Simple Twin Paradox Resolution Using Rocket Man Traveling from Earth to Planet Claire

[wmikewells]

I worked through a simple twin paradox problem, and I think I have the correct solution. However, I wanted to confirm that my numbers are correct. There is nothing fancy about the solution, but it resolves the "paradox" completely for me. Every other resolution I have looked at either leaves some information out or just confuses me. Here are the situation, the states, the paradox, and the resolution.

Situation:
1. Earth and Planet Claire are 10 light years from each other.
2. Rocket Man will travel from Earth to Planet Claire.
3. State 1: Rocket Man is initially parked on Earth
4. State 2: Rocket Man has just completed an acceleration to reach 0.8 c in 1 second heading toward Planet Claire.
5. After acceleration, Rocket Man will coast the rest of the way until he reaches Planet Claire
6. State 3: Rocket Man is just about to decelerate in 1 second in order to land Planet Claire
7. State 4: Rocket Man has completed deceleration and has come to rest on Planet Claire

Simplifications:
1. The 1 second acceleration and deceleration is ignored in overall time measurements since they are insignificant given the years it takes to complete the full trip.
2. Information delay and the Doppler Effect are ignored; it is assumed there is a way to record and communicate times effectively.
3. All clocks are synchronized at State 1
4. Gravity on Earth and Planet Claire is ignored.

Paradox: How can Rocket Man arrive at Planet Claire with less time on his clock when the clock on Planet Claire must run more slowly than Rocket Man's clock from Rocket Man's viewpoint?

Table 1 - Snapshots of All Measurements from All Viewpoints at the Four States

State 1 - Rocket Man Parked on Earth (Everyone agrees)
____Rocket Man's Viewpoint
________Length between Planets___10 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______0 years
____Earth's Viewpoint
________Length between Planets___10 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______0 years
____Planet Claire's Viewpoint
________Length between Planets___10 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______0 years

State 2 - Just after the One Second Acceleration
____Rocket Man's Viewpoint
________Length between Planets____6 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______8.0 years
____Earth's Viewpoint
________Length between Planets___10 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______0 years
____Planet Claire's Viewpoint
________Length between Planets___10 light years
________Rocket Man's Clock________0 years
________Earth's Clock_____________0 years
________Planet Claire's Clock_______0 years


State 3 - Just Before the One Second Deceleration
____Rocket Man's Viewpoint
________Length between Planets____6 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________4.5 years
________Planet Claire's Clock_______12.5 years
____Earth's Viewpoint
________Length between Planets____10 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________12.5 years
________Planet Claire's Clock_______12.5 years
____Planet Claire's Viewpoint
________Length between Planets____10 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________12.5 years
________Planet Claire's Clock_______12.5 years


State 4 - Rocket Man Has Landed on Planet Claire (Everyone agrees)
____Rocket Man's Viewpoint
________Length between Planets____10 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________12.5 years
________Planet Claire's Clock_______12.5 years
____Earth's Viewpoint
________Length between Planets____10 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________12.5 years
________Planet Claire's Clock_______12.5 years
____Planet Claire's Viewpoint
________Length between Planets____10 light years
________Rocket Man's Clock________7.5 years
________Earth's Clock_____________12.5 years
________Planet Claire's Clock_______12.5 years

Calculations and Solution:
1. The conversion factor for time and distances when traveling at 0.8 c is 0.6 = (1 - (0.8 c)^2)^0.5
2. From the viewpoint on Earth and Planet Claire, the time for Rocket Man to travel to Planet Claire is 12.5 years = 10 light years / 0.8 c
3. From the viewpoint of Earth and Planet Claire, the time Rocket Man's clock will advance during the trip is 7.5 years = 12.5 years * 0.6
4. From the viewpoint of Rocket Man, the time the clocks on Earth and Planet Claire will advance during the coasting trip (State 2 to State 3) is 4.5 years = 7.5 years * 0.6
5. Consequently, the clock on Planet Claire will advance 8.0 years from Rocket Man's viewpoint when Rocket Man accelerates, and Earth's clock from Rocket Man's viewpoint will advance 8.0 years when Rocket Man decelerates. I didn't calculate these values using General Relativity. I just inferred them.

Question:
1. Are these numbers correct, or do I need to go back to the drawing board?

 

[yuiop]

Calculations and Solution:
1. The conversion factor for time and distances when traveling at 0.8 c is 0.6 = (1 - (0.8 c)^2)^0.5
2. From the viewpoint on Earth and Planet Claire, the time for Rocket Man to travel to Planet Claire is 12.5 years = 10 light years / 0.8 c
3. From the viewpoint of Earth and Planet Claire, the time Rocket Man's clock will advance during the trip is 7.5 years = 12.5 years * 0.6
4. From the viewpoint of Rocket Man, the time the clocks on Earth and Planet Claire will advance during the coasting trip (State 2 to State 3) is 4.5 years = 7.5 years * 0.6
5. Consequently, the clock on Planet Claire will advance 8.0 years from Rocket Man's viewpoint when Rocket Man accelerates, and Earth's clock from Rocket Man's viewpoint will advance 8.0 years when Rocket Man decelerates. I didn't calculate these values using General Relativity. I just inferred them.

Question:
1. Are these numbers correct, or do I need to go back to the drawing board?

Looks correct to me :)

 

[HallsofIvy]

Although you are using the word "paradox" in a very unusual way and nothing you say has anything to do with the "twin paradox".

 

[wmikewells]

Looks correct to me :)

Thanks for the review. I have a starting point now for further exploration.

 

[wmikewells]

Although you are using the word "paradox" in a very unusual way and nothing you say has anything to do with the "twin paradox".

I know. I simplified the classic twin paradox to make it more manageable for me. I could have given Rocket Man a Rocket Man II twin that stayed on Earth while Rocket Man made the trip to Planet Claire and then returned. I think the return trip would be a duplicate of the outgoing trip (the numbers would be the same). So I just reduced the problem to address the outgoing trip. I figured if I could solve that, I could make sense of the twin paradox.

Also, I have read that acceleration is not required to create an age difference. However, I have yet to get to that point in my understanding of SR to make sense of that statement. So, I made headway on what I could understand.

 

[vin300]

5. Consequently, the clock on Planet Claire will advance 8.0 years from Rocket Man's viewpoint when Rocket Man accelerates, and Earth's clock from Rocket Man's viewpoint will advance 8.0 years when Rocket Man decelerates. I didn't calculate these values using General Relativity. I just inferred them.

Question:
1. Are these numbers correct, or do I need to go back to the drawing board?

When the rocket man has completed acceleration, neglecting the 1 second time as you said, the clock on planet Claire from rocket man's viewpoint will lag behind by vx/c^2, where x is the contracted length. Plugging in, it will be -0.8*6= -4.8 years. Also, IMO, the Earth clock from the rocket man's viewpoint will have advanced 12.5*0.6 =7.5 years when he decelerates.

 

[wmikewells]

When the rocket man has completed acceleration, neglecting the 1 second time as you said, the clock on planet Claire from rocket man's viewpoint will lag behind by vx/c^2, where x is the contracted length. Plugging in, it will be -0.8*6= -4.8 years. Also, IMO, the Earth clock from the rocket man's viewpoint will have advanced 12.5*0.6 =7.5 years when he decelerates.

What does that do to the clocks in my table above?

 

[vin300]

According to the time dilation formula, after the rocket man has completed acceleration, he will see the time on planet claire's clock to be -4.8 years. Adding this to the time on Earth's clock as seen by the rocket man after he has completed the journey, the time on planet claire's clock as seen by the rocket man when he has reached planet claire is -4.8+4.5= -0.3 years. However, this does not look correct to me because when the rocket man reaches planet claire, the time shown by the latter's clock is 12.5 years.

 

[GeorgeDishman]

I didn't calculate these values using General Relativity. I just inferred them.

The numbers are correct. The other way to calculate them is to assign coordinates to events and apply the Lorentz Transforms. You don't need GR, SR can handle changes of frames and even slower accelerations if you don't mind integrating.

 

[GeorgeDishman]

According to the time dilation formula, after the rocket man has completed acceleration, he will see the time on planet claire's clock to be -4.8 years. Adding this to the time on Earth's clock as seen by the rocket man after he has completed the journey, the time on planet claire's clock as seen by the rocket man when he has reached planet claire is -4.8+4.5= -0.3 years. However, this does not look correct to me because when the rocket man reaches planet claire, the time shown by the latter's clock is 12.5 years.

The change on Claire's clock as a result of the acceleration is also due to the relativity of simultaneity. The t=0 event on Claire 5 light years away becomes 6 years in the past and 8 light years away in the new coordinates with Claire moving towards Earth.

 

[GeorgeDishman]

Also, I have read that acceleration is not required to create an age difference. However, I have yet to get to that point in my understanding of SR to make sense of that statement.

Just have the rocketman travel at constant speed starting a little way from earth. He synchronises his clock with Earth time as he passes. He can sync with another ship returning to Earth as they pass at Claire and that ship compares it's clock when it passes Earth. You then have the twins scenario without accelerations.

 

[vin300]

How does the clock on planet Claire advance 8 years from the rocketman's viewpoint when he accelerates?

 

[GeorgeDishman]

How does the clock on planet Claire advance 8 years from the rocketman's viewpoint when he accelerates?

Think of a spacetime diagram. The acceleration looks like a curve in his worldline. The hypersurface he calls "now" is perpendicular to his worldline at any instant so it rotates as his line curves, the line he calls "now" sweeps over the worldline of the distant planet Claire.

If you were in the back seat of a car looking out the side, distant trees seem to flow past you. If the car turns away from them, they seem to move faster but it is only because the perpendicular to your motion is rotating.

In spacetime, the negated sign for time means the rocketman sees distant clocks advance if he accelerates towards them rather than away.

 

[wmikewells]

The numbers are correct. The other way to calculate them is to assign coordinates to events and apply the Lorentz Transforms. You don't need GR, SR can handle changes of frames and even slower accelerations if you don't mind integrating.

Thanks for the confirmation. I'll see if I can figure this out.

 

[wmikewells]

Just have the rocketman travel at constant speed starting a little way from earth. He synchronises his clock with Earth time as he passes. He can sync with another ship returning to Earth as they pass at Claire and that ship compares it's clock when it passes Earth. You then have the twins scenario without accelerations.

I'll have to play around to see if I can get that to work.