What is the speed and the total distance traveled...?

Question

what is the speed for one of the twins to travel in the outer space to age a day while the other twin will age a 1000 years on the earth? and what is the total distance traveled?

Answer 1

Hi Nick!

 

This deals with the special relativity consequence of time dilation.  This means -- described in a way appropriate for this problem -- that an observer in one reference frame (say, the Earth) will see a clock in another reference frame that is moving with respect to the Earth (say, the ship) as running more slowly.  Thus, from this point of view, one second on the ship's clock ticks off more slowly than one second on the Earth clock.  Thus, time on the ship clock, to the Earth observer, is stretched out, or *dilated.*

 

This means that, between the same two events, the observer on Earth will measure more seconds (and thus a longer time) on the Earth clock than he will observe on the ship clock.  The relationship between these can be summarized in the time dilation expression:

 

Δt' = Δt/γ

where the delta-t with the ' (prime) mark represents the time interval measured by the Earth observer on the ship clock, the the unprimed delta-t represents the time measured by the Earth observer, over the same interval, on the Earth clock.

 

Here,  γ is the Lorentz factor, defined as:

 

γ = 1/[sqrt(1 - (v/c)²)]

 

where v is the speed of the "moving" frame (like the rocket) compared to the "rest" frame (the Earth).  Because v must be less than (or equal to) c, γ is guaranteed to be greater than (or equal to) 1.  This makes it so Δt' is always less than Δt, as we described above (the Earth observer always measures a shorter time interval between events on the rocket clock compared to the Earth clock).

 

I take it the question means to ask what the speed v would have to be for the Earth observer to see 1000 years pass on his own clock while only seeing a day pass on the rocket clock.  Thus, we can set up the time dilation relationship (with the time intervals in days) as:

 

1 = 365,250/γ

 

[note: if we take 365.25 days in a year, there are 365,250 days in 1000 years]

 

Knowing that the speed of light is c = 3.0 x 10⁸ m/s, the preceding equation can be used to solve for v (which is inside the Lorentz factor γ).  Once you know v, the speed with which the rocket is traveling from the Earth reference frame, you can determine how far it travels in 1000 Earth years.

 

[note:  v will be very close to the speed of light to get this kind of dilation]

 

See if you can work it out from there.  If you have any other questions, or would like to check an answer, please let me know.

 

PS:  the source of the twin paradox, of course, is that an observer on the rocket sees the EARTH clock as running slower, and measures 1000 years passing on his ship for one day on Earth.  Who is right?  If the ship were always traveling at v, the answer would be NEITHER.  This is relativity, after all, so each one is "right" from his point of view.

 

The symmetry is usually broken by saying that there is not total symmetry between the frames, because the rocket does NOT travel at v forever.  It accelerated to get going, and will accelerate more to turn around, and then stop when it returns.  So its reference frame is not, as we say, "inertial" (non-accelerating).  This means the Earth frame "wins" in this case.

 

You could say that the rocket actually sees the EARTH accelerate, and this would be just another relative observation.  But one bedrock notion, even in relativity, is that events happen, or don't.  All observers will agree on the outcome and number of events, even if they do not agree about the order or location of the events.  The ship's engines fire; it accelerates; there is no denying that.  So the symmetry IS broken, and the Earth frame wins.