Time Dilation Paradox: Interstellar Race

[CakeOrDeath?]

Hi, I asked a theoretical question here a month or two ago and someone was able to help me. I have another:

Imagine an interstellar race. Starship 1 heads from Earth toward Alpha Centauri at a constant rate close to the speed of light, then makes a loop and returns home. Ship 2 is faster, and therefore has been given a handicap and leaves after Ship 1, but it is still fast enough to make up that distance and reach Alpha Centauri before Ship 1.

Wouldn't Ship 2 have experienced greater time dilation and therefore return further into Earth's future than the slower moving Ship 1, despite reaching Centauri first? This seems like a paradox.

I'm still trying to get a grasp on this subject, so thanks in advance.

 

[bcrowell]

Wouldn't Ship 2 have experienced greater time dilation and therefore return further into Earth's future than the slower moving Ship 1, despite reaching Centauri first? This seems like a paradox.

If "further into Earth's future" refers to time as measured by clocks on earth, then time dilation is irrelevant here. You only need to worry about time dilation if you want to compare times measured in two different frames of reference.

 

[CakeOrDeath?]

Yes, I'm referring to clocks on earth. I'm not sure I understand your response. Wouldn't a person on a near-light speed ship remain young relative to someone on earth?

 

[roel]

Yes one would, CakeorDeath. But to answer your original post, you already stated that ship 2 would arrive before ship 1, and dilation effects just the clock on the ships in relation to the one on Earth (hence the difference in time past on Earth and on the ship). This meaning ship 2 arrives before 1 and there is no paradox.

 

[yuiop]

If you don't mind I would like to make the experiment a little simpler so that it might be easier to understand what is going on. Let us say Ship 1 and 2 leave at the same time. Ship 2 is going much faster. When Ship 1 gets to AC, both ships simultaneously (in the Earth frame) turn around and head back to Earth. Ship 2 actually went a long way past AC. Both ships return to Earth 10 years after they left. Which ship has traveled the furthest into Earth's future? Neither, as they have both traveled 10 years into Earth's future just as anyone that remains on Earth has. On the other hand, Ship 2's passengers have experienced the least elapsed proper time relative to time on Earth, which is probably what you are getting at. Thinking of time dilation in terms of traveling into the future is just confusing.

<EDIT> As Janus points out, the round trip time would actually be less than 10 years as AC is only 4.3 ly away.

 

[K^2]

What you can end up with is one ship taking less time to travel by ship's time, and the other taking less time by Earth's time. There is still no paradox, though.

 

[CakeOrDeath?]

If you don't mind I would like to make the experiment a little simpler so that it might be easier to understand what is going on. Let us say Ship 1 and 2 leave at the same time. Ship 2 is going much faster. When Ship 1 gets to AC, both ships simultaneously (in the Earth frame) turn around and head back to Earth. Ship 2 actually went a long way past AC. Both ships return to Earth 10 years after they left. Which ship has traveled the furthest into Earth's future? Neither, as they have both traveled 10 years into Earth's future just a s anyone that remains on Earth has. On the other hand, Ship 2's passengers have experienced the least elapsed proper time relative to time on Earth, which is probably what you are getting at. Thinking of time dilation in terms of traveling into the future is just confusing.

This is for a science fiction story, which is why I worded it that way. Unfortunately, my fiction mind works better than my science mind. Your explanation makes sense to me, though.

May I ask another question? My story revolves around the idea of someone leaving and then returning roughly 40 years into Earth's future, while staying relatively young. Is Alpha Centauri theoretically far enough away for this to happen at high speeds, or is it necessary for my character to travel further away?

I really appreciate your help.

 

[Janus]

May I ask another question? My story revolves around the idea of someone leaving and then returning roughly 40 years into Earth's future, while staying relatively young. Is Alpha Centauri theoretically far enough away for this to happen at high speeds, or is it necessary for my character to travel further away?

I really appreciate your help.

No, it isn't. The time that passes on Earth would be the time that it takes the ship to travel to Alpha C and Back. Since Alpha C is 4.3 ly distant, The ship would have to travel at ~ 20% of c, and at that speed, the difference in time between ship and Earth would be less than a year.

If you wanted your astronaut to age 1 yr while 40 pass on Earth, he would have to travel at 99.9687% of c and travel to a star ~20 ly away.

Is it absolutely necessary for the age difference to be due to Relativity? How about a form of suspended animation? Say that 20% of c is the best that can be managed. Since this still means that it would take ~20 yrs ship time to reach Alpha C, the passengers are placed into suspended animation for the trip out and back.

 

[CakeOrDeath?]

Is it absolutely necessary for the age difference to be due to Relativity? How about a form of suspended animation? Say that 20% of c is the best that can be managed. Since this still means that it would take ~20 yrs ship time to reach Alpha C, the passengers are placed into suspended animation for the trip out and back.

I actually had considered this. The idea is that the pilot becomes sort of a Chuck Yaeger of spaceflight, proving that such a speed is possible for humans to withstand. In that case, suspended animation is a possible means of achieving this. However, I'd hoped to use relativity as a plot device because I think it's so interesting. Perhaps the pilot could be in SA for part of the trip and then pop out of it and experience further effects via relativity.

 

[JesseM]

I actually had considered this. The idea is that the pilot becomes sort of a Chuck Yaeger of spaceflight, proving that such a speed is possible for humans to withstand.

This is itself a sort of problematic idea, since it suggests that the scientists who thought up this test don't understand or don't believe in relativity. In relativity, there is no absolute notion of "speed", speed can only be defined relative to a given frame of reference, and all frames are equally valid...so, this implies any speed is equally easy to "withstand" since there is some frame where the Earth is already traveling at that speed right now (the only issue that would make it dangerous to travel really fast relative to the Earth is that you'd also be traveling much faster relative to the stray hydrogen atoms in interstellar space, and relative to the average rest frame of the cosmic microwave background radiation...this would just be a question of proper shielding though).

 

[CakeOrDeath?]

This is itself a sort of problematic idea, since it suggests that the scientists who thought up this test don't understand or don't believe in relativity. In relativity, there is no absolute notion of "speed", speed can only be defined relative to a given frame of reference, and all frames are equally valid...so, this implies any speed is equally easy to "withstand" since there is some frame where the Earth is already traveling at that speed right now (the only issue that would make it dangerous to travel really fast relative to the Earth is that you'd also be traveling much faster relative to the stray hydrogen atoms in interstellar space, and relative to the average rest frame of the cosmic microwave background radiation...this would just be a question of proper shielding though).

I actually have a plan of sorts for dealing with the radiation. That part might need a bit of "suspension of disbelief," but I'm okay with this as it's a key part of the plot. To address your first point, I probably worded it incorrectly when I said "prove." I mean they set out to actually do something that is theoretically possible, but currently limited by technology. "Why climb the mountain? Because it's there." That sort of thing. In fact, I had a 7th grade science teacher tell us that it's "impossible" to travel at those velocities because of the effects on the body.

 

[soothsayer]

In fact, I had a 7th grade science teacher tell us that it's "impossible" to travel at those velocities because of the effects on the body.

*facepalm*

 

[bobc2]

Imagine an interstellar race. Starship 1 heads from Earth toward Alpha Centauri at a constant rate close to the speed of light, then makes a loop and returns home. Ship 2 is faster, and therefore has been given a handicap and leaves after Ship 1, but it is still fast enough to make up that distance and reach Alpha Centauri before Ship 1.

Wouldn't Ship 2 have experienced greater time dilation and therefore return further into Earth's future than the slower moving Ship 1, despite reaching Centauri first? This seems like a paradox.

Here's a space-time diagram that you should keep in mind. It doesn't present both of your ships, but rather presents the standard twin paradox for twin A staying on Earth (Red line) and twin B (blue lines) traveling to Alpha Centauri. You can apply the same basic concept to resolve your perceived paradox for the two ships. I can do the space-time diagrams for both ships if you really need that--it is just a little more work.

Rather than present a symmetric space-time diagram, we show the hyperbolic calibration curves with the proper times identified. The actual total proper time (and time shown on his personal clock) for the twin who makes the round trip (blue lines) will be the sum of the proper times shown on the two hyperbolic calibration curves associated with his trip. The twin remaining at home (red line) will see his own clock advance consistent with the calibration curves that calibrate proper distances (and time) for both twins during twin B's trip out to Alpha Centauri.

A triangular inequality applies as shown in the sketch below. The lines on the screen are longer for the B trip as compared to the straight line for A. However, using the hyperbolic calibration curves you can see that when the B twin returns to rejoing his A twin, the A twin is much older than the B twin. Notice that if B rode a pulse of light out to Alpha Centauri and back, then no time would have lapsed for him--he would not have aged (of course light speed is not possible for him).

The thin green 45-degree lines depict photon world lines (particle moving at speed of light).

By the way I've noticed in some posts that people have attributed B's slower clock speed to the deceleration and acceleration when approaching and leaving Alpha Centauri. That has nothing to do with it. Just keep your eyes on the calibration curves.

 

[JesseM]

By the way I've noticed in some posts that people have attributed B's slower clock speed to the deceleration and acceleration when approaching and leaving Alpha Centauri. That has nothing to do with it.

That's a perfectly good (and common) way of explaining it. If you had two paths through 2D Euclidean space, and one was a straight-line path between two points while the other was a path with a bend in it between the same pair of points, would you object to someone saying that the bend is the reason the second path has a greater distance, since in Euclidean geometry a straight line is always the shortest distance between two points? This is directly analogous to the fact that in SR spacetime, a straight line through spacetime (i.e. an inertial worldline) always is always the path that has the greatest proper time between two points, so any non-straight path between the same two points will have a shorter proper time (for more on the analogy between SR and 2D Euclidean geometry, see [post=2972720]this post[/post]). This explanation is not inconsistent with an explanation involving calibration curves (which in 2D Euclidean geometry would just look like circles rather than hyperbolas, since the idea of such a curve is to show the set of all paths from a given point with the same distance), it's just a different way of looking at the problem.

 

[bobc2]

That's a perfectly good (and common) way of explaining it. If you had two paths through 2D Euclidean space, and one was a straight-line path between two points while the other was a path with a bend in it between the same pair of points, would you object to someone saying that the bend is the reason the second path has a greater distance, since in Euclidean geometry a straight line is always the shortest distance between two points? This is directly analogous to the fact that in SR spacetime, a straight line through spacetime (i.e. an inertial worldline) always is always the path that has the greatest proper time between two points, so any non-straight path between the same two points will have a shorter proper time (for more on the analogy between SR and 2D Euclidean geometry, see [post=2972720]this post[/post]). This explanation is not inconsistent with an explanation involving calibration curves (which in 2D Euclidean geometry would just look like circles rather than hyperbolas, since the idea of such a curve is to show the set of all paths from a given point with the same distance), it's just a different way of looking at the problem.

Thanks for your clarity on that, JesseM.