# The Time Dilation Equation (TDE) of Special Relativity

The traveling twin, upon reaching the turnaround point, can just as easily decide to put himself in her place instead of his own.

No, he CANNOT do that. On his entire outbound inertial leg, he MUST use the Time Dilation Equation (TDE), which tells him that she is ageing gamma times slower than he is. Otherwise, he is violating the fundamental assumption of Special Relativity, that any light pulse moves at 186,000 miles per second in any inertial reference frame, according to the people stationary in that inertial frame.

No, he CANNOT do that. On his entire outbound inertial leg, he MUST use the Time Dilation Equation (TDE), which tells him that she is ageing gamma times slower than he is. Otherwise, he is violating the fundamental assumption of Special Relativity, that any light pulse moves at 186,000 miles per second in any inertial reference frame, according to the people stationary in that inertial frame.

If the traveling twin explicitly uses the stay-home twin's reference frame, then he would also have to use her clocks and measuring sticks to measure the speed of light. In that case, he would conclude the speed of light is the same as what she would measure it to be.

Why would he WANT to use her frame? His measurements say that her clocks aren't synchronized. You seem to have an answer in search of a question.

Why would he WANT to use her frame? His measurements say that her clocks aren't synchronized. You seem to have an answer in search of a question.

In the case of Halc's reasoning, he prefers to use a frame where her age does not change back and forth if the distant twin decides to travel back and forth like the alien riding his bike. Personally, I have no problem with it.

Similarly, the stay-home twin could choose to say that her brother thinks her clock ticks slower than his on both legs of his journey. Then she would have to conclude that, according to him, her age does change during his turnaround. Note that she KNOWS her age does not change in that way. She simply makes a calculation that has no physical effect on her age whatsoever.

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Then she would have to conclude that, according to him, her age does change during his turnaround. Note that she KNOWS her age does not change in that way.

She would have no way to sense it if her age DOES change that way.

She would have no way to sense it if her age DOES change that way.

"if her age DOES change in that way"

SR says that information cannot travel faster than the speed of light, so her biological clock would have no way of knowing that her age was supposed to change (forward and backward!) every time her brother went back and forth like the alien does on his bicycle. Come on.

My point was that, at each instant of her (the home twin's) life, her brain is in some specific state. None of those states can be changed by what he (the traveling twin) does.

The fact that she AND he both conclude (via the time dilation equation) that whenever they themselves are inertial, they MUST conclude that the other twin is ageing gamma times slower than they themselves are. So SHE concludes that, during his entire trip, he is ageing at half her rate of ageing (when his speed relative to her is 0.866 ly/y). And HE concludes that SHE is ageing half as fast as he is, during the two inertial legs of his trip. But he finds at their reunion that she is twice as old as he is. The ONLY way he can explain that is to conclude that she instantaneously aged by the missing amount during his instantaneous turnaround. And that's the only way WE can explain it!

My point was that, at each instant of her (the home twin's) life, her brain is in some specific state. None of those states can be changed by what he (the traveling twin) does.

Right, so it is kind of silly to say that her age might be changing back and forth in her own inertial reference frame, but she might be unaware of it. And that is what you were implying when you said, "She would have no way to sense it if her age DOES change that way." Her age clearly doesn't change that way in her own reference frame!

The fact that she AND he both conclude (via the time dilation equation) that whenever they themselves are inertial, they MUST conclude that the other twin is ageing gamma times slower than they themselves are. So SHE concludes that, during his entire trip, he is ageing at half her rate of ageing (when his speed relative to her is 0.866 ly/y). And HE concludes that SHE is ageing half as fast as he is, during the two inertial legs of his trip. But he finds at their reunion that she is twice as old as he is. The ONLY way he can explain that is to conclude that she instantaneously aged by the missing amount during his instantaneous turnaround. And that's the only way WE can explain it!

Yes, if he uses the two different inertial frames in which he is stationary (outbound and inbound), he would have to conclude her age advances during the turnaround, just to fill in the missing time which he cannot explain without RoS. But if he understands SR, he could just use RoS to conclude that her age advances when he turns around and heads back toward her.

But there is no rule that says he must use those inertial frames in the first place. He could choose to use the inertial frame in which she is stationary, and then he would conclude she was older than him the whole time, the same way she does.

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I (Mike Fontenot) said:

My point was that, at each instant of her (the home twin's) life, her brain is in some specific state. None of those states can be changed by what he (the traveling twin) does.

Right, so it is kind of silly to say that her age might be changing back and forth in her own inertial reference frame

I never said that. Her age increases instantaneously by a large amount during the traveling twin's (his) instantaneous turnaround, ACCORDING TO HIM, NOT ACCORDING TO HER.

[...] if he uses the two different inertial frames in which he is stationary (outbound and inbound), [...]

He has ONE reference frame during his trip: during his outbound leg, he concludes she is ageing gamma times slower than he is. Then, during his instantaneous turnaround, he concludes that she instantaneously gets MUCH older. And finally, during his inbound leg, he again concludes she is ageing gamma times slower than he is. The amount of her sudden ageing (according to him) during his turnaround is exactly what is necessary so that they will agree about their respective ages at their reunion.

[...] he would have to conclude her age advances during the turnaround, just to fill in the missing time which he cannot explain without RoS. But if he understand SR, he could just use RoS to conclude that her age advances when he turns around.

There IS a simple equation that he can use to compute the amount of her sudden ageing during his instantaneous turnaround. It's not necessary for the simple scenario with only a single turnaround by the traveling twin (him) ... i that case, it can just be inferred from the fact that the two twins MUST agree at the reunion about their respective ages. But if he makes MULTIPLE instantaneous velocity changes during his trip, it is necessary to use a simple equation for each velocity change (or, alternatively, it can be determined graphically). Here is the simple equation. First, we compute the his velocity change delta_v:

delta_v = v2 - v1,

where v2 is his velocity AFTER his instantaneous velocity change (negative when directed TOWARD her), and v1 is his velocity BEFORE his instantaneous velocity change (positive when directed AWAY FROM her):

Then, in the present example, with

v1 = 0.866 ly/y

and

v2 = -0.866 ly/y,

we compute

delta_v = (-0.866) - (0.866) = -1.732 ly/y.

And then, the instantaneous age change by the home twin (her), according to the traveling twin (him), is

delta_CADO = ( -D ) ( delta_v ) = ( -34.64 ) ( -1.732 ) = 60.0 years,

when their separation (according to her) is 34.64 ly.

(The acronym "CADO" just stands for "Current Age of the Distant Object", where the "Distant Object" is the home twin).

So the delta_CADO equation says that she instantaneously gets 60 years older, according to him, when he instantaneously changes his velocity wrt her from 0.866 ly/y (going AWAY from her) to -0.866 ly/y (going TOWARD her). Very easy!

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Then, in his previous post, Neddy said:

But there is no rule that says he must use those inertial frames in the first place. He could choose to use the inertial frame in which she is stationary, and then he would conclude she was older than him the whole time, the same way she does.

No! He can't "choose" to do that. She has HER perspective, and he has HIS perspective. He can DETERMINE her perspective, but that's NOT his perspective. He HAS his perspective ... it ISN'T a choice he makes. That's easiest to understand when analyzing his outbound inertial frame. You can imagine that actual people (including HIM) are RESIDENT in that inertial frame. They can set up the reference frame by using the fact that any light pulse in that frame travels at 186,000 miles per second. That allows them to synchronize all the clocks in that frame, using light pulses and yardsticks. So whenever he is inertial, he has no choice but to use those synchronized clocks to determine the current age of the distance person (the home twin). One of those synchronized clocks will be momentarily co-located with the home twin, at each instant of his life during his outbound inertial trip (and likewise during his inbound inertial trip). So her current age, when he is age "T", is just what the particular clock, in his array of clocks, that happens to be momentarily co-located with her, reads "T".

He has ONE reference frame during his trip: during his outbound leg, he concludes she is ageing gamma times slower than he is. Then, during his instantaneous turnaround, he concludes that she instantaneously gets MUCH older. And finally, during his inbound leg, he again concludes she is ageing gamma times slower than he is. The amount of her sudden ageing (according to him) during his turnaround is exactly what is necessary so that they will agree about their respective ages at their reunion.

That is 2 different inertial frames, plus one acceleration. The outbound inertial frame is not the same as the inbound inertial frame. In the basic twin scenario only the velocity signs are different, but that still makes for two different inertial frames.

Imagine two extremely long trains, one heading "east" relative to the home twin, and the other heading "west" relative to her. The traveling twin can be imagined to jump on one train, ride it for awhile, then jump to the other train and ride it back to meet his sister again. The two trains are two different inertial frames. The jump from one train to the other was an acceleration.

That's easiest to understand when analyzing his outbound inertial frame. You can imagine that actual people (including HIM) are RESIDENT in that inertial frame. They can set up the reference frame by using the fact that any light pulse in that frame travels at 186,000 miles per second. That allows them to synchronize all the clocks in that frame, using light pulses and yardsticks. So whenever he is inertial, he has no choice but to use those synchronized clocks to determine the current age of the distance person (the home twin). One of those synchronized clocks will be momentarily co-located with the home twin, at each instant of his life during his outbound inertial trip (and likewise during his inbound inertial trip). So her current age, when he is age "T", is just what the particular clock, in his array of clocks, that happens to be momentarily co-located with her, reads "T".

That further illustrates that the outbound inertial frame is different than the inbound inertial frame. Each of my trains would say the array of clocks on the "other" train is not synchronised.

No! He can't "choose" to do that. She has HER perspective, and he has HIS perspective. He can DETERMINE her perspective, but that's NOT his perspective. He HAS his perspective ... it ISN'T a choice he makes.

I know what you mean, but it makes no physical difference. You might consider yourself to be the traveling twin, and I might consider myself to be the stay home twin, and we make different calculations. Then I want to see what it's like to be the traveling twin, so I make the same calculation you made. Meanwhile, neither of us was ever moving at a velocity of v=0.866c relative to one another, it's all calculations anyway.

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Neddy said:

"But there is no rule that says he must use those inertial frames in the first place. He could choose to use the inertial frame in which she is stationary, and then he would conclude she was older than him the whole time, the same way she does."

I (Mike Fontenot) respond:

One doesn't CHOOSE to "use" a frame. Your frame is determined by your actions, and you are stuck with it. She CHOSE to remain at home, where she and her twin were born. He CHOSE to instantaneously change his speed so that he moved away from her right after they were born, at a constant speed of 0.866 ly/y. As soon as he instantaneously changed his speed relative to her, he was stationary in a different inertial frame from hers. At that point, they each have a (different) perspective, and they're stuck with it as long as they don't make any other changes in speed. Specifically, she says that he is ageing gamma times slower than she is. He says that she is ageing gamma times slower than he is. And the amazing thing is, THEY ARE BOTH RIGHT. That's just the way Special Relativity IS. The point is, once he chose to move away from her at that constant speed, he couldn't "choose to use her frame" ... they are each stuck with the frame that they chose.

The home twin (she) IS stationary in her inertial reference frame. She didn't JOIN it. Just by HER being inertial, SHE is automatically stationary in that inertial frame, and it tells HER that HE is ageing more slowly than SHE is, by the factor gamma.

Likewise, The traveling twin (he) IS stationary in his inertial reference frame. He didn't JOIN it. Just by HIS being inertial, HE is automatically stationary in that inertial frame, and it tells HIM that SHE is ageing more slowly than He is, by the factor gamma.

The point is, NEITHER of them CHOSE their perspective AFTER they made their movement choices ... they were GIVEN that perspective merely because of the way they CHOSE to move or not move ... i.e., because of WHICH inertial frame they chose to be stationary in. After he decides to move away from her at constant speed, he CAN'T "choose to use her frame". She says he is ageing slower than she is. He says she is ageing slower than he is.

That is 2 different inertial frames, plus one acceleration.

I view it as ONE frame ... HIS frame during his entire trip. His frame isn't her frame, and vice-versa. Once you specify the scenario, he will never "be in her frame" ... they each have their own frame.

The significance of that is easiest to see by thinking about her inertial frame and his inertial (outbound) frame. They each say the other is ageing more slowly than they themselves are. And the light pulses in those two frames GUARANTEE that each of them is right. I.e., HE can't be wrong, when he says she is ageing more slowly than he himself is: if he were wrong, then the speed of a light pulse in his frame COULDN'T be 186,000 miles per second. And if THAT were true, Special Relativity would be wrong. And she can make the same argument, with the roles reversed. I don't believe that Special Relativity is wrong. So they are each right that they are ageing faster during their non-accelerated state than their twin, strange as that sounds.

HE can't be wrong, when he says she is ageing more slowly than he himself is: if he were wrong, then the speed of a light pulse in his frame COULDN'T be 186,000 miles per second.

If he chooses to use the clocks and rods which are stationary with respect to her, rather than the ones which are stationary with respect to him, the speed of light is still 186,000 miles per second by those measurements. Above you claim otherwise. Would you like to try calculating it?

If you agree to light speed still being 186,000 miles per second in the above case, then you could instead argue that her clocks are out of synch and her measuring rods are length-contracted in the traveling twin's rest frame. That would be true and I would agree it is a good reason for him to choose his own clocks and rods instead of hers. But it still doesn't mean he cannot choose to use hers anyway.

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I view it as ONE frame ... HIS frame during his entire trip.

I'm actually more concerned about this than what we are currently discussing. If I accept your idea that everyone should use the clocks and measuring rods that are stationary with respect to them, that means that the traveling twin has to use a different set of clocks and rods on the inbound leg of the journey than the ones he used on the outbound leg. Surely that should lead to you saying they are two different reference frames, right?

Once you accept that the outbound and inbound legs of his journey are two different inertial frames, it should become clear to you that his declaration that her age increases during the turnaround is due to simultaneity in the inbound frame being so different to the simultaneity in the outbound frame. You make it sound like he can only explain her age advancing by saying it has to do so in order for the numbers to add up at the end of the journey, and saying the whole journey was one inertial frame is just bonkers.

You make it sound like he can only explain her age advancing by saying it has to do so in order for the numbers to add up at the end of the journey, and saying the whole journey was one inertial frame is just bonkers.

I never said his whole journey was one inertial frame, I said his whole journey was HIS frame. And that frame consisted of an outbound inertial leg, followed by an instantaneous velocity change, followed by an inbound inertial leg.

If he chooses to use the clocks and rods which are stationary with respect to her, rather than the ones which are stationary with respect to him, the speed of light is still 186,000 miles per second by those measurements. Above you claim otherwise. Would you like to try calculating it?

He and his fellow inertial people (I'll refer to them later as his "helper friends") knew that any light pulses in their inertial frame moves at 186,000 miles per second. They USED those light pulses (and their yardsticks) to synchronize all their clocks. Once they had done that, they can communicate among each other, and conclude that the clocks stationary in the home twin's (her) frame, which are flying by them, are NOT synchronized. Specifically, one of his helper friends happens to be momentarily co-located with her at some instant, and that helper friend can SEE that her clock has a different reading than the other clocks stationary in her frame. (Those other clocks stationary in her frame have also been examined by other of his helper friends, and found to be unsynchronized).

If you agree to light speed still being 186,000 miles per second in the above case, then you could instead argue that her clocks are out of synch and her measuring rods are length-contracted in the traveling twin's rest frame. That would be true and I would agree it is a good reason for him to choose his own clocks and rods instead of hers. But it still doesn't mean he cannot choose to use hers anyway.

He doesn't choose to use them, because he KNOWS they are unsynchronized ... he'd be an idiot to use them.

All of the arguments I've made above could be made in support of the home twin's inertial frame. She will come to the same conclusions he came to, except as regards HER inertial frame. She'll say she and HER helper friends have done (on their frame) all the the work that the traveler (he) and his helper friends did on their frame. She'll say HER clocks are all synchronized, and it is HIS clocks that are unsynchronized. AND THEY WILL BOTH BE RIGHT IN THEIR STATEMENTS! That sounds nutty, BUT THAT IS JUST THE WAY SPECIAL RELATIVITY IS!

Been a while since I've chimed in.

The traveling twin, upon reaching the turnaround point, can just as easily decide to put himself in her place instead of his own.
It seems rather pointless to choose one frame (presumably the one in which he is stationary) on the outbound leg and then switch to her frame for the return journey. It still requires doing a frame rotation (Lorentz transform), an added complication on top of what is required if one just sticks to a single inertial frame the entire time.
I agree with the statement though. It would be 'just as easy' (and no more or less wrong) to do that as it would to do it Mike's way. My point was that it was easier (and no more correct) if the choice was not to do frame switches at all.

On his entire outbound inertial leg, he MUST use the Time Dilation Equation (TDE)
He would be using the TDE regardless of the arbitrary frame of choice. Nobody says otherwise.

the fundamental assumption of Special Relativity, that any light pulse moves at 186,000 miles per second in any inertial reference frame, according to the people stationary in that inertial frame.
The premise is more accurately stated that "any light pulse moves at 186,000 miles per second in any inertial reference frame". Anybody in any frame, stationary or not, inertial or not, should not be disputing that if they're accepting SR. There are very much those that dispute it, and that would result in a different theory, no less valid.
Yes, the TDE follows from the two premises of SR, and does not follow if the premises are denied.

Why would he WANT to use her frame?
I gave examples where one would very much want to use a different frame. You seemed to agree with them by your lack of response concerning why those examples were wrong, and that the observer in question is obligated to use only the one frame rather than the one I suggested.

In the case of Halc's reasoning, he prefers to use a frame where her age does not change back and forth if the distant twin decides to travel back and forth like the alien riding his bike.
Agree, and since any inertial frame has this desirable property, any arbitrary choice of one of them will do.

Then she would have to conclude that, according to him, her age does change during his turnaround. Note that she KNOWS her age does not change in that way.
I agree with Mike here. If she knows that Bob is using Mike's method, then she would know that her age does change in that way, in the same way that I can know that my age, according the bicycling twin who is using Mike's method, is currently wildly swinging from way-before birth to way-after-death.
It's nothing but an abstraction in the alien's mind after all, nothing physical. Physically, there's just my worldline which has no actual 'current age' unless you're a presentist, and a presentist interpretation is incompatible with Mike's assertions.

My point was that, at each instant of her (the home twin's) life, her brain is in some specific state. None of those states can be changed by what he (the traveling twin) does.
An 'instant' is a frame dependent thing. At each event (not a frame dependent thing), anything at that event is in some specific state. Brains are not treated different than any other state of matter. So agreeing, with qualifications. You seem to treat brains and conclusions as something privileged, when in fact SR is not a theory about brains.

And that's the only way WE can explain it!
This is of course a crock. There are plenty of explanations of the twins scenario that don't do it that way.

My point was that, at each instant of her (the home twin's) life, her brain is in some specific state. None of those states can be changed by what he (the traveling twin) does.
Don't understand this assertion. I can say to my sister (while pacing back and forth) that there's a spider in her hair, and that will very much change her brain state, albeit somewhat in the future of the making of that statement since it takes time for my vocalization to reach her. If you're talking about instant (faster than light) causation, well, you seem to assert that as well in your topics, so you can't have it both ways and deny it here.

One doesn't CHOOSE to "use" a frame
That's what's under contention. We say one can. If you deny it, you need to specify how a different choice violates SR, else it's just you asserting something you can't back.

All of the arguments I've made above could be made in support of the home twin's inertial frame. She will come to the same conclusions he came to, except as regards HER inertial frame. She'll say she and HER helper friends have done (on their frame) all the the work that the traveler (he) and his helper friends did on their frame. She'll say HER clocks are all synchronized, and it is HIS clocks that are unsynchronized. AND THEY WILL BOTH BE RIGHT IN THEIR STATEMENTS! That sounds nutty, BUT THAT IS JUST THE WAY SPECIAL RELATIVITY IS!

Right, they are both correct. When the traveling twin is 20 years old, the TDE tells him that his sister is currently 10 years old. However, he is also passing very close to one of the clocks in her synchronous array and that clock displays t=40 at the same time that he is 20 years old. So he could easily put himself in her position and say she is currently 40 years old instead of 10. It makes no difference, and it is not wrong as long as he states that he is using the reference frame in which she is stationary. In that frame she is 2x older than him throughout the entire journey, (not for the case where he stops and waits awhile at the turnaround point, but for the simpler case where he keeps moving the entire time).

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So he could easily put himself in her position and say she is currently 40 years old instead of 10.
[...]

The only way he can "put himself in her position" is to change his velocity and become stationary wrt her. And he doesn't do that, after his original departure.

As long as he remains stationary in HIS inertial reference frame (the one that is moving away from her at 0.866 ly/y), he MUST accept the fact that the clocks in his inertial frame were synchronized purely by using the assumption that the speed of a light pulse is 186,000 miles per second. So if his clocks aren't synchronized (which they CAN'T be, if he agrees with her), the speed of a light pulse in his frame CAN'T be 186,000 miles per second. And if the speed of a light pulse in his frame ISN'T 186,000 miles per second, then Special Relativity ISN'T correct. I believe Special Relativity IS correct. That means that the current age of his home twin (according to HIM) is what his "helper friend" (who is stationary in his frame [and who therefore shares his "NOW" moment] and who happens to be momentarily co-located wrt her at the instant that he wants to know her current age) says it is. There is just no "wiggle-room" here.