Time paradox

I get the impression that the concept of the sequence of inertial frames to represent the turnaround is not being understood.

The times assigned by blue in the succession of frames you describe are just coordinate times; they don't have any physical meaning by themselves.

Should there be any limit in smallness and number of the worldline increments used in the sequence of inertial frames? Of course not.

So, it is incorrect to regard this scenario as requiring a non-inertial analysis. This is nothing more than a generalization of the accepted practice of using a single turnaround point when analyzing the final age differences of the twins.

They are not even coordinate times (on the red worldline) since they violate the one-to-one requirement of coordinate charts.

The time assigned by each individual frame in the succession of blue's frames is a coordinate time; the time assignments of each individual frame are one-to-one. The problem only arises if you try to put together a single non-inertial frame whose coordinate assignments along a given worldline (such as red's) agree with those of the succession of inertial frames; as you point out, you can't do that globally without violating the one-to-one requirement.

It looks to me like bobc2 didn't actually intend to construct a single non-inertial frame in this way...

...but to me that's really a side issue; even if one is careful *not* to make any claims about a single non-inertial frame, it's still true that you can't get anything physically meaningful just by looking at coordinate times of events along red's worldline in the succession of blue's inertial frames.

The time assigned by each individual frame in the succession of blue's frames is a coordinate time; the time assignments of each individual frame are one-to-one. The problem only arises if you try to put together a single non-inertial frame whose coordinate assignments along a given worldline (such as red's) agree with those of the succession of inertial frames; as you point out, you can't do that globally without violating the one-to-one requirement.

You are exactly correct. That is what I've been trying to get across. Regarding the collection of individual inertial frames as representing a single non-inertial frame is definitely not the way to understand this.

That is a philosophically based idea. Einstein cautioned against these kinds of ideas which trap you into solipsism.

Regarding the collection of individual inertial frames as representing a single non-inertial frame is definitely not the way to understand this.

That is a philosophically based idea. Einstein cautioned against these kinds of ideas which trap you into solipsism.

The monitor may allow you to define for us how you would describe or define criteria for identifying the real world of existence in the context of physical theory--I'm not sure.

No, your brother will not receive many messages from you at the moment of his about-face. His about-face will not cause him to receive any messages from you. What's going to happen is that for the first half of his trip, he will receive messages from you at a slower rate than he sends them (1/R as I said in my first post to you), then for the second half of his trip he will receive messages from you at a faster rate than he sends them (R).

So for your example of your brother traveling at 0.5c, we can use the Relativistic Doppler formula to calculate what R is:

√((1+β)/(1-β)) = √((1+0.5)/(1-0.5)) = √((1.5)/(0.5)) = √3 = 1.732

And 1/R is the reciprocal, 0.57735.

This means that he will see your yearly messages coming to him slower than his during the first half of the trip. In fact it will take 1.732 years before he sees your first message.

And for the last half his trip, he will see your messages arriving more often than once per year according to his clock. It will only take 0.57735 years between each of your messages.

Now without knowing how long the trip will take, we can average these two numbers:

(1.732+0.57735)/2 = 2.30935/2 = 1.154675

This is the final ratio of your two clocks when he returns. How ever many years it took him according to his clock, yours will be 1.154675 times that amount.

So let's say your brother travels away at 0.5c for 13 years and then takes 13 years to get back at the same speed. Here is a spacetime diagram to show what is happening according to your rest frame. I show you as a thick blue stationary line with dots every year and your messages going out as thin blue lines traveling at c. I show your brother as a thick black line traveling at 0.5c with black dots every year and his messages coming back to you as thin black lines:



Now let's see how the previous calculations based on Relativistic Doppler fit in with this diagram. First off, I said that the rate at which your brother receives your yearly messages take 1.732 of his years. Can you see that on the graph? For example, at about his year 12, just before he turns around, he is just receiving the message you sent at your year 7. Can you follow that? If we divide 12 by 7 we get 1.714 which is about right. (We don't expect it to be exact because he didn't receive your message exactly at his year 12.)

Furthermore, if you look at your year 12, you can see that you are just receiving his message from year 7. It's symmetrical.

Now you should be able to see that after he turns around, he starts receiving your messages faster than one per year. In fact, from about his year 19 (you'll have to count his dots) to when you meet at his year 26, he will have received your messages from year 18 to 30. That is a ratio of the differences of (26-19)/(30-18) = 7/12 = 0.583, close enough to 0.577.

And in a similar manner, you can see that from his year 14 (just after he turns around) until you meet (12 years of messages from him), you will see them from your year 23 to your year 30 (7 years) and the reciprocal ratio applies.

Now that we can look at a spacetime diagram, we can see that the reason why the two of you age differently is because your brother sees these two ratios for half of his total trip time each but you see the smaller ratio for three-quarters of the time and the higher ratio for just one-quarter of the time. This means you are seeing him age less for a longer time while he sees you age less for half the time.

The last thing we want to notice is that ratio of your final age difference is 30/26 or 1.1538, very close to the actual 1.154675. (Again, these numbers are not exact because we're eyeballing them off the diagram.) This ratio is the famous value of gamma which is also the time dilation factor which shows in the diagram as the ratio of the coordinate time for your brother compared to his actual time on his clock. Can you see that?

Now I want to show you what the exact same information presented in the first diagram looks like in two more diagrams based on the IRF's in which your brother is at rest, first during his outbound leg and then during his inbound leg. First the outbound leg:



Notice how your brother's time is not dilated during the outbound leg (because he is at rest) but yours is. Note also that he has to travel at a higher speed than 0.5c (look up "veloctiy addition" in wikipedia to see that this higher speed is 0.8c) when he turns around and therefore now has more time dilation than you have. Nevertheless, all the signals between the two of you continue to travel at c and arrive at exactly the same times according to your own clocks as they did before. Does this all make sense to you?

Finally the diagram for the IRF in which your brother is at rest during the inbound leg:



This is very similar to the previous diagram so I won't go into any more explanation except that I want to point out that when your brother turns around, in no case does that have any bearing on what you see, until some time later and even then, each diagram shows accurately what you actually see and what your brother actually sees during the entire scenario.

Any questions?

Awesome diagrams and explanation! thank you for this!

Thanks George! I mean, I really want to thank you, in all of the threads I've already posted here, I've never seen such a good and detailed answer as yours. I'm new in special relativity and you explained everything so carefully I could understood almost completely. I thought I had understood before, but I didn't know the explanation had nothing to do to what I suppose it was correct. There should be more guys like you here in PF. I want to thank everyone who answered this thread but the george's answer was phenomenal (at least for me). Now I think I'm startin g to understand special relativity.

And yes, I do have some questons, I would appreciate if you could help me again

How it would be the diagram if we take the referential frame as the whole trip of my brother (I mean, my brother is at rest in the whole time). I'm not being able to "close" the graphic. If my brother is a straight line, my lines can't be together as they are smaller than his. Is this right?

I already addressed this in post 50, but that is exactly what you were doing in claiming that the red guy's time was going backwards. If you talk about a sequence of inertial frames then his clock goes forwards at all times and in all frames. If you talk about a sequence of "3D worlds" then you are talking about a 4D non-inertial frame, and mathematically that frame cannot cover the red guy's worldline.

Reference please?

the observer cannot know of anything but himself, for he is continually advancing forward along his worldline at the speed of light at the apex of his backward light cone. At that apex point he has no knowledge of the world of his hyperplane of simultaneity, and by the positivist’s claim he must consider himself to be a solipsist—the only known existing entity.

Here are a couple of Einstein quotes on solipsism from The Library of Living Philosophers Volume VII – Albert Einstein: Philosopher-Scientist, Paul Arthur Schillp,Editor

Here are a couple of Einstein quotes on solipsism

Sorry to be blunt, but this is hogwash. The observer has information coming in from his past light cone, and that information tells him about the existence of other objects. The information is time-delayed, but so what? It's still perfectly good information about the existence of other objects.

Edit: On reflection, it's even worse than that. You (bobc2) are arguing for a "block universe" interpretation of SR. But on that interpretation, "objects" don't exist in 3-D worlds

...they exist in 4-D spacetime. So knowledge of *any* event on an object's worldline counts as knowledge of the object's existence, since the object *is* its worldline. So not only are you incorrectly stating the opposing view, you aren't even consistently applying your own view.

You are still not getting it. I am not saying the red guy's clock goes backward for the red guy sitting at rest in his own frame of reference (see sketch at the bottom of my post #32). The red guy always sees his clock moving forward as he moves along his world line in his positive X4 (time) direction.

But if you were to make a list of the RED clock readings in the order they are presented in the blue guy's frame each time he (blue--referring to my earlier sketch at the bottom of post #32) boosts to his next inertial frame, then you would see those clock readings getting smaller and smaller as blue advances along his worldline (with blue's own clock readings moving forward in time). So, each time blue is coasting in a new inertial frame you note the clock reading on the red worldline at the intersection of the blue X1 axis with the red worldline (you can of course calculate this using the Lorentz transformation between red and blue coordinates, where red's frame is the same as black's except displaced along the black X2 axis). And of course you will perform a new Lorentz transformation for each new period of blue coasting in a new inertial frame.

So, we are not trying to manufacture some single coordinate transformation for a curvalinear non-inertial frame at all. We could talk about using something like Rindler coordinates, etc., but that's not at all what I've been trying to convey.