Simple resolution to the twin paradox?

Question

There are a ton of videos that cover the twin paradox. They're all quite involved and don't really get to the heart of the matter, which is the fundamental asymmetry between the two observers (as explained by: https://youtu.be/nRuVGOm7560 ).

I have a very simple resolution of the paradox, but it's so simple that I think it's incorrect. I'm wondering where the flaw in my logic is. Does the explanation below explain something that countless Youtubers (including Fermilab and Minutephysics) have tried to explain? I have my doubts since my explanation is too simple, so please let me know what you think!

Here's the setup: Alice and Bob are twins and are standing next to each other in space. There is a planet far away. Bob, Alice and the planet are not moving relative to each other. Bob accelerates by using a rocket, travels to the far away planet, then accelerates in the opposite direction by using a rocket, retraces his steps and finally rejoins Alice. For the sake of simplicity, let's assume both Alice and Bob live in a special universe where there are no objects apart from the Alice and Bob, Bob's rocket and the far away planet.

Result: Alice is now older.

Question: From Alice's perspective, Bob accelerated, travelled far away and then returned. From Bob's perspective, it is Alice that accelerated, travelled far away then returned. It seems like a perfectly symmetric situation, but Bob has nonetheless aged more slowly. Why is that?

Source of asymmetry (absurdly simple, so possibly incorrect): Bob and Alice disagree on the distance that Bob covered. Due to length contraction, Bob thinks he travelled a shorter distance.

Let's put some numbers:

• The planet is originally 18.8 light years away
• Bob accelerates very quickly to 0.9c
• As soon as Bob reaches 0.9c, the far away planet is around 8.2 light years away from his perspective since 8.2 ~= 18.8 * sqrt(1-(0.9c)^2/c^2)
• From Bob's point of view, the round trip was 16.4 light years
• From Alice's point of view, the round trip was 37.6 light years
• In other words, Bob's odometer reads 16.4 light years but Alice clearly saw Bob travel to a planet 18.8 light years away and back
• Since bob travelled at 0.9c and his odometer reads 16.4 light years, only 18.2 years have passed for him.
• For Alice, since Bob travelled 37.6 light years, a total of 41.8 years have elapsed

Let's assume that Bob accelerated twice. Originally, Bob was stationary. Then, Bob started moving away from Alice. He went from 0 to 0.9c. let's assume he did so in 1 second. Then, when he reached the planet, he had to change the direction of his velocity so that he could return to Alice. Let's assume he also did that in 1 second.

Let's do the calculations from Bob's perspective:

• Bob was stationary the whole time, Alice and the planet are the ones that moved
• During the initial acceleration, from his perspective, the far away planet moved from 18.8 light years away to 8.2 light years away in the span of 1 second
• For the "first leg of the voyage", Bob observed the planet come towards him at 0.9c. The planet travelled 8.2 light years from Bob's perspective
• At the same time, Alice was moving away from Bob. Alice travelled X light years (I'm not sure how to calculate X) in the same time that it took the planet to reach Bob.
• When "the planet reaches Bob", Alice goes from being X light years away to 8.2 light years away, so Alice travelled (X - 8.2) light years in 1 second.
• Then Alice travels the 8.2 light years to be reunited with Bob

Could someone tell me if this is accurate from Bob's perspective? Also, how do I calculate X? Is it simply (lorentz factor * 18.8) = 43.1 ?

Answer 1

I'm sorry, Jacques, but as you suspected your explanation is incorrect. The twin paradox seems paradoxical because time dilation is entirely symmetrical, so both Bob and Alice might be expected to be time dilated in the frame of the other. However- and this is the resolution of the paradox- the time dilation formula of SR applies only to motion in an inertial frame, and since Bob is not moving inertially the symmetry condition does not apply.

At the same time, length contraction is also symmetrical in the same way that time dilation is, so your explanation simply replaces one paradox (why does Bob age less?) with another (why does Bob travel further?).

The twin paradox is a consequence of the hyperbolic geometry of spacetime. If you follow a curved or kinked path between two events, the elapsed time along that path will always be less than that along a straight path.

Answer 2

The calculation you performed clearly shows that distance depends on the observer. This does not resolve the twin paradox since if Alice does the math instead of Bob's perspective then you would get the opposite.

You almost resolve the paradox by simply acknowledging who accelerates and when. At first both would argue that the other one is accelerating and are moving away. This is a symmetric situation. The asymmetry comes when any one of them starts to accelerate in the opposite direction. There will now only be one of them who experiences a force. This resolves who is the one that will age more compared to the other.

Answer 3

From Alice's perspective, Bob accelerated, travelled far away and then returned. From Bob's perspective, it is Alice that accelerated, travelled far away then returned.

No. For both Alice and Bob, it is Bob who changed velocity.

Bob does not have a single, unique, inertial frame during his trip. Alice does.

It is this fundamental mistake of failing to understand this fact that leads to the false claim that there is symmetry between Alice and Bob, and to the mistaken appearance of paradox.

Virtually all of the proposed paradoxes involving SR disappear if one solves what will happen in one inertial frame, and then performs Lorentz Transformations to find what will happen in another inertial frame. The Lorentz transformation will change the space-time coordinates of events, but will not not fundamentally change what happens. If when Alice and Bob meet and their watches differ, that same difference will appear in all inertial frames. If two objects collide in one frame, they will collide in all inertial frames. If they don't collide in one inertial frame, they will not collide in any frame, etc. etc.

Answer 4

Ronald Hatch proved to us (and it is re-proven every single day with GPS satellites) that the twin paradox simply does not exist. And it does not have anything to do with reversing course or an inertial or non-inertial frame. Look at Hatch's paper Relativity and GPS to see the full answer.

We all know that the clocks on GPS satellites are adjusted to run at a different rate than earth bound clocks, to adjust for gravitational and kinetic time dilation. Other satellites and intercontinental ballistic missiles rely on GPS for their own position information. We would think that if a satellite is moving around the earth following closely behind a GPS satellite, then there would be no difference in their clock rate relative to the GPS satellite. Likewise if a satellite was moving around the earth in the opposite direction, there would be a large adjustment of the clock rate relative to the GPS satellite. But this does NOT happen.

On the first page of the paper he says "it is very important to note that the GPS satellites clock rate and the receivers clock rate are NOT adjusted as a function of their velocity relative to one another. Instead they are adjusted as a function of their velocity with respect to the chosen frame of reference - in this case the earth-centered, non-rotating, (quasi) inertial frame."

What this means is that all of these flying objects have their clocks adjusted relative to a single time on earth and not against each other as would be indicated by the common interpretation of special relativity and by the twin paradox. So the entire question about the twin paradox simply does not occur.