Symmetry Argument: Resolving the Twin Paradox Contradiction

[Gear300]

Precisely 2 years off, because it takes 2 years for the light from space station's clock to reach your eye. But, since your clocks are synchronized and you're motionless wrt the space station, you just add 2 years to correct for light propagation and conclude that time on the station = time shown by your wristwatch.

At the moment of takeoff in your reference frame, the clock on the space station reads TL, the clock as you see it reads TL - 2 years.
At the moment you pass by the station, its clock will read TL + 2 / 0.6 = TL + 3.33 years. Your own clock will read TL + (2 / 0.6) * sqrt(1-0.6*0.6) = TL + 2.67 years.

I edited the post a bit...but altogether, the time you read on the space station's clock would appear the same compared to what the inhabitants read, right?...even if the space station clock was going slower than your clock in your frame?

 

[hamster143]

If you're at the same point as the inhabitants of the station, you'll see exactly the same reading as them.

 

[Gear300]

If you're at the same point as the inhabitants of the station, you'll see exactly the same reading as them.

I see...the point of my confusion is that while moving with a relative velocity of 0.6c, you would observe slower time intervals for the space station clock compared to your clock's time intervals...but when you read the time while in the space station, it turns out to have a larger reading than yours. Would this be because the star cruiser had to accelerate before it hit 0.6c?

 

[hamster143]

I see...the point of my confusion is that while moving with a relative velocity of 0.6c, you would observe slower time intervals for the space station clock compared to your clock's time intervals...but when you read the time while in the space station, it turns out to have a larger reading than yours. Would this be because the star cruiser had to accelerate before it hit 0.6c?

There's a difference between what you SEE and what the time on the station IS in your reference frame.

When your cruiser takes off, you SEE the reading equal to TL - 2 years. (Agreed?) When you arrive at the station, you see the reading of TL + 3.33, even though only 2.67 years have passed by your clock. So if you were to look at the station through the telescope during the trip, you'd SEE their clock ticking FASTER than yours ... but that would be because you're moving toward them - basic Doppler effect. Even though their clock is 'really' ticking slower than yours.

 

[JesseM]

Alright...apparently, I hit another wall.

Let us say a space station X is stationary relative to where you are and has a clock that is synchronized with your time. Because ss X is quite a distance (2 light years) from where you are, if you were to observe its time using a telescope, the reading would be some time interval C off.

These problems are easier to follow if you use specific numbers...if the space station X is 2 light years away from you in your frame, and its clock is synchronized with yours in this frame, then obviously when you look through the telescope it will be 2 years behind!

You are in a star cruiser and are preparing for take off. Time T_L is the take off time - at that time, you record the time in ss X as T_L + C.

It would be T_L - C, not + C...when you look through a telescope you don't see things as they will be in the future, but rather how things were in the past, since the light from past events is just reaching you now. So, just to have some specific numbers to work with, say that T_L = 10 years...in this case, at that moment you see the synchronized clock on the station reading 8 years in your telescope. But keep in mind that there's a difference between what you see visually and what's actually happening in your frame...in your frame immediately before takeoff, when you are still at rest relative to the station, both your clock and the station's clock "actually" read 10 years since they are synchronized in this frame, in spite of the fact that you only see the station clock reading 8 years. On the other hand, once you are in motion at 0.6c relative to the station you have a different rest frame, and because of the relativity of simultaneity it's no longer true in this second frame that the station clock was synchronized with your clock the moment after takeoff...specifically, the relativity of simultaneity says that if two clocks are synchronized and a distance D apart in their own rest frame, then in a frame where they are moving at speed v, they will be out-of-sync by vD/c^2. So from the perspective of a frame moving at 0.6c relative to the station, at the moment before you took off your clock and the station clock were out-of-sync by (0.6c)(2 light years)/c^2 = 1.2 years, so if your clock read 10 years at that moment the station clock read 11.2 years at the same moment (though this frame would still agree with the prediction that you'd see the station clock reading 8 years at the moment your clock read 10 years). Assuming the acceleration was brief, the moment after takeoff (when you come to rest in this frame) your clock still reads 10 years and the station clock still reads 11.2 years in this frame.

Also, note that in this frame the distance between the takeoff point and the station is less than 2 light years due to Lorentz contraction, instead it's only 2*\sqrt{1 - 0.6^2} = 2*0.8 = 1.6 light years away. And once your ship accelerates and comes to rest in this frame, the station is moving towards you at 0.6c, so it'll take 1.6 light years/0.6c = 2.666... years to reach you. Since you were at rest in this frame, your clock ticked forward by 2.666... years in this time, so your clock reads 10 + 2.666... = 12.666... years when you pass the station. On the other hand, the station's clock was slowed down by a factor of 0.8 so it only ticked forward by 0.8*2.666... = 2.1333... years during this time, so when you pass the station the station's clock reads 11.2 + 2.1333... = 13.333... years.

As it turns out, this is exactly the same thing you'd predict if you instead analyzed the situation from the perspective of the station's frame. In the station frame, at the moment of takeoff both your clock and the station's clock read 10 years, and the distance is 2 light years, so your ship will take 2/0.6 = 3.333... years to reach the station. The station's clock is ticking at normal speed in this frame since it's at rest, so the station's clock will read 10 + 3.333... = 13.333... years when you pass it. On the other hand, your clock was slowed down by a factor of 0.8 in this frame, so your clock only ticks forward by 0.8 * 3.333... = 2.666... years during the journey, so your clock will read 10 + 2.666... = 12.666... years when you pass the station. Thus you can see that both frames predict that your clock reads 12.666... years and the station's clock reads 13.333... years when you pass the station, despite the fact that they disagree about whose clock was ticking slowly during the journey. In general, it is always true in relativity that different frames will agree on local events like what two clocks read at the moment they pass one another; if they didn't, you could get genuine physical contradictions (imagine one clock was connected to a bomb that would explode when the clock reached a certain time--if different frames could disagree on what time that clock read when it passed a second clock, they could disagree on whether the bomb would go off when it was right next to the second clock and destroy it, or whether it would go off when it was far away from the second clock, leaving the second clock unharmed!)

 

[Gear300]

When your cruiser takes off, you SEE the reading equal to TL - 2 years. (Agreed?) When you arrive at the station, you see the reading of TL + 3.33, even though only 2.67 years have passed by your clock. So if you were to look at the station through the telescope during the trip, you'd SEE their clock ticking FASTER than yours ... but that would be because you're moving toward them - basic Doppler effect.

Alright...I'm seeing it now. That was a good explanation.
An additional question: Since the time-reading error was in the distance and speed of light, was this what allowed for the statement that information cannot travel faster than light?

It would be T_L - C, not + C...when you look through a telescope you don't see things as they will be in the future, but rather how things were in the past, since the light from past events is just reaching you now. So, just to have some specific numbers to work with, say that T_L = 10 years...in this case, at that moment you see the synchronized clock on the station reading 8 years in your telescope. But keep in mind that there's a difference between what you see visually and what's actually happening in your frame...in your frame immediately before takeoff, when you are still at rest relative to the station, both your clock and the station's clock "actually" read 10 years since they are synchronized in this frame, in spite of the fact that you only see the station clock reading 8 years. On the other hand, once you are in motion at 0.6c relative to the station you have a different rest frame, and because of the relativity of simultaneity it's no longer true in this second frame that the station clock was synchronized with your clock the moment after takeoff...specifically, the relativity of simultaneity says that if two clocks are synchronized and a distance D apart in their own rest frame, then in a frame where they are moving at speed v, they will be out-of-sync by vD/c^2.

I think I'm beginning to see how this would turn out...this would be an implied consequence of the principle of relativity, right?

 

[JesseM]

I think I'm beginning to see how this would turn out...this would be an implied consequence of the principle of relativity, right?

Don't quite understand what you mean...what specific aspect of what I was saying do you think would be an implied consequence of the principle of relativity?

 

[Gear300]

Don't quite understand what you mean...what specific aspect of what I was saying do you think would be an implied consequence of the principle of relativity?

It might be better not to give mind to that statement.