Re: Where is the astronaut?

Dear Ron,
This is a great question which hinges on the different perspectives of the 
two observers.  First, let's be careful with the numbers we use.  Let's 
assume the traveler is moving at 70% of the speed of light, that is, v = 
0.7c , where "c" is the speed of light.  An important quantity that is 
involved in relativistic calculations is called "gamma" and is written as 
the Greek letter.  (This will not come out as a real gamma in the 
text, but it is supposed to be.)  The value of gamma is determined by

gamma = 1 / sqrt(1 - v^2 / c^2) ,

if v = 0.7c, then gamma = 1.4 .  (Don't think that gamma is always equal 
to 2v/c , it just happens to be for v = 0.7c .)

If the traveler flies off for 20 years on your clock, then he thinks he 
has been traveling for 20years / gamma = 20years / 1.4 = 14.3 years on his 
clock.

You think he is now at a distance of d = v t = (0.7c)(20years) = 14cyears 
= 14 lightyears away.

The traveler thinks he is at a distance of d' = v t' = (0.7c) (14.3years) 
= 10cyears = 10 lightyears away.  (Where I have used primes (') to 
indicate quantities as measured by the traveler.)

How can this be?  Well, let's remember that not only are times changed for 
the traveler, but lengths are, too.  We say that time is dilated and 
length is contracted.  A length of 10 lightyears for the traveler actually 
corresponds to a longer distance in your frame of reference.  By how 
much?  By a factor of gamma!  So that by your measurement the traveler is 
at a distance of d = gamma d' = (1.4)(10 lightyears) = 14 lightyears!  
Just where you thought he should be!

Let's think about all of this again, from each of the different 
perspectives.  You think the traveler is moving at 0.7c for a time of 20 
years and ends up at a distance of 14 lightyears.  That's not surprising.  
What does the traveler think?  He thinks he is moving at 0.7c (both of you 
measure the motion of the other in the same way and get the same result) 
but thinks he travels for 14.3 years and has only gone 10 lightyears, 
however, he also thinks that the space through which he is traveling has 
shortened.  The distance now to his endpoint has decreased so that instead 
of 14 lightyears away, it only seems to be 10 lightyears away from where 
he started, and so it only takes him 14.3 years to get there.

The real surprising result in all of this is that the two observers, you 
and the traveler, measure quantities to have different values and it all 
comes about because of the different ways you measure them.


Well, I hope I have answered your question, Ron.  If you would like some 
more information, please let us know.

Sincerely,

Jim Guinn