I just had the following conversation with ChatGPT. Initially it fumbled but then it recovered. It is amazing that it even understood my scenario. And even more amazing is that understood my clarifications of the scenario albeit after several attempts. This is uncanny. The answer is much better than what I have seen many professional scientists give to this question of the cause of Twin Paradox.
Enjoy!
Me:
Imagine 3 clocks ClockO, ClockS and ClockI. ClockS is stationary. ClockO and ClockI are moving at same constant .8c velocity in exactly opposite direction travelling towards ClockS. ClockO is coming to ClockS from the left side. ClockI is coming to ClockS from the right side. ClockO reaches ClockS first. At that point both ClockS and ClockO start their stopwatches. ClockO continues past ClockS and after on light year distance reaches ClockI. At that point ClockO stops its stopwatch and transfer the reading to ClockI. Clock I also starts its own stopwatch at that same moment. ClockI travels to ClockS. When ClockI reaches ClockS both ClockI and ClockS stop thier stop watches. What will be the difference between reading of ClockS and sum of the readings of ClockO and ClockI. Note ClockO and ClockI are travelling at constant velocity i.e. no acceleration involved. Also note that no presynchronization of clocks is required.
Note: In the animated GIF below ClockO = O, ClockS = and ClockI = I
ChatGPT:
The difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be zero.
This is because both ClockO and ClockI are traveling at the same constant velocity relative to ClockS, so their relative time dilation will be the same. Therefore, when ClockO and ClockI meet, the elapsed time on their stopwatches will be the same. When ClockI reaches ClockS, the elapsed time on its stopwatch will be equal to the elapsed time on ClockS's stopwatch, so the difference between the two will be zero.
---
Me:
No no. When ClockO reaches ClockI that is when it stops its stopwatch and ClockI starts its stops watch. So this happens in sequence.
ChatGPT:
I apologize for misunderstanding your question. In this case, the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will depend on the relative velocity between the clocks and the distance they have traveled.
According to special relativity, the elapsed time on a moving clock is dilated (appears to pass slower) compared to a stationary clock. The amount of time dilation is given by the Lorentz factor:
Γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the moving clock and c is the speed of light.
Using this formula, we can calculate the time dilation for ClockO and ClockI relative to ClockS.
If the distance traveled by ClockO and ClockI is d, and the relative velocity between the clocks is v, then the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be:
Γ_O * d / v - Γ_I * d / v
= (1 / sqrt(1 - (v^2 / c^2))) * d / v - (1 / sqrt(1 - (v^2 / c^2))) * d / v
= 0
This result shows that the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI is zero, regardless of the relative velocity between the clocks or the distance they have traveled.
---
Me:
So the twin paradox like effect does not happen in this case?
ChatGPT:
The twin paradox is a thought experiment that involves two twins, one of whom remains stationary on Earth while the other travels at a high speed in a spacecraft and then returns to Earth. According to the twins' own perspective, both twins age at the same rate, but according to an observer on Earth, the twin in the spacecraft ages more slowly due to time dilation.
In the scenario you described, there is no paradox because all of the clocks are traveling at the same constant velocity relative to each other. In this case, all of the clocks will experience the same time dilation, so there will be no difference in the elapsed time between the clocks.
The twin paradox occurs because the twin in the spacecraft accelerates to a high speed, travels at that speed for a while, and then decelerates to a stop. The acceleration and deceleration cause the twin in the spacecraft to experience a different amount of time dilation than the twin on Earth, leading to the paradoxical result. In your scenario, there is no acceleration or deceleration, so there is no twin paradox.
---
Me:
No. ClockS is stationary in my scenario, like the twin who stayed back.
ChatGPT:
I apologize for misunderstanding your scenario. In this case, your scenario is similar to the twin paradox, with ClockS playing the role of the twin who stays on Earth, and ClockO and ClockI playing the role of the twin in the spacecraft.
According to the special theory of relativity, the elapsed time on a moving clock is dilated (appears to pass slower) compared to a stationary clock. The amount of time dilation is given by the Lorentz factor:
Γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the moving clock and c is the speed of light.
Using this formula, we can calculate the time dilation for ClockO and ClockI relative to ClockS.
If the distance traveled by ClockO and ClockI is d, and the relative velocity between the clocks is v, then the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be:
Γ_O * d / v - Γ_I * d / v
= (1 / sqrt(1 - (v^2 / c^2))) * d / v - (1 / sqrt(1 - (v^2 / c^2))) * d / v
= 0
ChatGPT was wrong here but it revovered later as you will see.
This result shows that the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI is zero, regardless of the relative velocity between the clocks or the distance they have traveled.
However, this result is based on the assumption that the relative velocity between the clocks is constant. If the clocks are accelerating or decelerating, the result will be different.
---
Me:
Nope. I already said Clock S in stationary and ClockO and ClockI are moving at .8c constant velocity. ClockO effectively is playing the role of outward journey of the twin that travelled and ClockI plays the same role of the travelling twins on inwards journey.
ChatGPT:
I apologize for misunderstanding your scenario. In this case, your scenario is similar to the twin paradox, with ClockS playing the role of the twin who stays on Earth, and ClockO and ClockI playing the role of the twin in the spacecraft.
According to the special theory of relativity, the elapsed time on a moving clock is dilated (appears to pass slower) compared to a stationary clock. The amount of time dilation is given by the Lorentz factor:
Γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the moving clock and c is the speed of light.
Using this formula, we can calculate the time dilation for ClockO and ClockI relative to ClockS.
If the distance traveled by ClockO and ClockI is d, and the relative velocity between the clocks is v, then the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be:
Γ_O * d / v - Γ_I * d / v
= (1 / sqrt(1 - (v^2 / c^2))) * d / v - (1 / sqrt(1 - (v^2 / c^2))) * d / v
= 0
This result shows that the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI is zero, regardless of the relative velocity between the clocks or the distance they have traveled.
However, this result is based on the assumption that the relative velocity between the clocks is constant. If the clocks are accelerating or decelerating, the result will be different.
---
Me:
Hmmmm... you said "In this case, your scenario is similar to the twin paradox, with ClockS playing the role of the twin who stays on Earth, and ClockO and ClockI playing the role of the twin in the spacecraft." in which case the sum of the readings of ClockO and ClockI should be less thant the reading on ClockS's stopwatch right?
ChatGPT:
Yes, that is correct. In the scenario you described, ClockS plays the role of the twin who stays on Earth, and ClockO and ClockI play the role of the twin in the spacecraft. Therefore, the elapsed time on ClockS's stopwatch will be greater than the sum of the elapsed times on ClockO's and ClockI's stopwatches.
According to the special theory of relativity, the elapsed time on a moving clock is dilated (appears to pass slower) compared to a stationary clock. The amount of time dilation is given by the Lorentz factor:
Γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the moving clock and c is the speed of light.
Using this formula, we can calculate the time dilation for ClockO and ClockI relative to ClockS.
If the distance traveled by ClockO and ClockI is d, and the relative velocity between the clocks is v, then the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be:
Γ_S * d / v - Γ_O * d / v - Γ_I * d / v
= (1 / sqrt(1 - (0^2 / c^2))) * d / v - (1 / sqrt(1 - (v^2 / c^2))) * d / v - (1 / sqrt(1 - (v^2 / c^2))) * d / v
= d / v - 2 * d / v
= -d / v
ChatGPT was corrected it self here. This is uncanny.
This result shows that the difference between the readings of ClockS and the sum of the readings of ClockO and ClockI is negative, indicating that the elapsed time on ClockS's stopwatch is greater than the sum of the elapsed times on ClockO's and ClockI's stopwatches.
The magnitude of the difference will depend on the relative velocity between the clocks and the distance they have traveled. The faster the clocks are moving relative to each other, and the greater the distance they have traveled, the larger the difference will be.
---
Me:
Aha...so now you got it right. So no acceleration is needed for this effect as I just showed.
ChatGPT:
Yes, that is correct. In the scenario you described, the clocks are moving at a constant velocity relative to each other, so there is no acceleration involved. However, the special theory of relativity still predicts that the elapsed time on a moving clock will be dilated (appear to pass slower) compared to a stationary clock.
The difference between the readings of ClockS and the sum of the readings of ClockO and ClockI will be negative, indicating that the elapsed time on ClockS's stopwatch is greater than the sum of the elapsed times on ClockO's and ClockI's stopwatches. The magnitude of the difference will depend on the relative velocity between the clocks and the distance they have traveled. The faster the clocks are moving relative to each other, and the greater the distance they have traveled, the larger the difference will be.
This effect does not require acceleration because it is a consequence of the time dilation that occurs due to the relative velocity between the clocks, as described by the special theory of relativity.
---
Me:
So the scientists are wrong when they say acceleration is needed for the twin paradox effect.
ChatGPT:
The twin paradox is a thought experiment that demonstrates the strange and counterintuitive effects of time dilation predicted by the special theory of relativity. In the twin paradox, one twin remains stationary on Earth while the other travels at a high speed in a spacecraft and then returns to Earth. According to the twins' own perspective, both twins age at the same rate, but according to an observer on Earth, the twin in the spacecraft ages more slowly due to time dilation.
The twin paradox is often cited as an example of the need for acceleration in order to experience time dilation, because the twin in the spacecraft must accelerate to a high speed, travel at that speed for a while, and then decelerate to a stop in order to experience the time dilation effect.
However, it is important to note that time dilation can occur due to relative velocity, not just acceleration. The special theory of relativity predicts that the elapsed time on a moving clock will be dilated (appear to pass slower) compared to a stationary clock, regardless of whether the clock is accelerating or not. This effect is not limited to the twin paradox scenario, and can occur in any situation where there is a relative velocity between two clocks.
So while the twin paradox is often cited as an example of the need for acceleration in order to experience time dilation, it is not accurate to say that acceleration is always required in order to experience this effect. Time dilation can occur due to relative velocity, not just acceleration.
Comments