agree with that bit, but the post 14 comment says that this negative (proper -1g) only lasts until he is 27, which indicates that either he stops accelerating altogether at 27, or he accelerates in a non-negative direction, both of which contradict the picture and your post 18.

All that is pretty obvious just from the picture. Her age, relative to his CMIF, continues to increase as long as his acceleration is negative (that is, towards the distant inertial object).

Why does it matter whether it's our perspective or the alien's perspective
We don't have a perspective. We don't have a worldline or a current position in any thought experiment. We're the narrator, and it is the pragmatic purpose of the narrator (Mike) which is being served by the narrator's choice of frame. I've said this repeatedly, but y9u keep asking.

that time goes back and forth at a distance when someone (human or otherwise) accelerates
Acceleration is not required at all. It is after all an abstract choice, a mental task, not a physical one. All one has to do is keep choosing different frames if that's what you want accomplished.

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.
Yes, Penrose (another narrator, not anybody actually accelerating) has the same pragmatic purpose as does Green and you. But somebody pacing back and forth typically has no reason to do a continuous frame change like that, as was also illustrated by my real world examples which you continue to ignore.

And no, special relativity does not demand that you performs such frame changes. I've given far simpler explanations of the twins thing which do not involve frame rotations at all, nor does it depend on what any of the participants conclude since their role is simply that of a meat-clock.

The original source of the TDE and the LCE is indeed the Lorentz equations.
Both TDE and LCE equations have a more general form which you should be using, but don't. The crazy results you get from using the simplified version should clue you in on that, but instead you just dig your hole deeper.

In the more general form (but still for 1D motion only along the one axis along which the contraction is being measured, is something like:

L = x=0-L L0/(γ(vx))

In other words, if the nonzero velocity of the object is not everywhere identical (which it cannot be for an accelerating object) at the moment in question in the frame of choice, then one has to integrate the contraction over its length. You're not doing that, and thus getting all the results that you cannot bring your self to admit are absurd .

From post 18:
OK, it is this last statement that seems to contradict what you said in post 14:

Yes, you are right that in my post #14 I said:

"His acceleration TOWARDS her only lasts from when he is 26 until he is 27."

That is incorrect. Sorry, I missed that mistake I made. Thanks for pointing it out. He continues to accelerate towards her until he is 28, which is the end of the part of the diagram that we are interested in.

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"Why does it matter whether it's our perspective or the alien's perspective"

And you (Halc) said:

"We don't have a perspective. We don't have a worldline or a current position in any thought experiment. We're the narrator, ...

When I used the phrase "our perspective", I merely meant that I am regarding myself as the person on Earth who is asking the question: "How old is that alien on that VERY distant planet in that VERY distant constellation?" as I ride a bicycle, toward and away from him, instead of the alien doing the back and forth motion, and asking the equivalent question about us on Earth. Brian Greene chose the alien to be the one who accelerates (and asks that question), rather than someone on Earth, for the scenario, because he knew we would be familiar with those historical events that were passing back and forth very quickly on Earth, spanning several hundred years, whereas we don't know anything about the alien's history (or his present).

Mike;
their own experimental measurements CONFIRM the correctness of the TDE.
...
SHE instantaneously gets older during HIS instantaneous turnaround.

This analysis is based on the initial conditions given, triangular spacetime path, 2 observers, and relative velocity of .866.

You don't mention any measurements to verify anything.

Since she remains in a constant velocity frame, she is the ref. frame (no td or lc).
Her clock emits time encoded light signals at a constant rate of 1per yr.
His speed is .866 relative to her. Begin in fig.1.
1.
Doppler effect is the observers perception that a frequency changes, when there is a relative velocity between the source and detector.
He receives 5 signals outbound in his 1st 20 yr, but knows that is a doppler effect.
He changes direction in zero time, his clock rate remains constant.
He receives 75 signals inbound in his 2nd 20 yr, but knows that is a doppler effect. He counts 80 yr for her vs 40 yrs for him.
2.
The clocks can be designed to respond to an incoming light signal with a time encoded signal emission.
If he made measurements out bound for td purposes, the last one would be at 1.4. He assigns (1.4+20)/2=10.7 as simultaneous with her 5.4 and a td of .50.
If he had to establish his green Bx axis (aka axis of simultaneity), he would have to continue outbound for another 20 yr to receive the return signal (?). He is denied that information when he reverses. The inbound trip is symmetrical with the outbound trip. He has no history of her from A10 to A70.
3.
If he made additional measurements around 10.7, he could establish her t=40 as simultaneous with his t=20. He thus concludes her clock had 40 annual events while his clock had 20. There is no missing time for her. At his t=29.3 he can predict her clock reads 80 and his reads 40 at reunion, on the condition that nothing changes for the remainder of his trip
His description of events is in fig.2, based on his measurements, not your imaginings.
She moves away at .866 until At=5.4, and returns at .866 at At=74.6
She remains at a constant distance of 9.28 from At=5.4 to At=74.6.
You can disagree with it, but know that it's the result of an imaginary/illogical experiment.

[...]
Brian Greene chose the alien to be the one who accelerates (and asks that question), rather than someone on Earth, for the scenario, because he knew we would be familiar with those historical events that were passing back and forth very quickly on Earth, spanning several hundred years, whereas we don't know anything about the alien's history (or his present).

Even though we can't know anything about an alien on that planet (or even if there IS such a planet there, RIGHT NOW), we DO know that if we alternately move back and forth in that direction, at bicycle speed, then the current time there, ACCORDING TO US, is changing (alternately increasing toward the future, and decreasing toward the past) by several hundred years, for each of our cycles. Special Relativity DEMANDS that.

Brian Greene chose the alien to be the one who accelerates (and asks that question), rather than someone on Earth, for the scenario, because he knew we would be familiar with those historical events that were passing back and forth very quickly on Earth, spanning several hundred years, whereas we don't know anything about the alien's history (or his present).
So assuming we did know all the relevant events, including the alien going back and forth on the bicycle, the question to ask seems to be, according to the alien, when does the Earth clock read Nov 7, 2023, noon, GMT? I'm asking at what time it was noon today according to the watch worn by the alien. Assume a long-lived alien that lives for centuries, and that his watch always shows his time elapsed since birth.

I ask this because the method I used to explain the twins thing can easily answer the question, but yours seems incapable.

Suppose nobody is in the ship that goes out and back, only a clock that accumulates 40 years compared to one back home that accumulates 80 years. How do you explain the discrepancy? The difference is, there is nobody on the rocket who 'MUST CONCLUDE' anything. Clocks don't make conclusions. There's nobody back home concluding anything either. Just the clocks. If the explanation requires a passenger that must conclude something, the explanation fails to explain things like GPS which doesn't have anybody along up there to do any concluding.

Straight from SR: It's frame dependent, and one frame is as valid as any other, so just pick any well behaved coordinate system, and stick with it. By well-behaved, I mean every event in spacetime has one and only one coordinate.
Picking a coordinate system where our Nov 7 event coexists with some part of the alien worldline would be a better choice than otherwise, relevant since I don't think our alien expects to live 20 billion years.
The coordinate system being well-behaved, there can be but one event on the alien worldline that has the same time coordinate as the Nov 7 event I specified, so the time on the alien watch at that event is the answer.

It's why I asked the question the way I did, to illustrate that your choice (the CMIF of a non-inertial alien worldline) isn't a well behaved coordinate system since it assigns more than one valid coordinate to distant events. There is no unique answer, so no answer that is more correct than the others. I find that to be a deficiency in your methods. Another deficiency is your dependency on the alien 'concluding' something, another detail which SR does not mention in any way. Your methods don't work for muons, neutrinos, and other objects which don't have HF's or other concluders along for the ride. (I think I have a new favorite word!)

Brian Greene [...] picks an alien about as far from us as is possible
This is special relativity we (Greene in particular) are talking about here. There is no concept of 'as far as possible' in SR. Not in GR either for that matter, but the inertial-frame trick that Greene is using doesn't work for a real accelerating object 10 GLY away.

I still stand by this statement I made previously:

"Even though we can't know anything about an alien on that planet (or even if there IS such a planet there, RIGHT NOW), we DO know that if we alternately move back and forth in that direction, at bicycle speed, then the current time there, ACCORDING TO US, is changing (alternately increasing toward the future, and decreasing toward the past) by several hundred years, for each of our cycles. Special Relativity DEMANDS that."

In the above quote, "we" is me, and Brian Greene, Richard Muller, Sean Carroll, and several other well-known physicists who participated in that NOVA show. Feel free to believe otherwise, though. It's pretty obvious that you and I will never agree about much of anything. Let's quit wasting our time.

I can agree that it is customary in variations of the andromeda scenario to place ones self as one of the parties involved, as an aid to the language used in the descriptions. Hence the usage of the word 'we' in your post.

Even though we can't know anything about an alien on that planet (or even if there IS such a planet there, RIGHT NOW),"
But we can because 'we're are also the narrator of the thought experiment, and we put the alien out there in some fictional SR universe, so 'we' know about both parties. Sure, the two parties in the fictional universe don't know about each other since light takes 10 billion years (in at least one frame) to go from one to the other, so I do agree with the statement in the way it was intended.

we DO know that if we alternately move back and forth in that direction, at bicycle speed, then the current time there, ACCORDING TO US, is changing (alternately increasing toward the future and decreasing toward the past)
Assuming that 'ACCORDING TO US' is equivalent to 'relative to our CMIF', then I agree. The method (not be well behaved) is still incapable of answering my question, and you seem to admit that. A CMIF is also completely useless in the real universe at such distances, but we're not talking about an alien in the real universe. It would be interesting to attempt to apply the concept of the movement of either party affecting the current state of the other in a real-universe scenario.

I can agree that it is customary in variations of the andromeda scenario to place ones self as one of the parties involved, as an aid to the language used in the descriptions. Hence the usage of the word 'we' in your post.

But we can because 'we're are also the narrator of the thought experiment, and we put the alien out there in some fictional SR universe, so 'we' know about both parties. Sure, the two parties in the fictional universe don't know about each other since light takes 10 billion years (in at least one frame) to go from one to the other, so I do agree with the statement in the way it was intended.

Assuming that 'ACCORDING TO US' is equivalent to 'relative to our CMIF', then I agree. The method (not be well behaved) is still incapable of answering my question, and you seem to admit that. A CMIF is also completely useless in the real universe at such distances, but we're not talking about an alien in the real universe. It would be interesting to attempt to apply the concept of the movement of either party affecting the current state of the other in a real-universe scenario.

Can you repeat your question, referred to above in the sentence that I've highlighted in red?

Can you repeat your question, referred to above in the sentence that I've highlighted in red?
the question to ask seems to be, according to the alien, when does the Earth clock read Nov 7, 2023, noon, GMT? I'm asking at what time it was noon today according to the watch worn by the alien.

Short story, in the SR universe, when does the the non-inertial alien consider our now to be his now?
I mean, each time he makes a circle, the time here varies somewhere say from about 1950 when he's bicycling away to about 2150 when he's cycling towards us. That 'wavering spotlight' so to speak crosses 2023 about twice every 10 of his seconds, making for an awful lot of 'correct' answers. The lack of one of them being 'the' correct one is the issue being highlighted.

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I'll respond to your above post shortly (I didn't realize until just now that you had posted something new), but for the moment, I want to discuss something else:

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Halc, you are confused. Allow me to explain.

In the scenario under discussion, you and I together are cast as "the home twin" in the twin paradox, who is permanently inertial, and considered to be stationary in the scenario. Therefore, we consider the alien to be like the traveling twin in the twin paradox (except that he has never been co-located with us, and never will be). Like the traveling twin, he is changing his speed with respect to us, but he is doing it repeatedly, back and forth. When we apply the time dilation equation (TDE) to him (which we can do, because we are ALWAYS inertial), we conclude that he is ALWAYS ageing more slowly than we are, except in those isolated instants where his velocity wrt us is exactly zero. In those instants, we conclude that his and our ageing rates are momentarily equal (but that only happens at isolated instants, and the ageing rates are finite, so it doesn't change our ages or his age). Because his velocity (when he IS moving) is always so small (just bicycle speed, about 10 miles per hour), the gamma factor is only a tiny bit larger than 1.0 (with MANY zeros after the decimal point before the first non-zero digit), and so his ageing is only VERY slightly less than ours then. Hence, for your question "when are our and his NOW moments the same?" (i.e., when are ours and the alien's ageing rates the same, according to US), the answer is "ONLY when he is momentarily stationary wrt us". At all other times, he is ageing a VERY tiny bit slower than we are.

Short story, in the SR universe, when does the the non-inertial alien consider our now to be his now?

I would word that differently. I would say "at each instant in his life, he can determine the date and time here at that instant in his life". To do that, he only needs to know how far away we are at that instant, and what our velocity relative to him is.

Halc said:
I mean, each time he makes a circle, the time here varies somewhere say from about 1950 when he's bicycling away to about 2150 when he's cycling towards us. That 'wavering spotlight' so to speak crosses 2023 about twice every 10 of his seconds, making for an awful lot of 'correct' answers. The lack of one of them being 'the' correct one is the issue being highlighted.

They are ALL correct!

Mike;
You will never reach your destination.

In the scenario under discussion, you and I together are cast as "the home twin" in the twin paradox, who is permanently inertial, and considered to be stationary in the scenario. Therefore, we consider the alien to be like the traveling twin in the twin paradox (except that he has never been co-located with us, and never will be). Like the traveling twin, he is changing his speed with respect to us, but he is doing it repeatedly, back and forth. When we apply the time dilation equation (TDE) to him (which we can do, because we are ALWAYS inertial), we conclude that he is ALWAYS ageing more slowly than we are, except in those isolated instants where his velocity wrt us is exactly zero.
Which we cannot do, because we (the ones cast as the home twin) cannot measure the alien at all, so we can conclude nothing of the sort.
Still the TDE is not a function of distance, so my son pacing in the next room counts. We can apply the TDE to him and yes, conclude (objectively even) that he's aging more slowly than sedentary us.
Hence, for your question "when are our and his NOW moments the same?"
I never asked that question. I wouldn't even know what such a question asks since all moments in both our respective worldlines are 'now' moments for both of us .

Short story, in the SR universe, when does the the non-inertial alien consider [one specific moment here] to be his now?
I would word that differently. I would say "at each instant in his life, he can determine the date and time here at that instant in his life"
Again, you changed my question to a completely different one.

They are ALL correct!
That answer seems to actually be a reply to what I asked. Yes, they're all correct. The specific event I selected has thousands (millions likely) of valid and quite different coordinates according to the alien, different times, different x,y,z even. It illustrates what I've been talking about, which is a coordinate system that does that vs one that assigns a single set of coordinates to any given event.

Mike: there is only ONE correct answer to that question, for ANYONE stationary in the GIVEN inertial frame.
My reply: This is equivalent to the cop pulling you over and asking if you know how fast you were going. By your assertion, there is only ONE correct answer to that question, which is: "I was stationary officer. It was the road going too fast. Give the ticket to the road."

Do you agree? Are you indeed not guilty of exceeding the speed limit?

You may not think much of my credentials, but here's a list of the physicists who signed on to Brian Greene's Nova TV show about the alien riding back and forth on a bike, on a planet 10 billion lightyears from Earth:

James Levin
David Albert
Alan Guth
Max Tegmark
Peter Galison
William Phillips
Steve Jefferts
S. James Gates, Jr
Sean Carroll
David Kaiser

Those people are professors at our most prestigious universities (and/or at NIST, the National Institute of Science and Technology in Boulder, Colorado). So go argue with them, not me.

You may not think much of my credentials, but here's a list of the physicists who signed on to Brian Greene's Nova TV show about the alien riding back and forth on a bike, on a planet 10 billion lightyears from Earth:
The Nova show illustrates the Andromeda effect, and I have no problem with any of what they say, unless they say that one is obligated to use such a coordinate system regardless of the situation at hand. Your argument from authority falls flat.
If you have a quote from the Nova programs that makes such an assertion, then you can ask Greene et al the same question, which you have still evaded.

What do you say when asked how fast you were driving? What do you actually conclude about your driving speed (asked that way just in case you choose to lie to the cop). What would Greene say if asked a similar question? Are you so afraid of such a simple question that you have to hide behind all these names from prestigious universities? (I only recognized two).
I noticed the other topic also ground to a halt when the questions got too simple.

Was Copernicus wrong when he said the sun doesn't revolve around the earth? Sure, Copernicus was not familiar with special relativity theory, but if he was, would he be obligated to conclude that he was wrong way back in the day and in fact it is the sun that is doing the motion, and not around the earth either, but around Copernicus to be exact.

[...]
The Nova show illustrates the Andromeda effect, and I have no problem with any of what they say, unless they say that one is obligated to use such a coordinate system regardless of the situation at hand.

You ARE obligated to conclude that if the alien suddenly changes his velocity in the direction toward us, he MUST conclude that we here on Earth suddenly get older (by the product of the distance from us to him [according to us] multiplied by the amount of the velocity change). The reason that is required is that it is essentially the same situation as in the standard twin paradox, and in that case, if the traveling twin DOESN'T conclude that his home twin instantaneously gets older when he reverses course, he won't be able to explain his home twin's age at their reunion.

You ARE obligated to conclude that if the alien suddenly changes his velocity in the direction toward us, he MUST conclude that we here on Earth suddenly get older (by the product of the distance from us to him [according to us] multiplied by the amount of the velocity change). The reason that is required is that it is essentially the same situation as in the standard twin paradox, and in that case, if the traveling twin DOESN'T conclude that his home twin instantaneously gets older when he reverses course, he won't be able to explain his home twin's age at their reunion.

The traveling twin, upon reaching the turnaround point, can just as easily decide to put himself in her place instead of his own. In that case, he would say the same thing she says. Namely, that she was already older than he was before he reversed course, and that neither of their ages has to change during the turnaround. But he should explicitly say that he is using her inertial frame of reference for the whole exercise, rather than using either of the two inertial frames of reference that he occupies himself during his outbound and inbound journey.

In practice, the alien would find this much more difficult to do. But theoretically there could be a clock located next to the alien, stationary with respect to the earth, and synchronised to the earth clock. In that case, the alien could put himself in the earth's place by saying the current time on earth "now" is whatever time is shown on that clock, regardless of the alien's movement back and forth relative to that clock. This is what phyti does not seem to like, the idea that synchronised clocks lead to "instant knowledge" of what time it is some distance away, even in the purely SR case where there is no gravity or expanding universe, etc.