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wmikewells
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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?
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?