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Relativity Conundrum

aggle-rithm

Ardent Formulist
aggle-rithm
Joined
Jun 9, 2005
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Austin, TX
I've been thinking about a variation on the twins paradox and gotten myself all confused. What if, I thought to myself, there was a spaceship capable of making it to Alpha Centauri in 18 years, Earth time. An astronaut on the spaceship would experience the time as being two years. An observer on the base at Alpha Centauri would also think the trip took two years, although the message saying that it actually left Earth would take four years to get there.

What would happen if the observers on Earth sent two messages when they saw the ship reach the halfway point: One to Alpha Centauri, saying "the ship is halfway there", and one to the ship saying, "tell Alpha Centauri you're halfway there"?

Here's what I think would be the chronology (Earth/Alpha Centauri time, assuming both are in the same inertial frame), although I am probably wrong:

1. The ship takes off, Earth sends a message to Alpha Centauri saying so.
2. Four years later, Alpha Centauri gets the message that the ship has taken off.
3. Five years after takeoff, Alpha Centauri gets a message from the ship saying it's halfway there.
4. Six years after takeoff, the ship arrives at Apha Centauri.
5. Nine years after takeoff, Earth sends the message that the ship is halfway there.
6. Thirteen years after takeoff, Alpha Centauri receives the message from Earth saying the ship is halfway there.
7. Eighteen years after takeoff, Earth gets message from Alpha Centauri that the ship has arrived.

Am I close?

ETA: I've already spotted a flaw. Communication from Earth to the ship at the halfway point isn't instantaneous. Grrrr.
 
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OK, now I've really confused myself.

I assumed that the halfway point for Earth would be the halfway point for the spaceship, but it isn't. By the time the Earth sends the message, the ship is already over halfway there. By the time it gets the message, it is well over halfway there.

Meanwhile, the ship may be experiencing everything coming from the Earth as being slowed down, but the messages still have to be moving at the speed of light. So the message would have to take more than two years to get to the ship, and by that time the ship has reached Alpha Centauri.

I need a drink.
 
If you're trying to better understand relativity and the twin paradox, here's my advice: Don't worry about when observers see events based on how long it takes light to reach their eyeballs. It adds an unnecessary complication that can be considered separate from relativity. Assume that all your observers in different reference frames automatically take into account the time it takes light to reach them when they make their observations. Once you figure that part out, then you can figure out what they actually see.

Also, in regards to your opening paragraph, the observer on Alpha Centauri would think the trip took 18 years, same as the Earth observer. Earth and Alpha Centauri are in the same reference frame.
 
aggle-rithm said:
OK, now I've really confused myself.

I assumed that the halfway point for Earth would be the halfway point for the spaceship, but it isn't.
Click to expand...

Depends on what you mean. Let me make what boooeee said a little more explicit. In special relativity, there is a distinction between what you see and what you observe. The analogy I like to make is with sound. Let's say you're watching a loud jet fly overhead, and you detect sound originating from behind the jet. You know that sound travels at a finite speed, and so it only seems like it's coming from the jet: when it was emitted, the jet was in a different place than it is now. You hear sound coming from well behind the jet, but taking into account how fast the jet is moving and how fast sound travels, you observe the sound coming from the jet.

Now this same distinction will exist in relativity. What you see is not necessarily what you observe. In particular, the standard formulas for length contraction and time dilation give you quantities that you observe, not what you see.. From the standpoint of thought experiments, what you observe is actually much easier to figure out than what you see.
 
boooeee said:
Also, in regards to your opening paragraph, the observer on Alpha Centauri would think the trip took 18 years, same as the Earth observer. Earth and Alpha Centauri are in the same reference frame.
Click to expand...

Yes, but wouldn't the observer on Alpha Centauri experience a sped-up version of the ship's flight because of Doppler effects?
 
aggle-rithm said:
Yes, but wouldn't the observer on Alpha Centauri experience a sped-up version of the ship's flight because of Doppler effects?
Click to expand...

They would see a sped-up trip compared to earth, but they would observe exactly the same thing.
 
Alpha Centauri is only about 4 light years away (more precisely 4.37), so your numbers are off. If the trip only takes 2 years ship-time, then it would take about 4.8 years earth-time (the speed would be about 0.91c). Alpha Centauri is roughly the same frame as earth, so they will also observe the trip taking 4.8 years, the same way they observe light taking 4.37 years to travel there. Of course, they can't see the light until it arrives, so 4.37 years after the ship leaves in earth time, they will see the ship leave earth, and over about a year will see the ship as it crosses the entire distance, and arrives.

For the next bit, I'll assume the distance is exactly 4 light years, and the ship takes exactly 5 light years, so a speed of 0.8c. This would lead to the ship-time being 3 years exactly.

If the ship sends a signal at it's halfway point, which is the physical halfway point between earth and AC (2 light years from either), then the signal will show up at the 4.5 year mark (earth time) at AC. The ship is emitting this at 1.5 ship-years, or 2.5 earth-years.

If the earth sends a signal when they know the ship is halfway, i.e. 2.5 earth-years after the ship leaves, then it arrives at 6.5 earth-years after launch at AC, 1.5 earth-years after the ship has arrived.

If the earth sends a signal when the see the ship at the halfway point, they need to wait 2.5 years for the ship to arrive, then 2 years for light from the ship to get back, then 4 years to get the signal to AC, so the signal arrives 3.5 earth-years after the ship has arrived.

Earth sees the ship arrive when light gets there from AC, so they see it 5 earth-years for the trip plus 4 years for the light = 9 earth-years after the ship has left. During this 9 years, they see the entire 5 year trip.
 
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