Time paradox

jaumzaum

I'm studying special relativity and I can't understand the following.

Let's imagine me and my twin brother making the following experiment: I stay in Earth and my brother travels in a spaceship with velocity 0.5c. For moving referentials the time passes more slowly, but for the first principle of special relativity, the Physics laws are the same for any inertial referential. So for me, my brother are in slow motion (I became older). But for my brother, he became older. When he arrives here, who will be older?

DaleSpam

You will be older.

As you say, the first postulate of relativity is that the laws of physics are the same in all INERTIAL frames. Your brothers frame is not inertial.

phinds

This has been discussed on this forum approximately 9,487 times. Do a forum search for the "twin paradox".

Drakkith

Huh. I am approximately 10 minutes older than my twin sister. Given 7 months and 1 week of gestation, I wonder what speed I would have needed to travel to get that difference in age over that time period. My mother must have had one hell of a pregnancy with all the relativity going on inside her.

Mordred

Roflmao maybe you should ask her

nitsuj

Given phinds' point, I surprised your variation of the twin paradox hasn't been asked / discussed.

ghwellsjr

There is a very simple way to answer this question. It relies just on the first principle of special relativity that says that you and your brother will each see the other ones clock ticking slower than their own by exactly the same amount during the outbound portion of the trip, plus the fact that light from different sources travels at the same speed (without identifying what that speed is, which is Einstein's second postulate). On the way back, your brother will see your clock ticking faster than his as soon as he turns around. This leads to the conclusion that if your brother travels away from you at the same speed that he travels back to you, then his observation of the ratios at which your clock ticks compared to his during each half of his trip will be reciprocals of each other.

Hermann Bondi's book, Relativity and Common Sense, pages 77 to 80, describes the process to show that this is true.

So whatever ratio your brother sees of your clock compared to his on the way out plus its reciprocal on the way back added together and divided by two gives us the final average ratio of your clock compared to his when you reunite, since the times for his two halves of the trip are the same because he is traveling at the same speed over the same distance (although we aren't specifying what that speed, distance or time are). That average ratio is always greater than one.

Let's say that that ratio is R for the return trip and 1/R for the outbound trip. Adding these together and dividing by two gives us (R + 1/R)/2 or (R²+R)/2R. For any value of R greater than 1 this evaluates to a number greater than 1. Try it and see.

jaumzaum

You are right. I've just read the twin paradox at wikipedia. They say like 20 times that the explanation beacause there is no contradition is that the event is not symmetrical, as only my brother has experienced acceleration.

But isn't acceleration relative too? When I say my brother is accelerating at acceleration a, shouldn't he say the same of me?

LastOneStanding

George, I know this is largely a matter of taste, but I think the relativistic Doppler effect is the least clear way of explaining the twin paradox. It takes a lot of work when first explaining relativity to get the point across that when we talk about events, we're not discussing when these events appear to happen in different reference frames but when they actually happen, taking into account the relativity of simultaneity. I think emphasizing what the twins actually see is unnecessarily confusing and risks giving the false impression that time dilation is just some kind of visual trick. Since the times when clock ticks are seen are governed by the relativistic Doppler effect, which is just the non-relativistic Doppler effect plus time dilation, you are ultimately just starting with time dilation, adding in the lag effect on finite propagation of light (i.e. the non-relativistic component of the Doppler effect), and then subtracting the lag effect out again (when showing how each twin would use their observations to calculate their sibling's age). I think that's a lot of needless clutter.

Personally, I find the clearest explanation is to not bother with any talk of when various age milestones are seen in each twins' frame and just focus on when they happen in each frame—i.e. the spacetime diagram approach. It's easy as pie to show that, while time dilation is indeed symmetric on each leg of the trip (separately!), relativity of simultaneity means that when the traveling twins makes her about-face, her brother ages a large amount in her frame in a very small amount of time—instantaneously in the limit of an instantaneous turnaround—and this more than makes up the difference. Everyone has their pedagogical preference, but I really think not futzing around with super telescopes does a much better job of showing precisely how the asymmetric aspect of the twins' experience (one inertial reference frame vs. two) directly leads to the correct calculation in both frames.

DaleSpam

No. Acceleration* is not relative. Your brother and you can each measure your acceleration using accelerometers without any reference to the other. If you do so then unambiguously your accelerometer reading will be 0 but his will not.


*The type of acceleration which is not relative is called "proper acceleration". There is also a type of acceleration called "coordinate acceleration" which is relative to some specified coordinate system. Coordinate acceleration cannot be measured by an accelerometer and doesn't have any physical effects, only proper acceleration does. So usually when people just say "acceleration" they mean "proper acceleration" which is not relative.

LastOneStanding

No, acceleration is not relative because you can feel it. Either you or your brother felt a force due to the acceleration and whoever did is simply the one who accelerated—no fooling about. All inertial observers will agree on whether or not something is accelerating and anyone who disagrees is, by definition, not an inertial reference frame.

Nugatory

No. It's somewhat surprising, but acceleration (change in velocity) is not relative although velocity is. You can measure your own acceleration without reference to anything else: If you are standing on a scale, the weight it measures will increase if you're accelerated upwards (think of high-G-force rocket launches); if you're holding a spinning gyroscope it will resist acceleration; and so forth.

We can build black boxes called accelerometers which display measure the acceleration they're being subjected to, and you can't do the same thing with speed (think about how an automobile speedometer "knows" that the car is moving relative to the roadway).

jaumzaum

I think I've got what you mean. Pretend I'm sending messages to my brother year by year, and he does the same . At the moment of the about-face my brother will receive many messages of mine, and when he arrives in Earth, I will be older.

But does't it mean that in the accelerated referential of the spaceship (at the moment of the about-face) see the light would aprroximating with a velocity bigger than c?

I know that c is constant for intertial referentials, I don't know how it works for accelerated referentials. Is it possible?

Nugatory

It works the same way - at any particular moment the accelerated twin can choose a reference frame in which he is at rest; it's just that the math gets more complicated because in the next moment he won't be at rest in that frame. That's why we try to work with inertial frames when we can, and why most textbook examples are worked in inertial frames.
(One unfortunate side effect of this tendency is that it's easy to get the impression that special relativity only works for inertial frames, and you need general relativity to handle acceleration . Although widely repeated, that is not true - you only need GR if the spacetime is not flat).

DaleSpam

As Nugatory mentioned, for an object in any state of motion there exists, at each instant in time, a specific inertial reference frame where the object is (at least momentarily) at rest. This is called the momentarily co-moving inertial reference frame (MCIRF). The MCIRF is an inertial reference frame so light moves at c in it, but the object may only be at rest in it for one instant.

There are also non-inertial reference frames, such as a rotating referece frame. In non-inertial frames the usual laws of physics take different forms unless you write them using tensors. Specifically, this means that unless you use tensors then you may get that the speed of light ≠ c in a non-inertial frame. Again, consider a rotating reference frame, in such a frame even nearby planets or the moon may be moving faster than c.

Nugatory

[This is an interesting point worth further discussion. It's also a digression from the original thread, so if it looks like it's going to take on a life of its own some harried, overworked, and underappreciated moderator (is there any other kind?) might want to split it out into a thread of its own]

As L1S says, this is a matter of taste, and we all know that de gustibus non disputandum est... but there is much pleasure to be had in discussion, as opposed to dispute.

My experience has been that there are two basic approaches to SR thought experiments: Start with the actual observable physical behavior of the light signal, as ghwellsjr does; and start with the spacetime picture and Lorentz transforms to construct consistent histories of events in each reference frame, as L1S does.

I find that many people naturally gravitate towards one style or the other, and find the other one somehow sneakily unsatisfying and unconvincing.

For example, I've never found the light behavior explanations to be gut-level satisfying; I feel as as if I could do something just a bit more clever with my moving mirrors and light sources I could somehow subvert the experiment. (This suspicion may be what's motivating the posters who show up asking whether relative effects are just an "optical illusion"). I prefer t work through the spacetime diagram and Lorentz transforms to satisfy myself that no matter how I manipulate the experiment, it all has to come out just as relativistic doppler and similar phenomena say it will.

On the other hand, I also know from endless friendly discussions that there are people who find the coordinate-based description to be completely non-fundamental; it's all full of coordinate artifacts and abstract mathematical relationships meaningful only if they connected to some real physical observers.

My personal opinion on the subject:
1) You don't really understand until you're comfortable using either style of description. (It's worth noting that Einstein, and just about any serious stdent of relativity after him, are effortlessly fluent in both styles).
2) When solving problems for yourself, use whichever style you're most comfortable with. When reading someone else's analysis in the style that you don't prefer, consider transforming it to the one that you do prefer. It's good practice for you and may help someone else understand.
3) When explaining to someone else, start with the style that you're most comfortable with. But be alert for signs that it's not working, and be prepared to switch to the other style. This doubles your chances of getting the magical "Aha - now I get it!" moment that is the goal of all explanation.

ghwellsjr

It's not a matter of taste--the relativisitic Doppler effect is the problem that "other" explanations need to explain.
No, you've got it backwards. What each observer sees is what is actually happening. These are all local events. The assignment of coordinates to remote events according to different reference frames are not "when they actually happen".
Time Dilation is not visual so it cannot be a visual trick. It's a mathematical calculation and dependent on the chosen frame and changes with each arbitrarily selected frame. How can it be visual? Relativistic Doppler is visual, at least in the sense that it is the result of transmitted light, but it never tricks us.
You could say that the relativistic Doppler effect is the time dilation plus the changing lag effect on the propagation of the light but since both of these are arbitrarily determined by the selected reference frame, neither one can be said to be governing.

Also, the twins don't need to do any calculation, they just watch their siblings age (or their clocks) and when they return, they each agree on what actually happened. We need to do some calculation to determine what they will see, but that's a different matter and it's very easy because it doesn't involve any understanding of Special Relativity or any other theory. We don't have to learn about synchronizing clocks or defining an Inertial Reference Frame or what the Lorentz Transformation is all about.
It's not like we have an either/or situation here, we can show both the relativistic Doppler and the time dilation on the same spacetime diagram which is what I do all the time. You can do a search on "diagram" with my name and find lots of them.
Yes, if you keep the two legs separate in two different Inertial Reference Frames, one for the traveling twin's outbound leg and one for the inbound leg, but when you try to conflate them into one spacetime diagram, you're in for all kinds of complications that are totally unnecessary and do not provide a single bit of insight into what is happening. Then please show me how you actually draw the spacetime diagrams according to your pedagogical preference.