The last statement. Alice does not perform her experiment twice according to Charlie.

He has no way of determining Alice's performing the experiment while he is on his journey, because the experiment is done at a space-like separation. So he cannot observe her doing it twice. He could of course calculate when she was doing it and would find a unique event but two different descriptions of that event depending on whether he used the coordinates of his moving frame of reference or the coordinates of his frame after arrival at Bob's station. But knowing how to do his Lorentz transformations, he could always find out that this was just a single event in spacetime.

K. Kassner,

You say: "He has no way of determining Alice's performing the experiment while he is on his journey, because the experiment is done at a space-like separation."

Why should he have a way to determine? The predictions of the relativity lie to us?

But, for avoiding doubt, let's say that the remote station sends two ships, one driven by Danny with destination Alice's station, one driven by Charlie with destination Bob. Since Alice's lab is more distant, Danny starts his flight before Charlie, but at some time they will be co-moving, i.e. have the same time-axis. According to the two ships' time, while Charlie is on his way to Bob, Danny reaches Alice's station just in time for witnessing her experiment. That would be, by the moving clock, definitely before Charlie begins his decceleration for landing on Bob's station.

Lorenz transforms hide a logical contradiction.

Please read the paper, may be helpful for you.

A new approach to the correction of

Galilean transformation.

Dear Sofia!

My fragmentary thoughts about the problem you mentioned:

Time becomes only visible by processes and changes. Time shows a identical flow under identical conditions on every place in the universe. But an universal time (or normal-time) does not exist. We have to assume that time is determined by the material-energetical conditions. It is impossible to compare independent and spaced apart processes by using of an assumed normal-time or a transformed time. Alice and Bob are making their experiments. Charlie can only compare the measurments and results after.

The Lorentz-transformations are mathematical descriptions under the assumption of an axiomatic defined mathematical space. But the real space is caused and determined by matter. Theoretical physics made the mistake to equate mathematical and real space. In my opinion is Einsteins theory of relativity only a first step to approach to the actual relativity. Einstein do not mentioned real and observed findings for his theory. Experiments appear later parallels between reality and Einsteins trains of thought. Thats all.

Best Wishes! Hans

Charlie should conclude that Alice's experiment and Bob's experiment are spatially separated. Spatially separated events can be labeled as A before B or B before A or as simultanious depending on coordinat system. A lot of confusion would be avoided if the notion of simultanious events was completely droppen and the notion of before/after was restricted to situations where one event is in the forward or backward light cone of another. For spatially separated events one may say that the events have certain time coordinates in a certain coordinate system, but the difference in time coordinate is of no special interest.

Dear Sofia

I think this paper will be useful for your question so you can used references of this paper too.

Regards 

Dear Prof. Saeed,

Thank you for the reference, but I fail to see what this paper gives me. To say it otherwise, the paradox of which I speak is real, or is there some hidden mistake?

You see, it's a not so good custom of people to reply to questions by "read this paper". I accepted to read in the past, jst because I found tiresome to argue, and I found myself that I wasted time reading a paper that gave me NOTHING.

Bottom line, if you believe that the paper if of help, please tell me in a few lines what is the help. I notice that the papers speak of two scalings in the same process. So what? I mean, how that helps in my case?

You see, some users advise me that in absence of contact with Alice and with Bob, in real time - i.e. at the measurement time(s) - I should not believe to the predictions of the theory - i.e. that Alice measured twice. I can't accept that. General relativity deals with accelerating/deccelerating frames and its results are correct. We cannot dismiss the general relativity because a difficult problem appeared. The problem should have a solution.

With kind regards!

Dear users,

I see different advices saying, in essence, not to accept the paradoxical predictions of the General Relativity (GR) appearing here, with all sort of motivations.

The GR works! Its predictions were found correct in all sort of cases. The GR is typical for accelerated systems.

GR is not quantum mechanics (QM). The QM predicts only what is measured. So, one could argue that Charlie can say nothing about an experiment he didn't do, or wasn't present when done. But GR is not QM, it makes predictions both for the case that Charlie was present or not.

Then, my question is whether the contradiction to which I point is a real anomaly, or there is some mistake in the logic?

Re the original question:

Nope.

Charlie's clock cannot run backwards (this would require a speed faster-than-light.) It is ever advancing.

Thus - wrt Charlie's clock: T(C@Bob) < T(C@Alice) < T(C@Bob-again)

Even if Charlie's clock is lagging a bit on the way to Alice, it is still advancing - if even slower that normal - on the way back to Bob.

Even if Charlie were traveling at light Speed and stopping for Zero time,

T(C@Bob-again) > T(C@Alice = T(C@Bob)

Where's the paradoxon?

U. Dreher,

I suggest you to READ the question and the problem exposed in the text before answering.

OK - step one:

'According to the clock of his ship, Alice performs her experiment before Bob' implies faster-than-light travel.

Did I really miss something?

@Wechsler: "Why should he have a way to determine? The predictions of the relativity lie to us?"

I have covered that already in my answer. Of course he can calculate the predictions of  relativity. These predictions say that there is a single event and that in one frame this event happens before he does the calculation, whereas in another frame that event happens just as he does his calculation. It is still a single space-time event.

A contradiction could arise only, if these different time orderings could be used for a causality violation. But that is not possible, because the pairs of events under discussion are space-like. They cannot influence each other, hence their different temporal order in two different frames is not usable for any operational contradiction.

My answer contained this already.

@Wechser: Do you happen to read German?

If so, on my web page

http://wase.urz.uni-magdeburg.de/kassner/srt/crashcourse/perry_rhodan.html

I have discussed a very similar paradoxon. Briefly, if you look at a very distant place, such as Andromeda, even the speed by which the earth goes around the sun is sufficient to propel distant events by years to the future or past - which is not really paradox, of course, as long as you cannot move faster than light. (It only looks so for our Newtonian intuition.)

But here is the argument. Andromeda is about 2.5 million light years away, and the earth moves around the sun at 1/10000 of the speed of light. Let us say there is a supernova explosion in Andromeda right now, when the earth is moving neither towards Andromeda nor away from it. A quarter of a year later, let the earth move away from Andromeda, then the explosion will not yet have been happened! It will be 2.5 million/10000 years = 250 years in the future! But half a year later, when the earth moves towards Andromeda, it will already have happened 250 years in the past. None of this is a contradiction, because simultaneity at a distance is arbitrary within the time interval that a light signal takes from there to us and back So for Andromeda everything that happens between almost 2.5 million years before and almost 2.5 milllion years after now may be considered simultaneous to what happens here and now from the point of view of some frame of reference. If you change the frame of reference, the positions of events in spacetime do not change but your description of them changes. So you may get two different times for the same event. It is still a single event. There is no paradox.

U. Dreher,

Yes, you missed the problem itself. First of all, why faster-than-light travel?

Let's go to the equations - they refer to the attached picture. I denote by x, t, the coordinates in the rest-frame of Alice and Bob, and by x', t' the coordinates in Charlie's ship. Let xA and xB be Alice's and Bob's positions in the rest-frame, t0 the time of the experiment in the rest-frame, and t'A, t'B the experiment times by the ship-clock. I suppose you know the notations β = V/c, and γ = 1/sqrt(1 - β2)

(1) t'B = γ(t0 - β xB/c); t'A = γ(t0 - β xA/c).

(2) t'A - t'B = -γβ (xA - xB)/c .

Since Alice is further located than Bob, (xA - xB) > 0, therefore (t'A - t'B) < 0. By the ship time-axis, Alice ends her experiment before Bob.

Now, I have no idea what is

==> "wrt Charlie's clock: T(C@Bob) < T(C@Alice)< T(C@Bob-again)",

It's not advisable to use notations that you don't explain. Also, my question doesn't speak of a Bob-again. neither am I intereted when would Charlie land on Alice's station. My problem appears when Charlie lands on Bob's station (with no "again"), and I recommend to read the problem before answering.

Best regards!

@K. Kassner,

I like very much your last explanation, the one with Andromeda. It is very simple and clear. I think that this is the explanation of the puzzle.

Unfortunately, I don't read German, I tried to learn but didn't have time to perseverate. The page in Internet that you indicated has no translation in English? Won't you bother to translate it?

Now, there is a point in your explanation, on which I'd like to have with you a some deeper talk. Now it's very late, I hope to talk with you tomorrow.

Thanks for your help.

Dear Prof. Kassner,

I see from your profile-page that general relativity belongs to your domain of research. So, I would like to ask some questions:

Please see, general relativity is widely used. It is not taken as a "change the frame of reference" - you probably wanted to say change of the inertial frame. The space-ship that deccelerates still represents one non-inertial frame. What would happen if the observer wouldn't know that Alice performed her experiment only once?

I agree with you that we have no paradox here, just an anomaly because of the use of a non-inertial frame. Though, should we conclude that judging according to non-inertial frame may be misleading and for avoiding that, one should do all the judgement according to one single inertial frame?

With thanks in advance!

I have translated the Perry-Rhodan page to English. To do it fast, I used Google translator and corrected the errors afterwards. I hope, I have not missed too many. Here is the link:

http://wase.urz.uni-magdeburg.de/kassner/srt/crashcourse/perry_rhodan_engl.html

There are two more objects ,the  inverse of length and the mass which vary as the time from one inertial frame to other inertial frame. .Therefore, when we consider the change in time, we should have to consider the variation of these quantities also. This consideration may resolve the problem arising with the time in the theory of relativity.