A question about the equivalence principle.

yuiop

It does track. See the attached chart showing two rockets with Born rigid acceleration. Each time signals (black diagonal lines) are emitted simultaneously in one MCIRF, they are received simultaneously in a subsequent MCIRF. In the chart green curves represent equal proper time for the rocket clocks, red lines represent lines of equal velocity relative to the LF (or subsequent MCIRFs) and the blue lines represent the worldlines of the rockets in the LF. Note the constant nature of the redshift. Signals sent every 3 ticks from the rear rocket are received every 4 ticks by the leading rocket.
You have the reasoning reversed. The MCIRFs reveal that there is no relative velocity in the reference frames of the accelerating rockets. You are right that simultaneously at given instant (other than t=0) in the LF, the velocities of the front and rear rockets are different, but this simultaneity does not apply to the accelerating reference frame. We can for example show that if there is no length contraction and the rockets are accelerating identically in the LF, that differential time dilation is still occurring in the rocket reference frame.
Here I meant LF, as in pick one MCIRF and stick with it, rather than keep switching to subsequent MCIRFs.
I just meant you could pick any arbitrary MCIRF as a LF. There is nothing particularly special about the LF. The rockets are still accelerating even in the LF.
This is the bizarre aspect. The inertial observer attributes the redshift to classical Doppler shift due to velocity differential between emission and reception and yet the time dilation observed by the accelerating observers is real (as in physical) and an observer in the rear rocket really ages slower than an observer in the front rocket.
.. because nature does not give us much choice and there is no natural way to have synchronised clocks and static coordinate system in an accelerating reference frame or in a gravitational field.
Yes, it is still an accelerating system and I never claimed to make it an inertial reference frame. I only claimed it gave us a way to have a sensible coordinate system and a way to synchronise clocks that gives a static reference system with coordinate axes that are not changing over time. Can you suggest another way to synchronise clocks that are running at different rates?
It does work for other velocities. Have another look at the posted chart. The rear rocket sends signals every 3 ticks and the front rocket receives those signals every 4 ticks, consistently even as the relative rocket velocities constantly change over time. If we speed up the rear clock by a factor of 4/3 then the front observer will see the rear clock ticking at the same rate as his own clock for all time.
Again, if you look at the attached chart you see that the front rocket sends a signal when the rocket velocities in the LF are approximately 0.69c and the rear rocket receives that signal when the rocket velocities are approximately 0.81c in the LF and there are no problems with the rear rocket overtaking the front rocket. (Another curious aspect is that a light signal sent from left of the origin can never catch up with the accelerating rockets, even though they never attain light speed. That should bake your noodle! ).

P.S. I think Peter has a pretty good handle on it all in post #81.

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harrylin

That looks bizarre because it's wrong. Indeed the redshift of accelerating rockets is almost purely (semi)classical Doppler shift, insofar as there is no or negligible difference in acceleration. I see no reason for the claim that in reality it's not so, as I also stressed in post #65:
It depends on your choice of inertial reference system if you deem that the clock in the rear ages slower or not; according to the launch frame's POV they age equally.

yuiop

Are you talking about the case with no length contraction or the case with length contraction as used in in Rindler coordinates or Born rigid acceleration? In the latter case there is most definitely significant difference in proper acceleration as measured on board the rockets and a significant difference in coordinate acceleration as measured simultaneously in the launch frame
They only age equally from the launch frame's POV if there is no length contraction. Even then, the rear rocket observers will see blue shift of clocks at the front of the rocket and any experiment they carry out will convince them that the rear clocks are tangibly running slower than the front clocks. In this thread we are mainly discussing the equivalence between measurements in an artificially accelerated system and a gravitational system, so we are more interested in what the accelerated observers measure. Also, the case where there is length contraction such that the accelerated observers consider themselves to be at constant distance from each other, is more relevant to a typical gravitational field such as that of the Earth. When we stand on top of a tower and look towards the base, we generally consider the height of the tower to remain constant.

P.S. Some reservations about the none length contracting case have occurred to me. I will try to analyse that in detail later.

harrylin

I thought that you were talking about the case with no length contraction as seen from the launch pad frame. Else a little correction is needed, as I also hinted at in my post #65 which I also cited again. I thought (and still think) that you were not talking about that small effect of length contraction when you made your claim about "really slower aging", as that redshift is very small compared to "the redshift" that you discussed. If I misunderstood you, please clarify.
Only if, as I pointed out, he is fooling himself into thinking that he his not accelerating. However, that would not be reasonable for someone in a rocket with firing rocket engines - as you also seemed to realise in your answer in post #73. For some reason that escapes me, you replaced "fooled"(=not real) by "real" (=true) between that post and post #86. Someone's instrument reading is not necessarily identical to "what really happens", nor does a smart rocket pilot accept everything at face value.

stevendaryl

I'm not sure exactly what you are claiming. The differential aging of someone in the front and rear of a rocket is real. We can make it operational as follows:

1. Take a pair of clocks to the rear of the rocket.
2. Set them to the same time, t=0.
3. Move one of the clocks to the front of the rocket.
4. Wait a year.
5. Move it back to the rear.
6. Compare the two clocks.

The prediction is that if we allow length contraction (Born rigid acceleration) then the moving clock will be ahead of the clock that was always in the rear by a factor of 1+gL/c². So it's not simply some kind of illusion.

yuiop

Generally when I talk about accelerating rockets in this thread I am talking about the the length contraction version and when I am talking about the more unusual and perhaps less useful none length contracting version I usually make it clear that I am talking about that version. In my last post I mentioned that I intend to analyse the none length contraction version more closely as that might be interesting.
When Einstein introduced the equivalence idea he described comparing measurements in a closed accelerating box so that the observers inside would be unaware of whether they were stationary in a gravitational field or accelerating artificially in flat space. Without the luxury of being able to look out the window he would no be aware of his rocket engines fireing away. In both cases he would measure proper acceleration and redshift of signals from below him and in a small enough enclosure whereby tidal effects are negligable, he would be "fooled", in the sence that he would be uncertain as to whether he was being artificially accelerated in flat space or stationary in a gravity field. In both the artificially accelerated case and when stationary in a gravity field, clocks lower down really and unambiguously run slower than clocks higher up. No one is being fooled about whether the clocks run at different rates or not.

Austin0

And to what do you attribute the different rates if differential velocity is ruled out??

yuiop

I guess the inertial observer would attribute part of the differential rates to differential velocity, but the Rindler observers on board the rockets would not because as far as they are concerned the rockets are stationary with respect to each other and they would have to attribute the differential clocks rates to a real or pseudo force field.

harrylin

Your operation is quite different from the one I commented on, and I have not analysed yours. I thought that Yuop was discussing clocks in two rockets, and thus I assumed a similar situation as Bell's spaceships.
In a case of two rockets such as presented by Bell, according to the launch frame observation the two clocks will age identically, and that observation is as valid as any other one; and note that the Doppler redshift will be nearly the same as in a case with length contraction. If instead we consider a single rocket as viewed from the launch frame then there will be a small effect due to length contraction (I did not calculate it or analyse from all perspectives, but at first sight it gives a slight slowdown of the rear clock according to all observers). You did not reply my question to you if indeed you were talking about the (much bigger?) effect of Doppler redshift.

Austin0

I think we are all talking now about the length contracted case. either as a single rocket or two rockets with the expected contraction effected through differential acceleration.

So there are three questions;

1) How much dilation would be effected purely through length contraction ?[which i think would have to be calculated from the launch frame , not momentarily comoving frames]
]
2) Would this dilation factor increase over time with greater velocities?.

3)how would this figure compare with the expected relative dilation in the accelerating system as calculated using the Rindler coordinates?

1+gL/c². yes i think they are talking about this factor as being actual dilation , not apparent Doppler dilation

harrylin

I was still editing my answer when you answered, as just after answering I got the impression that although I wasn't commenting on the calculations, someone (perhaps me) may have made an error related to the numbers. But if so, I haven't yet figured out where...

In any case, answers to your questions will also clarify that point for me! (now I have now no time to look at it myself).

stevendaryl

Well, the difference between the two rocket case and the one-rocket case is length contraction. In the two-rocket case (with identical accelerations), the clocks will always show the same time in the "launch" frame, but the front clock will run ahead of the rear clock in the instantaneous comoving frame of the rear clock.

Well, sort of. I thought you were saying that the differential aging was a kind of illusion, which I interpreted as saying that they were really the same age. The relative age of distant twins (or clocks--I forget which we're talking about) is a frame-dependent quantity, but I wouldn't call that an illusion.

It's not a small effect, when you consider the case of the rocket accelerating for long periods of time. As I have pointed out in a different post, the time difference between the times on the front and rear clocks can be broken down into two contributions:

Let e₁ be the event at which the rear clock shows time T₁. Let e₂ be the event at the front clock that is simultaneous with e₁, according to the "launch" frame. Let T₂ be the time on the front clock at event e₂. Let e₃ be the event at the front clock that is simultaneous with e₁ in the comoving frame of the rocket. Let T₃ be the time on the front clock at event e₃.

Let δT₁ = T₂ - T₁.
Let δT₂ = T₃ - T₂.

δT₁ is purely due to length contraction; it's equal to 0 if there is no length contraction (the two-rocket case).

δT₂ is an additional contribution due to relativity of simultaneity; what's simultaneous in the launch frame is not simultaneous in the comoving frame.

δT₁ starts off zero and gradually gets bigger and bigger, growing without bound, if the two clocks continue accelerating.

δT₂ starts off nonzero, and approaches a maximum value.

The total discrepancy between the two clocks, as viewed by the comoving frame of the rocket, is the sum of the two δT = δT₁ + δT₂. That sum grows at a constant rate of gL/c₂; that is, δT/T₁ = gL/c² at all times.

The two effects, length contraction and relativity of simultaneity, are both important in explaining the discrepancy between the two clocks. Relativity of simultaneity is the dominant effect soon after launch, and length contraction is the dominant effect long after launch.

stevendaryl

I think those questions have already been answered. The effect due to length contraction starts off zero, and increases without bound. Long after launch, the ratio of the time on the front clock to the time on the rear clock approaches the value 1+gL/c², as measured in the launch frame.

Well, "actual" versus "apparent" is a fuzzy distinction. The time difference is real, in the operational sense that I gave: If you synchronize two clocks in the rear, take one clock to the front and let it sit for a year, and then bring it back to the rear, the clock that was in the front will show more elapsed time. And the difference will be exactly what the Doppler shift showed.

harrylin

Instead I was saying that the pseudo gravitational field is a kind of illusion, and I simply tried to clarify in post #87 that as the inertial observer attributes the redshift at low relative velocity to classical Doppler shift, an observer in the rear rocket cannot be said to really age slower by this red shift factor than an observer in the front rocket - and thus there is nothing "bizarre" going on here.
Thanks for the analysis with which I agree (only what you call "relativity of simultaneity", I call Doppler shift). It's interesting to see that at very high speeds the effect is mainly attributed to length contraction, indeed I had not realised that.

yuiop

If we have twins at the front of the rocket (initially the same age) and one free falls to the rear of the rocket and some time later the other free falls to the rear of the rocket, the twin that spent the most time at the rear of the rocket will have physically aged less than the twin that spent the most time at the front of the rocket. When we compare twins side by side and observe differential ageing, that is as real as it gets, as far as time dilation is concerned.

harrylin

Surely we all agree on that; it's different from the case that you discussed in which their ages are not compared side by side. Why did you find that case bizarre?