Twin paradox with acceleration

In summary, the spaceship with one twin on board leaves Earth and accelerates at a constant rate for 5 years in the frame of the ship, then decelerates for 5 years. This process is repeated twice. When the twin returns, he is 40 years old. It is assumed that they travel at relativistic speeds. The acceleration in the frame of the ship is given as 30 m/s^2 and it is known that time dilation depends on the velocity attained, which in turn depends on the acceleration. The answer to the question of how old the twin who stayed on Earth is will be a function of the acceleration in the reference frame of the ship. To solve this problem, one could write the velocity of the ship
  • #1
BlackyTheCat
4
0
A spaceship leaves earth. One twin stays back, the other is on the ship. The ship accelerates for 5 years with a constant acceleration (the 5 years are in the reference frame of the ship), then it decelerates for 5 years. Then, it turns around and does the same thing again. All the accelerations and decelerations are at the same rate. When he returns, he is 40 years old. How old is the twin who stayed on Earth?I know the following equations that could be relevant:
a = (1-v2/c2)3/2/(1+v*u/c2)*a'
Where a is the acceleration in the resting frame, a' in the moving frame, v is the velocity of the frame that is moving, and u is the velocity of the object moving within the moving frame.
Also, time dilatation formula.I know how to solve the normal twin paradox with time dilatation. Here, I assume that I should use the acceleration formula above somehow. However, I am at a loss at how. I do not know any velocities. Also, even if I could for example calculate a, I don't see how it would help. I could get gamma from it I guess, but for that I would already need gamma before, so it would be a bit redundant.

I see that if I solve the problem for one step (one acceleration or deceleration), I am done, since it is symmetric though.
 
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  • #2
BlackyTheCat said:
A spaceship leaves earth. One twin stays back, the other is on the ship. The ship accelerates for 5 years with a constant acceleration (the 5 years are in the reference frame of the ship), then it decelerates for 5 years. Then, it turns around and does the same thing again. All the accelerations and decelerations are at the same rate. When he returns, he is 40 years old. How old is the twin who stayed on Earth?I know the following equations that could be relevant:
a = (1-v2/c2)3/2/(1+v*u/c2)*a'
Where a is the acceleration in the resting frame, a' in the moving frame, v is the velocity of the frame that is moving, and u is the velocity of the object moving within the moving frame.
Also, time dilatation formula.I know how to solve the normal twin paradox with time dilatation. Here, I assume that I should use the acceleration formula above somehow. However, I am at a loss at how. I do not know any velocities. Also, even if I could for example calculate a, I don't see how it would help. I could get gamma from it I guess, but for that I would already need gamma before, so it would be a bit redundant.

I see that if I solve the problem for one step (one acceleration or deceleration), I am done, since it is symmetric though.

Are you sure you have all the information? If the acceleration is small, so that speeds are non-relativistic, then the twins would be approximately the same age.
 
  • #3
PeroK said:
Are you sure you have all the information? If the acceleration is small, so that speeds are non-relativistic, then the twins would be approximately the same age.
Yes, that is all the information given. I am assuming that they travel at relativistic speeds, since our problem sets for the last 2 weeks were on this topic, and the course is a 2nd year theoretical physics class, so I think the non-relativistic case would be silly to ask for. (Yes, the problem statements are often set terribly and there is implicit information like that you should consider relativistic speeds and alike...)
 
  • #4
BlackyTheCat said:
Yes, that is all the information given. I am assuming that they travel at relativistic speeds, since our problem sets for the last 2 weeks were on this topic, and the course is a 2nd year theoretical physics class, so I think the non-relativistic case would be silly to ask for. (Yes, the problem statements are often set terribly and there is implicit information like that you should consider relativistic speeds and alike...)

Okay, but that's not the point. The point is that the time dilation will depend on the velocity attained, which depends on the acceleration. If the acceleration is low, time dilation will be small, and if it's high enough, time dilation could be very great indeed.

Alternatively, your answer will be a function of ##a##, the acceleration in the reference frame of the ship.
 
  • #5
PeroK said:
Okay, but that's not the point. The point is that the time dilation will depend on the velocity attained, which depends on the acceleration. If the acceleration is low, time dilation will be small, and if it's high enough, time dilation could be very great indeed.

Alternatively, your answer will be a function of ##a##, the acceleration in the reference frame of the ship.

Okay, so I was now confused too and checked the problem statement again. Turns out, they changed the problem statement about an hour ago, now stating the acceleration a' in the ships frame of reference is: a'=30 m/s^2. This seems to me like a large enough acceleration to cause some time dilatation over time.

My idea right now is to write v'(t) as a function of a' (v'=a'*t), and then plug this into the formula for the acceleration that I gave before. Then I'd have a time dependent acceleration of the ship in the Earths frame of reference. Does this sound about right?

I don't see, however, how I would then get to the velocity of the ship in the resting frame in order to get some gamma out of that...
 
  • #6
BlackyTheCat said:
Okay, so I was now confused too and checked the problem statement again. Turns out, they changed the problem statement about an hour ago, now stating the acceleration a' in the ships frame of reference is: a'=30 m/s^2. This seems to me like a large enough acceleration to cause some time dilatation over time.

My idea right now is to write v'(t) as a function of a' (v'=a'*t), and then plug this into the formula for the acceleration that I gave before. Then I'd have a time dependent acceleration of the ship in the Earths frame of reference. Does this sound about right?

I don't see, however, how I would then get to the velocity of the ship in the resting frame in order to get some gamma out of that...

Try to get a formula for v in terms of a' and t.

You can get a formula relating t and ##\tau## the ship's proper time from that.
 
Last edited:
  • #7
PeroK said:
Try to get a formula for v in terms of a' and t.

You can get a formula relating t and ##\tau## the ship's proper time from that.
Thanks a lot, I was able to solve it now!
 

Related to Twin paradox with acceleration

1. What is the twin paradox with acceleration?

The twin paradox with acceleration is a thought experiment in physics that explores the effects of special relativity on twins who are moving at different velocities and experience different rates of acceleration. It is often used to illustrate the principles of time dilation and the relativity of simultaneity.

2. How does the twin paradox with acceleration work?

In the thought experiment, one twin stays on Earth while the other travels at a high speed away from Earth and then returns. The traveling twin experiences acceleration during the turnaround, while the stationary twin does not. Due to the principles of special relativity, the traveling twin will have aged less than the stationary twin upon their reunion.

3. Why is it called a paradox?

The twin paradox with acceleration is called a paradox because it seems to contradict common sense and everyday experience. In the thought experiment, the traveling twin experiences less time, even though they are the ones in motion. This goes against our intuition that time should pass at the same rate for all observers.

4. What is the significance of the twin paradox with acceleration?

The twin paradox with acceleration is significant because it highlights the implications of special relativity, a fundamental theory in physics. It demonstrates how time and space are relative, and how the rate of time can be affected by acceleration and velocity. This thought experiment has also been used to develop and confirm the principles of special relativity.

5. Is the twin paradox with acceleration a real phenomenon?

While the twin paradox with acceleration is a thought experiment, its principles are based on real phenomenon. Time dilation and the relativity of simultaneity have been observed and confirmed through experiments and observations, such as the famous Hafele-Keating experiment. However, the twin paradox with acceleration itself has not been observed in real life, as it requires traveling at immense speeds and experiencing significant accelerations.

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