# The Twin Paradox and the Equivalence Principle

by haroldholt
P: 2,082
 Quote by DaleSpam The problem mentioned here isn't a problem of the gravitational explanation, but one of poorly specified non inertial coordinate systems.
My other point stands. The gravitational explanation isn't an explanation at all,
it's a calculation. If in inertial coordinates, the invariant interval is given by:

ds2 = dt2 - dx2/c2

and you change coordinates to X,T, then it follows that in terms of X and T

ds2 =
(A2 - C2/c2) dT2
+ 2(AB - CD/c2) dX dT
+ (B2 - D2/c2) dX2

where A = ∂t/∂T,
B = ∂t/∂X
C = ∂x/∂T
D = ∂x/∂X

This is just calculus, there is no new insight about gravitation or the way it affects clocks required.
P: 4,021
 Quote by stevendaryl I've always been slightly annoyed by the "gravitational time dilation" explanation for the twin paradox for two reasons. First, it's kind of circular. You say that the accelerating twin is equivalent to a stationary twin in a gravitational field, and then you apply the formula for gravitational time dilation. But how do you know that clocks in a gravitational field experience gravitational time dilation? By transforming the problem to an equivalent case involving acceleration in flat spacetime, and using SR. You can skip the "gravitation" step and just use SR, plus calculus to transform to a noninertial coordinate system, if that's more convenient.
So, you don't like invoking the word "gravity" and the EP, but instead stick to the SR non-inertial frame as such, where clocks run at different rates too.
 Quote by stevendaryl Second, it has bizarre consequences. Let me describe a variant of the twin paradox: There are two planets many light-years apart. Two twins were separated at birth and one went to live on one planet, and the other went to live on the other planet. The two planets are at rest relative to each other, and the twins are the same age, according to the reference frame of the planets. When one twin reaches age 20, he accelerates to nearly the speed of light. and travels to the other twin. When he arrives, he has only 21 years old, but the other twin is 40 years old. The traveling twin explains this using gravitational time dilation: during the time of acceleration, the distant twin aged rapidly, gaining at least 20 years during the brief time of acceleration. What's wrong with that explanation? Well, imagine that traveling twin, after accelerating, changes his mind; he decelerates and returns to his own planet. If the whole process of accelerating and decelerating was quick (much less than one year) then the twins will still be approximately the same age. But what that means is that the distant twin gained 20 years during acceleration, and then lost 20 years during deceleration. Gravitational time dilation can work to youthen a distant twin as well as age him prematurely.
Yes, but don't this bizarre consequences appear in the non-inertial frame of the travelling twin in any case, even if you skip that gravity part, as you suggest above?
P: 2,082
 Quote by A.T. So, you don't like invoking the word "gravity" and the EP, but instead stick to the SR non-inertial frame as such, where clocks run at different rates too. Yes, but don't this bizarre consequences appear in the non-inertial frame of the travelling twin in any case, even if you skip that gravity part, as you suggest above?
If you use noninertial coordinates, as Dale says, there is no such thing as the coordinate system of the accelerated observer. Instead, you have several "coordinate charts", and you have to piece together several of them to compute elapsed time for a clock. One of the criteria for being a legitimate coordinate chart is that the same spacetime point cannot have two different coordinates.

It's actually like trying to compute the length of a path on the surface of the Earth using maps. If you look at a flat map, distances are completely distorted near the North Pole and South Pole (the North Pole itself is stretched out into a line, instead of being just a point). It's not very good for computing distances. If you look at maps for smaller regions, say 100 km x 100 km, then a flat map gives a good approximation to distances. So a way to compute the length of a very long route over the Earth is to start with a 100x100 map of the starting point. Compute the length up until the route passes out of the region. Then get another map, and figure out which point on the second map corresponds to the exit point on the first map. Keep adding up the distances until the route passes outside of the region for the second map. Then get a third map, etc.

So you can compute the total distance as follows:
Let P1 be the starting point. Find some map that covers P1.
Let P2 be the last point on the first map. Pick a second map that covers P2. Let P3 be the last point on the second map. Etc.

Let Dn be the distance between Pn and Pn+1, as measured using the nth map. Then D = sum over n of Dn.

The same sort of thing works to find elapsed time for a complicated path through spacetime.
 HW Helper Thanks P: 4,840 A nice explanation of the twin paradox using the GR point of view is in the old book "Relativity, Thermodynamics, and Cosmology" by Tolman. If you go to Google Books at books.google.com/?hl=EN and search "Tolman Cosmolgy" you will find the book. The explanation starts on page 194.
Mentor
P: 17,224
 Quote by stevendaryl For sufficiently complicated situations, there is no "coordinate system of the traveling twin" from which can "explain" the differential ages.
No, I said there was no unique coordinate system. Meaning there are a variety of ways of constructing a non inertial coordinate system, and you have to specify which one you are using in a lot of detail.

Any valid coordinate chart you choose can "explain" it correctly, but the explanation will differ. In none of them will the other twin age backwards.
 HW Helper Thanks P: 4,840 Here’s a link to a derivation of the rate of an arbitrarily accelerating clock as observed by an arbitrarily accelerating observer assuming that the clock and observer are always moving along the same line. //sdrv.ms/JPxY5H The derivation is tedious, but the final result (equation 18) is quite simple. The rate of ticking of the clock according to the observer is just a product of a time dilation factor due to relative speed (u) and a factor due to the acceleration felt by the observer (g). (How u and g are defined for the accelerating clock and observer is discussed in the derivation.) The factor due to the acceleration is interesting, it has the appearance of “gravitational time dilation” due to a uniform gravitational field of strength g.
P: 21
 Quote by stevendaryl If you use noninertial coordinates, as Dale says, there is no such thing as the coordinate system of the accelerated observer.
That doesn't sound right. Shouldn't anyone be able to figure out how old the home twin is?
P: 2,082
 Quote by Underwood That doesn't sound right. Shouldn't anyone be able to figure out how old the home twin is?
There are two different, but related questions about the age of a distant twin: (1) How old is that twin right now? (2) How old is that twin when such and such event happens? (where the event might be specified by: the two twins reunite).

To be able to talk about how old a twin is "right now" requires a choice about a time coordinate that covers both twins. For an inertial frame, there is a unique "best" time coordinate, but for an accelerated twin, there is no unique choice of a time coordinate.
P: 2,082
 Quote by DaleSpam No, I said there was no unique coordinate system. Meaning there are a variety of ways of constructing a non inertial coordinate system, and you have to specify which one you are using in a lot of detail.
But the fact that there are a variety of ways to construct a noninertial coordinate system means to me that none of them is really telling things "from the point of view of the accelerated twin". The Rindler coordinates come close, but they cannot be used for a twin who "turns around" and starts accelerating in the opposite direction. Once you start using coordinate charts, you're really abandoning the idea of understanding things from any observer's point of view. The charts aren't necessarily from anyone's "point of view".
P: 21
 Quote by stevendaryl To be able to talk about how old a twin is "right now" requires a choice about a time coordinate that covers both twins. For an inertial frame, there is a unique "best" time coordinate, but for an accelerated twin, there is no unique choice of a time coordinate.
If I was on a rocket trip, I would know that my sister back home always had some age at any time on my trip. If the rocket driver told me that she didn't have some age right then, I wouldn't believe him. I wouldn't believe anybody who told me that.
Thanks
P: 3,676
 Quote by Underwood If I were on a rocket trip, I would know that my sister back home always had some age at any time on my trip. If the rocket driver told me that she didn't have some age right then, I wouldn't believe him. I wouldn't believe anybody who told me that.
A wise undergraduate physics student once said that the way to resolve all paradoxes of special relativity is to look for the evil and treacherous phrase "at the same time".... Sometimes it's well-hidden, but if you keep looking.... Here that evil and treacherous concept is hiding in the text that I've bolded above.

Yes, of course your sister always has some definite age, just as her wristwatch will always record some definite time since you and she parted. However, different observers on different rocket ships moving at different speeds will have different ideas about what that age is; and that's not especially remarkable because they all have different ideas of what's "right then". They're checking her watch and her age at different times so of course they get different answers.

None of this has anything to do with what your sister experiences. She's happily living her life back at home, celebrating a birthday once a year and keeping track of her age in her home reference frame, unconcerned that you and the other rocket observers are seeing time pass at different rates.
P: 298
 Quote by lalbatros It is possible to script another scenario where the "second twin" doesn't return home and just sends "somebody else" back and in such a way that there is no acceleration anywhere in the scenario. In this way, the geometry appears to contain the full effect.
Yes, an incoming triplet can transfer the time on the outgoing triplet's clock to hir own and carry that back to the stay at home triplet. No acceleration is necessary for the effect.
P: 2,082
 Quote by Underwood If I was on a rocket trip, I would know that my sister back home always had some age at any time on my trip. If the rocket driver told me that she didn't have some age right then, I wouldn't believe him. I wouldn't believe anybody who told me that.
Well, if it makes you happy to think that your sister always has a definite age, no matter how far away she is, just pick a coordinate system, any coordinate system, and always use that one. In relativity discussions, people always assume that the traveling twin picks a coordinate system in which he is at rest. He doesn't have to--he can continue to use the Earth's coordinate system, if he likes. The point of relativity is that the traveling twin can use any coordinate system he likes to do physics.

You might think that your sister's actual age is whatever age she has, according to the Earth's coordinate system. However, if you use a different coordinate system, you would compute a different age for your sister "right now" and you would never run into any contradiction.

Suppose you are billions of miles away from your sister. While you are away, your sister celebrates her 20th birthday--call that event esister--and you celebrate your 20th birthday--call that event eyou. Then the question is: is esister before, after, or at the same time as eyou?

You can say that esister is definitely before eyou if it is possible to send a signal from the first to the second. If you receive a video transmission of her birthday party, and it arrives before you celebrate your birthday, then you know for sure that her birthday happened earlier.

You can say that esister is definitely after eyou if it is possible to send a signal from the second to the first. If she receives a video transmission of your birthday party, and it arrives before she celebrates her birthday, then you know for sure that your birthday happened earlier.

But if it is not possible to send a signal in either direction in time--your video of birthday arrives after her birthday, and vice-verse--then the two events are called "spacelike separated". According to some coordinate systems, your sister's birthday was first. According to some other coordinate systems, your birthday was first. According to yet other coordinate systems, the two events happened at the same time. As far as the physics is concerned, there is no "right" answer; any one of those coordinate systems are just as good as any other.
P: 21
 Quote by stevendaryl Well, if it makes you happy to think that your sister always has a definite age, no matter how far away she is, just pick a coordinate system, any coordinate system, and always use that one.
I don't think you understood what I was saying. If I was on a rocket trip, every time I thought about my sister back home, I would know that she was somewhere right then, and I would know that she was doing something right then. I would want to know how to figure out how old she was right then, because I would want to imagine what she might be doing right then. If the captain told me that the navigator says she is 2 years old right then, and that the driver says she is 30 years old right then, and if the captain told me that they were both right, I would never believe that. She has some age right then, and it's not just some choice or pick that people on that rocket can make. I might not know how to figure out what her age is, but she has some particular age right then, and everyone on that rocket should figure the same age for her, I'm sure of that.
Thanks
P: 3,676
 Quote by Underwood I don't think you understood what I was saying. If I was on a rocket trip, every time I thought about my sister back home, I would know that she was somewhere right then, and I would know that she was doing something right then. I would want to know how to figure out how old she was right then, because I would want to imagine what she might be doing right then. If the captain told me that the navigator says she is 2 years old right then, and that the driver says she is 30 years old right then, and if the captain told me that they were both right, I would never believe that. She has some age right then, and it's not just some choice or pick that people can make. I might not know how to figure out what her age is, but it's some real age, I'm sure of that.
You're still using that evil and treacherous phrase "right then" :)

If the captain, the driver, and you are all using the same reference frame, as you likely are because you're all on the same ship, then you will all be talking about the same thing when you say "right then", so you all will have the same answer for the "How old is she and what is she doing right now?" question (see, I slipped those evil and treacherous words in too; otherwise the question would be well and thoroughly unanswerable).

However, someone else in another spaceship could go zooming by at a high rate of speed relative to you, so is using a different reference frame. As he passes your ship, you call out "How old is my sister right now?", and he shouts his answer back to you. It could be different from the answer that you got from your ship's crew - and they'd both be right.